Università degli Studi di Salerno [607245]
Università degli Studi di Salerno
FACOLTÀ DI INGEGNERIA
CORSO DI LAUREA IN INGEGNERIA CIVILE
Tesi di Laurea
in
TECNICA DELLE COSTRUZIONI II
“APPLICATION OF SOME CLASSIC
CONSTITUTIVE THEORIES TO THE
NUMERICAL SIMULATION OF THE
BEHAVIOR OF PLAIN CONCRETE ”
RELATORE
Ch.mo P rof. Ing.
Ciro Faella
CORRELATORI CANDIDATO
Ch.mo P rof. Ing. Antonio Caggiano
Guillermo Etse Matr.: 06201000025
Dott. Ing. Enzo Martinelli
Ing. Paula Folino
A
NNO ACCADEMICO 2007/2008
Abstract
In past years, the methods of analysis and design for concrete structures
were mainly based on elasticity combined with various classical procedures
as well as on empirical formule devel oped on the basis of a large amount of
experimental data. Such approaches are still necessary and desirable and
continue to be the most convenient and effective methods for ordinary
design.
However, the rapid development of mo dern numerical analysis techniques
and high-speed digital computers has provided structural engineers with powerful tools for complete nonlinear analysis of concrete structures.
Indeed, stress and strain response of c oncrete structures can be efficiently
reproduced by using the finite-element method and performing an
incremental inelastic analysis. Th e increasing use of fully three-
dimensional finite-element analysis in reinforced and prestressed concrete
structures motivates to developm ent of sophisticated constitutive
formulations when the structural response is to be predicted beyond the
linear elastic limit.
The present Thesis deals with the desc ription and validati on of a concrete
material model based on non-associated plasticity models which can be
used for an easy and robust numerical implementation. All the analysis are
performed by means of the “Constitut ive Driver Interactive Graphics”
program (namely Co.Dri.) and by Conc rete Damage-Plasticity constitutive
model implemented in Abaqus (gener al-purpose nonlinear finite element
analysis program).
A comparison of the numerical predictions with experimental tests
available within the scientif ic literature is also presented. Actual limits and
further developments of the proposed models are finally outlined in this job.
CONTENTS
I
1 INTRODUCTION ……………………………………….….….1
1.1 COSTITUENTS OF CONCRETE MATERIAL ………….…….…….1
1.1.1 PORTLAND CEMENT ……………………………………..….….…..3
1.1.2 AGGREGATES ……………………………………………..…………3
1.1.3 WATER ……………………………………………………..….…….4
1.2 BASIC FEAUTERES OF CONCRETE BEHAVIOR ……………… 4
1.2.1 NONLINEAR STRESS -STRAIN BEHAVIOR …………………………….5
1.2.2 DIFFERENT RESPONSES IN TENSION AND COMPRESSION …………..6
1.2.3 MULTIAXIAL COMPRESSIVE LOADING ……………………………..7
1.2.4 VOLUME EXPANSION UNDER COMPRESSIVE LOADING ……………..9
1.2.5 STRAIN SOFTENING ……………………………………………….11
1.2.6 STIFFNESS DEGRADATION …………………………………………13
1.3 CONSTITUTIVE MODELING OF CONCRETE MATERIALS ……14
1.3.1 EMPIRICAL MODELS ………………………………………………14
1.3.2 LINEAR ELASTIC MODEL ………………………………………….15
1.3.3 NONLINEAR ELASTIC MODEL ……………………………………..15
1.3.4 PLASTICITY BASED MODEL ………………………………………..17
1.3.5 STRAIN SOFTENING AND STRAIN SPACE PLASTICITY ……………..18
1.3.6 FRACTURING AND CONTINUUM DAMAGE MODELS ……………….19
1.3.7 MESOMECHANIC ANALYSIS OF CONCRETE BEHAVIOR ……………20
1.3.8 MICROPLANE MODELS ……………………………………………21
CONTENTS
II
1.4 OBJECT OF THE THESIS ………………………………………….21
REFERENCES OF THE FIRST CHAPTER………………………22
2 BASIC EQUATIONS …………………………….…………30
2.1 STRESS AND STRESS TENSOR ……………………….…………30
2.1.1 PRINCIPAL STRESSES AND INVARIANTS OF THE STRESS TENSOR …33
2.1.2 STRESS DEVIATION TENSOR AND ITS INVARIANTS ………………..34
2.1.3 HAIGH -WESTERGAARD STRESS -SPACE ……………………………36
2.2 YIELD AND FAIL URE CRITERIA ………………….…………….40
2.2.1 YIELD CRITERIA INDEPENDENT OF HYDROSTATIC PRESSURE ……40
2.2.1.1 The Tresca Yield Criterion ……………………….…….41
2.2.1.2 The von Mises Yield Criterion…………………………..43
2.2.2 FAILURE CRITERION FOR PRESSURE -DEPENDENT MATERIALS …..45
2.2.2.1 The Mohr-Coulomb Criterion …………………………..49
2.2.2.2 The Drucker-Prager Criterion ………………………….53
2.3 LINEAR ELASTIC ISOSTROPIC STRESS -STRAIN
RELATION …………………………………………………………56
2.4 STRESS -STRAIN RELATION FOR WORK -HARDENING
MATERIALS ………………………………………………..……..59
2.4.1 PLASTIC POTENTIAL AND FLOW RULE ……………………………60
2.4.2 INCREMENTAL STRESS -STRAIN RELATIONSHIP ……….……….…62
2.4.3 SOFTENING BEHAVIOR …………………………….….………….66
CONTENTS
III
2.5 INTEGRATION SCHEME FOR ELASTO -PLASTIC
MODELS ……………………. ……………….. …………………… ………………….66
2.5.1 GENERAL DESCRIPTION OF A GENERAL ELASTOPLASTIC
INTEGRATION ………………………………………………..……67
REFERENCES OF THE SE COND CHAPTER……………………72
3 CO.DRI. INTERACTIVE GRAPHICS ………………74
3.1 USER OPTIONS ……………………………………. ……………….. …………….74
3.2 EXPERIMENTAL DATABASE ………………… ……………….. …………..78
3.2.1 IDENTIFICATION OF EXPERIMENTS ………………………. ……………..78
3.2.2 TEST APPARATUS ………………………………………. ………………..79
3.3 CONSTITUTIVE MODELS ……………………. ……………….. ……………81
3.3.1 ASSOCIATED VON MISES PLASTICITY MODEL …………………………..81
3.3.1.1 Quadratic hardening/softening function …………………….81
3.3.1.2 Simo hardening/softening function ………………… ………….82
3.3.1.3 Simo Modified hardening/softening law………….. …………84
3.3.1.4 Calibration and validation of the von Mises model ……..85
3.3.2 NON-ASSOCIATED DRUCKER -PRAGER PLASTICITY MODEL (TWO
PARAMETERS )……………………………………………… ………90
3.3.2.1 Calibration and validation of Drucker-Prager model ….91
3.3.3 NON-ASSOCIATED DRUCKER -PRAGER PLASTICITY MODEL (THREE
PARAMETERS ) ………………. ……………………………………………………98
3.3.3.1 Calibration and validation of Drucker-Prager
model …………………… ………………… ……………….. …………….103
3.3.4 NON-ASSOCIATED BRESLER -PISTER PLASTICITY MODEL ………….108
3.3.4.1 Calibration and validatio n of Bresler-Pister
model …………………… ………………… ……………….. …………….110
CONTENTS
IV
REFERENCES OF THE TH IRD CHAPTER………………….…116
4 CONSTITUTIVE MODELS A V AILABLE IN
ABAQUS ………………………………………………………..119
4.1 YIELD AND FAILURE SURFACE ………………………………123
4.2 HARDENING /SOFTENING LAWS …………………………….130
4.2.1 COMPRESSIVE BEHAVIOR ……………………………………..…130
4.2.2 TENSILE BEHAVIOR …………………………………..………….131
4.3 NONASSOCIATED FLOW LAW ………………………………..133
4.4 CALIBRATION AND VALIDATION OF DAMAGE PLASTICITY
MODEL ………………………………………………………..….134
REFERENCES OF THE F OURTH CHAPTER…………………..140
5 FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
..…………………………………..142
5.1 SOME CLASSICAL FA ILURE CRITERIA ..……………………143
5.1.1 LEON FAILURE CRITERION ……………………………………….143
5.1.2 HOEK AND BROWN FAILURE CRITERION …………………………146
5.1.3 WILLAM AND WARNKE (THREE PARAMETERS ) FAILURE
CRITERION ……………………………………………………….149
5.1.4 WILLAM AND WARNKE (FIVE PARAMETERS ) FAILURE
CRITERION ……………………………………………………….153
5.1.5 OTTOSEN FOUR -PARAMETER MODEL …………………………….156
5.1.6 HSIEH – TING – CHEN FOUR -PARAMETER MODEL ………….…….159
CONTENTS
V
5.1.7 EXTENDED LEON MODEL (ELM) PROPOSED BY ETSE ………………162
5.2 APPLICATION OF PLASTICITY BASED MODELS TO PASSIVE
CONFINEMENT ……………………….. …………………… ………………..166
5.2.1 STEEL CONFINEMENT ………………….. ……………………………………169
5.2.2 FRP CONFINEMENT ………………………. ………………… …………………178
REFERENCES OF THE FIRST CHAP TER………………. ……………181
6 SUMMARY AND CONCLUSION ……………….. ………..184
7 APPENDIX : “CODRI.F” ………………………. ……………..186
INTRODUCTION
11. INTRODUCTION
This chapter is subdivided into three sections:
– the first section deals with the princi pal components of concrete material;
– the second section contains the principal features of the mechanical behavior of
concrete under ordinary typica l solicitations in the field of the civil engineering.
The nonlinear behavior of concrete material and its being a composite material
in nature is treated.
– the las t part of the in tro duction pres ents var ious developmen ts in the field of the
constitutive modeling of concrete o n differ ent approaches such as elasticity,
plasticity, continuum damage mechanics, plastic fracturing, microplane models,
etc.
1.1 COSTITUENTS OF CONCRETE MATERIAL
Concrete has been the most common building material for many years and
the same trend is expected for the coming decades. Reinforced concrete
structures and infrastructures are quite common throughout the developed
world and are more and more frequent in developing countries; the greater
number of buildings for various uses and purposes are made on concrete as
well as bridges, massive dams, nuc lear power plants and so on.
In pre-historic times, some form of concrete using lim e-based binder may
have been used (Stanley [100]), but modern concrete using Portland cement
dates back to mid-eighteenth century, with the patent by Joseph Aspdin in
1824. Traditionally, concrete is basically a composite natural consisting of the
dispersed phase of aggregates (rangi ng from its maximum size coarse
aggregates down to the fine sand par ticles) embedded in the matrix of
cement paste. This is a “Portland cement concrete” with the four constituents:
INTRODUCTION
2- Portland cement;
– water;
– stone;
– and sand.
Fig. 1. 1 – The basic components of concrete: aggregates (stone and sand), Portland
cement, and water (Chen and Liew [32]).
These basic components remain in current concrete but other constituents
are now often added to modify its fresh and hardened properties. The
quality of concrete in a structure is determined not only by the proper
selection of its constituents and their proportions, but also by appropriate
techniques of production, transportation, placing, compacting, finishing, and curing of the concrete of the actu al structure, often at the job site.
The constituents of modern concrete have increased from the basic four
(Portland cement, water, stone and sa nd) to include both chemical and
mineral admixtur es . These admixtures have been in use for decades, first in
special circumstances, but have now be en incorporated in more and more
general applications for improving technical and performance cost-
effectiveness.
INTRODUCTION
31.1.1 PORTLAND CEMENT
In the past, Portland cement was restricted to that used in ordinary concrete
and is often called ordinary Portland cement . There is a general trend
towards grouping all cement types as Portland cement, included those
blended with molten iron slag or po zzolan such as fly ash (also called
pulverized fuel ash ), and silica fume into cements of different sub-classes
rather than special cements. This appr oach has been adopted in Europe (EN
197–1 [41]) but the American practice subdivides them into two separate
groups (American Society for Testing a nd Materials provides rules for both
Portland cement within ASTM C150 [3] and blended cements in ASTM
C595 [4]).
Raw materials for manufacturing Po rtland cement basically consist
calcareous and siliceous (generally clay -based) materials. Mixture is heated
to a high temperature (1400°-1600° C) within a rotating kiln to produce a
complex group of chemicals, collectively called “cement clinker” .
Further details about manufacturing process, the formation of these
chemicals and their reactions with wa ter are well beyond the scopes of this
Thesis and can be found in various sp ecific textbooks (e.g., Hewlett [53]).
1.1.2 AGGREGATES
Aggregates in concrete are usually gr ouped according to their size in fine
and coarse aggregates. The separation is based on materials passing or retained on the nominally 5 mm sieve (No. 4 sieve after ASTM D2487 [5]).
Fine aggregates basically consist in sand, while coarse aggregates are
represented by small stones. Traditiona lly, aggregates are derived from
natural sources in the form of river gr avel or crushed rocks and river sand.
INTRODUCTION
4Fine aggregates produced by crushing rocks to sand sizes are referred as
manufactured sands. Aggregates derive d from special synthetic processes
or as a by-product of other pr ocesses are also available.
In most concrete mix, volume fraction of aggregates is about twice the
volume of cement paste matrix. Hence, the physical properties of concrete
are dependent on the corresponding properties of the aggregates.
1.1.3 WATER
Water is basically needed for the hydratio n of cement, but not all is used for
this purpose only. Part of the water is aimed to provide workability during
mixing. This latter usage can be reduc ed by the introduction of chemical
admixtures, e.g., plasticiser s (Chen and Liew, [32]).
Where possible, potable water is used. Other sources may contain
impurities introducing u ndesirable effects on properties of fresh and
hardened concrete. A good list of conc rete mixtures is given in the PCA
Manual (Kosmatka & Pana rese, [64]). ASTM C9 4 [2] and BS 3148 [19]
both provide guidance on acceptance cr iteria for water of questionable
quality in terms of expected conc rete strength and setting time.
Seawater should not be used as mixing water for reinforced concrete due to
the presence of chloride and its eff ect on corrosion of steel reinforcement
(Chen and Liew, [32]).
1.2 BASIC FEAUTERES OF CONCRETE BEHAVIOR
Mechanical behavior of concrete is ve ry complex, being largely determined
by the structure of the component rela ted issues, such as water-to-cement
ratio, cement-to-aggregate ratio, shape and size of aggregates, the kind of
cement used, and so on. The present dissertation deals with stress-strain
behavior of an average ordinary conc rete. The physic-chemical structure of
INTRODUCTION
5the material is ignored and the rules of material behavior are developed on
the basis of continuum mechanics. U nder this standpoint the material is
basically assumed homogeneous and isotropic.
Concrete is a brittle material; its stress-strain behavi or is affected by micro-
and macro-cracks developing within the material body during the loading
process. Furthermore, c oncrete is affected by a large number of micro-
cracks, especially at interfaces between aggregates and mortar, even before the application of external load. Th ese initial microcracks are caused by
segregation, contraction, or thermal expansion in the cement paste (Chen &
Han, [31]). Under applie d loading, further devel opment of micro-cracks
may occur at the aggregate-cement inte rfaces, which is the weakest link in
the composite system. The progression of these cracks, which are initially
invisible, become visible when the cr acks occur with the application of
external loads and contribute to the overall nonlinear stress-strain behavior.
1.2.1
NONLINEAR STRESS -STRAIN BEHAVIOR
A typical stress-strain curve for conc rete in uniaxial compression tests is
shown in Fig.1.2 and three fundame ntal deformation stages can be
observed even in this simple te st (Kotsovos & Newman, [65]):
– the first stage corresponds to a stress in the region up to 30% of the
maximum compressive stress f’c. At this stage, cracks initially
existing in concrete remain nearly unchanged. Hence, the stress-
strain behavior is assumed linearly elastic. Therefore, 0.3f ’ c is
usually proposed as the limit for elas tic constant range of concrete;
– beyond this limit, the stress-stain cu rve begins diverting from the
original straight line. Stress between 30% and about 75% of f’c
characterizes the second stage, in which bond cracks start to increase
INTRODUCTION
6in length, width, and number; ma terial comes out as micro-cracks
develop within nonlinearity;
– after further load increases, in the third stage, the progressive failure
of concrete is classically caused by cracks through the mortar (Chen
& Han, [31]). These crac ks form a crack zone or “internal damage”;
at this load-like deformations may be localized in the damage zone
and the nonlinear behavior is very pronounced. Finally, the load
reaches the value of peak, loading the concrete specimen to failure.
Fig. 1.2 – Typical uniaxial compressive stress -strain curve (Domingo Sfer et al, [94]).
Although the above discussion deals only with the uniaxial compression
case in pre-peak region (tests in for ce-control), three deformation stages
can also be qualitatively identified in the same loading cases, the linear
elastic stage, the inelastic stage, and the so-called “localized stage”.
1.2.2 DIFFERENT RESPONSES IN TENSION AND COMPRESSION
Figure 1.3 shows a typical uniaxial tens ion stress-strain curve. In general
INTRODUCTION
7the limit of elasticity is observed to be about 60 to 80% of the ultimate
tensile strength. Beyond this level, bond microcracks start growing.
As the uniaxial compressive state tends to arrest the preexisting cracks in
the concrete material, the tension state of stress tends rather to promote the
opening of the same cracks. This is one of the reasons why the behavior of
concrete in tension is quite brittle in nature. In addition, the aggregate-
mortar interface has a significantly lowe r tensile strength than mortar. This
is the primary reason for the lower tens ile strength of conc rete materials.
Fig. 1.3 – Uniaxial tensile stress -strain curve (Hurlbut, [55]).
1.2.3 MULTIAXIAL COMPRESSIVE LOADING
A typical stress-strain behavior for concrete under multiaxi al loading condi-
tions is shown in Fig. 1.4 (Hurlbut, [55]).
The results are obtained from tests on cylindrical specimens. Concrete
cylinders are submitted to c onstant lateral pressures, σ2 = σ 3. The axial
load, is imposed in terms of strain ε1. Figure 1.4 shows the relationship
between axial stress σ1 and both axial and transverse strains, εz and εr
INTRODUCTION
8respectively, for various values of confining pressure σ2 = σ3.
The confining pressure significantly affects the deformation behavior of the
specimen. At first, the axial strain at failure (peak value) increases with
confining pressure. However, compar ed to the uniaxial compression case,
larger strains develop in c onfined concrete specimens.
Fig. 1.4 – Stress-strain curves under multiaxial compression (Hurlbut, [55]).
Softening behavior can be obser ved for specimen in unconfined
compression or under low levels of lateral confinement. When lateral
confinement attains a critical value the so-called “softening zone”
disappears and the stress-strain rela tion is increasing up to the ultimate
strain.
Furthermore, the maximum value of stre ss increases as confining pressure
is applied. Figure 1.4 shows that th e uniaxial strength for the unconfined
specimen is about 19 MPa, but it hugely increases as lateral confinement is
applied. Consequently, concrete and various ge otechnical materials are
classified as “pressure-d ependent materials”.
INTRODUCTION
9
1.2.4 VOLUME EXPANSION UNDER COMPRESSIVE LOADING
The volumetric strain (namely, the trace of strain tensorz y x vεεεε ++ = )
plotted against the uniaxial compression stress is shown in Fig.1.5
(Domingo Sfer et al., [94]). When concrete specimen is subjected to
increasing uniaxial compression, its-apparent-Poisson’s ratio start to continuously and significantly increas e beyond its well established elastic
value, as plotted in figure 1.7 (Domi ngo Sfer et al., [94]). On attaining a
certain stress level, called critical st ress (0.75 to 0.90 of the ultimate
uniaxial compressive stress), the volume of the concrete starts to increase rather than continuing to decrease. This inelastic behavior is due to the
composite nature of concrete.
Fig. 1.5 – Volumetric strain vs. stress, unde r uniaxial compression (Domingo Sfer et al.,
[94]).
Indeed, experimental tests, performed by Shah and Chandra [95], point out
that cement paste itself does not expand under compression loads.
Hardened paste specimens continue to consolidate at an increasing rate
with increased load (figure 1.6).
INTRODUCTION
10
Fig. 1.6 – Mechanical proprieties of Portland cement pastes with water-cement ratios
equal 0.40, 0.47, and o.54 (Shah and Chandra, [95]).
Shah and Chandra [14] observed that increasing the volume fraction of
aggregates significantly reduces the percen tage values of that critical stress.
Similarly, increasing the size of aggreg ate particles or reducing the strength
of bond between aggregate and past e makes concrete more inelastic.
Fig. 1.7 – Poisson’s ratio vs. stress, under uni axial compression (D. Sf er et al., [94]).
INTRODUCTION
11Hence, volumetric expansion is obs erved only when the cement paste is
mixed up with aggregates, consequently the composite nature of concrete
is primarily responsible for the volume dilatation.
1.2.5 STRAIN SOFTENING
Engineering materials like concrete, as well as other natural elements on
concern for engineering purposes like rocks, and soils exhibit a significant
strain-softening behavior beyond the peak stress. Figure 1.8 shows typical
uniaxial compressive stress-strain cu rves obtained from strain-controlled
tests. Each of these curves has a sharp descending branch beyond the peak
of failure stress.
Fig. 1.8 – Uniaxial compressive stress-strain curve for concrete (Wischers, [107]).
INTRODUCTION
12It is generally agreed that softening branch of a stress-strain curve does not
reflect a material property, but rath er represents the response of the
structure formed by the specimen together with its complete loading
system (van Mier, [102]). This ar gument is supported by compression tests
of specimens of different heights. Th e test results in terms of stress and
strain are shown in Fig. 1.9 where the descending branches of the stress-
strain curves are not identical but have slopes decreasing with increasing
specimen heights (van Mier, [102]). On the other hand, however, if the
post-peak displacement rather than st rain is plotted against stress, the
stress-displacement curves are almost identical, regardless of the specimen
heights.
Fig. 1.9 – Influence of specimen height on uniax ial stress-strain curve (van Mier, [102]).
This phenomenon can be explained as follows. Since the post-peak strain is localized in a small region of the specimens (van Mier, [102]). When we
calculate the strains for each specimen, we are using different heights to
divide the same value of displaceme nt (van Mier’s experimental tests
INTRODUCTION
13[102]). This will result in different stra in values. These strain values are not
real in every point of continuous body , but represent some average strains
along the heights of the specimens.
Consequently, as the post-peak deformation is localized, the descending
branch of the stress-strain curve ca nnot be considered as a material
property.
1.2.6 STIFFNESS DEGRADATION
Figure 1.10 shows a typical uniaxial compressive stress-strain curve of
concrete under cyclic loading. As can be seen, the unloading-reloading
curves are not straight-line segments, but loops of changing size with decreasing average slopes.
Fig. 1.10 – Cyclic uniaxial compressive st ress-strain curve (Sinha et al., [98])
Assuming that average slope is the slope of a straight line connecting the
two turning points of one cycle and that the material behavior upon
INTRODUCTION
14unloading and reloading is linearly elas tic (outlined line in Fig. 1.10), then
the elastic modulus (or the slope) degr ades with increasing straining. This
stiffness degradation behavi or is somehow related to damage , which is
significant throughout the post-p eak range (Sinha et al., [98]).
1.3 CONSTITUTIVE MODELING OF CONCRETE MATERIALS
The intensive investigations carried out in recent years have led to a better
understanding of the constitutive behavior of concre te under various
loading conditions. Many theories propos ed in literature for the prediction
of the concrete behavior such as em pirical models, linear elastic, nonlinear
elastic, plasticity based models, mode ls based on endochronic theory of
inelasticity, fracturing models, continuum damage mechanics models,
micromechanics models, etc., are di scussed in the following sections.
1.3.1 EMPIRICAL MODELS
Models in which the material constitu tive law is, derived through a series
of experimental observations, are called empirical model . The experimental
data is then used to propose functions describing the mate rial behavior by
curve fitting. Many empirical uniaxial a nd biaxial stress-strain relations are
available in the literature. Stress-st rain relations specific for ascending
branch and for different kind of load ing are available in the literature:
– compression stress case: Desayi and Krishan [39], Saenz [38], Smith
and Young [99], the European C oncrete Committee (CEB) [41],
Attard and Setunge [6], Richard a nd Abbott [89], Popov ics [7], etc.;
– stress-strain relations for reinforced concrete in tension: e.g.,
Carreira and Chu, [23];
– confined concrete: e.g., Mander et al. [78], Attard & Setange [6],
etc.;
INTRODUCTION
15- biaxial stress-strain relation: Gerstle [49], Chen [30], etc.;
– triaxial stress conditions: Chen [30] , Shina et al. [98], etc. .
1.3.2 LINEAR ELASTIC MODEL
In linear elastic models concrete is treated as linear elastic until it reaches
ultimate strength and subsequently it fail s in a brittle way. For concrete in
tension, since the failure strength is small, linear elastic model is quite
accurate and sufficient to predict th e behaviour of conc rete up to failure
(Babu et al., [7]). Linear elastic stre ss-strain relation using index notation
can be written as:
kl ijkl ijCεσ= (1.4)
where Cijkl represents material stiffness.
Since concrete falls unde r the category of pressu re-sensitive material
whose general response under imposed load is highly nonlinear and
inelastic, this simple linear elastic c onstitutive law is often inappropriate
when the concrete is subject to exte rnal load characterized by elevated
confinements.
1.3.3 NONLINEAR ELASTIC MODEL
Concrete under multiaxial compressive stress states exhibit significant
nonlinearity and linear elastic models fail in these situations. Significant
improvements can be made in this situation using nonlinear constitutive
models. There are two basic approaches followed for nonlinear modelling namely
secant form ulation (total stress-strain) and tangential formulation
(incremental stress-strain ), (Babu et al., [7]).
Incremental stress-strain relation using index notation can be written in the
following form (Gerstle, 1981 [49]):
()kl klt
ijkl ij d C dε ε σ = (1.5)
INTRODUCTION
16where Cijkl t is the tangent material stiffness.
The secant formulation is a simple extension of linear elastic models
formulated by assuming functional relations as in the following form:
()kl ij ijFεσ= (1.6)
The elastic material defined by Eq. (1.6 ) is termed Cauchy elastic material
(Chen and Han, [31]) Secant formulations are load-path independent. It generally comes
approved in literature what the mechanic al behavior of bodies that suffer
irreversible (plastic) strains is func tion of their load history. Concrete
material falls in this circumstance. For this reason which the secant
formulation is applicable primarily for monotonic or proportional loading situations.
In the (linear and nonlinear) elasticity based models, a suitable failure
criterion is incorporated for a complete description of the ultimate strength
surface. Failure can be defined as the ultimate load capacity of concrete
and represents the boundary of the work-hardening region. Many failure
criteria are available in th e literature for normal, high strength, light weight
and steel fibre concrete (Mohr-Coulomb criterion, [31]; Drucker-Prager,
[31]; Chen and Chen, [27]; Ottosen, [83]; Hsieh-Ting-Chen, [54]; Willam
and Warnke, [106], Menetrey and W illam, [79]; Sankara subrsmanian and
Rajasekaran, [91], Fan and Wang, [44], etc. The most commonly used failure criteria are defined in stress space by a number of constants varying
from one to five independent control parameters (Babu et al., [7]). Accumulate plastic (irreversible) deform ations occur in a general concrete
body when certain level of external load-actions are reache d. Elastic based
analysis doesn't contemplate, for genesis, the generation of plastic
deformations. If we remove the extern al load, using these models my body
returns in the original configuration. For this reason that, the elastic models
INTRODUCTION
17result to be inadequate to the math ematical modeling of the mechanical
behavior of concrete.
1.3.4 PLASTICITY -BASED MODEL
The classical theory of plasticity was originally developed for metals. The
deformational mechanisms of metals are quite different from those of
concrete, however, from a macroscopic point of view, they have some
similarities, particularly before failure (Chen and Han, [3 1]). For example,
concrete exhibits a nonlinear stress-strain behavior durin g loading and has
a significant irreversible strain upon unloading. Especially under compressive loadings with confining pressure, concrete may show some
ductile behavior. The irreversible deformations of concrete are induced by microcracking and may be treated by the theory of plasticity (Chen and
Han, [31]). Any plasticity model must invo lve three basic assumptions:
(i) an initial yielding surface, within the stress space, defining the
stress level at which plas tic deformation begins;
(ii) a hardening rule defining the yielding surface evolution after
beginning of plastic deformations;
(iii) a flow rule, which is related to a pl astic potential function, gives an
incremental plastic stress-strain relation.
In plasticity theory the total strain in crement tensor is assumed to be the
sum of the elastic and plastic strain increment tensors:
p
ije
ij ij d d dεεε+= (1.7)
The relationship between in cremental stress and incremental strain can be
formulated as in the following form:
klep
ijklijd C dεσ= (2.105)
INTRODUCTION
18The coefficient tensor in parentheses represent the elastic-plastic stif fness
tensor in t erms of tangent moduli . The formulation to research the previous
tensor is treated in detail in various texts regarding the plasticity theory
(e.g., Chan and Han [31], Lubliner [77], Desay and Siriwardane [37]).
There are many researchers who have us ed plasticity alone to characterize
the concrete behavior (e.g. Chen a nd Chen [27]; Willam and Warnke [106];
Bazant [12]; Dragon and Mroz [40]; Ko tsovos [66]; Ottosen [84]; Hsieh,
Ting, and Chen [54]; Fardis, Alibe, a nd Tassoulas [45]; Schreyer [93]; Yang
, Dafalias, and Herrmann [109]; Verm eer and de Borst [103]; Chen and
Buyukozturk [28]; Schreyer and Babcock [92]; Han and Chen [52]; Onate
et al. [82]; de Boer and Desenkamp [ 36]; Lubliner, Oliver et al [76];
Pramono and Willam [87]; Faruque and Chang [46]; Abu-Lebdeh and
V oyiadjis [1]; Karabinis and Kiousis [62]; Este and Willa m [42]; Menetrey
and Willam [79]; Feenstra and de Bo rst [47]; Balan, Filippou, and Popov
[8]; Jiang and Wang [58]; Li and Ansari [73]; Grassl et al. [51]). The main
characteristic of these models is a plasticity yield surface that includes
pressure sensitivity, load-path sensitiv ity, non-associative flow rule, and
hardening/softening work.
1.3.5 STRAIN SOFTENING AND STRAIN SPACE PLASTICITY .
The stress-strain response after peak (strain softening) depend on many
factors like test equipment, test proce dure, sample dimensions and stiffness
of the machine, etc. (Lubliner [77]).
Classical plasticity theories are developed in stress space where stress and
its increments are treated as indepe ndent variables. Even though stress
space formulation is commonly accepted in engineering practice this approach has some inherent disa dvantages (Babu et al. [7]):
(i) for strain softening materials, th ere is no clarity in defining the
INTRODUCTION
19criteria of loading-unloading.;
(ii) for many structural materials, the slope of the uniaxial stress-strain
curve becomes zero at the ultim ate strength point (peak) where
the stress space formulation may not offer reliable results.
These disadvantages of stress space form ulation can be eliminated with the
help of strain space formulation. The basic formulation of strain space
plasticity have been discussed in th e literature (e.g., Chen and Han [31];
Il’Yushin [56]; Naghdi and Trapp [81]; Casey and Naghdi [2 4]; Pekau et al.
[86]; Kiousis [63]; Mizono and Hatana ka [80]; Barbagelata [9]; Stevens
[101]; Iwan and Yoder [57] ; Dafalias [35]); Runesson et al. [90]; and Lee
[71]; etc.).
1.3.6 FRACTURING AND CONTINUUM DAMAGE MODELS
These models are based on the concep t of propagation of microcracks,
which are present in the c oncrete even before the application of the load.
Damage based models are often used to describe the mechanical behavior
of concrete in tension.
In the earlier class of models, plastic deformation is de fined by usual flow
theory of plasticity and the stiffne ss degradation is modelled by fracturing
theory. A second class of models is based on the use of a set of state
variables quantifying the internal damage resulting from a certain loading
history (Babu et al. [7]). The fundame ntal assumption in these models is
that the local damage in the material can be represented in the form of internal damage variables. Then, the tangential stiffness tensor of the
material is directly related to the internal damage.
The models of this category can describe progressive dama ge of concrete
occurring at the microscopic level, thr ough variables defined at the level of
the macroscopic stress-strain relationshi p. In 1980s, it was established that
INTRODUCTION
20damage mechanics could model accurate ly the strain-softening response of
concrete (Krajcinovic [67] & [68], Lemaitre [69] & [70], Chaboche [25] &
[26]).
Various damage models such as el astic damage, plastic damage are
available in the literature (e.g., Ju [59], Lee and Fenves [72]) or damage
models which use the endochronic theory with continuum damage
mechanics (V oyiadjis [104] & [105]), Wu and Komarakulna nakorn [108]).
1.3.7 MESOMECHANIC ANALYSIS OF CONCRETE BEHAVIOR
Heterogeneous materials like concrete require different levels of
observations to fully understand the mechanism governing their response
behaviors when they are subjected to complex loading cases that activate
non-linear responses. This is partic ularly when traditional macroscopic
models, based on continuou s concept, need observations at meso and,
moreover, micro levels to accurately evaluate and distinguish the rate
sensitivity of the different constituents as well as their influences in the
overall behavior.
Several authors have al ready recognized the importance of mesostructure
evaluations proposing various mesostructural models for concrete (Granger
et al., [50], Lopez et al., [74], Zhu and Tang, [111], Ciancio et al. [33], Etse
et al. [43], Lorefice et al. [75], Caballero et al. [20], etc.). Three main features char acterize these models:
– it includes a non-regular array of pa rticles representing the largest
aggregates;
– a homogeneous matrix modelling th e behavior of mortar plus small
aggregates;
– and the interfaces betw een the two phases.
INTRODUCTION
21The mesomechanic level of observation combined with a plasticity theory
allows to numerically evaluate the influence of the composite
mesostructure and a good characterizat ion of mechanical behavior of
concrete. The disadvantage of the meso mechanic model is the complexity
of theory.
1.3.8 MICROPLANE MODELS
Micromechanical models attempt to de velop the macroscopic stress-strain
relationship from the mechanics of the microstructure. The microplane
model, first proposed by Budianski (1949), for metals in the name of slip
theory of plasticity and later extended to concrete and other geomaterials
like rocks and soils (Bazan t et al. [14], Pande and Sharma [85], Gambarova
and Floris [48], Carol et al. [ 22], Caner et al. [21], etc.).
Unlike the other constitutive models , which characterize the material
behavior in terms of second orde r tensors, the microplane model
characterize in terms of stress and st rain vectors. The macroscopic strain
and stress tensors are determined as a summation of all these vectors on
planes of various orientations (Mic roplanes). The main advantage of
microplane models is its conceptual clarity as the model is formulated in
terms of vectors while the disadvant age in the micropl ane model is the
complexity of theory and the huge computational work.
1.4 MAIN AIMS AND SCOP ES OF THE THESIS
In this Thesis, the application of same classical plasticity-based models to the numerical simulation of the behavi or concrete is discussed in some
detail. Emphasis is placed on the underlying concepts of the yield surface,
the hardening rule, and the flow rule which are suitable for modelling the
overall concrete behavior.
INTRODUCTION
22REFERENCES OF THE FIRST CHAPTER
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BASIC EQUATIONS AND PROCEDURES
302. BASIC EQUATIONS
This chapter deals with the formulation of constitutive equations for general
hardening/softening materials, approached through the “incremental theory” or
“flow theory” of plasticity and a typical algorithm for integrating the constitutive
equations is presented in the last part of the chapter.
2.1 STRESS AND STRESS TENSOR
Stress is defined as the intensity of internal forces acting between particles
of a body on ideal internal surfaces. Let us consider a surface area ∆Ω in
the neighbours of a point P o with a unit vector n normal to the area ∆Ω as
shown in Fig. 2.1. Let Fn be the resultant force due to the action across the
area ∆Ω of the material from one side onto the other side of the cut plane n.
Then the stress vector at point P o associated with the cut plane n is defined
by:
∆Ω ∆Ω n n n n F F F F
0lim
→=nt (2.1)
The state of stress at a point defines the stress vector tn as a function of the
normal direction n.
Fig. 2.1 Continuous body.
BASIC EQUATIONS AND PROCEDURES
31Since we can make an infinite number of cuts through a point, we have an
infinite number of values of tn which, in general, are different from each
other. This infinite number of values of tn characterizes the state of stress at
that point. Fortunately, there is no need to know all the values of the stress
vectors on the infinite number of planes containing the point. If the stress
vectors t1 t2 and t3 on three mutually perpendicular planes are known, the
stress vector on any plane containing this point can be found from
equilibrium conditions at that point.
Fig. 2.2 – Internal forces of continuous body.
Figure 2.3 shows an element OABC with the stress vectors t x ty and tz and tn
acting on its faces OBC, OAC, OAB, and ABC, respectively. Stress vector
tx (ty , tz) represents the stress acting across the cut plane normal to axis x
(y, z) from the negative side onto the positive side.
The unit vector n can be written in the component form:
n = (n x, ny, nz ), (2.2)
and the direction cosines n i are given by:
ni = cos (ei, n). (2.3)
BASIC EQUATIONS AND PROCEDURES
32Let A be the area of ∆ABC. Then the area of perpendicular to the iBaxis,
denoted by A i, is given by:
Ai = A n i . (2.4)
From equilibrium of the body OABC, we get:
tn A = tx Ax +ty Ay +tz Az (2.5)
and using eq. (2.4), we obtain the wellBknow Cauchy’s theorem:
tn = tx nx +ty ny +tz nz. (2.6)
Fig. 2.3 – Stress vectors acting on arbitrary plane n and on the coordinate planes.
In general:
σx,τxy,τxz components of tx
τyx , σy,τyz components of ty
τzx,τzy,σz components of tz
and in the compact tensorial form:
tn = σσ σσ :: :: n (2.7)
where σij denotes the jBth component of the stress vector acting on the iBth
coordinate planes.
BASIC EQUATIONS AND PROCEDURES
33The nine quantities σij required to define the three stress vector tx ty and tz,
are called the components of the stress tensor, which is given by:
=
z zy zxyz y yxxz xy x
ij
σ τ ττσ ττ τ σ
σ (2.8)
It can be shown that the stress tensor ijσ is symmetric (ji ijσ σ=) by means
of considerations of moments equilibrium on a material element.
2.1.1 PRINCIPAL STRESSES AND INVARIANTS OF THE STRESS TENSOR
Suppose that the direction n at a point P o in a body is so oriented that the
shear components of the stress vector tn vanish (Sn =0) and t n = σ n.
The plane n is then called a principal plane at the point, its normal direction
n is called the principal direction, and the scalar normal stress σ is called
the principal stress. At every point in a body, there exist at least three
principal directions. From the definition, we have:
tn = σ n (2.9)
Substituting for tn from Eq. (2.7) leads to:
σσ σσ : : : : n = σ n (2.10)
which implies the following three equations:
(σxBσ) n x +τxy ny+τxz nz=0
τxy nx +(σyBσ) n y+τyz nz=0 (2.11)
τxz nx +τyz ny+(σzBσ) n z=0.
These three linear simultaneous equations are homogeneous for n x, ny and
nz. In order to have a nonBtrivial solution, the determinant of the
coefficients must vanish:
0
σ B σ τ ττσ B σ ττ τ σ B σ
z yz xzyz y xyxz xy x
= (2.12)
BASIC EQUATIONS AND PROCEDURES
34so that this requirement determines the value of σ. There are, in general,
three roots, σ1, σ2 and σ3. Since the basic equation was tn = σ n, these three
possible values of σ are the three possible magnitudes of the normal stress
corresponding to zero shear stress.
Expanding Eq. (2.12) leads to the “characteristics equation”:
03 22
13= − + −I I Iσ σ σ (2.13)
where
B I1 = sum of diagonal terms of σij;
B
z yzyz y
z xzxz x
y xyxy x
2σ ττ σ
σ ττ σ
σ ττ σ+ + = I (2.14)
B I3 = determinant of σij.
It can be easily shown that:
3 2 3 1 2 1 2σσ σσ σσ + + =I
3 2 1 3σσσ=I
where σ1, σ2 and σ3 are the roots of Eq. (2.12), namely the principal stress
values.
Quantities I 1, I2, I3 are the invariants of the stress tensor, their values are
constant regardless of rotation of the coordinates axis.
2.1.2 STRESS DEVIATION TENSOR AND ITS INVARIANTS
It is convenient in material modeling to decompose the stress tensor into
two parts, one called the spherical or the hydrostatic stress tensor and the
other called the stress deviator tensor . The hydrostatic stress tensor is the
tensor whose elements are pδij where p is the mean stress defined as
follows:
) (31
31) (31
31
3 2 1 1 σ σ σ σ σ σ σ + + = = + + = = I pz y x kk ( 2.15)
The components of the stress deviator tensor sij are defined by subtracting
BASIC EQUATIONS AND PROCEDURES
35the spherical state of stress from the actual state of stress. We have:
ij ijs p+ =δ ijσ ( 2.16)
ij ij p s δ−= ijσ ( 2.17)
The components of the stress deviator tensor are given by:
+
=
z yz xzyz y xyxz xy x
z yz xzyz y xyxz xy x
s s ss s ss s s
p 0 00 p 00 0 p
σ τ ττσ ττ τ σ
( 2.18)
hence:
.z yz xzyz y xyxz xy x
z yz xzyz y xyxz xy x
s s ss s ss s s
p 0 00 p 00 0 p
σ τ ττσ ττ τ σ
=
−
( 2.19)
where that δij = 0 and s ij = σij for i ≠j. It is apparent that by subtracting a
constant value from the normal stresses σx, σy and σz no change in the
principal directions results. In terms of the principal stresses, the stress
deviator tensor ijs is:
.321
321
s 0 00 s 00 0 s
p 0 00 p 00 0 p
σ 0 00 σ 00 0 σ
+
=
( 2.20)
An equation similar to Eq. (2.19) can be considered to obtain the invariants
of the stress deviator tensor sij:
03 22
13= − + −J J s J sσ ( 2.21)
where J 1, J2 and J 3 are the invariants of the stress deviator tensor. The
invariants J 1, J2 and J 3 may be expressed in different forms in terms of the
components of S ij or its principal values, s 1, s2 and s 3, or alternatively, in
terms of the components of the stress tensor σij or its principal values, σ1,
σ2 and σ 3 . The following quantities can be defined:
B J1 = is the sum of diagonal terms of s ij ( )01=J ;
BASIC EQUATIONS AND PROCEDURES
36B ( )
[ ] [ ] [ ]( )2
3 22
3 12
2 12 2 2 2
332
222
11 2
612 2 221
21
σ σ σ σ σ στ τ τ
− − −+ + + + +
+ + == = yz xz xy ij ij s s s s s J
( 2.22)
B ki jk ijs s s J21
3=
It can be shown that the invariants J 1, J2 and J 3 are related to the invariants
I1, I2 and I 3 of the stress tensor σ ij through the following relations:
) 27 9 2 (271) 3 (310
3 2 13
1 322
1 21
I I I I JI I JJ
+ − =− ==
( 2.23)
2.1.3 HAIGH-WESTERGAARD STRESS -SPACE
Various geometric representations have been proposed for better pointing
out the stress state described in tensorial terms (see Chen and Han, 1988
[5]).
Among those representations, the HaighBWestergaard stressBspace is very
useful in studying plasticity theory and failure criteria (Lubliner, [13]).
Since the stress tensor σij has six independent components, they can be
considered as positional coordinates in a sixBdimensional space. However,
this is too difficult to deal with a sixBcomponent space. The simplest
alternative is to take the three principal stresses σ1, σ2 and σ3 as
coordinates, and, represent the stress state at a point in threeBdimensional
stressBspace. This space is called the HaighBWestergaard stress space. In
the principal stress space, every point having coordinates σ1, σ2 and
σ3, represents a possible stress state.
BASIC EQUATIONS AND PROCEDURES
37It is possible that two stress states at a point P differ by the orientation of
their principal axes, but not in the principal stress values and are
consequently represented by the same point in the threeBprincipal stress
space. This implies that this type of stress space representation is focused
primarily on the geometry of stress and not on the orientation of the stress
state with respect to the material body.
Fig. 2.4 – HaighBWestergaard stress space.
Consider the straight line ON (Fig. 2.4) passing through the origin and
forming the same angle with respect to each of the coordinate axes. Then,
for every point on this line, the state of stress is one for which σ1= σ2 = σ3.
Thus, every point on this line corresponds to a hydrostatic or spherical state
of stress, while the deviatoric stresses are equal to zero. This line is
therefore termed the “hydrostatic axis”. Furthermore, any plane
BASIC EQUATIONS AND PROCEDURES
38perpendicular to ON is called the “deviatoric plane”. Such plane can be
described by the following equation:
( ) ξ σ σ σ33 2 1= + + ( 2.24)
where ξ is the distance from the origin to the plane measured along the
normal ON.
pI= =
+ + =3 3 3 31 3 2 1 σ σ σξ ( 2.25)
Furthermore the particular deviatoric plane passing through the origin O:
( )03 2 1= + + σ σ σ ( 2.26)
is called the πBplane.
Fig. 2.5 – State of stress at a point projected on a deviatoric plane.
Let us consider an arbitrary state of stress at a given point with stress
components σ1, σ2 and σ3,this state of stress is represented by the point P =
BASIC EQUATIONS AND PROCEDURES
39(σ1, σ2, σ3) in the principal stress space in Fig. 2.4. The stress vector OP
can be decomposed into two components, the vector ON in the direction
n=
31,
31,
31 and the vector NP perpendicular to ON. Thus,
ξ= ON ( 2.27)
The components of vector NP are defined as follows:
( ) ) ; ( ; ;3 2 ; 1 3 2 1 s s s p p p ON OP NP = − − − = − = σ σ σ ( 2.28)
hence, the length ρ of vector NP is given by:
22
32
22
1 2J s s s = + + = ρ . (2.29)
The vectors ON and NP represent the hydrostatic components ( ijpδ) and
the deviatoric stress components ( ijs), respectively, of the state of stress
( ijσ) represented by point P in Fig. 2.4.
Figure 2.5, the axes σ1’, σ2’ and σ3’ are the projections of the axes ( σ1, σ2
and σ3) on the deviatoric plane, and NP is the projection of vector NP on
the same plane.
Developing some simple geometric considerations, we obtain:
123cos s =θ ρ (2.30)
Substituting for ρ from Eq. (2.29) into Eq. (2.30) results:
21
23cos
Js=θ
(2.31)
In a similar manner, the deviatoric stress components s 2 and s 3,we can also
be obtained in terms of the “lode angle” θ:
22
23
32cos
Js=−θπ
BASIC EQUATIONS AND PROCEDURES
4023
23
32cos
Js=+θπ (2.32)
The ξ,ρ,θ coordinates are called Haigh2Westergaard coordinates and
they can be used in alternative to the principal stresses or to the stress
tensor invariants.
In view of Eq. (2.20), (2.31), (2.32), and (2.24), the three principal stresses
of σ ij are given by:
( )
( )
+− +
=
32 cos32 coscos
32
2
321
π θπ θθ
σσσ
J
ppp
(2.33)
( )
( )
+− +
=
32 cos32 coscos
32
31
321
π θπ θθ
ρ
ξξξ
σσσ
(2.34)
2.2 YIELD AND FAILURE CRITERIA
Particular surfaces can be described within the stress space and it is
possible alternative representation to describe states of stresses material
resulting in yielding or failure.
2.2.1 YIELD CRITERIA INDEPENDENT OF HYDROSTATIC PRESSURE
The yield criterion defines the elastic limits of a material under combined
states of stress. In general, the elastic limit or yield stress is a function of
the state of stress, ijσ. Hence, the yield condition can generally be
expressed as:
0 2 1 = ,…..),k,kf(σij (2.35)
where k 1, k2… are material constants.
For isotropic materials, the values of the three principal stresses suffice to
BASIC EQUATIONS AND PROCEDURES
41describe the state of stress uniquely. A yield criterion therefore consists in a
relation of the form:
0 2 1 3 2 1 = ,…..),k,k,σ,σf(σ (2.36)
The three principal stresses can be expressed in terms of the combinations
of the three stress invariants ),J,J(I 3 2 1, where I1 is the first invariant of the
stress tensor, J2 and J3 are the second and third invariants of the deviatoric
tensor. Thus, one can replace Eq. (2.36) by:
0 2 1 3 2 1 = ,…..),k,k,J,Jf(I (2.37)
Furthermore, these three particular principal invariants are directly related
to HaighBWestergaard coordinates ),,(θρξ in the stress space:
0 2 1 = ,…..),k,k,,f( θρξ (2.38)
Yield criteria of materials should be determined experimentally. An
important experimental fact for metals, is that the influence of hydrostatic
pressure on yielding is not appreciable. The absence of a hydrostatic
pressure effect means that the yield function can be reduced to the form:
0 2 1 3 2 = ,…..),k,k,Jf(J (2.39)
The classical yield criteria used for metal are the Tresca and Von Mises
Criteria [Chen and Han, 1988 [5]).
2.2.1.1 The Tresca Yield Criterion.
The first yield criterion for a combined state of stress for metals was
proposed by Tresca (1864), who suggested that yielding would occur when
the maximum shearing stress at a point reaches a critical value k. In terms
of principal stresses:
k=
− − −3 2 3 1 2 121;21;21max σ σ σ σ σ σ (2.40)
where the material constant k may be determined from the simple tension
BASIC EQUATIONS AND PROCEDURES
42test. Then, 20σ=k , in which σ0 is the yield stress in simple tension.
Assuming the ordering of stresses to be σ1 ≥ σ2 ≥ σ3 and using the Eq.
(2.33), we can rewrite:
( ) [ ] ) 60 0 (32 cos cos
31) (21
22 1 ° ≤ ≤ = + − = − θ π θ θ σ σ k J (2.41)
obtaining the Tresca criterion in terms of θ,2J coordinates:
( ) 03sin 2 ) ( 0 2 , 2 = − + = σ πθ θ J J f (2.42)
or in terms of the variables ),,(θρξ :
( ) 03sin 2 ) ( 0 , = − + = σ πθ ρ θ ρ f , (2.43)
Fig. 2.6 – Tresca yield surfaces in principal stress space.
Since the hydrostatic pressure has no effect on the yield surface, Eq. (2.42)
or Eq. (2.43) must be independent by hydrostatic pressure p, the first
invariant I 1, or ξ. On the deviatoric plane, Eq. (2.42) or Eq. (2.43) is a
regular hexagon (Fig. 2.8), whose distance from vertices, from Eq. (2.43):
( )3sin 20
πθσρ
+= (2.44)
while, in a principal stress space, the equations represent the surface in
BASIC EQUATIONS AND PROCEDURES
43figure 2.6.
Fig. 2.7 – Yield criteria (Tresca and von Mises) in the plane stress state ( σ3 = 0).
2.2.1.2 The von Mises Yield Criterion.
The octahedral shear stress is a convenient alternative choice to the
maximum shear stress to formulate a yield criterion for materials which are
pressure independent. The von Mises yield criterion (1913) is based on this
alternative; it states that yielding begins when the octahedral shear stress
reaches a critical value k:
k J oct32322= = τ (2.45)
which, reduces to the simple form:
0 ) (2
2 2 = − =k J J f . (2.46)
Considering Eq. (2.22) and substituting into Eq. (2.46) the following
expression can be derived:
[ ][ ][ ]2 2
3 22
3 12
2 1 3 , 2 , 1 6 ) ( k f = + + = − − − σ σ σ σ σ σ σ σ σ. (2.47)
In a uniaxial tension test:
B σ1 = σ0 σ2 = 0 σ3 = 0;
B [ ][ ][ ]2 2
3 22
3 12
2 1 6k= + + − − − σ σ σ σ σ σ (2.48)
BASIC EQUATIONS AND PROCEDURES
44B [ ][ ][ ]2 2 2
02
0 6 0 0 0 0k= − + − + − σ σ
hence,
30σ=k (2.49)
Equation (2.46) represents a circular cylinder whose intersection with the
deviatoric plane is a circle of radius 22J =ρ :
2
222 2k J= = ρ
k2=ρ . (2.50)
Fig. 2.8 –von Mises yield surfaces in principal stress space.
Fig. 2.9 – Yield criteria in a deviatoric plane.
BASIC EQUATIONS AND PROCEDURES
45If the von Mises and Tresca criteria are made to agree for a simple tension
yield stress, graphically, the von Mises circle circumscribes the Tresca
hexagon as shown in Fig. 2.9. However, if the two criteria are made to
agree for the case of pure shear, the circle will inscribe the hexagon.
2.2.2 FAILURE CRITERION FOR PRESSURE -DEPENDENT MATERIALS
Failure of a material is usually defined in terms of strength limits. As in the
case of the yield criteria, a general form of the failure criteria can be given
by Eq. (2.35) for anisotropic materials and by Eq. (2.36) through (2.39) for
isotropic ones. Yielding of more ductile metals is not affected by
hydrostatic pressure, while failure behavior of many nonBmetallic
materials, such as soils, rocks, and concrete, is hugely influenced by
hydrostatic pressure.
The general shape of a failure surface 0 3 2 1=),J,Jf(I or 0=),,f( θρξ in a
threeBdimensional stress space can be described by its crossBsection with
the deviatoric planes and its meridians in the meridian planes (Figs. 2.4
and 2.10). The cross sections of the failure surface are the intersection
curves between this surface and a deviatoric plane which is perpendicular
to the hydrostatic axis with ξ = const. The meridians of the failure surface
are the intersection curves between this surface and a plane (the meridian
plane) containing the hydrostatic axis with θ = const (see & 2.1.3).
For an isotropic material the crossBsectional shape (deviatoric planes) of the
failure surface has a threefold symmetry [Chen and Han, 1988 [5]).
Therefore, when performing experiments, it is necessary to explore only
the sector θ = 0° to θ = 60°, the other sector being known by symmetry.
The regular ordering of the principal stresses is 3 2 1σ σ σ > > . With this
ordering, there are two extreme case:
1) 3 2 1σ σ σ > = (2.51)
BASIC EQUATIONS AND PROCEDURES
46Eq. (2.53) represents a stress state corresponding to a hydrostatic
stress state h h h
3 2 1 σ σ σ = = , with a further compressive stress
superimposed in one direction. If we substitute Eq. (2.51) into Eq. (
2.31):
21
23cos
21= =Jsϑ
3πϑ= (2.52)
so the meridian corresponding to θ = 60° is called the compression
meridian.
2) 3 2 1σ σ σ = > (2.53)
Eq. (2.53) represents a state stress corresponding to a hydrostatic
stress state h h h
3 2 1 σ σ σ = = , with a tensile stress superimposed in one
direction. In analog manner:
123cos
21= =
Jsϑ
0=ϑ (2.54)
and the meridian corresponding to θ = 0° is called the tensile
meridian.
Furthermore, the meridian determined by θ = 30° is sometimes called the
shear meridian. If we add to a hydrostatic stress state
) (21) (21) (21
3 2 3 1 2 1h h h h h hσ σ σ σ σ σ+ + + = = , a pure shear state
[ ] [ ]
− −h h h h
1 3 3 121; 0 ;21σ σ σ σ , we get :
23
23cos
21= =Jsϑ
BASIC EQUATIONS AND PROCEDURES
476πϑ= (2.55)
Fig. 2.10 – Failure criteria in the meridian planes (a) and in the deviatoric planes (b), for
concretes, (Chen and Han, 1988 [5]).
The failure function for concrete, and other frictional materials, is defined
by experimental data. The available experimental data clearly indicate the
essential features of a failure surface. It is largely accepted that concrete
can be described by a failure surface with curved meridians, indicating that
BASIC EQUATIONS AND PROCEDURES
48the hydrostatic pressure results in increasing shear capacity of the material
itself (Fig 2.10a). A pure hydrostatic loading cannot cause failure for
ordinary stresses, while an elevated hydrostatic stress state, quite
uncommon in civil engineering, can cause failure of material. The value of
ctρρincreases with increasing hydrostatic pressure. It is about 0.5 near the
πBplane and reaches a value of about 0.8 for elevated values of hydrostatic
pressure (Fig 2.10 b).
The shape of the failure of concrete, in the deviatoric plane (Fig. 2.10b),
changes from nearly triangular for tensile and small compression stresses to
a bulged shape (near circular) for higher compressive stresses. The
deviatoric sections are convex and θBdependent (Chen and Han, 1988 [5]).
Based on knowledge concerning the shape of the failure surface of concrete
materials, a variety of failure criteria have been proposed (Mao, 2002 [14]).
In Chapter 5 some of those theories will be discussed. Those criteria are
classified by the number of material constants appearing in the expression
as oneBparameter through fiveBparameter models.
OneBparameter models, as the von Mises or Tresca type of failure surface,
is used for pressure independent materials. Because of the limited tensile
capacity of concrete, the von Mises or Tresca surface is an unsuitable
failure model for concreteBlike materials.
Among twoBparameter models, the DruckerBPrager and MohrBCoulomb
surfaces are the simplest types of pressureBdependent failure criteria
(Lubliner 1990, [13]). We shall consider in more details in the following
discussion these simple and classical failure criteria. TwoBparameters
models with straight lines as the meridians are therefore inadequate for
describing failure surface of concrete in the highBcompression range.
BASIC EQUATIONS AND PROCEDURES
49The refined models, with three, four or five parameters, reproduce all the
important feature of the triaxial failure surface and give a close estimate of
relevant experimental data.
In the present Thesis, some classical models for the elasticBplastic analysis
of concrete, using one, two and three parameters models, will be described
and compared.
2.2.2.1 The Mohr-Coulomb Criterion.
Mohr's criterion (1900), may be considered as a generalized version of the
Tresca criterion. Both criteria are based on the assumption that the
maximum shear stress is the only decisive measure or impending failure.
However, while the Tresca criterion assumes that the critical value of the
shear is constant, Mohr’s failure criterion considers the limiting shears
stress τ in the plane , to be a function of the normal stress σ in the same
plane at the point:
) (σ τf= (2.56)
where )(σf is an experimentally determined function.
Fig. 2.11 – Tresca criterion on σ−τ plane.
BASIC EQUATIONS AND PROCEDURES
50
Fig. 2.12 – Mohr’s criterion on σ−τ plane.
In terms of Mohr's graphical representation of the state of stress, Eq (2.56),
means that failure of material will occur if the radius of the large principal
Mohr’s circle is tangent to the envelope curve )(σf , as shown in Fig. 2.12.
In contrast to the Tresca criterion (Fig. 2.11), it is seen that Mohr's criterion
allows for the effect of the mean stress or the hydrostatic stress.
The simplest form of the Mohr envelope )(σf is a straight line, illustrate
in Fig. 2.13. The equation for the straightBline envelope is known
Coulomb's equation (1776 [8]):
) tan(φ σ τ − =c (2.57)
in which c is the cohesion and φ is the angle of internal friction, both are
material constants determined by experiments. Failure criterion associated
with Eq. (2.57), will be referred as the MohrBCoulomb criterion. In the
special case of frictionless materials, for which φ = 0, this criterion reduces
to the maximum shear stress of Tresca.
BASIC EQUATIONS AND PROCEDURES
51
Fig. 2.13 – MohrBCoulomb criterion: with straight line as failure envelope.
From Eq. (2.57) and for 3 2 1σ σσ ≥ ≥ , the MohrBCoulomb criterion can be
written:
φ φσ σσ σ φ σ σ tan sin2) () (21cos ) (213 1
3 1 3 1
−+ + − = − c (2.58)
or rearranging:
1cos 2sin 1
cos 2sin 1
3 1 =−−+
φφσφφσc c (2.59)
if we define:
φφ
sin 1cos 2 '
−=cfc and φφ
sin 1cos 2 '
+=cft (2.60)
Eq. (2.59) is further reduced to:
1'3
'1= −
c tf fσ σ (2.61)
where ft’ is the strength in simple tension and fc’ is the strength in simple
compression.
BASIC EQUATIONS AND PROCEDURES
52
Fig. 2.14 – MohrBCoulomb criterion and DruckerBPrager criterion in principal stress
spase (a), and in a deviatoric plane (b).
It is sometimes convenient to introduce a parameter m , where:
φφ
sin 1sin 1
''
−+= =
tc
ffm (2.62)
then Eq. (2.61) can be written in the form:
' 3 1cf m = −σσ for 3 2 1σ σσ ≥ ≥ (2.63)
To demonstrate the shape of the threeBdimensional failure surface of the
MohrBCoulomb criterion, we again use Eq. (2.33) and rewrite Eq. (2.59) in
the following form:
0 cos sin3cos
33sin sin31) , , (
22 1 2 1
= − + ++ + =
φ φπθπθ φ θ
cJJ I J I f
(2.64)
or identically in terms of variables ),,(θρξ :
BASIC EQUATIONS AND PROCEDURES
530 cos 6 sin3cos3sin 3 sin 2 ) , , (
= − + ++ + =
φ φπθ ρπθ ρ φ ξ θ ρ ξ
cf
(2.65)
with 0° ≤ θ ≤ 60°.
In principal stress space, this gives an irregular hexagonal pyramid (Fig.
2.14a). Its meridians are straight lines, and its cross section in the πBplain is
an irregular hexagon (Fig. 2.14b). Only two characteristic lengths a
required to draw this hexagon, ρc and ρ t.
2.2.2.2 The Drucker-Prager Criterion.
As we have seen, the MohrBCoulomb failure criterion can be considered a
generalized Tresca criterion accounting for the hydrostatic pressure effect.
The DruckerB Prager criterion, formulated in 1952, is a simple modification
of the von Mises criterion, where the influence of a hydrostatic stress
component on failure is introduced by inclusion of an additional term in the
von Mises expression to give:
0 ) , ( 2 1 2 1 = − + =k J I J I f α (2.66)
and using HaighBWestergaard variables:
0 2 6 ) , ( = − + =k f ρ αξ ρ ξ (2.67)
where α and k are material constants. Eq. (2.66) reduces to the von Mises
criterion, when α is zero,.
The failure surface of Eq. (2.66) in principal stress space is clearly a rightB
circular cone. Its meridian and cross section on the πBplane are shown in
Fig. 2.15.
BASIC EQUATIONS AND PROCEDURES
54
Fig. 2.15 –DruckerBPrager criterion: (a) meridian plane, (b) π plane.
The MohrBCoulomb hexagonal failure surface is mathematically conB
venient only in problems where it is obvious which one of the six sides is to
be used. If this information is not known in advance, the corners of the
hexagon can cause considerable difficulties resulting in complications to
obtain a numerical solution. The DruckerBPrager criterion, as a smooth
approximation to the MohrBCoulomb criterion, can be made to match the
latter by adjusting the size of the cone. For example, if the DruckerBPrager
circle is assumed to fit the outer apices of the MohrBCoulomb hexagon, the
two surfaces coincide along the compression meridian ρc, where 3πθ=,
then the constants α and k are related to the constants c and φ:
) sin 3 ( 3cos 6,
) sin 3 ( 3sin 2
φφ
φφα
−=
−=ck (2.68)
The cone corresponding to the constants in Eq. (2.68) circumscribes the
hexagonal pyramid and represents an outer bound on the MohrBCoulomb
failure surface (Fig. 2.14). On the other hand, the inner cone passes through
BASIC EQUATIONS AND PROCEDURES
55the tension meridian ρt, where 0=θ , and will have the constants:
) sin 3 ( 3cos 6,
) sin 3 ( 3sin 2
φφ
φφα
+=
+=ck (2.69)
Other approximations is possible for 6πθ= along the shear meridian.
The material constants α and k can be determined from the given tensile
failure stress ft’ and compression failure strength fc’. Substituting stress
states:
B Uniaxial compression:
[ ] [ ] [ ]( )
= + + =− = + + =− = = =
− − −2' 2
3 22
3 12
2 1 2'
3 2 1 1'
3 2 1
31
610 0
ccc
f Jf If
σ σ σ σ σ σσ σ σσ σ σ
(2.70)
B Uniaxial tension:
[ ] [ ] [ ]( )
= + + == + + == = =
− − −2' 2
3 22
3 12
2 1 2'
3 2 1 13 2'
1
31
610 0
ttt
f Jf If
σ σ σ σ σ σσ σ σσ σ σ
(2.71)
into the failure condition or Eq. (2.66), one gets:
= − += − + −
0
30
3
'
''
'
kffkff
t
tc
c
αα
(2.72)
and solving for α and k leads to:
( )
( )
+
+−=+−=
3 33
'
'
' '' '' '' '
t
t
c tc tc tc t
ff
f ff fkf ff fα
(2.73)
BASIC EQUATIONS AND PROCEDURES
562.3 LINEAR ELASTIC ISOSTROPIC STRESS -STRAIN RELATION
Elastic materials completely recover their original shape and size after
removing the applied forces. For many materials, the elastic range also
includes a linear relationship between stress and strain. This linear
proportion or the stressBstrain relation in general form is given by:
kl ijkl ijCε σ= ~ (2.74)
where ijklC is the material elastic constant tensor . It may also be remarked
that Eq. (2.74) is the simplest generalization of the linear dependence of
stress and strain observed in the familiar Hooke's experiment in a simple
tension test, and consequently Eq. (2.74) is often referred to as the
generalized Hooke's law .
We need only two independent elastic constants to describe the behavior
for a homogeneous isotropic linear elastic material. This hypothesis for
concrete can be realistic, at a macroscopic level of investigations. Now Eq.
(2.74) can be written in matrix form as:
{}[]{}ε σ C= (2.75)
where the matrix []C is called the matrix of elastic moduli and in the
hypothesis of homogeneous isotropic linear elastic material, Eq. (2.75) in
explicit form becomes:
− +=
yzxzxyzyx
yzxzxyzyx
22 B 10 0 0 0 0022 B 10 0 0 00 022 B 10 0 00 0 0 B 10 0 0 B 10 0 0 B 1
) 2 1 )( 1 (
τττσσσ
γγγεεε
νννν ν νν ν νν ν ν
ν νE(2.76)
It can be shown that Eq. (2.76), when reduced to the twoBdimensional plane
BASIC EQUATIONS AND PROCEDURES
57stress case (γ yz=γxz=0 , τ xz=τyz=0 , σ z=0), take the following simple form:
− +=
xyzyx
xyyx
22 B 10 0 00 B 10 B 10 B 1
) 2 1 )( 1 (
τ0σσ
γεεε
νν ν νν ν νν ν ν
ν νE (2.77)
Fig. 2.16 – Plane stress case.
It is noted that in the plane stress space, the strain component εz is nonBzero
and takes the value:
) (1) ( y x y x zEε εννσ σνε +−− = + − = (2.78)
The plane stress relations given above are commonly used in many
practical applications. For instance, the analysis of thin, flat plates loaded
in the plane of the plate (xBy plane) are often treated as plane stress
problems (fig. 2.16).
The plane strain conditions (γ yz=γxz=0 , τ xz=τyz=0 , εz=0) are normally
found in elongated bodies of uniform cross section subjected to uniform
loading along their longitudinal axis (zBaxis), such as in the case of tunnels
BASIC EQUATIONS AND PROCEDURES
58(fig 2.17). Under the conditions of plane strain, Eq. (2.76) can be reduced
to the simple form:
− +=
xyyx
xyzyx
0
22 B 10 0 00 B 10 B 10 B 1
) 2 1 )( 1 (
τσσσ
γεε
νν ν νν ν νν ν ν
ν νE (2.79)
Fig. 2.17 – Plane strain case.
For this case, the stress component σz has the value:
) ( y x z σσνσ + = (2.80)
Analysis of bodies of revolution under axisymmetric loading is similar to
that for plane stress and plane strain conditions since this problem is also
twoBdimensional. In the usual notation, the nonzero stress components in
the axisymmetric case are σr, σθ, σz, τrθ and the corresponding strains εr,
εθ, εz, γrθ. Equation (2.76) can be reduced:
BASIC EQUATIONS AND PROCEDURES
59
− +=
ϑϑ
ϑϑ
γεεε
νν ν νν ν νν ν ν
ν ν
rzr
rzr
22 B 10 0 00 B 10 B 10 B 1
) 2 1 )( 1 (
τσσσ
E (2.81)
Fig. 2.18 – Axisymmetric case.
2.4 STRESS-STRAIN RELATION FOR WORK -HARDENING
MATERIALS.
Engineering materials usually exhibit a workBhardening behavior consisting
in an evolution of the initial yielding surface, consequently, stress increases
beyond the initial surface produces both elastic and plastic deformations. At
each stage of plastic deformation, a new yield surface, called the
subsequent loading surface , is established. Only elastic deformations and
no plastic ones develop as the point representing the state of stress tends
moving forward the inner part of such surface.
The classical approach used in plasticity is the incremental theory (or flow
theory ) (Chen and Han, 1988 [5]). In this theory the total strain increment
tensor is assumed to be the sum of the elastic and plastic strain increment
BASIC EQUATIONS AND PROCEDURES
60tensors:
p
ije
ij ij d d dε ε ε+ = . (2.82)
According to Hooke’s law, the total stress increment is determined by:
e
ij ijkl kl d C dε σ = (2.83)
where ijklC is the material elastic constant tensor.
2.4.1 PLASTIC POTENTIAL AND FLOW RULE
The flow rule is the necessary kinematic assumption postulated for plastic
deformation or plastic flow . It gives the ratio or the relative magnitudes of
the components of the plastic strain increment tensor p
ijdε.
As the elastic strain can be derived directly by differentiating the elastic
potential function Ω, for hyperelastic or Green elastic materials (Lubliner,
1990 [13]):
ijij
ijσσΩε∂∂=) (
(2.84)
von Mises proposed the similar concept of the plastic potential function ,
which is a scalar function of the stresses, )(ijgσ (Lubliner 1990, [13]). Then
the plastic flow equations can be written in the form:
ijp
ijgd dσλ ε∂∂= (2.85)
where λd is a positive scalar factor of proportionality, which is nonzero
only when plastic deformations develop.
)(ijgσ = constant
defines a surface of plastic potential within the nineBdimensional stress
space. The direction cosines of the normal vector to this surface at the point
ijσ on the surface are proportional to the gradient
ijg
σ∂∂.
In some cases the identity between the yielding function f and the plastic
potential g can be assumed. Hence, the following relationship can be
BASIC EQUATIONS AND PROCEDURES
61derived:
ijp
ijfd dσλ ε∂∂= . (2.86)
Equation (2.86) is called the associated flow rule because the flow rule is
connected or associated with the yield criterion, while relation (2.85) is
called a nonassociated flow rule .
The nonlinear volume change during plastic strain is a feature of concrete
materials. Experimental results indicate that under compressive loadings,
inelastic volume contraction occurs at the beginning of yielding and
volume dilatation occurs at about 75 to 90% of the ultimate stress (Shah
and Chandra, 1968 [15]). Inflection points are usually observed (see Fig.
1.5 in Chapter 1). This kind is not compatible with the associated flow rule
(Chen and Han, 1988 [5]) and a plastic potential is needed to define the
flow rule. For the sake of simplicity, a functional form of the DruckerB
Prager type can be assumed:
2 1 J I g+ =β (2.87)
where β is the dilation angle in the 2 1 J I− plane. Consequently, the flow
rule takes the following form:
∂∂
∂∂+∂∂
∂∂=
∂∂=
ij ij ijp
ijJ
Jg I
Igdgd dσ σλσλ ε2
21
1 (2.88)
and
+ =
∂∂+∂∂=∂∂=
2 2 1 2Jsd sJg
Igdgd dij
ij ij ij
ijp
ij βδ λ δ λσλ ε (2.89)
The incremental plastic volume change (inelastic dilatancy), is given by
λ β ε εd d tr dp
ijp
v 3= = (2.90)
To reflect the nonlinear volume chance, a functional form of β may be
defined according to the available experimental data. For the sake of
BASIC EQUATIONS AND PROCEDURES
62simplicity, a constant value will be assumed for β in the models presented
in the next chapters.
2.4.2 INCREMENTAL STRESS -STRAIN RELATIONSHIP
Loading surface is defined as the subsequent yield surface for an
elastoplastically deformed material; such surface defines the boundary of
the current elastic region. If a stress point lies within this region, no
additional plastic strains develop. On the other hand, if the state of stress is
on the boundary of the elastic region and in the successive time it remains
in the same region, additional plastic deformation will occur, accompanied
by a configuration change of the current loading surface. In other words,
the current loading surface or the subsequent yield surface will change its
current configuration when plastic deformation takes place. Thus, the
loading surface may be generally expressed as a function of the current
state of the stress, and some internal variables of material state such that
0 ,=k) , f(σp
ijijε (2.91)
where p
ijε is the measure of the plastic strain while ()p
ijk kε= is a general
hardening parameter.
For many applications, the quantities p
ijε and k are condensed in only one
parameter, called effective plastic strain pεand defined as:
p
ijp
ij p ε ε ε32= ; (2.92)
hence, Eq. (2.91) becomes:
0=) , f(σpijε . (2.93)
The total strain increment is assumed as the sum of the elastic and the
plastic one:
p
ije
ij ij d d dε ε ε+ =
BASIC EQUATIONS AND PROCEDURES
63The elastic strain increment can be obtained inverting the Eq. (2.83):
kl ijkle
ij d C dσ ε1−= (2.94)
and the plastic strains is obtained from the flow rule in Eq. (2.85).
Then the complete strainBstress relations for an elasticBplastic materials are
expressed as:
ijkl ijkl ijgd d C dσλ σ ε∂∂+ =−1 (2.95)
where λd is a undetermined factor whose value can be derived to comply
with the following conditions:
λd
< = < == = >
0 0 0 00 0 0
df and ) f(σ or ) f(σ ifdf and ) f(σ if
ij ijij (2.96)
The first expression of Eq. (2.96) is called consistency condition and
rendering explicit the same expression, we obtain:
0 ) , ( 0 ) , ( 01 1= = ⇒ =+ +k f and k f dfn
ijn n
ijnσ σ (2.97)
in which ktakes the place of pε of Eq. 2.93 (figure 2.19).
Fig. 2.19 – Graphical representation of Consistency condition.
BASIC EQUATIONS AND PROCEDURES
64The scalar λd can be determined directly from Eq. (2.96) as described in
the following relationships.
0=∂∂+∂∂=p
pij
ijdfdσσfdf εε (2.98)
From Eq. (2.82) we can rewrite:
e
ijp
ij ijd d dε ε ε= − (2.99)
and using Eq. (2.83):
( )p
kl kl ijkl ij d d C dε ε σ − = (2.100)
Substituting Eqs. (2.100) , (2.92) and (2.85) into the consistency condition
(2.98):
032=∂∂
∂∂
∂∂+
∂∂
∂∂= −
ijij p klkl ijkl
ijg gdf gd d Cσfdfσ σλε σλ ε (2.101)
and
032=∂∂
∂∂
∂∂+
∂∂
∂∂
∂∂
−
ijij p klijkl
ijkl ijkl
ijg gdf gCσfd d Cσf
σ σλε σλ ε (2.102)
resolving for λd:
ijij p klijkl
ijkl ijkl
ij
g g f gCσfd Cσf
d
σ σ ε σε
λ
∂∂
∂∂
∂∂−
∂∂
∂∂∂∂
=
32 (2.103)
Substituting Eq. (2.103) into Eq. (2.85):
tr
ijij p klijkl
ijkl ijkl
ij p
trσg
g g f gCσfd Cσf
d∂∂
∂∂
∂∂
∂∂−
∂∂
∂∂∂∂
=
σ σ ε σε
ε
32 (2.104)
then using Eq. (2.100) and rearranging the indices; the following definition
of the stress tensor increment is derived as follows:
BASIC EQUATIONS AND PROCEDURES
65
∂∂
∂∂
∂∂−
∂∂
∂∂∂∂
∂∂
− = kl
de depturstu
rspqkl
pq mnijmn
kl ijkl ij d
g g f gCσfCσg
σfC
d C d ε
σ σ ε σε σ
32 (2.105)
and in compact form:
[ ] klep
ijklklp
ijklijkl ij d C d C C dε ε σ = − = (2.106)
The coefficient tensor in parentheses represents the elastic2plastic stiffness
tensor in terms of tangent moduli .
For a given stress state and loading history, stress increment dσij can be
derived through Eq. (206) as a function of the strain increment dεkl. This
equation is needed in a numerical analysis of plasticity, such as finite
element analysis. If a nonassociated flow rule is used, the elasticBplastic
stiffness tensor is an unsymmetric tensor, while if we adopt an associated
flow rule the some tensor is symmetric as described in the following
elaborations.
The elastic2plastic stiffness tensor is symmetric if the following condition
is satisfied:
ep
klijep
ijklC C= (2.107)
using Eq. (2.105), we render explicit the symmetric condition (Eq. 2.107):
de depturstu
rspqij
pq mnklmn
klijde depturstu
rspqkl
pq mnijmn
ijkl
g g f gCσfCσg
σfC
Cg g f gCσfCσg
σfC
C
σ σ ε σσ σ ε σ
∂∂
∂∂
∂∂−
∂∂
∂∂∂∂
∂∂
−=
∂∂
∂∂
∂∂−
∂∂
∂∂∂∂
∂∂
−
3232
(2.108)
BASIC EQUATIONS AND PROCEDURES
66eliminating the terms that don't compete the determination of the symmetry,
Eq. (2.108) becomes:
pqij
pq mnklmn pqkl
pq mnijmn Cσg
σfC Cσg
σfC∂∂
∂∂=∂∂
∂∂ (2.109)
If gf≠ Eq. (2.109) is an invalid relation and the elasticBplastic stiffness
tensor is an unsymmetric tensor, while if gf= the symmetry is verified:
pqij
pq mnklmn pqkl
pq mnijmn Cσf
σfC Cσf
σfC∂∂
∂∂=∂∂
∂∂ (2.110)
Opportunely modifying the indices of Eq. (110), we obtain:
mnij
mn pqklpq pqkl
pq mnijmn Cσf
σfC Cσf
σfC∂∂
∂∂=∂∂
∂∂ (2.111)
and the subsequent condition:
pqkl
pq mnijmn pqkl
pq mnijmn Cσf
σfC Cσf
σfC∂∂
∂∂=∂∂
∂∂ (2.112)
we have so shown the symmetry condition of Eq. 107, when an associated
flow rule is adopted.
2.4.3 SOFTENING BEHAVIOR
As discussed in Section 1.1.5, axial compression tests on concrete
specimens usually results in softening behavior. Although, softening
behavior is a consequence of the structural features of the test specimen
rather than an intrinsic mechanical propriety of the material, in the present
Thesis the behavior in the postBfailure regime is treated as a particular case
of hardening behavior, where the subsequent yield surface becomes smaller
when the plastic strain take place.
2.5 INTEGRATION SCHEME FOR ELASTO -PLASTIC MODELS .
The incremental constitutive relation for a general elasticBplastic material
has been presented in Section 2.4.2. In particular, Eq. (2.106) relates the
BASIC EQUATIONS AND PROCEDURES
67stress increment dσij to the total strain increment dεkl. However, in a
numerical analysis, since a finite load increment instead of an infinitesimal
one is applied in each loadBstep, the relevant increments of stress and strain
have finite sizes. Therefore, the incremental constitutive relation (§ 2.4.2)
has to be integrated numerically. The algorithms to implement this
numerical integration play an important role and, a buy with the algorithms
for solving nonlinear simultaneous equations, constitute the core of an
elasticBplastic numerical analysis. An unsuitable algorithm may lead not
only to an inaccurate stress solution, but may also delay the convergence of
the equilibrium iteration, and even lead to divergence of the iteration. Since
stress computation is usually timeBconsuming, the global efficiency of an
algorithm is therefore essential.
2.5.1 GENERAL DESCRIPTION OF A GENERAL ELASTOPLASTIC
INTEGRATION
In matrix form, the stress increment ijdσ can be expressed in terms of
elastic strain increment,e
ijdε (as viewed in the section 2.4.2):
( )p
kl kl ijkle
kl ijkl ij d d C d C dε ε ε σ − = = (2.113)
or in terms of total strain increment ijdε, as
klep
ijklij d C dε σ = (2.114)
In a generic loadBstep ([m+1]th step), we already know, at the end of the
mth load step in which the equilibrium iteration has converged, the
following mechanical quantities:
ijmσ,ijmε, pmε (2.115)
We define a trial stress increment, assuming an elastic response of the
specimen:
kl ijkle
ijC ε ∆ σ ∆ = (2.116)
BASIC EQUATIONS AND PROCEDURES
68in which the total strain increment can be derived as follows:
klm
klm
kl ε ε ε ∆− =+1. (2.117)
klmε1+ is the total strain at (m+1)th step and klmε is the total strain at mth
loadBstep.
We assume that at the end of (m+1)th load step, the stress point is in elastic
state, satisfying the following condition:
0< + ) , σ f(pm e
ij ijmε σ ∆ ;
in such case the numerical procedure finishes and we go to the new loadB
step ([m+2]th step).
If it occurs that 0> + ) , σ f(pm e
ijijmε σ ∆ (inadmissible state) the stress state
enters into an elasticBplastic state in the (m+1)th step.
The most common procedure to integrate elasticBplastic models is
presented in the following integration algorithm:
1) At first stepBalgorithm, we calculate the following quantities:
e
kl klm e
ijmσklijkl
σijijgCσf
σ ∆ σ ∆ σ+ + ∂∂
∂∂ (2.117)
pmpf
εε∂∂ (2.118)
From Eq. (2.103) we can calculate the first value of λd:
e
ijm e
ijme
kl klm e
ijme
ijm
σijijσijij pmpσklijkl
σijijkl ijkl
σijij
g g f gCσfd Cσf
d
σ ∆ σ ∆σ ∆ σ ∆σ ∆
σ σεε σε
λ
+ ++ ++
∂∂
∂∂
∂∂−
∂∂
∂∂∂∂
=
32(2.119)
then the new value of the current total stress is:
) (1λ σ ∆ σ ∆ σ σdp
ije
ij ijm
ijm− + =+ (2.120)
BASIC EQUATIONS AND PROCEDURES
69where ) ( λ σ ∆dp
ij is the plastic corrector (see Figure 2.20) defined in Eq.
(2.100) as:
e
ijmσijklijklp
kl ijklp
ijgC d d C d
σ ∆ σλ ε λ σ ∆
+ ∂∂= = ) ( (2.121)
Fig. 2.20 – Graphical representation of the elastic predictor and the plastic corrector
(plastic return).
We can calculate the plastic strain increment, using Eq. (2.85) and the new
value of the yielding function can be finally derived:
()) d d σ f( fpm p
ije
ijijmλ ε λ σ ∆ σ ∆1,+− + = (2.122)
where
e
kl klmσklpm
pm gd
σ ∆ σλ ε ε
++
∂∂+ =1.
Generally, the yield function in Eq. (2.122), at first ( k=1) numerical
iteration assumes a negative value.
We seek that particularly value of λd that brings to zero the yield criterion
(Eq. 2.122). The expression 2.122 describes a function f of an only λd
independent variable .
In numerical analysis, Newton's method (also known as the Newton–
Raphson method) is one of the best known method for finding successively
better approximations to the zeros (or roots) of a realBvalued function.
BASIC EQUATIONS AND PROCEDURES
70Hypothesizing that 1+kdλ is a root of “ f” function ( 0 ) (1=+kd fλ ), we can
write (see figure 2.21):
1 110 ) ( ) ( ) () (+ ++
−−=−−=∂∂
k kk
k kk k
k
d dd f
d dd f d fddf
λ λλ
λ λλ λλλ (2.123)
Fig. 2.21 – An illustration of one iteration of Newton's method.
here, λdf
∂∂ denotes the derivative of the function f. Then by simple algebra
we can derive:
) () ( 1
kk
k k
ddfd fd d
λλλλ λ
∂∂− =+ (2.124)
We will arrest the algorithm process when:
∆
λλλ<
∂∂) () (
kk
ddfd f (2.125)
being ∆ a sufficiently small tolerance.
BASIC EQUATIONS AND PROCEDURES
71Finally, we can know the plastic strain increment
ijp
ijgd dσλ ε∂∂= and the
final plastic corrector ) ( λ σ ∆dp
ij .
Fig. 2.22 – Flow chart of integration scheme for elastoplastic models.
BASIC EQUATIONS AND PROCEDURES
72REFERENCES OF THE SECOND CHAPTER
[1] ABAQUS Theory Manual, ABAQUS, Inc.166 Valley Street Providence, RI 02909,
USA.
[2] ABAQUS Analysis User ’s Manual, ABAQUS, Inc.166 Valley Street Providence, RI
02909, USA.
[3] Bathe, K.2J. (1996), Finite element procedures. Prentice2Hall, Englewood, New
Jersey, USA.
[4] Chen, W. (1982), Plasticity in reinforced concrete. McGraw2Hill, London, England.
[5] W. F. Chen (Author), D. J. Han (Author), Plasticity for Structural Engineers,
October 1988, 606 pages.
[6] Chen, W.F., Constitutive Equations for Engineering Materials, Vol. 1: Elasticity
and Modeling, Elsevier Publications, 1994.
[7] Cook, R. D. Finite Element Modeling for Stress Analysis J. Wiley & Sons, New
York, 1995.
[8] Coulomb, C. A. (1776). Essai sur une application des regles des maximis et minimis
a quelquels problemesde statique relatifs, a la architecture. Mem. Acad. Roy. Div. Sav.,
vol. 7, pp. 3432387.
[9] Crisfield, M. A. Non2linear Finite Element Analysis of Solids and Structures, Vol. 12
2.
[10] C. S. Desai, H. J. Siriwardane, Constitutive Law for Engineering Materials,
Prentice2Hall (1984).
[11] Drucker, D. (1959). A definition of stable inelastic materials. Journal of Applied
Mechanics, 26:1012106.
[12] Etse, G. and Willam, K. (1996). Integration algorithms for concrete plasticity.
Engineering Computations, 13(8):38265.
[13] Jacob Lubliner, Plasticity Theory, Macmillan Publishing, New York (1990).
[14] Mao2hong Yu. Advances in strength theories for materials under complex stress
state in the 20th Century. School of Civil Engineering & Mechanics, Xian Jiaotong
University, Xian, 710049, China.
[15] SP Shah, S Chandra . “Critical Stress, Volume Change, and Microcracking of
Concrete”2 ACI Journal Proceedings, 1968 2 ACI.
BASIC EQUATIONS AND PROCEDURES
73[16] Shah, S., Swartz, S., and Ouyang, C. (1995). Fracture Mechanics of Concrete.
John Wiley & Sons, Inc.
[17] Simo, J.C. and Hughes, T.J.R. (1998). Computational inelasticity. Springer, Berlin,
Germany.
[18] Timoshenko, S. and Goodier J.N. Theory of Elasticity. McGrawll2Hill B.C. 1951.
[19] Zienkiewicz, O. and Taylor, R. (1994). The Finite Element Method, volume 1 & 2.
McGraw2Hill, London, England, 4th edition.
CO.DRI. INTERACTIVE GRAPHICS
743. CO.DRI. INTERACTIVE GRAPHICS
This chapter deals with the mechanical modeling of concrete using the program
CO.DRI. (“Constitutive Drive Interactive Graphics”), originally proposed by
Willam et al.[31]. With a stress/strain history input and material parameters, this
program calculates the resulting stress/strain response using one of the available
material models. The program contains four plasticity material models all described
and calibrated in this chapter.
Numerical simulation in the non-linear range of concrete behavior will be
carried out through the “Constitutive Drive Interactive Graphics” (Willam
and Iordache, [31]) program (also known as Co.Dri.) whose key features
are listed below:
1) interactive simulation of response behavior for plain concrete
under arbitrary input histories in the form of stress, strain
(displacement) and mixed control;
2) interactive sensitivity studies of four classical plasticity models
which have been incorporated into the constitutive driver;
3) interactive comparison with experimental results of 5 load history
tests on concrete which have been included in the data base for
verifications purposes.
3.1 USER OPTIONS
Co.Dri. is a “menu driven” interactive program which prompts the user
with a sequence of questions which are arranged as follows:
CO.DRI. INTERACTIVE GRAPHICS
75Window 1 : Selection of output type
Fig. 3.1 – Window 1 of Co.Dri.
Window 2 : Selection of Load History
Window 2 prompts the user to drive the constitutive model in stress, strain
or mixed control. In other words the user has three options for applying
boundary conditions, either in the form of the history of the three principal
stresses, the three principal strains or a combination of the axial strain with
lateral stress histories (mixed control).
Fig. 3.2 – Window 2 of Co.Dri.
The input histories are either directly defined by the user or by reference to
a particular load history in the database.
Window 3 : Selection of Stress State
Fig. 3.3 – Window 3 of Co.Dri.
From window 3 it’s possible to choose among the different options for
stress state as shown in figure 3.3.
Window 4 : Selection of Concrete Model
CO.DRI. INTERACTIVE GRAPHICS
76The window 4 requests the user to choose one of the four elastoplasticity
model or the Hooke elastic model which are currently incorporated in the
Constitutive Driver.
Fig. 3.4 – Window 4 of Co.Dri.
Window 5 : Selection of Experimental Test
Fig. 3.5 – Window 5 of Co.Dri.
Window 6 : Selection of Material Parameters
Fig. 3.6 – Window 6 of Co.Dri.
CO.DRI. INTERACTIVE GRAPHICS
77The figure 3.6 displays some of the characteristic features of the three
parameters Drucker-Prager (1952) plasticity model (ABAQUS Theory
Manual [1]) (we have for example taken the Drucker-Prager model,
nothing changes for the other models). The values of the material
proprieties are listed in table 3.1.
Table 3.1 – Material proprieties.
mat(1,1) = E Elastic module
mat(2,1) = u Poisson’s ratio
mat(3,1) = yo Hardening/Softening function parameter
mat(4,1) = alpha Yield function parameter
mat(5,1) = beta Plastic potential parameter
mat(6,1) = cp1
Hardening/Softening function parameter mat(7,1) = h
mat(8,1) = yi
mat(9,1) = iend Indicator for the hardening/softening function
mat(10,1) = cp2
Hardening/Softening function parameter mat(11,1) = yc
mat(12,1) = qc
mat(13,1) = ymax
mat(14,1) = qmax
mat(15,1) = yr
mat(16,1) = qr
mat(17,1) = z
Window 7 : Selection Name of Output File
The last information that asks us the interactive program is to declare the
desired name of the output file.
Fig. 3.7 – Window 7 of Co.Dri.
CO.DRI. INTERACTIVE GRAPHICS
783.2 EXPERIMENTAL DATABASE
In the experimental database, 5 tests are loaded. All experiments are stored
in a unified format and labeled to provide easy access using the
Constitutive Driver on one side and characterizing the type of experiment
on the other side. However, the identifier cannot contain all the detailed
characteristics of an experiment. Each experiment is documented by two
plots displaying the stress history and the strain history, the three principal
stresses and strains are recorded vs. time. It is understood, that in quasi-
static experiments 'time' has to be viewed as “pseudo-time” or number of
load-steps.
The database is composed of the load-history tests on concrete with a
nominal strength of f'c = 2.76ksi (19.03N/mm2). The experiments were
performed at the University of Colorado on cylindrical specimen using a
modified Hoek Cell by B.J. Hurlbut (1985, [20]). In those experiments, low
strength concrete was used.
3.2.1 IDENTIFICATION OF EXPERIMENTS
Each experiment is labeled by ten alphanumeric characters. These
descriptors refer to the testing device, the control of the applied load,
characteristics of the load history, the number of the experiment and the
number of repetitions if experiments with identical load histories were
repeated. For example, the label hc03mut.m11 designates an experiment as:
– hc…Hoek Cell.
– 03…Rounded uniaxial compressive strength of the concrete used is 3
ksi.
– m….Mixed boundary conditions: displacement control in the axial
direction and stress control in the radial direction.
– ut…Uniaxial Tension.
CO.DRI. INTERACTIVE GRAPHICS
79- m….Monotonic
– 1….experiment 1.
– 1….repetition of experiment with this particular load history, testing
device, concrete etc.
The experimental test data (Hurlbut, 1985 [20]) present in the database are
the following:
– hc03mut.m11: uniaxial direct tension test, NX specimen (no lateral
reading);
– hc03muc.c11: cyclic unconfined compression test, unloading and
reloading cycles in pre and post peak regime (no lateral reading);
– hc03mcc.c11: 100 psi (0.69 MPa) confined compression test,
unloading and reloading cycles in pre and post peak regime;
– hc03mcc.c21: 500 psi (3.45 MPa) confined compression test,
unloading and reloading cycles in pre and post peak regime;
– hc03mcc.c31: 2000 psi (13.8 MPa) confined compression test,
unloading and reloading cycles in pre and post peak regime;
3.2.2 TEST APPARATUS
The experiments were performed within a servo-controlled MTS
compression – tension test apparatus with 110 kip (where 1kip = 4.448 N ,
hence 110 kip = 0.4891 kN ) axial load capacity. The axial load can be
applied either in load or in displacement control. The triaxial compression
experiments were performed in a modified Hoek Cell on NX size
specimens (d = 54.74 mm, h = 100 mm) and confining pressure was
applied by manually operated pumps ( Hurlbut, 1985 [20]).
CO.DRI. INTERACTIVE GRAPHICS
80
Fig. 3.8 – Hoek triaxial cell.
The surfaces were covered with plastic filler material to avoid the intrusion
of the membrane into air voids due to confining pressure is applied during
the experiment.
Due to the application of the axial load with rigid steel end-caps, the
obtained data for the triaxial compression experiments have to be corrected
to eliminate the initial deformations produced by initial confinement. This
was done by assuming linear elastic material behavior during the first load
step when the confining pressure is applied. Hence, zero axial deformation
was assumed during this load step.
The uniaxial compressive strength of specimen was 2.76 ksi after 28 days
of curing time. The experimental data, as stored in the database are
corrected to eliminate experimental errors data in the initial phase of the
experiment. The adopted material properties for the computed
displacements are for Young's modulus 2.800.000 psi (19306 MPa) and 0.2
for Poisson's ratio.
CO.DRI. INTERACTIVE GRAPHICS
813.3 CONSTITUTIVE MODELS
3.3.1 ASSOCIATED VON MISES PLASTICITY MODEL
The von Mises Criterion (1913), also known as octahedral shear stress
theory, is often used to estimate yielding of ductile materials. The features
of the von Mises model, available in Co.Dri. Interactive Graphics, are:
– Yield criterion:
0
3) () , ( 22 = − =q yJ k J fn (3.1)
in which 2J is the second invariant of the deviatoric stress tensor and
p
ijp
ij q ε ε32= is the equivalent plastic strain.
– Isotropic Hardening/Softening law:
) (q y yn n=
3.3.1.1 Quadratic hardening/softening function.
The expression of quadratic hardening/softening law is the following:
q h q cp y q yn + + =2
021) ( (3.2)
The parameters y 0, cp and h characterize the geometry of curve (fig. 3.9).
Fig. 3.9 – Quadratic Hardening/Softening function (Encinas, 2007 [14]).
CO.DRI. INTERACTIVE GRAPHICS
82in which:
– ymax is the maximum dimension of yield surface (failure reached);
– qmax is the equivalent plastic strain as hardening law reaches the
maxim value;
– y0 is the initial dimension of yielding surface.
The parameters of the hardening law are calibrated using the experimental
test data of Hurlbut at el. (1985 [20]), available in the mentioned database.
The boundary conditions assumed in calibration are listed below:
==∂∂=
max maxmax0
) (0 ) () 0 (
y q yqqyy y
nnn
(3.3)
developing the expressions (3.3), we obtain:
− =−=
maxmaxmax 0 ) ( 2
q cp hqy ycp
(3.4)
3.3.1.2 Simo hardening/softening function.
The exponential expression proposed by Simo et al. (1998, [28]) is the
following:
( )( )q h e y y y q yq cp
i n + − − + =−1
0 0 1 ) ( (3.5)
calibrating the function parameters according to the experimental tests
present in the database of Co.Dri. (Hurlbut, 1985).
The boundary conditions used are the followings (figure 3.10):
==∂∂==
r r nnnn
y q yqqyy q yy y
) (0 ) () () 0 (
maxmax max0
(3.6)
CO.DRI. INTERACTIVE GRAPHICS
83
Fig. 3.10 – Exponential Simo Hardening/Softening function (Simo at el, 1998, [28]).
in which:
– y0 is the initial dimension of yield criterion;
– ymax is the maximum dimension of yield surface (failure reached);
– qmax is the equivalent plastic strain when the hardening law reaches
the maxim value;
– yr is the dimension of yield surface at the ultimate strength (residual
strength);
– qr is the equivalent plastic strain when hardening law reaches the
residual value;
From the third condition of Equation (3.6):
⇒ =∂∂0 ) (maxqqyn( )( ) 0 ) (1max 1
0 = + − − −−h c e y ypq cp
i (3.7)
Now, substituting Eq. (3.7) into the second condition of (3.6):
⇒ =max max) (y q yn
( )( )( )( )max 1max 1
0max 1
0 0 max ) ( 1 q c e y y e y y y ypq cp
iq cp
i − − + − − + =− − (3.8)
We define the following quantity:
CO.DRI. INTERACTIVE GRAPHICS
84( )( )[ ]max 1
max 10 max
01 1q cp
pie q cy yy y z−+ −−= − = (3.9)
Finally, we substitute Eq. (3.9) into the last condition of (3.6):
( ) [ ]rq cp qr cp
r q e cp e z y ymax 1 1
0 1 1− −− − + = (3.10)
The only unknown independent variable in the previous expression is cp1.
Eq. (3.10) is in implicit form, to search the root or solution ( cp1) of the
equation a numerical iteration is used.
3.3.1.3 Simo Modified hardening/softening law.
Fig. 3.11 – Modified Simo Hardening/Softening function (Encinas, 2007 [14]).
The Simo Modified expression proposed by Etse (2007, [14]) et al. can be
placed in the following form:
( )( )
( )
≤ <≤ ≤ + − − +=−−
r cqc q cp
ccq cp
i
nq q q per e yq q per q h e y y yq y21
0 0 0 1) ( (3.11a & b)
– y0 is the initial dimension of yield criterion;
– ymax is the maximum dimension of yield surface (failure reached);
– qmax is the equivalent plastic strain when the hardening law reaches
the maximum value;
CO.DRI. INTERACTIVE GRAPHICS
85- yc is the dimension of yield surface when the expression of
hardening/softening law changes (from 3.11a to 3.11b);
– qc is the equivalent plastic strain when the hardening law reaches the
yc value;
– yr is the dimension of yield surface at the ultimate strength (residual
strength);
– qr is the equivalent plastic strain when the hardening law reaches the
residual value.
We calibrate the function parameters using the Hurlbut experimental test
data (1985). The boundary conditions are the followings:
===∂∂==
r r nc c nnnn
y q yy q yqqyy q yy y
) () (0 ) () () 0 (
21max1max max 10 1
(3.12)
Now we can use the expression obtained in the section 3.3.1.2:
( )( )
( )( )[ ]
( ) [ ]
− − + =+ −−= − == + − − −
− −−−
cq cp qc cp
cq cp iq cp
i
q e cp e z y ye q cpy yy y zh cp e y y
max 1 1
0max 1
max0 max
0max 1
0
1 11 1 10 ) 1 (
(3.13)
and from the last relationship of (3.12), we seek the ultimate constant of the
hardening law:
( )qc qr cp
c re y y−=2 (3.14)
3.3.1.4 Calibration and validation of the von Mises model.
The one-parameter von Mises criterion is calibrated from a triaxial
compression test with radial confinement σr= – 0,69 MPa (figure 3.12),
such assumption is arbitrary since as a matter of principle, only one
CO.DRI. INTERACTIVE GRAPHICS
86parameter defines the von Mises criterion and its value could be different.
Generally for ductile materials, as metals, failure criteria are calibrated with
respect to tension tests.
The Modified Simo hardening/softening law is used and it’s calibrated with
five load paths, one test in direct-tension and four test in compression for
different values of lateral confinement pressure of the fc’ = 19.03 MPa
concrete.
Fig. 3.12 – von Mises failure criterion compared with Hurlbut test data (1985) in the
meridian plane.
Figure 3.12 shows a comparison of the von Mises failure criterion with
Hurlbut test data [20]. Each experimental test is plotted in the I1 – J20.5 plane
through the following transformations:
– direct tension: tf I=1
35 . 0
2tfJ=
– direct compression: cf I−=1
35 . 0
2cfJ=
– confined compression: l ccf f I21 − −=
35 . 0
2l ccf fJ−=
CO.DRI. INTERACTIVE GRAPHICS
87in which ft is the uniaxial tensile strength, fc is the uniaxial
compressive strength, fcc is the confined triaxial compressive
strength and fl is the constant lateral confinement.
The material parameters of the failure surface and the hardening law are
summarized in table 3.2:
Table 3.2 – Material parameters.
hc03mut.m11 hc03muc.c11 hc03mcc.c11 hc03mcc.c21 hc03mcc.c31
Failure
surface
parameter Yn [MPa] 25.04
Modified
Simo
hardening/
softening
law cp1 210799.64 1252.71 929.54 1557.25 196.22
h [MPa] -154027.24 -4695.74 -937.92 0 -211.26
yi [MPa] 28.22 36.32 29.18 25.04 29.69
cp2 -2589.88 -122.60 -17.08 – –
y0 [MPa] 7.48 8.41 6.72 1.51 0.16
yc [MPa] 8.31 8.05 19.17 – –
qc 0.000129 0.00602 0.01067 – –
Figure (3.13) to (3.17) show the performance of the von Mises model
compared with the results of the direct-tension, direct compression and
triaxial compression tests under various radial confinement pressures σr= –
0.69, -3.45, -13.78 MPa.
Fig. 3.13 – Comparison with Direct-Tension-Test Data (Hurlbut, 1985)
CO.DRI. INTERACTIVE GRAPHICS
88For the compression case, the failure criterion overestimates the strength
(figure (3.14)), according to the failure conditions of figure 3.12: the
compression test (hc03muc.c11) is below the compressive meridian of the
failure surface.
Fig. 3.14 – Comparison with Compression Test Data (Hurlbut, 1985)
Fig. 3.15 – Comparison with σr=-0.69 MPa Confined Triaxial Compression Test Data
(Hurlbut, 1985).
One-parameter von Mises criterion doesn't gather the different tensile and
compressive behavior. The model excessively overestimates the uniaxial
CO.DRI. INTERACTIVE GRAPHICS
89tensile strength (figure (3.13)). The initial yielding surface is never reached
from the internal stresses such that the behavior of the material is elastic-
linear up to ultimate load-step.
Fig. 3.16 – Comparison with σr=-3.48 MPa Confined Triaxial Compression Test Data
(Hurlbut, 1985).
The model predictions for confined compression test (hc03mcc.c11) with
radial constant confinement σr = -0.69 MPa (fig. 3.15) is excellent (the
model is just calibrated for this test). Figure 3.12 shows as the same
confined compression test (hc03muc.c11) is situated just along the
compressive meridian.
For medium lateral confinement ( σr = -3.48 MPa) and high lateral
confinement ( σr = -13.78 MPa), the failure criterion underpredicts the
respective strength of the tests (figures (3.16) and (3.17)), according to the
failure conditions of figure 3.12: the confined compression tests with
medium (hc03mcc.c21) and high confinement (hc03mcc.c31), respectively,
are above the compressive meridian of the failure surface.
CO.DRI. INTERACTIVE GRAPHICS
90
Fig. 3.17 – Comparison with σr=-13.79 MPa Confined Triaxial Compression Test Data
(Hurlbut, 1985).
Further details about the implementation of the von Mises model with
Co.Dri. can be found in the final Appendix.
3.3.2 NON-ASSOCIATED DRUCKER -PRAGER PLASTICITY
MODEL (TWO PARAMETERS ).
Drucker Prager model (1952), is typically used for soils, concrete and other
frictional materials. The features of model, available in Co.Dri. Interactive
Graphics, are:
– Yield criterion:
0
3) () , , (122 1 = − + =q yI J k J I fnα (3.15)
1I, is the first invariant of the stress tensor, 2J is the second invariant of the
deviatoric stress tensor and p
ijp
ij q ε ε32= is the equivalent plastic strain.
The parameters α and y f (peak value of ) ( q yn function) are the friction and
the cohesion, respectively, for the failure model, also known as material
CO.DRI. INTERACTIVE GRAPHICS
91constants. These parameters can be calibrated from experimental data as
described in the previous chapter of this thesis (§ 2.2.2.2).
– Isotropic Hardening/Softening law:
) (q y yn n=
The expressions of isotropic hardening/softening law, available in the
Drucker-Prager elastoplasticity model, are the same of the von Mises
model:
– Quadratic hardening/softening function:
q h q c y q yp n + + =2
021) (
– Simo hardening/softening function:
( )( )q h e y y y q yq cp
i n + − − + =−1
0 0 1 ) (
– Modified Simo hardening/softening function:
( )( )
( )
≤ <≤ ≤ + − − +=−−
r cqc q cp
ccq cp
i
nq q q per e yq q per q h e y y yq y21
0 0 0 1) (
The parameters of the hardening law are calibrated using the experimental
test data of Hurlbut at el. (1985), present in the database (§ 3.3.1).
– Plastic potential:
122 1) , , ( I J k J I gβ+ = (3.16)
The form of plastic potential is a Drucker-Prager type, in which the
frictional parameter α is opportunely modified to give β. In our models a
constant value of αβ≠ is assumed. The original plasticity model of
Drucker-Prager (1958) is with α = β in such case the model takes the
diction: “Associated Drucker-Prager plasticity model” not appropriate to
the concrete.
3.3.2.1 Calibration and validation of Drucker-Prager model.
CO.DRI. INTERACTIVE GRAPHICS
92The two-parameters Drucker-Prager failure criterion is calibrated using two
experimental test. Usually a direct tension test and a direct compression test
are used for this purpose (figure 3.18).
The Modified Simo hardening/softening law is used and calibrated with
respect to five load paths, one test in direct-tension and four tests in
compression at different levels of confinement of the fc’ = 19.03 MPa
concrete (Hurlbut, 1985).
Fig. 3.18 – Drucker Prager failure criteria compared with Hurlbut test data (1985) in the
meridian plane.
Figure 3.18 shows a comparison of the Drucker-Prager failure criterion
with Hurlbut test data [20]. Each experimental test is plotted in the I1 – J20.5
plane through the following transformations:
– direct tension: tf I=1
35 . 0
2tfJ=
– direct compression: cf I−=1
35 . 0
2cfJ=
CO.DRI. INTERACTIVE GRAPHICS
93- confined compression: l ccf f I21 − −=
35 . 0
2l ccf fJ−=
The material parameters of the failure surface, flow law and the hardening
law are summarized in table 3.3.
Table 3.3 – Material parameters.
hc03mut.m11 hc03muc.c11 hc03mcc.c11 hc03mcc.c21 hc03mcc.c31
Failure
surface
parameter α 0.4398
Yf [MPa] 4.53
Modified
Simo
hardening/
softening
law cp1 258167.76 1252.71 927.84 1557.25 132.68
h [MPa] -35898.1 -893.5 -238.2 0 -102.6
yi [MPa] 5.36 6.91 5.81 4.76 7.27
cp2 -3171.94 -122.60 -29.56 – –
y0 [MPa] 1.42 1.59 0.13 0.18 0
yc [MPa] 1.58 1.53 3.28 – –
qc 0.000105 0.00602 0.01066 – –
Plastic
Potential
parameter β 0.433 0.1625 0.13 0 -0.16
Figures from (3.19) to (3.23) show the predictions of the Drucker-Prager
plasticity model (two parameters) compared with the results of the direct-
tension experiment and those of triaxial compression tests with radial
confinements σr= -0.69, -3.45, and -13.8 MPa.
The model predictions for direct tension (fig. 3.19) and direct compression
(fig. 3.20) are admirable (the model is just calibrated for these tests). Figure
3.18 shows as the direct tension test (hc03mut.m11) and the direct
compression test (hc03muc.c11) are collocated just along the failure model.
CO.DRI. INTERACTIVE GRAPHICS
94
Fig. 3.19 – Comparison with Direct-Tension-Test Data (Hurlbut, 1985)
Fig. 3.20 – Comparison with Compression Test Data (Hurlbut, 1985)
For low lateral confinement ( σr = -0.69 MPa) the strength prediction of
model is acceptable (figure (3.21)), according to the failure conditions of
figure 3.18.
CO.DRI. INTERACTIVE GRAPHICS
95
Fig. 3.21 – Comparison with σr=-0.69 MPa Confined Triaxial Compression Test Data
(Hurlbut, 1985).
Fig. 3.22 – Comparison with σr=-3.48 MPa Confined Triaxial Compression Test Data
(Hurlbut, 1985).
The model overestimates the material response as the confinement pressure
increases. For medium and high lateral confinement ( σr = -3.48 and -13.79
MPa) the model overestimates the strength (fig. 3.22 and 3.23), according
to the failure conditions of figure 3.18 dealing with the failure envelope of
model with respect to the experimental test (hc03mcc.c21 and
hc03mcc.c31).
CO.DRI. INTERACTIVE GRAPHICS
96
Fig. 3.23 – Comparison with σr=-13.79 MPa Confined Triaxial Compression Test Data
(Hurlbut, 1985).
Fig. 3.24 – Comparison between associated and non-associated plasticity model for
direct compression case.
It’s interesting to observe resulting by consulting whether associated or
non-associated plasticity model. The concrete behavior is not compatible
with the associated flow rule (Chen and Han, 1988 [6]) and a non-
associated plastic potential is needed to define the flow rule. A non-
associated flow rule is adopted to control the amount of inelastic dilatancy:
CO.DRI. INTERACTIVE GRAPHICS
97αβ< ( non-associated flow rule)
and thus the lateral deformation behavior in stress-controlled environments
(from fig. 3.24 to fig.3.27).
Fig. 3.25 – Comparison between associated and non-associated plasticity model with
σr=-0.69 MPa Confined Triaxial Compression.
Fig. 3.26 – Comparison between associated and non-associated plasticity model with
σr=-3.45 MPa Confined Triaxial Compression.
CO.DRI. INTERACTIVE GRAPHICS
98
Fig. 3.27 – Comparison between associated and non-associated plasticity model with
σr=-13.79 MPa Confined Triaxial Compression.
Further details about the implementation of the Drucker-Prager model with
Co.Dri. can be found in the final Appendix.
3.3.3 NON-ASSOCIATED DRUCKER -PRAGER PLASTICITY
MODEL (THREE PARAMETERS ).
The analysis in the previous chapters point out that the behavior of concrete
and granular materials is too complex to be reproduced by plasticity models
initially conceived for ductile materials. However, under the loading
conditions of interest for engineers, rather simple constitutive models lead
to useful design information. These constitutive models are essentially
pressure-dependent plasticity models that have historically been popular in
geotechnics. However, more recently they have also been found to be
useful for modeling some composite materials such as concrete, that
exhibits significantly different yield behavior in tension and compression.
The model described here is an extension of the original Drucker-Prager
model (Drucker and Prager, 1952). The extension of interest includes the
CO.DRI. INTERACTIVE GRAPHICS
99use of noncircular yield surfaces in the deviatoric stress plane, and the use
of non-associated flow law as detailing described in this chapter. Such a
failure criterion can be expressed as follows:
0 tan =− − d pt β (3.17)
with,
⋅− − + =3111121
qr
K Kq t (3.18)
Fig. 3.28 – Failure criterion in the meridian plane.
where:
– 131
31I trace p − = − =ðððð
equivalent pressure stress;
– 2323J S S qij ij= = Mises equivalent stress;
– 31
331
227
29== J S S S rki jk ij third invariant of deviatoric stress;
– ij ij ijp S σ δ+ = , is the deviatoric part of the stress tensor ðððð
.
The model provides a noncircular section and associated inelastic flow in
the deviatoric plane, while separate dilation and friction angles in the
meridian plane. Input data parameters (Hurlbut et al., 1985) define the
CO.DRI. INTERACTIVE GRAPHICS
100shape of the failure and flow surfaces in the deviatoric plane as well as the
friction and dilation angles for the meridian plane.
Fig. 3.29 – Typical failure surface for the Drucker-Prager model (three parameters), in
the deviatoric plane (ABAQUS Theory Manual, [1]).
In this model we define a deviatoric stress measure t (Eq. (3.18)), in which
K is a material parameter. To ensure convexity of the yield surface,
1 778.0 ≤≤K . This measure of deviatoric stress is used because it allows
the matching of different stress values in tension and compression in the
deviatoric plane, thereby providing flexibility in fitting experimental results
when the material exhibits different failure values in triaxial tension and
compression tests. This function is sketched in Figure 3.29.
Since 13
=
qr in uniaxial tension, Kqt= in this case; 13
− =
qr in
uniaxial compression and qt= in this other case. When K = 1, the
dependence on the third deviatoric stress invariant is removed; the Mises
circle is recovered in the deviatoric plane: qt=.
CO.DRI. INTERACTIVE GRAPHICS
101The parameters β, d and K are the material constants and can be calibrated
from experimental data in the following procedure.
We consider the failure values of three experimental tests in terms of stress
invariants:
3 3 32 2 21 1 1
, ,, ,, ,
r q pr q pr q p
(3.19)
Substituting the first test of (3.19) into Eq. (3.17):
d pqr
K Kq = −
− − + βtan111121
13
11
1 (3.20)
Now, substituting the second test of (3.19) and Eq. (3.20) into Eq. (3.17),
we obtain:
βtan ) (111121 111121
1 23
11
13
22
2
p pqr
K Kqqr
K Kq
−=
− − + −
− − +
(3.21)
Resolving for βtan :
) (111121 111121
tan
1 23
11
13
22
2
p pqr
K Kqqr
K Kq
−
− − + −
− − +
=β (3.22)
Finally, substituting the third test (3.19) into the failure criterion (3.17) and
using Eq. (3.22), the only material parameter to determine is K:
0 tan111121
33
33
3 = − −
− − + d pqr
K Kq β (3.23)
An isotropic hardening/softening elastoplasticity model is formulated and
the yield surface can be expressed as follows:
()0 tan = − − q d p t β (3.24)
CO.DRI. INTERACTIVE GRAPHICS
102where )(qd is the Isotropic Hardening/Softening law. As for the previous
models (von Mises and Drucker-Prager) the expressions of isotropic
hardening/softening law are the followings:
– Quadratic hardening/softening function:
q h q cp d q d+ + =2
021) (
– Simo hardening/softening function:
( )( )q h e d d d q dq cp
i + − − + =−1
0 0 1 ) (
– Modified Simo hardening/softening function:
( )( )
( )
≤ <≤ ≤ + − − +=−−
r cqc q cp
ccq cp
i
q q q per e dq q per q h e d d dq d21
0 0 0 1) (
Equally to the previous models, the parameters of the hardening laws are
calibrated using the experimental test data of Hurlbut at el. (1985), loaded
in the database (§ 3.3.1) of Co.Dri..
Fig. 3.30 – Schematic of hardening for the linear model, in the meridian plane.
The plastic potential, also known as the flow potential, chosen in this
model is:
ψtanptg −= (3.25)
CO.DRI. INTERACTIVE GRAPHICS
103where ψ is the dilation angle (see β at Section 2.4.1) in the t–p plane. A
geometrical interpretation of ψ is shown in the t–p diagram of Figure 3.30.
Comparison of Eq. (3.17) and Eq. (3.25) shows that the flow is associated
in the deviatoric plane, because the yield surface and the flow potential
both have the same functional dependence on t.
However, the dilation angle ψ and the material friction angle β may be
different, so the model may not be associated in the t–p plane. For 0=ψ
the material is nondilational and if βψ=, the model is fully associated.
For βψ= and 1=K the original associated Drucker-Prager (1952) model
is recovered.
3.3.3.1 Calibration and validation of Drucker-Prager model.
The three-parameters Drucker-Prager failure criterion is calibrated using
three experimental test. For this reason, a direct tension test a direct
compression test and a confined triaxial compression test (radial stress σr=
-0.69 MPa) are used (figure 3.31).
Fig. 3.31 – Drucker Prager failure criterion (three parameters) compared with Hurlbut
test data (1985), in the meridian plane.
CO.DRI. INTERACTIVE GRAPHICS
104Figure 3.31 shows a comparison of the Drucker-Prager failure criterion
(three parameters) with Hurlbut test data [20]. Each experimental test is
plotted in the I1 – J20.5 plane through the following transformations:
– direct tension: tf I=1
35 . 0
2tfJ=
– direct compression: cf I−=1
35 . 0
2cfJ=
– confined compression: l ccf f I21 − −=
35 . 0
2l ccf fJ−=
The isotropic hardening/softening function is used through the Modified
Simo law, that is calibrated with five load paths, one test in direct-tension
and four test in compression at different levels of confinement of the fc’ =
19.03 MPa concrete (Hurlbut, 1985).
The material parameters defining the failure surface, the nonassociated
flow law and the hardening law are summarized in table 3.4:
Table 3.4 – Material parameters.
hc03mut.m11 hc03muc.c11 hc03mcc.c11 hc03mcc.c21 hc03mcc.c31
Failure
surface
parameter tgβ 2.1477
df [MPa] 5.4058
K 0.788
Modified
Simo
hardening/
softening
law cp1 287156.44 1419.13 2015.57 1769.34 200
h [MPa] -39414 -916.384 -256.79 0 -40.236
di [MPa] 6.05 7.52 5.96 5.36 6.65
cp2 -2996.57 -119.69 -25.83 – –
d0 [MPa] 0.38 1.24 0.77 -34.7 -29.5
dc [MPa] 1.61 1.49 3.95 – –
qc 0.0001126 0.006569 0.007844 – –
Plastic
Potential
parameter tgψ 2.14 1.15 0.75 0 -0.80
CO.DRI. INTERACTIVE GRAPHICS
105Figures from (3.32) to (3.36) show the predictions of the Drucker-Prager
plasticity model (three parameters) compared with the results of the direct-
tension experiments and the triaxial compression tests with radial
confinements σr= -0.69, -3.45, and -13.8 MPa.
Fig. 3.32 – Comparison with Direct-Tension-Test Data (Hurlbut, 1985)
Fig. 3.33 – Comparison with Compression Test Data (Hurlbut, 1985)
For direct tension (fig. 3.32) and direct compression (fig. 3.33), the
Drucker-Prager (three parameters) model predicts the mechanical behavior
CO.DRI. INTERACTIVE GRAPHICS
106in an admirable manner. Figure 3.31 shows as the direct tension test
(hc03mut.m11) and the direct compression test (hc03muc.c11) are
collocated just along the failure model.
According to the failure conditions of figure 3.31, for low constant
confinement ( σr = -0.69 MPa) the mechanical response of model is
acceptable (figure (3.34)).
Fig. 3.34 – Comparison with σr=-0.69 MPa Confined Triaxial Compression Test Data
(Hurlbut, 1985).
For medium and high lateral pressure ( σr = -3.48 and -13.79 MPa) the
model overestimates the confined compression strength (fig. 3.35 and
3.36), according to the failure conditions of figure 3.31 dealing with the
failure envelope of model with respect to the experimental tests
(hc03mcc.c21 and hc03mcc.c31). The model overestimates the material
response as the confinement pressure increases.
CO.DRI. INTERACTIVE GRAPHICS
107
Fig. 3.35 – Comparison with σr=-3.48 MPa Confined Triaxial Compression Test Data
(Hurlbut, 1985).
Fig. 3.36 – Comparison with σr=-13.79 MPa Confined Triaxial Compression Test Data
(Hurlbut, 1985).
Further details about the implementation of the Drucker-Prager model
(three parameters) with Co.Dri., can be found in the final Appendix.
CO.DRI. INTERACTIVE GRAPHICS
1083.3.4 NON-ASSOCIATED BRESLER -PISTER PLASTICITY MODEL
As observed in the previous paragraphs, failure models generally described
by straight line, in the meridian plane, are inadequate for describing the
failure of concrete in the high-confinement range.
The generalized Drucker-Prager surface proposed by Bresler and Pister
(1958 [4]) is a three-parameters model which assumes a parabolic
dependence of 2J on 1I, while the deviatoric sections are independent of
θ:
022
1 1 = − + −J cI bI a (3.26)
The model provides a circular section in the deviatoric plane and curved
meridian in2J,1I plane.
The parameters a , b and c are material constants and can be defined
according to experimental data.
If we consider the failure values of three experimental tests in terms of
stress invariants:
3
23
12
22
11
21
1
,,,
J IJ IJ I
(3.27)
Substituting the tests of (3.27) into Eq. (3.26), we obtain a
nonhomogeneous system of linear equations:
= − + −= − + −= − + −
000
3
23
2
13
12
22
2
12
11
21
2
11
1
J I c I b aJ I c I b aJ I c I b a
(3.28)
CO.DRI. INTERACTIVE GRAPHICS
109and, in matrix form, one obtains:
=
−−−
3
22
21
2
3
2
13
12
2
12
11
2
11
1
111
JJJ
cba
I II II I
(3.29)
and, consequently: b b b b a a a a
= M M M M
(3.30)
Now, inverting equation (3.30), the vector a a a a
of material constants
describing the failure criterion can be determined:
b b b b a a a a - 1 - 1 - 1 - 1 M M M M
= (3.31)
The yield surface, with an isotropic hardening/softening elastoplasticity
model, is the following:
() 022
1 1 = − + −J cI bI q a (3.32)
in which )(qa is the Isotropic Hardening/Softening law, the expression of
the same law, available in this model, are the same one of the other models
previously viewed:
– Quadratic hardening/softening function:
q h q cp a q a+ + =2
021) (
– Simo hardening/softening function:
( )( )q h e a a a q aq cp
i + − − + =−1
0 0 1 ) (
– Modified Simo hardening/softening function:
( )( )
( )
≤ <≤ ≤ + − − +=−−
r cqc q cp
ccq cp
i
q q q per e aq q per q h e a a aq a21
0 0 0 1) (
CO.DRI. INTERACTIVE GRAPHICS
110As in the case of the other introduced models, the parameters of hardening
law are calibrated using the experimental test data of Hurlbut at el. (1985),
present in the database (§ 3.3.1) of the numerical program.
The plastic potential for this model is:
2 1'J I b g− = (3.33)
in which 'b is the dilation angle in the 2 1J I− plane.
For this model, as for the others, the flow law is associated in the deviatoric
plane, while is nonassociated in the meridian planes. When 0'=b , the
material is nondilational. For bb=' and 0=c the original associated
Drucker-Prager (1952) plasticity model is recovered.
3.3.4.1 Calibration and validation of Bresler-Pister model.
The three-parameters Bresler-Pister failure criterion is calibrated using
three experimental test. For this reason, a direct tension test a direct
compression test and a high confined triaxial compression test (radial stress
σr=-13.79 MPa) are used (figure 3.37).
The Modified Simo law is used as isotropic hardening/softening function,
according the same law with five load paths, one test in direct-tension and
four test in compression at different levels of confinement of the fc’ = 19.03
MPa concrete (Hurlbut 1985).
Figure 3.37 shows a comparison of the Bresler-Pister failure criterion (three
parameters) with Hurlbut test data [20]. Each experimental test is plotted in
the I 1 – J20.5 plane through the following transformations:
– direct tension: tf I=1
35 . 0
2tfJ=
– direct compression: cf I−=1
35 . 0
2cfJ=
CO.DRI. INTERACTIVE GRAPHICS
111- confined compression: l ccf f I21 − −=
35 . 0
2l ccf fJ−=
Fig. 3.37 – Bresler and Pister failure criteria (three parameters) compared with Hurlbut
test data (1985), in the meridian plane.
Table 3.5 – Material parameters.
hc03mut.m11 hc03muc.c11 hc03mcc.c11 hc03mcc.c21 hc03mcc.c31
Failure
surface
parameter af [MPa] 2.81
b 0.4527
c [MPa-1] -0.00122
Modified
Simo
hardening/
softening
law cp1 287156.44 1172.76 1958.52 1769.34 250
h [MPa] -20524.19 -471.12 -160.87 0 0
ai [MPa] 3.15 4.08 3.19 2.81 3.03
cp2 -2998.130138 -107.906 -32.881157 – –
a0 [MPa] 0.19 0.56 0.17 -13.55 -16.54
ac [MPa] 0.83 0.81 1.93 – –
qc 0.0001126 0.00694 0.00784 – –
Plastic
Potential
parameter b’ 0.35 0.1875 0.15 0 -0.20
CO.DRI. INTERACTIVE GRAPHICS
112The material parameters of the failure surface, nonassociated flow and the
hardening law are summarized as in table 3.5.
Figures from (3.38) to (3.42) show the predictions of the Bresler-Pister
plasticity model (three parameters) compared with the results of the direct-
tension experiments and those of triaxial compression tests with radial
confinements σr= -0.69, -3.45, and -13.8 MPa.
Figure 3.37 shows as the tensile test (hc03mut.m11) is located just along
the failure meridian. For this reason, that the model predicts in an excellent
manner the mechanical response of uniaxial tensile case (Fig. 3.38).
Fig. 3.38 – Comparison with Direct-Tension-Test Data (Hurlbut, 1985)
The model prediction for direct compression is also excellent (Fig. 3.39).
Figure 3.37 shows that, as for the direct tension case, the direct
compression test (hc03muc.c11) is situated just along the failure model.
Figure 3.37 shows as the confined compression test (hc03mcc.c11) with
radial constant confinement σr = -0.69 MPa is located nearby the
CO.DRI. INTERACTIVE GRAPHICS
113compressive meridian. For this reason the model prediction is acceptable
for low confined test ( σr = -0.69 MPa, see fig. 3.40).
Fig. 3.39 – Comparison with Compression Test Data (Hurlbut, 1985)
Fig. 3.40 – Comparison with σr=-0.69 MPa Confined Triaxial Compression Test Data
(Hurlbut, 1985).
For medium lateral confinement ( σr = -3.48 MPa) the failure criterion
overestimates the strength (figure (3.41)), according to the failure
CO.DRI. INTERACTIVE GRAPHICS
114conditions of figure 3.37: the confined compression test with medium
confinement (hc03mcc.c21) is below the compressive meridian of the
failure surface.
Fig. 3.41 – Comparison with σr=-3.48 MPa Confined Triaxial Compression Test Data
(Hurlbut, 1985).
Fig. 3.42 – Comparison with σr=-13.79 MPa Confined Triaxial Compression Test Data
(Hurlbut, 1985).
CO.DRI. INTERACTIVE GRAPHICS
115Figure (3.42) shows the predictions of the Bresler and Pister model
compared with the the triaxial compression test with radial confinement
σr= -13.78 MPa: the model prediction for this load-case is excellent. Figure
3.37 shows as the ( σr= -13.78 MPa) confined compression test
(hc03mcc.c31) is situated just along the failure meridian.
Further details about the implementation of the Bresler-Pister model with
Co.Dri. can be found in the final Appendix.
CO.DRI. INTERACTIVE GRAPHICS
116REFERENCES OF THE THIRD CHAPTER
[1] ABAQUS Theory Manual, ABAQUS, Inc.166 Valley Street Providence, RI 02909,
USA.
[2] ABAQUS Analysis User ’s Manual, ABAQUS, Inc.166 Valley Street Providence, RI
02909, USA.
[3] Bathe, K.-J. (1996), Finite element procedures. Prentice-Hall, Englewood, New
Jersey, USA.
[4] Bresler B and Pister KS (1958), Strength of concrete under combined stresses, ACI
Mater. J., 55, 321–345.
[5] Chen, W. (1982), Plasticity in reinforced concrete. McGraw-Hill, London, England.
[6] W. F. Chen (Author), D. J. Han (Author), Plasticity for Structural Engineers,
October 1988, 606 pages.
[7] Chen, W.F., Constitutive Equations for Engineering Materials, Vol. 1: Elasticity
and Modeling, Elsevier Publications, 1994.
[8] Chinn, J. and Zimmermann, R. (1965). Behavior of plain concrete under various
high triaxial loading conditions. Technical Report, University of Colorado, Boulder,
USA.
[9] Cook, R. D. Finite Element Modeling for Stress Analysis J. Wiley & Sons, New
York, 1995.
[10] Crisfield, M. A. Non-linear Finite Element Analysis of Solids and Structures, Vol.
1-2.
[11] C. S. Desai, H. J. Siriwardane, Constitutive Law for Engineering Materials,
Prentice-Hall (1984).
[12] Desayi, P. and Krishnan, S., Equation for the stress-strain curve of concrete, ACI
J., Vol. 61(1964)345-350.
[13] Drucker, D. (1959). A definition of stable inelastic materials. Journal of Applied
Mechanics, 26:101-106.
[14] Encinas Galdo J. D. (2007), “Teorias constitutivas elastoplasticas y analisis
computacional del comportamiento mecanico del hormigon ”. Proyecto final de
carrera.
[15] Etse, G. (1992). Theoretische und numerische Untersuchung zum di_usen und
lokalisierten Versagen in Beton. PhD thesis, Universitat Karlsruhe, Karlsruhe,
Germany..
CO.DRI. INTERACTIVE GRAPHICS
117[16] Este, G., Willam, K.J., A fracture-energy based constitutive formulation for
inelastic behavior of plain concrete, 1994, Journal of Engineering Mechanics, ASCE
120, 1983-2011.
[17] Etse, G. and Willam, K. (1996). Integration algorithms for concrete plasticity.
Engineering Computations, 13(8):38-65.
[18] Etse, G. and Willam, K. (1999). Failure analysis of elastoviscoplastic material
models, ASCE-EM, 125(1):60-69.
[19] Gerstle, K.H., Simple formulation of biaxial concrete behavior, ACI Journal,
78(1981)62-68.
[20] Hurlbut, B. J., Experimental and Computational Investigation of Strain-Softening
in Concrete, MS thesis, University of Colorado, Boulder,. 1985.
[21] MD Kotsovos, JB Newman, Behavior of Concrete Under Multiaxial Stress by – ACI
Journal Proceedings, 1977 – ACI.
[22] Lee, J., and G. L. Fenves, Plastic-Damage Model for Cyclic Loading of Concrete
Structures, Journal of Engineering Mechanics, vol. 124, no.8, pp. 892–900, 1998.
[23] Lubliner, J., J. Oliver, S. Oller, and E. Oñate, A Plastic-Damage Model for
Concrete, International Journal of Solids and Structures, vol. 25, no.3, pp. 229–326,
1989.
[24] Jacob Lubliner, Plasticity Theory, Macmillan Publishing, New York (1990).
[25] D. Sfer, I. Carol, R. Gettu and G. Etse, "Study of the Behaviour of Concrete Under
Triaxial Compression", J. of Engng. Mech., V. 128, No. 2, pp. 156-163 (2002).
[26] SP Shah, S Chandra. Critical Stress, Volume Change, and Microcracking of
Concrete – ACI Journal Proceedings, 1968 – ACI.
[27] Shah, S., Swartz, S., and Ouyang, C. (1995). Fracture Mechanics of Concrete.
John Wiley & Sons, Inc.
[28] Simo, J.C. and Hughes, T.J.R. (1998). Computational inelasticity. Springer, Berlin,
Germany.
[29] Sinha B.P., Gerstle K.H., Stress-Strain Relations for Concrete under Cyclic
Loading, and Tulin L.G., Journal of ACI, Proc., 1964, 61(2), pp. 195-211.
[30] J.G.M. van Mier, Strain-softening of concrete under multiaxial loading conditions,
PhD. thesis, Eindhoven University of Technology, (1984).
[31] Willam, K. and Iordache, M.-M., (1996), „Constitutive Driver for Cohesive-
Frictional Materials,'' Proc. 4th-ASCE Materials Conference, Washinton D.C., Nov. 10-
14, 1996.
CO.DRI. INTERACTIVE GRAPHICS
118
[32] Zienkiewicz, O. and Taylor, R. (1994). The Finite Element Method, volume 1 & 2.
McGraw-Hill, London, England, 4th edition.
CONSTITUTIVE MODELS AVAILABLE IN ABAQUS
1194. CONSTITUTIVE MODELS AVAILABLE IN
ABAQUS
This chapter deals with the mechanical modeling of concrete using the finite
element analysis program ABAQUS (Student Edition). A constitutive model for
concrete, which uses the plasticity theory, is available in ABAQUS. Primarily a
detailed description of the model is treated while the last part of the chapter is
devoted to calibrate and validate a model on experimental tests available in
literature.
A large variety of materials of interest in engineering respond elastically at
least under reasonably limited values of stresses and strains. Elastic
behavior means that the deformation is fully recoverable: the specimen
return to its original shape as the load is removed. If the load exceeds some
limit (the “yield load”), the deformation is no longer fully recoverable.
Plasticity theory models the mechanical response of materials when
nonrecoverable deformations occur. The theory has been developed most
intensively for metals, but it is also applied to soils, concrete, rock, ice, etc.
(ABAQUS Manuals, [1] and [2]). These materials behave in very different
ways. For example, large values of pure hydrostatic pressure cause small
inelastic deformation in metals, but quite small hydrostatic pressure values
may result in significant, nonrecoverable volume change in a concrete
specimen. Nonetheless, the fundamental concepts of plasticity theory are
sufficiently general and models based on these concepts have been
successfully developed for a wide range of materials.
Most of the plasticity models in ABAQUS are based on the “incremental”
theory in which the mechanical strain rate is decomposed into an elastic
part and a plastic (inelastic) part. Incremental plasticity models are usually
CONSTITUTIVE MODELS AVAILABLE IN ABAQUS
120formulated in terms of (Lubliner, 1990 [20]):
– a yield surface , which generalizes the concept of “yield load” and it
can be used to determine if the material responds purely elastically at
a particular state of stress, temperature, etc;
– a flow rule , which defines the inelastic deformations that occur if the
material point is no longer responding purely elastically;
– evolution laws of the yield surface as the inelastic strains occur
(hardening/softening behavior) .
This chapter describes the “Concrete Damaged Plasticity” model provided in
ABAQUS (Student Edition) for the analysis of concrete and other quasi-brittle
materials.
The plasticity model of damage concrete is primarily intended to be used for
analysis of structures under cyclic and/or dynamic loading. The model is also
suitable for the analysis of other quasi-brittle materials, such as rocks, mortars
and ceramics; but it is the behavior of concrete that is used in the remainder of
this section to motivate different aspects of the constitutive theory. Under low
confining pressures, concrete behaves in a brittle manner; the main failure
mechanisms are cracking in tension and crushing in compression (ABAQUS
Manuals [1] and [2]).
Brittle behavior of concrete disappears when confining pressure is sufficiently
large to prevent crack propagation. In these circumstances failure is driven by
the consolidation and collapse of the concrete porous microstructure, leading
to a macroscopic response resembling the aim of a ductile material with
hardening work.
Modelling the behavior of concrete under large hydrostatic pressures is out of
the scope of the plastic-damage model available in ABAQUS (Student
Edition). The constitutive theory in this section aims to capture the effects of
irreversible damage associated with the failure mechanisms that occur in
CONSTITUTIVE MODELS AVAILABLE IN ABAQUS
121concrete under fairly low confining pressures (less than four or five times the
ultimate compressive stress in uniaxial compression loading, ABAQUS
Manuals [1] and [2]).
The principal features of concrete are summarized in the following
macroscopic properties (ABAQUS Manuals [1] and [2]):
– different yield strengths in tension and compression, with the initial
yield stress in compression being 10 or more higher time than the
initial yield stress in tension;
– softening behavior in tension as opposed to initial hardening
followed by softening in compression;
– different degradation of the elastic stiffness in tension and
compression; and
– stiffness recovery effects during cyclic loading;
The model assumes that the uniaxial tensile and compressive response of
concrete is characterized by damaged plasticity, as shown in Figure 4.1.
Under uniaxial tension the stress-strain response follows a linear elastic
relationship until the value of the failure stress0tσ is attained. The failure
stress corresponds to the onset of micro-cracking in concrete. Beyond the
failure stress the formation of micro-cracks is represented macroscopically
with a softening stress-strain response, which induces strain localization in
the concrete structure. Under uniaxial compression the response is linear
until the value of initial yield, 0cσ . In the plastic regime the response is
typically characterized by stress hardening followed by strain softening
beyond the ultimate stress,cuσ (Figure 4.1.b).
The plastic-damage model in ABAQUS is based on the models proposed
by Lubliner et al. (1989) [20] and by Lee and Fenves (1998) [19].
An additive strain rate decomposition is assumed for the plasticity model:
CONSTITUTIVE MODELS AVAILABLE IN ABAQUS
122p
ije
ij ij d d dε ε ε+ = (4.1)
where ijdε is the total strain rate, e
ijdε is the elastic part of the strain rate,
and p
ijdεis the plastic part of the strain rate.
Fig. 4.1 – Uniaxial response of concrete in tension (a) and compression (b), ABAQUS
Theory Manual [1].
The stress-strain relations are governed by scalar damaged elasticity:
CONSTITUTIVE MODELS AVAILABLE IN ABAQUS
123( ) ( ) ( )pl el pl eldââââââââ D D D D ââââââââ D D D D ðððð
− = − − = : : 10 (4.2)
where, el
0 D D D D
is the initial (undamaged) elastic stiffness of the material.
( )el eld0 1 D D D D D D D D
− = is the degraded elastic stiffness; and d is the scalar stiffness
degradation variable (in figure 4.1, d c is the stiffness degradation variable
for the compression case and d t is the stiffness degradation variable for the
tensile case), which can take values in the range from zero (undamaged
material) to one (fully damaged material).
Within the context of the scalar-damage theory, the stiffness degradation is
isotropic and characterized by a single degradation variable, d. Following
the usual notions of continuum damage mechanics, the effective stress is
defined (Lubliner, 1989 [20]) as:
( )pl elââââââââ D D D Dðððð
− =:0 (4.3)
The Cauchy stress is related to the effective stress through the scalar
degradation relation:
( )ðððð ðððð
d−=1 (4.4)
In the absence of damage, d=0, the effective stress is equivalent to the
Cauchy stress. It is convenient to formulate the plasticity problem in terms
of the effective stress using equation (4.4) to calculate the Cauchy stress.
Our prediction model has been calibrated not considering the part of the
model regarding the damage. Under such hypothesis the effective stress is
equal to the Cauchy stress and the stiffness degradation variable is null.
4.1 YIELD AND FAILURE SURFACE
The yield function, ( ) p l
ε~ , , , ,ðððð
F F= , represents a surface in effective stress
space, which determines the states of failure or damage. For the plastic-
damage model ( )0~≤ p l
ε , , , ,ðððð
F .
The plastic-damage concrete model uses a yield condition based on the
CONSTITUTIVE MODELS AVAILABLE IN ABAQUS
124yield function proposed by Lubliner et al. (1989) [20] and incorporates the
modifications proposed by Lee and Fenves (1998) [19] to account for
different evolution of strength under tension and compression. In terms of
effective stresses the yield function takes the form:
( ) ( ) ( )( ) 0~ ˆ ˆ ~311 ~
max max ≤ − − − + −−= p l p l p l
ε σ σ γ σ ε β ααεc p q F , , , ,ðððð
(4.5)
where α and γ are material constants;
131
31I trace p − = − =ðððð
is the effective hydrostatic stress;
2323J S S qij ij= =
is the Mises equivalent effective stress;
ij ij ijp S σ δ+ =
is the deviatoric part of the effective stress tensor ðððð
, and maxσ is the
algebraically maximum eigenvalue of ðððð
. The operator applied to the
quantity maxσis called “the Macauley bracket” (Lubliner et al., 1989 [20]):
)ˆ ˆ (21ˆ
max max max σ σ σ+ = (4.6)
The function () p l
ε β~ is given as:
( ) ) 1 ( ) 1 ()~()~( ~α αε σε σε β + − − = p l p l
p l
t tc c (4.7)
p l
cε~
and p l
tε~
are the equivalent plastic strains in compression and tension,
respectively:
dtt
c c∫=
0~ ~ p l p l
ε εɺ
and
dtt
t t∫=
0~ ~ p l p l
ε εɺ
.
CONSTITUTIVE MODELS AVAILABLE IN ABAQUS
125The effective plastic strain rates are given as:
p l p l
11~ ~ε εɺɺ − =c in uniaxial compression;
p l p l
11~ ~ε εɺɺ=t in uniaxial tension;
cσand tσ are the effective tensile and compressive stresses, respectively.
If we take the tensile and compression stress case:
– compression case:
c I p σ31
31
1= − =
c c J q σ σ= = =2
2313 3
0 ˆ
max= σ: 0 ) 0 0 (21ˆ
max = + = σ
0 ) 0 0 (21ˆ
max = − − = − σ
substituting these load conditions into the equation 4.5 we obtain:
( ) 011= − −−c c c σ σ α σα
then,
c cσ σ=
This means that the compressive stress state is automatically
contained into the yield condition (Eq. 4.5).
– tensile case:
t I p σ31
31
1− = − =
t t J q σ σ= = =2
2313 3
tσ σ=maxˆ: t t tσ σ σ σ= + =) (21ˆ
max 0 ) (21ˆ
max = − − = −t tσ σ σ
if we substitute these load conditions into the equation 4.5 we obtain:
( ) ( ) 0~
11= − + +−c t t t σ σ ε β σ α σα p l
resolving for () p l
ε β~:
( ) ( ) ( )ασσα ε β + − − =1 1~
tc p l
CONSTITUTIVE MODELS AVAILABLE IN ABAQUS
126proving the equation (4.7).
The coefficient α can be determined from the biaxial and uniaxial
compressive strength, bσ and cσ. For the cases of uniaxial and biaxial
compression 0max=σ and substituting these two load cases into Eq. (4.5),
we obtain:
– biaxial compression case
b I p σ32
31
1= − = b b J q σ σ= = =2
2313 3
0 ˆ
max= σ : 0 ) 0 0 (21ˆ
max = + = σ 0 ) 0 0 (21ˆ
max = − − = − σ
substituting the previous load conditions into the equation 4.5 we
obtain:
( ) 0 211= − −−c b b σ σ α σα
resolving for α, we obtain:
1 21
2−−
=−−=
cbcb
c bc b
σσσσ
σ σσ σα (4.8)
Typical experimental values of the ratio
cbσσ for concrete are in the range
from 1.10 to 1.16, yielding values of α between 0.08 and 0.12 (Lubliner et
al., 1989 [20]).
The coefficient γ enters the yield function only for stress states of triaxial
compression, when 0max<σ . This coefficient can be determined by
comparing the failure conditions along the tensile and compressive
meridians. By definition, the tensile meridian (TM) represents the points in
which stress states satisfying the condition 3 2 1 max σ σ σ σ= >= and the
compressive meridian (CM) is the locus of stress states such that
CONSTITUTIVE MODELS AVAILABLE IN ABAQUS
1273 2 1 max σ σ σ σ> == , where 3 2 1, ,σ σ σ are the eigenvalues of the effective
stress tensor.
It can be easily shown that,
( ) p qTM − =32
maxσ (4.9)
( ) p qCM − =31
maxσ (4.10)
along the tensile and compressive meridians, respectively, in the following
manner:
– tensile case, 3 2 1 max σ σ σ σ= >= :
32
312 1
1σ σ+− = − =I p
2 1 2 3 σ σ− = = J q
( )12 1
2 132
32
32σσ σσ σ =++ − = − p q
max 1σ σ= is the algebraically maximum eigenvalue of ðððð
.
– compressive case, 3 2 1 max σ σ σ σ> == :
32
313 1
1σ σ+− = − =I p
3 1 2 3 σ σ− = = J q
( )13 1
3 132
31
31σσ σσ σ =−+ − = − p q
max 1σ σ= is the algebraically maximum eigenvalue of ðððð
.
With 0max<σ substituting the expressions (4.9) and (4.10) into the failure
criterion (Eq. 4.5), the same criterion becomes:
( ) ( ) ( )TM p qcσ α α γ γ− = + − + 1 3 132, (4.11)
CONSTITUTIVE MODELS AVAILABLE IN ABAQUS
128( ) ( ) ( )CM p qcσ α α γ γ− = + − + 1 3 131. (4.12)
Let ( ) ( )CM TM c q q K = for any given value of the hydrostatic pressure p, with
0max<σ; then:
3 23
++=γγ
cK (4.13)
resolving for γ:
( )
1 21 3
−−=
cc
KKγ (4.14)
A value of 32=cK , which is typical for concrete (Lubliner et al.,1989 [20]),
gives 3=γ .
If 0max>σ and substituting the equations (4.9) and (4.10) into the failure
surface (Eq. 4.5), the yield conditions along the tensile and compressive
meridians reduce to:
( ) ( ) ( )TM p qcσ α α β β− = + − + 1 3 132, (4.15)
( ) ( ) ( )CM p qcσ α α β β− = + − + 1 3 131. (4.16)
Let ( ) ( )CM TM t q q K = for any given value of the hydrostatic pressure p
with 0max>σ ; then:
3 23
++=ββ
tK . (4.17)
The model provides a noncircular section when 1 <cK (figure 4.2). The
tensile meridian and the compression meridian are characterized by a
bilinear form, the change of inclination occurs at the biaxial compression
CONSTITUTIVE MODELS AVAILABLE IN ABAQUS
129state for the tensile meridian and at the uniaxial compression state for the
compression meridian (fig. 4.7).
Fig. 4.2 – Failure surface in a deviatoric plane, corresponding to different values of K c,
ABAQUS Theory Manual [1].
Fig. 4.3 – Failure surface in plane stress, ABAQUS Theory Manual [1].
CONSTITUTIVE MODELS AVAILABLE IN ABAQUS
130Typical failure surfaces are shown in Figure 4.2 in the deviatoric plane and
in Figure 4.3 for plane-stress conditions ( σIII = 0).
4.2 HARDENING/SOFTENING LAWS
An isotropically hardening/softening yield surface forms the basis of the
model for the inelastic response. The evolution of the yield (or failure)
surface is controlled by two hardening variables, p l
cε~and p l
tε~, linked to
failure mechanisms under compression and tension loading, respectively.
We refer to p l
cε~and p l
tε~as compressive and tensile equivalent plastic strains
(previously defined), respectively. The hardening/softening behavior is
represented from the laws:
==
)~()~(
p l p l
t t tc c c
ε σ σε σ σ (4.18)
were )~( p l
c c cε σ σ = is the isotropic hardening/softening law for the
compression case while )~( p l
t t tε σ σ = is the isotropic softening law for the
tensile case.
4.2.1 COMPRESSIVE BEHAVIOR
We can define the stress-strain behavior of plain concrete in uniaxial
compression outside the elastic range. Compressive stress data are provided
as a tabular function of inelastic strain, i n
cε~. The compressive inelastic
strain is defined as the total strain minus the elastic strain corresponding to
the undamaged material, e l i n
c c c ε ε ε~~ ~− = , where
0~
Ec
cσε= e l
, as illustrated in
Figure 4.4. Positive (absolute) values should be given for the compressive
stress and strain.
CONSTITUTIVE MODELS AVAILABLE IN ABAQUS
131
Fig. 4.4 – Definition of the compressive inelastic strain used for the definition of
compression hardening data, ABAQUS Theory Manual [1].
ABAQUS will issue an error message if the calculated plastic strain values
are negative and/or decreasing with increasing inelastic strain, which
typically indicates that the compressive damage curves are incorrect.
Hardening/Softening data are given in terms of an inelastic strain, i n
cε~,
instead of plastic strain, p l
cε~. ABAQUS automatically converts the inelastic
strain values to plastic strain values using the relationship
p l
cε~=
0) 1 (~
E ddc
cc
cσε−− i n
. In the absence of compressive damage ( dc=0)
i n
cε~= p l
cε~ as the models calibrate in this Thesis.
4.2.2 TENSILE BEHAVIOR
The post-failure behavior for direct tension is modelled defining the strain-
softening behavior for cracked concrete.
The specification of post-failure behavior generally means giving the post-
failure stress as a function of cracking strain, c r
tε~
. The cracking strain is
CONSTITUTIVE MODELS AVAILABLE IN ABAQUS
132defined as the total strain minus the elastic strain corresponding to the
undamaged material:
e l c r
t t tε ε ε~~ ~− =,
where
0~
Et
tσε= e l
as illustrated in Figure 4.5.
Fig. 4.5 – Illustration of the definition of the cracking strain used for the definition of
tension softening data, ABAQUS Theory Manual [1].
ABAQUS will issue an error message if the calculated plastic strain values
are negative and/or decreasing with increasing cracking strain, which
typically indicates that the tensile damage curves are incorrect.
CONSTITUTIVE MODELS AVAILABLE IN ABAQUS
133ABAQUS automatically converts the cracking strain values to plastic strain
values using the relationship p l
tε~=
0) 1 (~
E ddt
tt
cσε−− c r
. In the absence of
tensile damage c r
tε~= p l
tε~ as the work developed in this Thesis.
4.3 NONASSOCIATED FLOW LAW
The plastic-damage model assumes nonassociated potential flow,
ijijGd dσλ ε∂∂= p l
. (4.19)
In the deviatoric plane the inelastic flow is associated, while separate
dilation and friction angles can be defined in the meridian plane.
The flow potential G chosen for this model is the Drucker-Prager
hyperbolic (Lubliner et al., 1989 [20]) function:
( ) ψ ψ εσ tan tan2 2
0 p q Gt − + = (4.20)
where ψ is the dilation angle measured in the p–q plane at high confining
pressure; 0tσ is the uniaxial tensile stress at failure; and ε is a parameter,
referred as the eccentricity, that defines the rate at which the function
approaches the asymptote (figure 4.6). The flow potential tends to a straight
line as the eccentricity tends to zero; the default flow potential eccentricity
is 1.0=ε , which implies that the material has almost the same dilation
angle over a wide range of confining pressure stress values. Increasing the
value of ε provides more curvature to the flow potential, implying that the
dilation angle increases more rapidly as the confining pressure decreases.
This flow potential, which is continuous and smooth, ensures that the flow
direction is defined uniquely. The function asymptotically approaches the
linear Drucker-Prager flow potential at high confining pressure and
intersects the hydrostatic pressure axis at 90° (figure 4.6).
CONSTITUTIVE MODELS AVAILABLE IN ABAQUS
134Because plastic flow is nonassociated, the use of the plastic-damage
concrete model requires the solution of nonsymmetric equations.
Fig. 4.6 – Schematic Plastic Potential with Drucker-Prager exponential form,
ABAQUS Theory Manual [1].
4.4 CALIBRATION AND VALIDATION OF A DAMAGE
PLASTICITY MODEL .
Failure criterion contained into the damage plasticity model available in
ABAQUS is a four-parameters model, proposed by Lubliner et al, 1989
[20]. With the purpose to calibrate the failure surface, four experimental
tests are necessary:
– direct tension test: 0 0= = =III II t If σ σ σ ;
– direct compression test c III II I f−= = = σ σ σ 0 0 ;
– biaxial compression test b III II I f−= = = σ σ σ 0 ;
– confined triaxial compression test III II Iσ σ σ > = ;
CONSTITUTIVE MODELS AVAILABLE IN ABAQUS
135Material parameters of the failure surface and nonassociated flow laws are
calibrated according to the experimental data of Hurlbut et al. (1985) [17]),
as described in the previous sections, and summarized in table 4.1:
Table 4.1 – Material parameters.
hc03mut.m11 hc03muc.c11 hc03mcc.c11 hc03mcc.c21
Failure
surface
parameters f’c [MPa] 19.03
f’t [MPa] 2.77
α 0.0833
γ 2/3
Plastic
Potential
parameters Dilation
angle ψ [°] 55 50 40 0
Eccentricity 0 0 0 0
Fig. 4.7 – Lubliner failure criterion (four parameters) compared with Hurlbut test data
(1985) [17], in the meridian plane.
Figure 4.7 shows a comparison of the Lubliner failure curves with Hurlbut
test data [17]. Each experimental test is plotted in the I 1 M J20.5 plane through
the following transformations:
CONSTITUTIVE MODELS AVAILABLE IN ABAQUS
136- direct tension: tf I=1
35 . 0
2tfJ=
– direct compression: cf I−=1
35 . 0
2cfJ=
– confined compression: l cc f f I21 − −=
35 . 0
2l ccf fJ−=
in which ft is the uniaxial tensile strength, fc is the uniaxial
compressive strength, fcc is the confined triaxial compressive
strength and fl is the constant lateral confinement.
Figure (4.8) to (4.11) show the predictions of the Damage Plasticity model
available in Abaqus compared with the results of the direct-tension test,
direct compression test and the triaxial compression tests with radial
confinements σr= -0.69, -3.45, -13.78 MPa.
Fig. 4.8 – Comparison with Direct-Tension-Test Data (Hurlbut 1985 [17])
CONSTITUTIVE MODELS AVAILABLE IN ABAQUS
137The model predictions for direct tension (fig. 4.8) and direct compression
(fig. 4.9) are excellent (the model is just calibrated for these tests). Figure
4.7 shows as the direct tension test (hc03mut.m11) and the direct
compression test (hc03muc.c11) are situated just along the tensile meridian
and the compressive meridian, respectively.
Fig. 4.9 – Comparison with Compression Test Data (Hurlbut 1985 [17])
Figure 4.7 shows as the confined compression test (hc03mcc.c11) with
radial constant confinement σr = -0.69 MPa is located above the
compressive meridian. For this reason the model underpredicts the triaxial
compression strength for low confined test ( σr = -0.69 MPa, see fig. 4.10).
For medium lateral confinement ( σr = -3.48 MPa) the failure criterion
overestimates the strength (figure (4.11)), according to the failure
conditions of figure 4.7: the confined compression test with medium
confinement (hc03mcc.c21) is below the compressive meridian of the
failure surface.
CONSTITUTIVE MODELS AVAILABLE IN ABAQUS
138
Fig. 4.10 – Comparison with σr=-0.69 MPa Confined Triaxial Compression Test Data
(Hurlbut 1985 [17]).
Fig. 4.11 – Comparison with σr=-3.48 MPa Confined Triaxial Compression Test Data
(Hurlbut 1985[17]).
For high confinement pressure the model keeps in error because it accepts
only positive dilation angle while, as pointed out in various models in
CONSTITUTIVE MODELS AVAILABLE IN ABAQUS
139literature (e.g., Etse – Willam, 1994 [14]) or as seen in the analysis of the
previous chapter, the dilation angle for high confinement is often negative.
The prediction of high-confined compression test ( σr = -13.78 MPa) of
Hurlbut is shown in figure 4.12.
For high lateral confinement ( σr = -13.78 MPa) the model overestimates
the strength (fig. 4.12), according to the failure conditions of figure 4.7
dealing with the failure envelope of model with respect to the experimental
test (hc03mcc.c31).
Fig. 4.12 – Comparison with σr=-13.78 MPa Confined Triaxial Compression Test Data
(Hurlbut 1985 [17]).
CONSTITUTIVE MODELS AVAILABLE IN ABAQUS
140REFERENCES OF THE FOURTH CHAPTER
[1] ABAQUS Theory Manual, ABAQUS, Inc.166 Valley Street Providence, RI 02909,
USA.
[2] ABAQUS Analysis User ’s Manual, ABAQUS, Inc.166 Valley Street Providence, RI
02909, USA.
[3] Bathe, K.MJ. (1996), Finite element procedures. PrenticeMHall, Englewood, New
Jersey, USA.
[4] Chen, W. (1982), Plasticity in reinforced concrete. McGrawMHill, London, England.
[5] W. F. Chen (Author), D. J. Han (Author), Plasticity for Structural Engineers,
October 1988, 606 pages.
[6] Chen, W.F., Constitutive Equations for Engineering Materials, Vol. 1: Elasticity
and Modeling, Elsevier Publications, 1994.
[7] Chinn, J. and Zimmermann, R. (1965). Behavior of plain concrete under various
high triaxial loading conditions. Technical Report, University of Colorado, Boulder,
USA.
[8] Cook, R. D. Finite Element Modeling for Stress Analysis J. Wiley & Sons, New
York, 1995.
[9] Crisfield, M. A. NonMlinear Finite Element Analysis of Solids and Structures, Vol. 1M
2.
[10] C. S. Desai, H. J. Siriwardane, Constitutive Law for Engineering Materials,
PrenticeMHall (1984).
[11] Drucker, D. (1959). A definition of stable inelastic materials. Journal of Applied
Mechanics, 26:101M106.
[12] Etse, G. (1992). Theoretische und numerische Untersuchung zum di_usen und
lokalisierten Versagen in Beton. PhD thesis, Universitat Karlsruhe, Karlsruhe,
Germany..
[13] Este, G., Willam, K.J., A fractureMenergy based constitutive formulation for
inelastic behavior of plain concrete, 1994, Journal of Engineering Mechanics, ASCE
120, 1983M2011.
[14] Etse, G. and Willam, K. (1996). Integration algorithms for concrete plasticity.
Engineering Computations, 13(8):38M65.
[15] Etse, G. and Willam, K. (1999). Failure analysis of elastoviscoplastic material
models, ASCEMEM, 125(1):60M69.
CONSTITUTIVE MODELS AVAILABLE IN ABAQUS
141[16] Gerstle, K.H., Simple formulation of biaxial concrete behavior, ACI Journal,
78(1981)62M68.
[17] Hurlbut, B. J., Experimental and Computational Investigation of StrainMSoftening
in Concrete, MS thesis, University of Colorado, Boulder,. 1985.
[18] MD Kotsovos, JB Newman, Behavior of Concrete Under Multiaxial Stress by M ACI
Journal Proceedings, 1977 M ACI.
[19] Lee, J., and G. L. Fenves, PlasticMDamage Model for Cyclic Loading of Concrete
Structures, Journal of Engineering Mechanics, vol. 124, no.8, pp. 892–900, 1998.
[20] Lubliner, J., J. Oliver, S. Oller, and E. Oñate, A PlasticMDamage Model for
Concrete, International Journal of Solids and Structures, vol. 25, no.3, pp. 229–326,
1989.
[21] Jacob Lubliner, Plasticity Theory, Macmillan Publishing, New York (1990).
[22] Shah, S., Swartz, S., and Ouyang, C. (1995). Fracture Mechanics of Concrete.
John Wiley & Sons, Inc.
[23] Sinha B.P., Gerstle K.H., StressMStrain Relations for Concrete under Cyclic
Loading, and Tulin L.G., Journal of ACI, Proc., 1964, 61(2), pp. 195M211.
[24] Simo, J.C. and Hughes, T.J.R. (1998). Computational inelasticity. Springer, Berlin,
Germany.
[25] J.G.M. van Mier, StrainMsoftening of concrete under multiaxial loading conditions,
PhD. thesis, Eindhoven University of Technology, (1984).
[26] Zienkiewicz, O. and Taylor, R. (1994). The Finite Element Method, volume 1 & 2.
McGrawMHill, London, England, 4th edition.
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
1425. FAILURE CRITERIA FOR CONCRETE
UNDER TRIAXIAL STRESSES
This chapter presents a survey of the strength theories of concrete, under complex
stress states, pointing out the relationships among them. In the second part, the
chapter introduces the features of passive confinement with steel and FRP materials
within the framework of the theory of plasticity.
About one-hundred years ago (in 1900), the well-known Mohr-Coulomb
strength theory (see Chen and Han, 1988 [7]) was established for cohesive-
frictional materials. A considerable amount of theoretical and experimental
research on strength theory of materials under complex stress states has
been carried out in the 20th Century.
For many concrete structures, such as dams or nuclear power plants,
concrete is subjected to multiaxial stresses. Even though most of the
concrete design codes employ uniaxial strength for ultimate design, there is
a need for studying concrete strength under multiaxial stresses. This is
especially true nowadays when engineers are striving to design structures
more economically, as the uniaxial strength has not utilized the full
potential of concrete.
Concrete is a highly nonlinear material, so it is quite difficult to build a
mathematical strength model. Much of the research on concrete strength
under multiaxial stresses has been mainly numerically and experimentally
based, and various models based on experimental data have been proposed
in the past years.
Considerable efforts have been devoted to the formulation of strength
theories and to their correlation with test data, but no single model or
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
143criterion has emerged which is fully adequate. Hundreds of models or
criteria have been proposed. The most common failure criteria are defined
by a number of constants varying from one to five independent parameters,
e.g.:
– two parameters: Leon [24], Hoek and Brown [18], Etse [12], etc.;
– three parameters: Bresler and Pister [4], Willam-Warnke [37],
Menetrey-Willam [29], etc.;
– four parameters: Ottosen [32], Hsieh, Ting, and WF Chen [19], de
Boer et al. [9], etc.;
– five parameters: Willam-Warnke [37], Song-Zhao-Peng [36], Jiang
[21], etc..
5.1 SOME CLASSICAL FAILURE CRITERIA
Some of the above mentioned models will be discussed in the following
sections. Calibration of those models with respect to a set of experimental
result on concrete specimens under various level of confinement will be
also proposed.
5.1.1 LEON FAILURE CRITERION
The isotropic Leon (1935 [24]) failure criterion (two parameters) for
frictional materials like concrete and rocks was originally formulated as a
parabolic expression of Mohr's failure envelope, which may be cast in
terms of the major ( σI) and minor ( σIII) principal stresses as:
0 , ( ('2
'= −
++
−= =L
cIII I
L
cIII I
III I cfmfF Fσ σ σ σσ σ
)) ))ð)ð)ð)ð) (5.1)
with tension being positive, Lm and Lc represent the frictional parameter
and the cohesion of the Leon model.
For plasticity studies, yield and failure criteria are usually expressed in
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
144terms of stress invariants and using the Haigh-Westergaard coordinates (§
2.2). Using the following transformation relationships presented in section
2.1.3:
( )32 cos32
31cos32
31
π θ ρ ξ σθ ρ ξ σ
+ + =+ =
IIII
(5.2)
equation (5.1) can be transformed as follows:
0
32
'1 sin23cos 2 / 132
'sin 2 / 1 cos23
'2 2
= + −
−+
+
ξ θ θ ρθ θρ
c cc
fm
fmf
(5.3)
where θρξ,, are the Haigh – Westergaard coordinates (§2.1.3), which are
directly related to the stress tensor invariants:
2 / 3
2 321
2 / 27 3 cos23
J JJI
===
θρξ
(5.4)
where I1 is the first invariant of the stress tensor, J2 and J3 are the second
and third invariants of the deviatoric tensor (§ 2.1).
Equation (5.3) defines a failure criterion with curved meridians (figure 5.1)
and noncircular cross sections on the deviatoric planes.
Leon criterion is a two – parameters model, which implies that two load
condition cases are strictly required for its directed calibration:
– direct compression stress case:
'0c III II I f− = = = σ σ σ (5.5)
Substituting this load condition into the failure criterion as
expressed in Eq. (5.1), we have:
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
1450''2
''
= −
−+
L
cc
L
cccffmff (5.6)
resolving for the cohesion parameter cL:
L Lm c−=1 (5.7)
– direct tension stress case:
0'= = =III II t If σ σ σ (5.8)
Substituting into the failure surface as formulated in Eq. (5.1):
0''2
''
= −
+
L
ct
L
ctcffmff (5.9)
Fig. 5.1 – Leon failure criterion [24] (two parameters) compared with Hurlbut test data
(1985) [20], in the I 1-J20.5 plane.
Now, using the relation (5.7) into Eq. (5.9) and resolving for the frictional
coefficient m L, we obtain:
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
1460 1''2
''
= − +
+
L
ct
L
ctmffmff (5.10)
then,
+
−
=
''2
''
11
ctct
L
ffff
m (5.11)
Calibrating the Leon failure model with the experimental test data of
Hurlbut (1985 [20]), the values of the constant materials calculated with the
relations (5.11) and (5.7) are the followings:
143 . 0857 . 0
==
LL
cm
(5.12)
Figure 5.1, based on the comparison with respect to Hurlbut’s cylindrical
tests, shows that Leon criterion underpredicts the confined triaxial
compression strength, as the lateral confinement increases.
5.1.2 HOEK AND BROWN FAILURE CRITERION
A modification of Leon criterion was proposed by Hoek and Brown (1980
[18]):
0 , ( ('2
'= −
+
−= =HB
cI
HB
cIII I
III I cfmfF Fσ σ σσ σ
)) ))ð)ð)ð)ð) (5.13)
Both parabolic failure criteria are described by two strength parameters:
– the uniaxial compressive strength '
cf;
– and the friction parameter HBm;
whereby, the cohesion parameter is not an independent variable at ultimate
strength.
The model can be calibrated in same way of the previous Leon model,
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
147considering only the two followings elementary stress states:
– direct compression stress case :
'0c III II I f− = = = σ σ σ (5.14)
substituting Eq. (5.14) into the failure criterion Eq. (5.13), we
have:
02
''
= −
HB
cccff (5.15)
which implies that the cohesion parameter cHB:
1=HBc (5.16)
– direct tension stress case :
0'= = =III II t If σ σ σ (5.17)
substituting into the failure surface of Eq. (5.13):
0 1''2
''
= −
+
ct
HB
ct
ffmff (5.18)
and resolving for the frictional coefficient m HB, we obtain:
' '2'2'
t ct c
HBf ff fm−= (5.19)
The failure criterion may be expressed in terms of the mean normal stress
31I=σ , the deviatoric stress 22J =ρ , and the polar angle θ as:
0 1 cos23
'sin 2 / 1 cos23
'2 2
= −
+ +
+
θ ρ σ θ θρ
c c fm
f (5.20)
Similarly to Leon, Hoek and Brown (5.20) define a failure criterion with
curved meridians (figure 5.2) and noncircular cross sections on the
deviatoric planes.
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
148The constant materials calibrated in accord with Hurlbut [20] experimental
tests are the followings:
1844 . 6
==
LL
cm
(5.21)
Figure 5.2 shows that the Leon modified model performed by Hoek and
Brown produces some good results. The strength measures of envelope
model, in comparison with experimental tests, are acceptable for low and
middle levels of confinement. Only when high ranges of hydrostatic
pressure is applied to the specimens, that the model underpredicts the
triaxial strength (figure 5.2).
Fig. 5.2 – Hoek and Brown [18] failure criterion (two parameters) compared with
Hurlbut test data (1985) [20], in the I 1-J20.5 plane.
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
1495.1.3 WILLAM AND WARNKE (THREE PARAMETERS ) FAILURE
CRITERION
The Willam-Warnke [37] yield criterion is a function that is used to predict
when failure occur in concrete and other cohesive-frictional materials such
as rock, soil, and ceramics. The early version of the three – parameter
surface developed by Willam and Warnke retains the linear p–q relation,
but deviatoric sections exhibit lode angle θ – dependence:
meridiantensileq b p bmeridian e compressiv q a p a
00
1 01 0
= + += ++
(5.22)
where, a 0, a1, b0 and b 1 are material constants;
131
31I trace p − = − =ðððð
is the spherical stress;
2323J S S qij ij= =
is the Mises equivalent stress;
ij ij ijp S σ δ+ =
is the deviatoric part of the stress tensor ðððð
.
Since the two meridians must intersect the hydrostatic axis at the same
point, it fallows that:
A b a= =0 0 . (5.23)
The three parameters can be determined by three typical tests:
– Uniaxial tension: '
31
t tf p− = '
t tf q= °=0θ
– Uniaxial compression: '
31
c cf p= '
c cf q= °=60θ
– Confined compression: 32'
l cc
ccf fp+= l cc cc f f q− =' °=60θ .
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
150Substituting these load conditions into the failure criterion (Eq. 5.22), we
have:
meridian e compressivpp
aa
qq
ccc
ccc
−−=
10
11
(5.24)
then,
meridian e compressivpp
qq
aa
ccc
ccc
−−
=
−1
10
11
(5.25)
with A b a= =0 0:
meridiantensileqp bb
tt−−=0
1 (5.26)
Using the well-known Hurlbut [20] experimental tests, the constant
materials assume the following values:
MPa A813.3= 534 . 01−=a 067 . 11−=b (5.27)
Fig. 5.3 – Willam-Warnke [37] failure criterion (three parameters) compared with
Hurlbut test data (1985) [20], in the I 1-J20.5 plane.
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
151Once the two meridians have been determined from a set of experimental
data, the cross section can be constructed by connecting the meridians and
using appropriate curves.
If we use an appropriate function in a generic deviatoric section, Willam
and Warnke failure criterion, is convex and smooth everywhere. This is
achieved by using a portion of an elliptic curve proposed by Willam and
Warnke [37] (fig. 5.4). Due to the threefold symmetry, it is only necessary
to consider the part °≤≤° 60 0θ .
Fig. 5.4 – Elliptic approximation of Willam-Warnke failure criterion [34] (three
parameters), in the deviatoric sections.
The equation of the ellipse, in terms of polar coordinates (q, θ), can be
expressed in terms of the parameters qt and q c:
( )2 2 22 2 2 2
) 1 2 ( cos ) 1 ( 44 5 cos ) 1 ( 4 ) 1 2 ( cos ) 1 ( 2
− + −− + − − + −=e ee e e e eq qcθθ θθ (5.28)
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
152where
c tq q e/=
In which qt represents the semiminor axis and qc the semimajor axis of the
ellipse.
Two limiting cases of Eq. (5.28) can be observed. First, for 1 / = =c tq q e,
the ellipse degenerates into a circle (similar to the deviatoric trace of the
von Mises (1913) or Drucker-Prager (1952) models as in figure 5.5).
Second, when the ratio 5 . 0 / = =c tq q e , the deviatoric trace becomes nearly
triangular (similar to the deviatoric section of the Rankine criterion (1876),
see figure 5.5).
Fig. 5.5 – Schematic representations of failure criteria in the deviatoric planes, Chen
and Han (1988 [7]).
The triaxial failure criterion, approximated by the elliptic description in the
deviatoric region (Willam and Warnke 1975 [37]), generate a Cl-continuous
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
153surface with the complete elimination of corners in the deviatoric trace.
With the advantage that numerical applications whit plasticity based
models don't suffer convergence problems when the yield surface is smooth
and convex (Etse and Willam, 1994 [13]).
The triangular deviatoric curve has corners at the compressive meridians
(3πθ= in fig. 5.5), in which the first derivate of function is discontinue.
Therefore, both convexity and smoothness of the failure curve (Eq. 5.28)
can be assured for 1 / 5 . 0≤ =<c tq q e.
5.1.4 WILLAM AND WARNKE (FIVE PARAMETERS ) FAILURE
CRITERION
The Willam-Warnke five-parameter [37] model has curved tensile and
compressive meridians expressed by quadratic parabolas of the form:
meridiantensileq b q b p bmeridian e compressiv q a q a p a
00
2
2 1 02
2 1 0
= + + += + + +
(5.29)
where p is the spherical stress and q is the Mises equivalent stress defined
in the previous section; a 0, a1, a2, b0, b1, b2 are material constants.
The two meridians must intersect the hydrostatic axis at the same point
again and it follows that:
A b a= =0 0. (5. 30)
The five parameters can be determined by five typical tests:
– Uniaxial tension 0 0'= = > =III II t If σ σ σ :
'
31
t tf p− = '
t tf q= °=0θ
– Uniaxial compression '0c III II I f− = = = σ σ σ :
'
31
c cf p= '
c cf q= °=60θ
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
154- Biaxial compression '0b III II I f− = = = σ σ σ :
'
32
b bf p= '
b bf q= °=0θ
– Confined uniaxial compression '
cc III l II I f f− = − = = σ σ σ :
32'
l cc
ccf fp+= l cc cc f f q− =' °=60θ
– Confined biaxial compression ,
bc III II a I f f = = > =σ σ σ :
32' '
bc a
bcf fp+= '
bc a cf f q− = °=0θ
Using these load conditions into the failure criterion (Eqs. 5.22), we have:
meridiantensile
ppp
bbb
q qq qq q
bcbt
bc bcb bt t
−−−
=
210
222
111
(5.31)
inverting:
meridiantensile
ppp
q qq qq q
bbb
bcbt
bc bcb bt t
−−−
=
−1
222
210
111
(5.32)
While along the compressive meridian:
meridian e compressivA pA p
aa
q qq q
ccc
cc ccc c
− −− −=
21
22
(5.33)
then,
meridian e compressivA pA p
q qq q
aa
ccc
cc ccc c
− −− −
=
−1
22
21 (5.34)
In literature typical failure data are suggested, e.g., Chen and Han (1988
[7]):
1. Uniaxial compressive strength: '
cf
2. Uniaxial tensile strength: ' '1 . 0c tf f =
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
1553. Biaxial compressive strength: ' '15 . 1c bf f =
4. Confined biaxial compressive strength:
'95 . 1c bc f p − = '393 . 3c bc f q =
5. Confined uniaxial compressive strength:
'9 . 3c cc f p − = '239 . 4c cc f q =
Based on Mills et al. tests (1970 [30]), with MPa fc 3 . 22'= concrete, the
five parameters of the failure function are now determined and the values
of the constant materials are the followings:
MPa A107.4= 702 . 01−=b 1
2 00137 . 0−− = MPa b
467 . 01−=a 1
2 000693 . 0−− = MPa a
Fig. 5.6 – Willam-Warnke failure criterion [37] (five parameters) compared with Mills
et al. test data (1970) [30], in the I 1-J20.5 plane.
The failure criterion is approximated by the elliptic function in the
deviatoric plane (Willam and Warnke 1975 [37]), generating a Cl-
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
156continuous surface with the complete elimination of corners in the
deviatoric trace (figure 5.7).
Fig. 5.7 – Willam Warnke failure criterion in comparison with some experimental data.
5.1.5 OTTOSEN FOUR -PARAMETER MODEL
Ottosen (l977 [32]) suggested the following criterion for concrete material
involving all three stress invariants θ, 3 ,321J qIp = − = :
0 12= − − +p b q aq λ (5.35a)
where λ is a function of θ3cos :
( ) ( )
( ) ( )
≤ =
− −≥ =
=
−−
0 3 cos , 3 cos cos31
3cos0 3 cos , 3 cos cos31cos
2 1 21
12 1 21
1
θ θ θπθ θ θ
λ
for k R k k kfor k Q k k k
(5.35b)
In Eq. (5.35a,b) a, b, k 1 and k 2 are constant materials. For convenience in
the calibration of the model, we introduce the functions ()( )θ θ , , ,2 2k R k Q .
Equations (5.35a, b) define a failure surface with curved meridians and
noncircular cross sections on the deviatoric planes (figure 5.8). The
meridians described by Eq. 5.35a are quadratic parabolas which are convex
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
157if 0>a and 0>b. The cross sections have the geometric properties of
symmetry and convexity, and have changing shapes from nearly triangular
to nearly circular with increasing hydrostatic pressure. The model
encompasses several earlier models as special cases, e.g., the von Mises
(1913) model for 0==ba and .const=λ and the Drucker Prager (1952)
model if 0=a, 0≠band .const=λ
The four parameters in the failure criterion may be determined on the basis
of two typical uniaxial concrete tests (compression strength,'
cf, and tension
strength '
tf) and two typical biaxial an triaxial data:
1. Uniaxial compressive strength ( 1 3 cos3− = ⇒ =θπθ ): '
cf
2. Uniaxial tensile strength ( 13cos0 = ⇒= θ θ ): ' '1 . 0c tf f =
3. Biaxial compressive strength ( 13cos0 = ⇒= θ θ ): ' '15 . 1c bf f =
4. Confined uniaxial compressive strength ( 1 3 cos3− = ⇒ =θπθ ):
'9 . 3c cc f p − = '239 . 4c cc f q =
values suggested by Chen and Han, 1988 [7].
The steps of the calibration are the followings:
– substituting the uniaxial compression test into the failure surface
(Eq. 5.35a, b):
resolving for b:
cc c c
pq R k aqb112− += (5.36)
– substituting the confined compression test into Eq. 5.35a, b and
using Eq. 5.36: 0 112= − − +c c c c p b q R k aq 0 112= − − +c c c c p b q R k aq
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
1580 1112
12= −− +− +cc
cc c c
cc cc cc ppq R k aqq R k aq
inverting for a:
cc c c ccc cc cc cc c c cc c
p q p qp q R k p q R k p pa2 21 1
−− + −= (5.37)
– substituting the uniaxial compression test into Eq. 5.35a, b:
0 112= − − +t t t t p b q Q k aq
using eqs. (5.36) and (5.37) and inverting for k1, we obtain :
)() )( (
2 2 2 22 2 2 22 22 2 2 2
1
cc c t c c c cc t c c t c c cc cc t c cc c cc t t cc c c t t c cc c t c cc cc c t cc c ccc c c cc c tt c cc t c c c t cc c t c
p q p q R p q p q R p q p q R p q p q Rp q Q p q p q Q p q p q p q R p q p q Rp q p q p pp q p p q p p q p p q p
k
+ − + −+ − + −− − −− + + −
=
(5.38)
– finally, substituting the biaxial compression test into Eq. 5.35a, b:
0 112= − − +b b b b p b q Q k aq (5.39)
using eqs. (5.36) , (5.37) and (5.38) into Eq. (5.39) which reduces
to only k2 –dependence. Hence we have one implicit equation and
one incognita and the equation system becomes easily resolvable.
The values obtained for the parameters from Mills et al. test (1970 [30])
data are the following:
884 . 02=k 00122 . 01=k
10000010075. 0−= MPa a 00163.0=b
Figure 5.8 shows the comparison of the failure criterion with Mills et al.
triaxial data in meridian planes.
In general, the four-parameter failure criterion is valid for a wide range of
stress combinations. However, the expression for the λ-function is quite
involved. Hsieh et al. (1982) proposed a simpler form which can also fit the
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
159experimental data very well and presented in the next section.
Fig. 5.8 – Ottosen failure criterion (four parameters) compared with Mills et al. test data
(1970) [30], in the I 1-J20.5 plane.
5.1.6 HSIEH – TING – CHEN FOUR -PARAMETER MODEL
Hsieh et al. (1982 [19]) proposed a λ-function with the simple form
) cos( c b + = θ λ for ° ≤60θ , where b are c are constants. Replacing λ in
Eq. (5.35a) of Ottosen model by this expression the failure function is in
the following form:
0 1 ) cos (2= − − + +dp q c b aq θ (5.40)
a, b, c, d are material constants.
To determine the four material parameters, a, b, c and d, we use some of the
triaxial tests of Mills and Zimmerman (1970 [30]). As for the Ottosen
criterion, the parameters are determined from the following four failure
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
160states and in absence of experimental data some classical failure values are
proposed by Chen and Han (1988 [7]):
1. Uniaxial compressive strength ( 1 3 cos3− = ⇒ =θπθ ): '
cf
2. Uniaxial tensile strength ( 13cos0 = ⇒= θ θ ): ' '1 . 0c tf f =
3. Biaxial compressive strength ( 13cos0 = ⇒= θ θ ): ' '15 . 1c bf f =
4. Confined uniaxial compressive strength ( 1 3 cos3− = ⇒ =θπθ ):
'9 . 3c cc f p − = '239 . 4c cc f q =
Fig. 5.9 – Hsieh – Ting – Chen failure criterion (four parameters) compared with Mills
et al. test data (1985) [30], in the I 1-J20.5 plane.
To calibrate the failure criterion we use the following procedure:
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
161
=
− °− °− °− °
1111
60 cos60 cos0 cos0 cos
2222
dcba
p q q qp q q qp q q qp q q q
cc cc cc ccc c c cb b b bt t t t
(5.41)
inverting the nonhomogeneous system of linear equation (5.41), we obtain
the material constants:
=
− °− °− °− °
=
−
0.001627890.000340780.00085945 MPa 0.00000101
1111
60 cos60 cos0 cos0 cos1 -1
2222
cc cc cc ccc c c cb b b bt t t t
p q q qp q q qp q q qp q q q
dcba
(5.42)
In Fig. 5.9 the compressive =3πθ and the tensile ( )0=θ meridians are
shown. The failure criterion and the experiments of Launay and Gachon
(1970 [23]) are compared in the deviatoric plane in Fig. 5.10.
Fig. 5.10 – Comparison of Hsieh – Ting – Chen failure criterion with Launay and
Gachon triaxial data (1970) [23], in the deviatoric plane.
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
162The four-parameter criterion satisfies the convexity requirement for alls
stress conditions, it still has edges along compressive meridians (Fig. 5.10)
where continuous derivatives along the edges do not exist. Continuous
derivatives used in a general constitutive relation with a general load case
would result in a better convergence during iteration in a numerical
analysis. While for classical load cases (e.g., uniaxial compression case or
confined compressive case) such problems don't occur. Thus, smoothness
everywhere of the yield surface is a desirable property.
5.1.7 EXTENDED LEON MODEL (ELM) PROPOSED BY ETSE
To describe the triaxial concrete strength, the failure criterion by Leon
(1935 [24]) and its extension by Hoek and Brown (1980 [18]) is adopted by
Etse (1992 [12]).
The principal advantages of this criterion are the followings (Etse and
Willlam, 1994 [13]):
– it retains simplicity while not sacrificing accuracy;
– it provides continuous transition between failure in direct tension and
triaxial compression;
– it reduces calibration of the failure criterion to two strength
parameters that are readily available from uniaxial-tension and
uniaxial-compression data.
The isotropic Leon (1935) failure criterion for concrete and the subsequent
modification by Hoek and Brown (1980) presented in the sections 5.1.1
and 5.1.2 respectively have the followings expressions in terms of Haigh –
Westergaard coordinates:
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
163- Leon model:
0
32
'1 sin23cos 2 / 132
'sin 2 / 1 cos23
'2 2
= + −
−+
+
ξ θ θ ρθ θρ
c cc
fm
fmf
(5.43)
– Hoek and Brown model:
0 1 cos23
'sin 2 / 1 cos23
'2 2
= −
+ +
+
θ ρ σ θ θρ
c c fm
f (5.44)
The main disadvantage of both the Leon and the Hoek-Brown criteria are
the corners in the deviatoric trace complicating the numerical
implementation to the simulation of mechanical behavior of concrete (Fig.
5.12).
Considering the tension and compression meridians of Hoek and Brown
model (Eq. 5.44), respectively, when ( )0=θ and =3πθ
:
0 132
' ' 232
= −
+ +
t
c ct
fm
fρ σρ (5.45)
0 1
6 ' ' 232
= −
+ +
c
c cc
fm
fρσρ (5.46)
Etse approximates the triaxial failure criterion of Hoek and Brown with the
elliptic description of the Willam and Warnke model (1975 [37]) in the
deviatoric region in order to generate a Cl-continuous surface. A smooth
surface is obtained by replacing the deviatoric radius vector with the
following function proposed by Willam and Warnke:
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
164( )2 2 22 2 2 2
) 1 2 ( cos ) 1 ( 44 5 cos ) 1 ( 4 ) 1 2 ( cos ) 1 ( 2
− + −− + − − + −=e ee e e e e
cθθ θρ θ ρ (5.47)
and where the eccentricity is defined by the ratio c t e ρρ/= . The polar
coordinate ()θρ defines the elliptic variation of the model (to see section
5.1.3).
Fig. 5.11 – Deviatoric View of ELM (Etse 1992).
Fig. 5.12 – Deviatoric Sections of Hoek and Brown model (1980).
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
165Hence, the discontinuous failure surface (Eq. 5.44) may be approximated
by the following smooth approximation as a function of the three scalar
invariants σ, ρ, and θ:
( ) ( )0 1
6 ' ' 23) , , (2
= −
+ +
=θ ρσθ ρθ ρ σ
c cfm
fF (5.48)
It is worth noticing that for e = 1 , the influence of the polar angle θ
disappears, and the deviatoric shape of the failure surface becomes circular
along the line of the generalized Drucker – Prager criterion (1952). The
eccentricity must satisfy the condition 1 5.0 ≤<e.
Fig. 5.11 illustrates the deviatoric sections of the smooth failure criterion at
various levels of hydrostatic pressure. The difference between the smooth
failure criterion as compared to the original Leon-Hoek and Brown criteria
is best appreciated in the deviatoric planes. (difference between Fig. 5.11
and Fig. 5.12).
Fig. 5.13 – ELM failure criterion (Etse 1992 [12]) compared with Hurlbut test data
(1985) [20], in the I 1-J20.5 plane.
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
166The smooth failure criterion of Etse describes a C1-continuous curvilinear
trace as opposed to the highly polygonal shape of the Hoek and Brown
criterion in the deviatoric region.
5.2 APPLICATION OF PLASTICITY BASED MODELS TO
PASSIVE CONFINEMENT
In the present section, the influence of passive confinement is studied in the
field of the elastoplasticity models. The models presented in the previous
chapters (see §3 & §4) have shown that the same models are suitable to
describe the mechanical behavior in case of constant – active confinement.
In the present section, it is aimed to show if the elastoplasticity models are
also capable to describe the response of specimens in passive confinement.
Concrete cylinders enclosed by steel and by fibre reinforced polymers
(FRP), are representative examples of passively confined structures.
Fig. 5.14 Stresses acting on (a) the confining material (steel or CFRP) and (b) the
enclosed concrete, Grassl (2002) [17].
Triaxial compression stress states are usually activated by lateral expansion
when the specimens are passively confined. The amount of lateral
expansion determines the confinement stress, which is called passive
confinement. The confining material, in the circumferential direction,
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
167activates radial stresses, as shown in Fig. 5.14. It is assumed that only the
concrete core is loaded in the axial direction. Additionally, the confining
stress is assumed to be constant in the circumferential direction, provided
that both the concrete and the confining material have a homogenous
surface.
By means of these simplifications, the relation of the tσ stresses, in the
enclosing material, to the radial stresses acting on the concrete specimen,
rσ , may be derived by stress equilibrium in the lateral direction as:
t d rt r σ ϕ ϕ σπ
2 sin
0= ∫ (5.49)
so that the lateral stress, rσ , results as:
t rDtσ σ2= (5.50)
in the case of passive confinement with transversal and discontinuous
armature, the Eq. (5.50) becomes:
ts
rsDAσ σ2= (5.51)
in which sAis the transversal area and s is the step.
Furthermore, the radial strainrε:
rr
r∆ε= (5.52)
is equal to the axial strain of the confining material (steel or FRP):
r trr
UUεπ∆π ∆ε = = =22 (5.53)
The main difference between steel and FRP as confining materials (see Fig.
5.15) is:
– the ratio of the maximum stress;
– the difference of the elasticity modulus and
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
168- the post-peak behavior.
The behavior of the confining steel was idealized by an elastic–plastic
stress–strain relation, where the Young’s modulus is assumed to be E = 200
GPa and the yield stressyσ, which correspond to a stress–stiffness ratio of
Ey y/σε= .
Fig. 5.15 Difference of mechanical behavior between steel and FRP, Grassl (2002) [17].
The response of the FRP is idealized by means of an elastic-brittle stress–
strain relation with the Young’s modulus that is assumed to be E (varying
according to the type of FRP material: Carbon FRP (160 – 300 GPa),
Aramid FRP (50 – 90 GPa), Glass FRP (25 – 80 GPa) and the ultimate
stress is uσ, which results Eu
uσε=.
The plasticity models are founded upon three fundamental assumptions
(Chen and Han, 1988 [7]):
(i) an initial yielding surface, within the stress space, defining the
stress level at which plastic deformation begins;
(ii) an isotropic hardening/softening rule defining the yielding surface
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
169evolution after beginning of plastic deformations;
(iii) a flow rule, which is related to a plastic potential function, gives an
incremental plastic stress-strain relation.
Prediction models introduced in the previous chapters (§3 and §4) are
appropriate only when constant confinement is applied to the cylindrical
specimens. The reasons for these features can be listed in the following
manner:
1) isotropic hardening/softening law, characterizing the same models, is
strongly dependent on the level of lateral confinement, for this
reason that in the chapters regarding the plasticity based models, the
hardening/softening law is calibrated in accord with various confined
test. Then, for levels of confinement which the experimental tests are
not available, a linear interpolation of the isotropic law, by means of
the value of confinement, is used.
2) As for the hardening/softening law the dilation angle of plastic flow
(see chapter 3 & 4) depends on the level of lateral confinement.
Assuming decreasing values as confinement increases thin to assume
negative values for high hydrostatic pressure.
5.2.1 STEEL CONFINEMENT
Figure 5.15 shows the mechanical behavior in uniaxial tension of steel
material. It is worth noticing that if Ey y/σε= quantity is sufficiently
small, we can hypothesize that concrete cylinders enclosed by steel can be
studied as constant – confined compression tests with the following lateral
constant stress:
y rDtσ σ2= (5.54)
in which yσ is the yield and/or failure uniaxial stress of steel.
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
170Typical value of yield stress is MPay350=σ which correspond to a yield
strain of 00175 . 0 /= =Ey yσ ε .
Fig. 5.16 – Typical uniaxial compressive stress-strain curve (Domingo Sfer et al, 2002
[35]).
Comparing the yield strain value of the considered steel with the radial
strain of a typical uniaxial compressive test we observe that these values
are comparable.
This implicates that when we confine a cylindrical specimen with steel, all
the capacity strength of confining material is mobilized. For this reason that
we don't commit reasonable errors if we consider a constant confinement to
predicts the failure condition when confining steel material is used.
Various authors have been proposed, according to experimental test-data,
empirical or semi-empirical approaches to describe the strength of a
confined concrete, e.g.:
– Richart et al., 1928 [33] (see Fig. 5.17):
l c ccf f f1 . 40+ = (5.55)
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
171- Newman and Newman, 1972 [31] (see Fig. 5.18):
+ =86 . 0
00 7 . 3 1
cl
c ccfff f (5.56)
– Mander et al., 1988 [28] (see Fig. 5.19):
− + + − =
0 00 2 94 . 7 1 254 . 2 254 . 1
cl
cl
c ccff
fff f (5.57)
where ccf and 0cf are the confined and unconfined concrete strength,
respectively and lf is the lateral confinement, based on equilibrium (Eq.
5.54):
y lDtf σ2= (5.58)
Fig. 5.17 – Comparison between Richart formula (1928) with experimental steel
confinement data available in literature.
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
172
Fig. 5.18 – Comparison between Newman and Newman formula (1972) with
experimental steel confinement data available in literature.
Fig. 5.19 – Comparison between Mander et al. formula (1988) with experimental steel
confinement data available in literature.
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
173Figures from 5.17 to 5.19 show the comparison of these empirical formule
with some steel-confined concrete (cylindrical specimens).
The comparison with some available tests has shown that the relationships
proposed by Richart et al. [33], Newman and Newman [31] and Mander
[28] to describe the mechanical properties of concrete subjected to triaxial
compression capture in a excellent mode the steel-confined compressive
strength (see Figs. 5.17, 5.18 and 5.19).
Figures from 5.20 to 5.26 show the comparison of classical failure criteria
for concrete, with some steel confined (cylindrical tests). Note that all
failure criteria have been normalized by '
cf (uniaxial compressive
strength).
Fig. 5.20 – Comparison between Drucker – Prager criterion (1952) with experimental
steel confinement data available in literature.
The Drucker-Prager model overestimates the material response as the
confinement pressure increases (Fig. 5.20) according to the results obtained
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
174in the previous Chapter in which we had found that failure models
generally described by straight line, in the meridian plane, are inadequate
for describing the failure of concrete in high-confinement range (see § 3.3.2
and 3.3.3).
Fig. 5.21 – Comparison between Bresler – Pister (1958, [4]) criterion with experimental
steel confinement data available in literature.
The comparison between Bresler and Pister model with some available
tests has shown that the same model is appropriate to describe the
mechanical properties of concrete subjected to steel-confined triaxial
compression (Fig. 5.21).
Figure 5.22, based on the comparison with respect to concrete cylinders
enclosed by steel, shows that Leon criterion underpredicts the confined
triaxial compression with steel-passive confinement, as lateral confining
steel-material increases.
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
175
Fig. 5.22 – Comparison between Leon criterion (1935 [24]) with experimental steel
confinement data available in literature.
Fig. 5.23 – Comparison between Hoek and Brown (1980, [18]) criterion with
experimental steel confinement data available in literature.
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
176Figure 5.23 shows that the Leon modified model (performed by Hoek and
Brown, 1980) produces some good results. The strength measures of
envelope model, in comparison with experimental tests, are acceptable for
low and middle levels of steel confinement. Only when high ranges of
hydrostatic pressure produced by steel confinement is applied to the
specimens, that the model underpredicts the triaxial strength (figure 5.23).
Fig. 5.24 – Comparison between Willam Warnke (1975, [37]) criterion with
experimental steel confinement data available in literature.
Willam-Warnke (five parameters), Ottosen (four parameters) and Hsieh et
al. (four parameters) are models characterized by the same features to
modeling the confined triaxial compressive strength (from figure 5.24 to
5.26). The strength values of the failure criteria, in comparison with
experimental tests, are acceptable for low and middle levels of steel
confinement. The models overestimate the triaxial strength as high ranges
of confining steel is applied to the specimens (figures 5.24, 5.25 and 5.26).
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
177
Fig. 5.25 – Comparison between Ottosen (1978, [32]) criterion with experimental steel
confinement data available in literature.
Fig. 5.26 – Comparison between Hshie et al. (1982, [19]) criterion with experimental
steel confinement data available in literature.
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
1785.2.2 FRP CONFINEMENT
The response of FRP is represented by elastic-brittle stress–strain relation
(figure 5.15). For this reason passive confinement, realized through FRP,
produces a radial stress gradually increasing as the radial strain of specimen
increases.
Plasticity based models with isotropic hardening/softening rule and
constant dilation angle of plastic potential as introduced in the previous
chapters result to be inadequate to predict the mechanical response of
specimens in FRP passive confinement.
Nevertheless the incremental elastic – plasticity algorithm introduced in the
Chapter 2 (Basic Equations and Procedure), can be improved adding a new
return – cycle in the calculation scheme (figure 5.27).
Fig. 5.27 – Flow chart of elastoplastic procedure modified to simulate the FRP passive
confinement.
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
179The new algorithm inserted to valley of the incremental elastoplastic
relationships (for more details see chapter 2) is the following:
– we calculate the radial strain at the first step of incremental
elastoplastic calculation I
rε ∆, with the radial confinement equal
to the calculated value of the previous elastoplastic cycle 1−i
rσ;
– now, we may be derived by stress equilibrium in the lateral
direction a new radial action in the following manner:
I
r FRPI
rEDtε ∆ σ ∆2= (5.59)
then, the new radial confinement is:
I
ri
r σ ∆ σ+−1 (5.60)
– with the previous level of confinement (Eq. 5.60), we again do the
incremental elastoplastic iteration and we obtain the subsequent
lateral strain II
rε ∆. It is opportune to observe that:
I
rII
ri
rI
ri
r ε ∆ ε ∆ σ σ ∆ σ < ⇒ > +− − 1 1 (5.61)
– we may be derived by stress equilibrium in the lateral direction a
new radial confinement in the following manner:
II
r FRPII
rEDtε ∆ σ ∆2= (5.62)
then, the subsequent radial confinement is:
II
ri
r σ ∆ σ+−1 (5.63)
with:
II
ri
rI
ri
r σ ∆ σ σ ∆ σ+ > +− − 1 1 (5.64)
In this way to proceed, we can observe that:
III
rII
r σ ∆ σ ∆<
IV
rIII
r σ ∆ σ ∆>
V
rIV
r σ ∆ σ ∆<
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
180etc.
– the algorithm will finish when:
( )14
11
∆σ ∆σ ∆ σ ∆≤−
−−
IN
rIN
rIN
r (5.65)
being ∆ a sufficiently small tolerance.
And we will go to the next incremental elastoplastic cycle with
IN
ri
ri
r σ ∆ σ σ+ =−1.
In this way we can use the plasticity based models (originally used for
predicting the behavior of concrete under constant confined compression)
to the numerical simulation of non-constant confined compression using
FRP passive confinement.
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
181REFERENCES OF THE FIRST CHAPTER
[1] Abu-Lebdeh TM and Voyiadjis GZ (1993), Plasticity-damage model for concrete
under cyclic multiaxial loading, J. Eng. Mech., 119-7, 1465–1484.
[2] Ahmad, S.H., Shah, S.P. Complete triaxial stress–strain curves for concrete, J.
Struct. Div. Proc. ASCE 108 (1982) 728–742.
[3] Bathe, K.-J. (1996), Finite element procedures. Prentice-Hall, Englewood, New
Jersey, USA.
[4] Bresler B and Pister KS (1958), Strength of concrete under combined stresses, ACI
Mater. J., 55, 321–345.
[5] Buyukozturk O, Nilson AH, and Slate FO (1971), Stress-strain response and
fracture of a concrete model in biaxial loading, ACI Mater. J., 68-8, 590–599.
[6] Cedolin L, Crutzen YRJ, and Dei Poli S (1977), Triaxial stress-strain relationship
for concrete, J. Eng. Mech. Div., 103-3, 423–439.
[7] Chen, W.-F., and Han, D. J. (1988). Plasticity for structural engineers. Springer-
Verlag, New York, N.Y.
[8] Dafalias YF and Popov EP (1975), A model of nonlinearly hardening materials for
complex loading, Acta Mech., 21, 173–192.
[9] de Boer R and Desenkamp HT (1989), Constitutive equations for concrete in failure
state, J. Eng. Mech. Div., 115-8, 1591–1608.
[10] Drucker, D. (1959). A definition of stable inelastic materials. Journal of Applied
Mechanics, 26:101-106.
[11] Eid, R., “Behavior of Circular Reinforced Concrete Columns Confined with
Transverse Steel Reinforcement and FRP Sheets,” ISIS Canada 11th Annual
Conference, May 2006, Calgary, Canada.
[12] Etse, G. (1992). "Theoretische und numerische Untersuchung zum diffusen und
lokalisierten Versagen in Beton," PhD thesis, University of Karlsruhe, Karlsruhe,
Germany.
[13] Etse, G. and Willam, K. (1994). Fracture energy formulation for inelastic behavior
of plain concrete. ASCE-EM, 120:1983-2011.
[14] Etse, G. and Willam, K. (1996). Integration algorithms for concrete plasticity.
Engineering Computations, 13(8):38-65.
[15] Faruque MO and Chang CJ (1990), A constitutive model for pressure sensitive
materials with particular reference to plain concrete, Int. J. Plast., 6-1, 29–43.
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
182[16] Geniev GA et al (1978), Strength of Lightweight Concrete and Porous Concrete
under Complex Stress State in Russian, Moscow Building Press.
[17] Grassl Peter (2002) Modeling of dilation of concrete and its effect intriaxial
compression. Chalmers University of Technology, Department of Structural
Engineering, Concrete Structures, SE-412 96 Gloteborg, Sweden
[18] Hoek, E., and Brown, E. T. (1980). "Empirical strength criterion for rock masses."
J. Geotech. Engrg., ASCE, 106(9), 1013-1035.
[19] Hsieh SS, Ting EC, and Chen WF (1982), A Plasticity-fracture model for concrete,
Int. J. Solids Struct., 18-3, 181–197.
[20] Hurlbut, B. (1985). "Experimental and computational investigation of strain-
softening in concrete," MSc thesis, University of Colorado, Boulder, Colo.
[21] Jiang JJ and Wang HL (1998), Five-parameter failure criterion of concrete and its
application, Strength Theory, Science Press, Beijing, New York, 403–408.
[22] Labbane M, Saha NK, and Ting EC (1993), Yield criterion and loading function
for concrete plasticity, Int. J. Solids Struct., 30-9, 1269–1288.
[23] Launay, P., and Gachon H. (1972). "Strain and ultimate strength of concrete under
triaxial stress." Am. Concrete Inst. Spec. Publ. 34, paper 13, American Concrete
Institute (ACI), D..etroit, Mich.
[24] Leon, A. (1935). "Uber die Scherfestigkeit des Betons." Beton und Eisen, Berlin,
Germany, 34(8) (in German).
[25] Li JK (1997), Experimental research on behavior of high strength concrete under
combined compressive and shearing loading in Chinese, China Civil Engrg. J., 30-3,
74–80.
[26] Li, Y.-F., and Fang, T.-S. 2004. A constitutive model for concrete confined by steel
reinforcement and carbon fiber reinforced plastic sheet. Structural Engineering and
Mechanics, 18: 21–40.
[27] Lubliner Jacob, Plasticity Theory, Macmillan Publishing, New York (1990).
[28] Mander JB, Priestley MJN, Park R. Theoretical stress-strain model for confined
concrete. J Str Eng ASCE 1988;114(8):18041826.
[29] Menetrey P and Willam KJ (1995), Triaxial failure criterion for concrete and its
generalization, Colloid J. USSR, 92-3, 311–318.
[30] Mills LL and Zimmerman RM (1970), Compressive Strength of plain concrete
under multiaxial loading conditions, ACI Mater. J., 67-10, 802–807.
FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES
183[31] Newman, K., & Newman J.B, 1972, Failure theories and design criteria for plain
concrete, Proceedings, International Engineering Materials Conference
[32] Ottosen NS (1977), A failure citerion for concrete, J. Eng. Mech., 103-4, 527–535.
[33] Richart, F. E., Brandtzaeg, A., and Brown, R. L., A Study of the Failure of
Concrete under Combined Compressive Stresses, Bulletin No. 185, Engineering
Experimental Station, University of Illinois, Urbana, 1928, 104 pp.
[34] Schreyer HL (1989), Smooth limit surfaces for metals: Concrete and geotechnical
materials, J. Eng. Mech. Div., 115-9, 1960–1975.
[35] Sfer, D., Carol, I., Gettu R. and Etse G., "Study of the Behaviour of Concrete
Under Triaxial Compression", J. of Engng. Mech., V. 128, No. 2, pp. 156-163 (2002).
[36] Song YP and Zhao GF (1996), A general failure criterion for concretes under
multi-axial stress, China Civil Eng. Journal, 29-2, 25–32.
[37] Willam KJ and Warnke EP (1975), Constitutive model for the triaxial behavior of
concrete, Int. Assoc. Bridge. Struct. Eng. Proc., 191–31.
[38] Wu HC (1974), Dual failure criterion for plain concrete, J. Eng. Mech. Div., 100-
6, 1167–1181.
[39] Yang BL, Dafalias YF, and Herrmann LR (1983), A bounding surface plasticity
model for concrete, J. Eng. Mech. Div., 111-3, 359–380.
[40] Zienkiewicz, O. and Taylor, R. (1994). The Finite Element Method, volume 1 & 2.
McGraw-Hill, London, England, 4th edition.
SUMMARY AND CONCLUSION
1846. SUMMARY AND CONCLUSIONS
Numerical methods along with sophisti cated material models provide a
powerful tool for the analyses of concrete structures. Among the large
number of constitutive models for c oncrete, plasticity-based models are
commonly used for a wide range of applications.
In the first part of this work, five constitutive models for plain concrete,
formulated in the framework of plasticity theory, are discussed:
– von Mises (one parameter) plasticity model (1913);
– Drucker-Prager (two paramete rs) plasticity model (1952);
– Drucker-Prager (three paramete rs) plasticity model (1952);
– Bresler-Pister (three parame ters) plasticity model (1958);
– Lubliner et al. (four parameters) plasticity model (1989) available in
ABAQUS (student edition).
The main steps of the development a nd validation of constitutive models
for concrete are:
– the development of a robust and efficient algorithmic formulation of
the material models; an elastoplasticity integration scheme
(incremental stress-strain relationship) is used for the numerical
integration of the evolution equations. The non-linear system of
equations, are solved by means of Newton-Raphson algorithm;
– the model performances are investigat ed with respect to experimental
SUMMARY AND CONCLUSION
185data available within literature. Th e influence of material and model
parameters on the numerical results is investigated through these
experiments. It is found that some of calibrated models are capable
to describe concrete be havior for a broad range of loading conditions
while other models (e.g. von Mises mo del) result to be less suitable
for these purposes.
In the second part, plasticity models are used to study the behavior of
concrete in compression on passively confined structures. For passively
confined structures, the behavior differs significantly for steel and FRP-
confined cylinders.
When we confine a cyli ndrical specimen with steel, all the capacity
strength of confining material is m obilized, in failure regime. For this
reason that we can consider a cons tant-confinement case to predict the
failure condition of concre te specimen when confining steel material is
used. Consequently, the plasticity m odels introduced, in the previous
chapters, result to be adequate.
Instead, plasticity-based models with is otropic hardening/softening rule and
constant dilation angle of plastic pot ential as introduced in the previous
chapters result to be inadequate to predict the mechanical response of
specimens in FRP passive confinement.
APPENDIX:
“CODRI.F”
CODRI.F
187c–––––––––––––––––––––––-c
c ELASTIC-PLASTIC CONSTITUTIVE MODEL c c FOR FRICTIONAL-COHESIVE MATERIALES c c–––––––––––––––––––––––-c c CONTROL TYPE: MIXED – STRAIN CONTROL – STRESS CONTROL c
c STATES: PLANE STRAIN – PLANE STRESS – AXISYMMETRIC STATE c
c–––––––––––––––––––––––-c
c Variables c
c–––––––––––––––––––––––-c c c c sal – output type c c = 1 sigma vs. epsilon c c = 2 square root(J2) vs. I1 c
c con – control type c
c = 1 strain c c = 2 mixed c c = 3 stress c c est – indicative of state c c = 1 plane strains c c = 2 plane stresses c
c = 3 axisymmetric c
c mod – indicative of model c c inc – number of increment c c ninc – number of total increments c c mat – material proprieties c c e – recorded strains c
c s – recorded stresses c
c de – recorded strain increment c c ds – recorded stress increment c c dem – calculated strain increment c c dsm – calculated stress increment c c em – calculated strains c c sm – calculated stresses c
c x – recorded volumetric stress c
c y – recorded deviatoric stress c c xm – calculated volumetric stress c c ym – calculated deviatoric stress c c c c–––––––––––––––––––––––-c
c Version: 08.03.08 c
c Modified by: A.Caggiano c c–––––––––––––––––––––––-c program codri c–––––––––––––––––––––––-c c Principal Program c–––––––––––––––––––––––-c
implicit none
character arcsal*32 integer*4 sal,con,est,mod,inc,ninc,csal,esal,i real*8 s,e,sm,em,ds,de,dsm,dem,sigini,mat,x,y,xm,ym,q dimension s(3,1500),e(3,1500),sm(3,1500),em(3,1500), . ds(3),de(3),dsm(3),dem (3),sigini(4),mat(17)
c––Initial zero rese tting of calulated stresses and strains
sm(1,1)=0.d0 sm(2,1)=0.d0 sm(3,1)=0.d0 em(1,1)=0.d0 em(2,1)=0.d0
em(3,1)=0.d0
c––Menu call menu(sal,con,est,mod) c––Imputh dates call entrada(s, e,mat,ninc)
csal=7 esal=1
do while(esal.ne.0)
c––entry of the name of output file write(*,'(" Output File: ",\)') read (*,'(a32)')arcsal open(unit=csal,file=arcsal,status='NEW',iostat=esal) end do
rewind(csal)
if (sal.eq.1) then write(csal,'(12a13)')'DEF 1','TEN 1', . 'DEF 2','TEN 2', . 'DEF 3','TEN 3', . 'DEFM 1','TENM 1', . 'DEFM 2','TENM 2',
. 'DEFM 3','TENM 3'
elseif (sal.eq.2) then write(csal,'(4a13)') 'X','Y','Xm','Ym'
endif if (sal.eq.1) then write(csal,'(12(E13.4))')e(1,1), s(1,1),
CODRI.F
188. e(2,1), s(2,1),
. e(3,1), s(3,1), . em(1,1), sm(1,1), . em(2,1), sm(2,1), . em(3,1), sm(3,1) elseif (sal.eq.2) then x=(s(1,1)+s(2,1)+s(3,1))/3.d0
y=((s(1,1)-s(2,1))**2+(s(2,1)-s(3,1))**2+(s(3,1)
. -s(1,1))**2)/6. y=sqrt(y) xm=(sm(1,1)+sm(2,1 )+sm(3,1))/3.d0
ym=((sm(1,1)-sm(2,1))**2 +(sm(2,1)-sm(3,1))**2+
. (sm(3,1)-sm(1,1))**2)
ym=ym/6.
ym=sqrt(ym) write(csal,'(4(E13.4))')x,y,xm,ym endif do inc=2,ninc sigini(1) = sm(1,inc-1) sigini(2) = sm(2,inc-1)
sigini(3) = 0.d0
sigini(4) = sm(3,inc-1) c––Calculation of the imputh incr eases of stresses and strains
do i=1,3 ds(i)=s(i,inc)-s(i,inc-1) de(i)=e(i,inc)-e(i,inc-1)
end do
call modelos(dsm,dem,ds,de,sigini,mat,inc,con,est,mod,q) c––Calculation of the output increases of stresses and stra ins using the mo
dels do i=1,3 sm(i,inc)=sm(i,inc-1)+dsm(i) em(i,inc)=em(i,inc-1)+dem(i)
end do
c––Output St ress-strain
if(sal.eq.1)then write(csal,'(12(E13.4))')e(1,inc), s(1,inc), . e(2,inc), s(2,inc), . e(3,inc), s(3,inc),
. em(1,inc), sm(1,inc),
. em(2,inc), sm(2,inc), . em(3,inc), sm(3,inc) c––Output square root(J2) vs. I1
elseif(sal.eq.2)then x=(s(1,inc)+s(2,inc)+s(3,inc))/3.d0 y=((s(1,inc)-s(2,inc))**2+(s(2,inc)-s(3,inc))**2+(s(3,inc)
. -s(1,inc))**2)/6.
y=sqrt(y) xm=(sm(1,inc)+sm(2,inc)+sm(3,inc))/3.d0 ym=((sm(1,inc)-sm(2,inc))**2 +(sm(2,inc)-sm(3,inc))**2+
. (sm(3,inc)-sm(1,inc))**2) ym=ym/6.
ym=sqrt(ym)
write(csal,'(4(E13.4))')x,y,xm,ym endif end do c––Closing of output file
close(unit=csal)
write(*,*)
c––End of the program write(*,*)'PROGRAMA FINALIZADO' write(*,*) end program c–––––––––––––––––––––––-c c SUBROUTINES
c–––––––––––––––––––––––-c
subroutine modelos(dsm,dem,ds,de,sigini,mat,inc,con,est,mod,q) c–––––––––––––––––––––––-c c Calculation of the output increases of stresses and strains using the models c–––––––––––––––––––––––-c implicit none
real*8 dsm,dem,ds,de,sigini,mat,sal,q
integer*4 inc,con,est,mod,i dimension dsm(3),dem(3),ds(3),de(3),sigini(4),mat(17) c––Linear Elastic Model (imod=1) if (mod.eq.1)then call elalin(dsm,dem,ds,de,sigini,mat,inc,con,est) c––von Mises Model (imod=2)
elseif(mod.eq.2)then
call vonMises(dsm,dem,ds,de,sigini,mat,inc,con,est,q) c––No-associated Linear Drucker-Prager Model [two parameters model](imod=3)
elseif(mod.eq.3)then call linealdp (dsm,dem,ds,de,sigini,mat,inc,con,est,q) c––No-associated Linear Drucker-Prager Model [three parameters model](imod=
CODRI.F
1894)
elseif (mod.eq.4)then call dp3(dsm,dem,ds,de,sigini,mat,inc,con,est,q) c––No-associated Bresler and Pister [three parameters model](imod=5)
elseif (mod.eq.5)then call bp(dsm,dem,ds,de,sigini,mat,inc,con,est,q) c––Error message
else
write(*,*) stop ' NONEXITENT MODEL' endif end subroutine c–––––––––––––––––––––––-c
subroutine menu(sal,con,est,mod)
c–––––––––––––––––––––––-c c Options menu c–––––––––––––––––––––––-c implicit none integer*4 sal,con,est,mod write(*,'(7(a65,/))')
+'ÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛ',
+'ÛÛ ÛÛ', +'ÛÛ ÛÛ', +'ÛÛ OPTIONS FOR THE CONTROL ÛÛ', +'ÛÛ ÛÛ', +'ÛÛ ÛÛ',
+'ÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛ'
write(*,'(7(a65,/))') +'ÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛ', +'ÛÛ ÛÛ', +'ÛÛ OUTPUT ÛÛ', +'ÛÛ (1) Sigma vs. epsilon ÛÛ', +'ÛÛ (2) square root (J2) vs. I1 ÛÛ',
+'ÛÛ ÛÛ', +'ÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛ' write(*,'(16x," ",\)') read (*,*) sal write(*,*) write(*,'(8(a65,/))')
+'ÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛ',
+'ÛÛ ÛÛ', +'ÛÛ TYPE OF CONTROL ÛÛ', +'ÛÛ (1) Strains ÛÛ', +'ÛÛ (2) Mixed ÛÛ', +'ÛÛ (3) Stresses ÛÛ', +'ÛÛ ÛÛ',
+'ÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛ'
write(*,'(16x," ",\)') read (*,*) con write(*,*) write(*,'(8(a65,/))') +'ÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛ',
+'ÛÛ ÛÛ',
+'ÛÛ STATES ÛÛ', +'ÛÛ (1) Plane strain ÛÛ', +'ÛÛ (2) Plane stress ÛÛ', +'ÛÛ (3) Axisymmetric ÛÛ', +'ÛÛ ÛÛ',
+'ÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛ'
write(*,'(16x," ",\)') read (*,*) est write(*,*) write(*,'(14(a65,/))') +'ÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛ', +'ÛÛ ÛÛ',
+'ÛÛ MODELS ÛÛ',
+'ÛÛ (1) Linear Elastic ÛÛ', +'ÛÛ (2) von Mises (one parameter) ÛÛ', +'ÛÛ (3) Linear D.-P. (t wo parameters) ÛÛ',
+'ÛÛ (4) Linear D.-P. (three parameters) ÛÛ', +'ÛÛ (5) Bresler and Pister (three parameters) ÛÛ',
+'ÛÛ ÛÛ',
+'ÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛ' write(*,'(16x," ",\)') read (*,*) mod write(*,*) end subroutine c–––––––––––––––––––––––-c
subroutine entrada(s,e,dmt,ninc)
c–––––––––––––––––––––––-c c Reading inputh dates c–––––––––––––––––––––––-c implicit none real*8 s, e, dmt
CODRI.F
190integer*4 ninc, cdat, edat, cmat, emat, ii, i
character arcdat*32,lineas*80 dimension s(3,1500),e(3,1500),dmt(17),lineas(9) cdat=5 edat=1 do while(edat.ne.0) write(*,'(" Database Experiment: ",\)')
read (*,'(a32)')arcdat
open (unit=cdat,file=arcdat ,status='OLD',iostat=edat)
end do rewind(cdat) do i=1,9 read (cdat,'(a80)',io stat=edat)lineas(i)
if (edat.ne.0) stop 'Error en encabezado de archivo de datos' end do write(*,'(9(a80))')lineas c––Reading of stresses, strains and number of increases
ii=1 do read(cdat,'(6e13.0)',iostat=edat)
+ s(1,ii),e(1,ii),s(2,ii),e(2,ii),s(3,ii),e(3,ii)
if (edat.gt.0) stop 'Error en archivo de datos de ensayos' if (edat.lt.0) exit ii=ii+1 end do ninc=ii-1
write(*,'(" Number of recorded increments: ",\)')
write(*,*)ninc close(unit=cdat) write(*,*) c––Reading of material proprieties cmat=10 emat=1
do while(emat.ne.0)
write(*,'(" File of Material Parameters: ",\)') read (*,'(a32)')arcdat open (unit=cmat,file=arcdat ,status='OLD',iostat=emat)
end do rewind(cmat)
do i=1,17
read (cmat,'(e13.0)',iostat=emat)dmt(i) if (emat.gt.0) stop 'Error en archivo datos de materiales' if (emat.lt.0) exit write(*,'(1x,e13.5)')dmt(i) end do close(unit=cmat)
write(*,*)
end subroutine c–––––––––––––––––––––––-c c–––––––––––––––––––––––-c c COMMON SUBRUTINES IN THE CONSTITUTIVE MODELS c c–––––––––––––––––––––––-c
c–––––––––––––––––––––––-c
c–––––––––––––––––––––––-c subroutine elast(mat,ee,dd) c–––––––––––––––––––––––-c c Elastic matrix c c–––––––––––––––––––––––-c
c mat – material proprieties c
c ee – stiffness matrix c c dd – inverse stiffness matrix c c e – Young's module c c poi – Poisson's ratio c c–––––––––––––––––––––––-c implicit none
real*8 mat, ee, dd, e, poi
integer*4 i, j dimension mat(17),ee(4,4),dd(4,4) c––Material Parameters e=mat(1) poi=mat(2)
c––Definition elements of the stiffness matrix
do j=1,4
do i=1,4 ee(i,j) = 0.d0 dd(i,j) = 0.d0 end do end do
ee(1,1)=e*(1.-poi)/(( 1.+poi)*(1.-2*poi))
ee(1,2)=e*poi/((1.+poi)*(1.-2*poi))
ee(2,1)=ee(1,2) ee(2,2)=ee(1,1) ee(3,3)=e/(2.*(1.+poi)) ee(1,4)=ee(1,2)
CODRI.F
191ee(2,4)=ee(1,2)
ee(4,1)=ee(1,2) ee(4,2)=ee(1,2) ee(4,4)=ee(1,1) c––Definition elements of the inverse stiffness matrix
dd(1,1)=1/e dd(1,2)=-poi/e
dd(2,1)=dd(1,2)
dd(2,2)=dd(1,1) dd(3,3)=2.*(1.+poi)/e dd(1,4)=dd(1,2) dd(2,4)=dd(1,2) dd(4,1)=dd(1,2)
dd(4,2)=dd(1,2)
dd(4,4)=dd(1,1) end subroutine c–––––––––––––––––––––––-c c–––––––––––––––––––––––-c subroutine predel(sigi,epsi,sigfin,sigini,sigp,ds,de, . ee,dd,poi,con,est)
c–––––––––––––––––––––––-c
c Elastic predictor c–––––––––––––––––––––––-c c ds – Imposed stress increment c de – Imposed strain increment c sigi – Elastic stress increment (elastic predictor)
c epsi – Strain increment
c sigp – Plastic stress increment (plastic corrector) c sigfin – Final stress c sigini – Initial stress c ee – Elastic matrix c dd – Inverse elastic matrix c poi – Poisson's ratio
c con – type of control
c est – tensional state c–––––––––––––––––––––––-c implicit none real*8 ds,de,epsi,sigi,sigp,sigfin,sigini,ee,dd,poi integer i,j,con,est
dimension de(3),ds(3 ),epsi(4),sigi(4),
* sigp(4),sigfin(4),sigini(4),
* ee(4,4),dd(4,4) c STRAIN CONTROL if(con.eq.1)then c imposed strains epsi(1)=de(1)
epsi(2)=de(2)
epsi(3)=0.d0 if (est.eq.2)then epsi(4)=-poi/(1-poi)*(de(1)+de(2)) !plane stress elseif(est.eq.1)then epsi(4)=0.d0 !plane strain
elseif(est.eq.3)then
epsi(4)=de(3) !axisymmetry else stop ' ERROR al seleccionar ESTADO TENSIONAL' endif c elastic predictor
do i=1,4
sigi(i)=0.d0 do j=1,4 sigi(i)=sigi(i)+ ee(i,j)*epsi(j)
end do end do c final stress
do i=1,4
sigfin(i)=sigini(i)+sigi(i) end do c MIXED CONTROL elseif(con.eq.2)then !PLANE STRAIN
if(est.eq.1)then
c elastic predictor sigi(1)=ds(1)+sigp(1) c imposed stain epsi(2)=0.d0 epsi(3)=0.d0 epsi(4)=de(3)
epsi(1)=sigi(1)/ee(1,1)
do j=2,4 epsi(1)=epsi(1)-ee(1,j)*epsi(j)/ee(1,1) end do c elastic predictor do i=2,4
CODRI.F
192sigi(i)=0.d0
do j=1,4 sigi(i)=sigi(i)+ ee(i,j)*epsi(j)
end do end do !PLANE STRESS else if(est.eq.2)then
c elastic predictor
sigi(2)=0.d0 + sigp(2) c imposed strain epsi(1)=de(1) epsi(3)=0.d0 epsi(4)=de(3)
epsi(2)=sigi(2)/ee(2,2)-ee(2,1)*epsi(1)/ee(2,2)
do j=3,4 epsi(2)=epsi(2)-ee(2,j)*epsi(j)/ee(2,2) end do c elastic predictor do i=3,4 sigi(i)=0.d0
do j=1,4
sigi(i)=sigi(i)+ ee(i,j)*epsi(j)
end do end do sigi(1)=0.d0 do j=1,4
sigi(1)=sigi(1)+ ee(1,j)*epsi(j)
end do
!AXISYMMETRIC else if(est.eq.3)then c elastic predictor sigi(1)=ds(1)+sigp(1) sigi(2)=ds(2)+sigp(2)
sigi(3)=sigp(3)
c imposed strain epsi(4)=de(3) do i=1,3 epsi(i)=0.d0 do j=1,3
epsi(i)=epsi(i)+dd(i,j)*(sigi(j)-ee(j,4)*epsi(4))
end do end do c elastic predictor sigi(4)=0.d0 do i=1,4 sigi(4)=sigi(4)+ee(4,i)*epsi(i)
end do
else stop ' ERROR to select the TENSIONAL STATE' endif c Final stress do i=1,4
sigfin(i)=sigini(i)+sigi(i)
end do C STRESS CONTROL elseif(con.eq.3)then c elasti predictor sigi(1)=ds(1)+sigp(1)
sigi(2)=ds(2)+sigp(2)
sigi(3)=0.d0 +sigp(3) if (est.eq.2)then sigi(4)=0.d0+sigp(4) !plane stress elseif(est.eq.1)then sigi(4)=poi*(ds(1)+ds(2))+ sigp(4) !plane strain
elseif(est.eq.3)then
sigi(4)=ds(3)+sigp (4) !axisymmetric
else
stop ' ERROR to select TENSIONAL STATE' endif do i=1,4 epsi(i)=0.d0
do j=1,4
epsi(i)=epsi(i)+ dd(i,j)*sigi(j)
end do end do do i=1,4 sigfin(i)=sigini(i)+sigi(i) end do
else
stop ' ERROR to select TYPE OF CONTROL'
endif end subroutine c–––––––––––––––––––––––-c subroutine incten(dsm,ds,sigfin,sigini,poi,con,est)
CODRI.F
193c–––––––––––––––––––––––-c
c Calculated increments of stress c–––––––––––––––––––––––-c c dsm – calculated increment of stress c ds – imposed increment of stress c sigfin – final stress c sigini – initial stress
c poi – Poisson's ratio
c con – type of control c est – tensional state c–––––––––––––––––––––––-c implicit none real*8 dsm,ds,sigfin,sigini,poi
integer con,est,i
dimension dsm(3),ds(3),sigfin(4),sigini(4) C–- STRAIN CONTROL if(con.eq.1)then dsm(1)=sigfin(1)-sigini(1) dsm(2)=sigfin(2)-sigini(2) dsm(3)=sigfin(4)-sigini(4)
C–- MIXED CONTROL
elseif(con.eq.2)then !PLANE STRAIN if(est.eq.1)then dsm(1)=ds(1) dsm(2)=sigfin(2)-sigini(2)
dsm(3)=sigfin(4)-sigini(4)
!PLANE STRESS else if(est.eq.2)then dsm(1)=sigfin(1)-sigini(1) dsm(2)=0.d0 dsm(3)=sigfin(4)-sigini(4) !AXISYMMETRY
else if(est.eq.3)then
dsm(1)=ds(1) dsm(2)=ds(2) dsm(3)=sigfin(4)-sigini(4) else stop ' ERROR to select the TENSIONAL STATE'
endif
c STRESS CONTROL elseif(con.eq.3)then dsm(1)=ds(1) dsm(2)=ds(2) if (est.eq.2)then dsm(3)=0.d0 !plane stress
elseif(est.eq.1)then
dsm(3)=poi*(ds(1)+ds(2)) !plane strain elseif(est.eq.3)then dsm(3)=ds(3) !axisymmetry else stop ' ERROR to select the TENSIONAL STATE'
endif
else stop ' ERROR to select the TYPE OF CONTROL'
endif end subroutine c–––––––––––––––––––––––-c
c––––––––––––––––––––––––
c FUNCTIONS c–––––––––––––––––––––––– c======================================================================= real*8 function norma(vector) c–––––––––––––––––––––––– c Vector norm
c––––––––––––––––––––––––
implicit none real*8 vector dimension vector(4) norma= dsqrt(vector(1)*vector(1)+ + vector(2)*vector(2)+
+ vector(3)*vector(3)+
+ vector(4)*vector(4) ) return end c–––––––––––––––––––––––– c–––––––––––––––––––––––-c c LINEAR ELASTIC MODEL c
c–––––––––––––––––––––––-c
c Variables c c–––––––––––––––––––––––-c c c c inc – number of increase c c est – indicative of state c
CODRI.F
194c = 1 plane strain c
c = 2 plane stress c c = 3 axisymmetric c c con – control type c c = 1 strain c c = 2 mixed c c = 3 stress c
c mat – material proprieties c
c poi – Poisson's ratio c c ee – stiffness matrix c c dd – inverse stiffness matrix c c ds – Imposed stress increment c c de – Imposed strain increment c
c dsm – Calculated stress increment (output) c
c dem – Calculated stra in increment (output) c
c epsi – Strain increment c c sigi – Elastic stress increment c c sig – Final stress state c c sigini – Initial stress state c c sigp – plastic corrector (null) c
c c
c–––––––––––––––––––––––-c c–––––––––––––––––––––––-c subroutine elalin(dsm,dem,ds,de,sigini,mat,inc,con,est) c–––––––––––––––––––––––-c c Calculation of the stress and strain increments c
c–––––––––––––––––––––––-c
implicit none real*8 de,ds,dem,dsm,mat,ee,dd,e, * sigp,epsi,sigi,sig,poi,sigini integer*4 inc, con, est, i, j dimension de(3), ds(3), dem(3), dsm(3), mat(16), sigini(4), * sig(4), sigp(4), sigi(4), epsi(4),
* ee(4,4), dd(4,4) c––Material proprieties e=mat(1) !stiffness module poi=mat(2) !Poisson's ratio c––Elastic matrix call elast(mat,ee,dd)
c––Plastic corrector (null)
sigp(1) = 0.d0 sigp(2) = 0.d0 sigp(3) = 0.d0 sigp(4) = 0.d0 c––Elastic stress and strain increments call predel(sigi,epsi,sig,sigini,sigp,ds,de,ee,dd,poi,con,est)
c––Calculated stress an d strain increments
call incten(dsm,ds,sig,sigini,poi,con,est)
dem(1)=epsi(1) dem(2)=epsi(2) dem(3)=epsi(4) end subroutine
c–––––––––––––––––––––––-c
c VON MISES PLASTICITY MODEL c c–––––––––––––––––––––––-c c Variables c c–––––––––––––––––––––––-c c c
c inc – number of increment c
c est – indicative of state c c = 1 plane strain c c = 2 plane stress c c = 3 axisymmetric c c con – control type c c = 1 strain c
c = 2 mixed c
c = 3 stress c c mat – material proprieties c c e – Elastic module c c poi – Poisson's ratio c c cp – Hardening/Softening parameter of quadratic function c
c cp1 – Hardening/Softening parameter of Simo's function c
c cp2 – Hard./Soft.parameter of Modified Simo's function c
c h – Hard./Soft.parameter of Simo and Mod Simo's function c c yi – Hard./Soft.parameter of Simo and Mod Simo's function c c iend – indicator for the hardening/softening function c c = 1 Hard./Soft. Quadratic c c = 2 Hard./Soft. exponential SIMO c
c = 3 Hard./Soft. modified exponential SIMO c
c y0 – Hard./Soft.parameter of Q-S-MS function c c ymax – Hard./Soft.paramet er of Q-S-MS function c
c qmax – Hard./Soft.paramet er of Q-S-MS function c
c yc – Hard./Soft.parameter of Q-S-MS function c
c qc – Hard./Soft.parameter of Q-S-MS function c
CODRI.F
195c yr – Hard./Soft.parameter of Q-S-MS function c
c qr – Hard./Soft.parameter of Q-S-MS function c
c ee – Elastic matrix c
c dd – Inverse elastic matrix c c func – yield function c c ds – Imposed stress increment (input) c
c de – Imposed strain increment (input) c
c dsm – Calculated stress increment (output) c c dem – Calculated stra in increment (output) c
c epsi – strain increments c
c sigi – Elastic stress increment (elastic predictor) c c sigini – Auxiliar y stress state c
c sig – Calculated stress c
c rm – Gradient function of plastic potential c
c rn – Gradient function of yield function c c en – plastic corrector direction c c dsigp – variation of plastic corrector c c sigp – plastic corrector at anterior iteration c c sigpnew – plastioc corrector: actual iteration c c norma1 – norm of dsigp c
c norma2 – norm of sigpnew c
c itr – number of iteration (stress control) c c maxitr – maxim number of iterations (stress control) c c maxitc – maxim number of ite rations (palstic corrector) c
c errrel – maxim relative error between two iterations c c minmin – tolerance to the zero c
c c
c–––––––––––––––––––––––-c c–––––––––––––––––––––––-c subroutine vonMises(dsm,dem,ds,de,sigini,mat,inc,con,est,q) c–––––––––––––––––––––––-c c Calculation of stress and strain increments c c for Associated Plastic Potential c
c–––––––––––––––––––––––-c
implicit none real*8 de, ds, dem, dsm, mat, ee, dd, dq,cp, * sigp,sigpnew,dsigp,epsi,sigi,sig, e,sigini,ymax,qmax, * poi, alpha, beta, func, yi, q, dlam, cp1, yc, qc,yr, * norma, norma1, norma2, minm in, errrel, cp2, h, y0,qr
integer*4 inc,con,est,itr,maxitr,maxitc,i,j,iend dimension de(3),ds(3),dem(3),dsm(3),mat(16), * ee(4,4),dd(4,4),epsi(4),sigi(4),sig(4), * sigp(4),sigpnew(4),dsigp(4),sigini(4) c––Tolerances minmin=1./10.**14 errrel=0.0001
maxitr=100
maxitc=200 c––Material and model data e = mat(1) poi = mat(2) y0 = mat(3)
alpha= mat(4)
beta = mat(5) cp1 = mat(6) h = mat(7) yi = mat(8) iend = mat(9)
cp2 = mat(10)
yc = mat(11) qc = mat(12) ymax = mat(13) qmax = mat(14) yr = mat(15) qr = mat(16)
c––Elastic Matrix
call elast(mat,ee,dd) c––Plastic corrector reset sigp(1) = 0.d0 sigp(2) = 0.d0 sigp(3) = 0.d0
sigp(4) = 0.d0
if(inc.eq.1)q=0 c––START TO THE ITERATION (fo r mixed and stress control)
do itr=1,maxitr call predel(sigi,epsi,sig,sigini,sigp,ds,de,ee,dd,poi,con,est) c––Flow Condition call vmfunc(sig,y0,yi,h,q,cp1,func,iend,cp2,yc,qc,cp)
c––Calculated stress in elastic regime
if(func.le.minmin)exit c––Plastic Corrector call vonres(epsi,sig,ee,q,dlam,y0,yi,h,cp1,iend, + sigpnew,dq,cp2,yc,qc,cp) c–- Final stress update
CODRI.F
196sig(1)=sig(1)-sigpnew(1)
sig(2)=sig(2)-sigpnew(2) sig(3)=sig(3)-sigpnew(3) sig(4)=sig(4)-sigpnew(4) c Exit condition for strain control if(con.eq.1) exit c Exit condition for stress and mixed control
do i=1,4
dsigp(i)=sigpnew(i)-sigp(i) end do norma1=norma(dsigp) norma2=norma(sigpnew) do i=1,4
sigp(i)=sigpnew(i)
end do if(dabs(norma1/norma 2).lt.errrel) exit
c––END OF ITERACION (for stress and mixed control) end do if(itr.ge.maxitr) write(*,*)' the model does not converge', . ' using the stress control'
c––State variable update
q=q+dq c––Real stress increments call incten(dsm,ds,sig,sigini,poi,con,est) c––Real strain increments dem(1)=epsi(1)
dem(2)=epsi(2)
dem(3)=epsi(4) end subroutine c–––––––––––––––––––––––-c subroutine vonres(epsi,st,ee,q,dlam,y0,yi,h,cp1,iend, + sigp,dq,cp2,yc,qc,cp) c–––––––––––––––––––––––-c
c Function :von Mises
c Plastic corrector for asso ciated plastic potential
c Non-linear Hardening/Softening c –––––––––––––––––––––––c implicit real*8 (a-h,o-z) integer*4 iend
dimension epsi(4),ee(4,4),sigp(4)
dimension sn(4),st(4),en(4),rn(4) c––Plastic potential gradient call vmgrad (st,rn) c––Plastic poten tial direction
call vmdir (rn,en,ee) c––Initial value for the plastic multiplier
call vmdlmt (st,rn,en,q,h,y0,yi,cp1,func,dlma,iend,cp2,yc,qc,cp)
c––Development of non-linear consistence condition iter=1 dlma=0.d0 10 call vmdlam(st,rn,sn,en,dq,q,h,y0,yi,cp1,dlma,dlam,iend, + cp2,yc,qc,cp)
if(dabs((dlam-dlma)/dlam).gt.1.d-9.and.iter.le.50)then
dlma=dlam call vmgrad(sn,rn) call vmdir(rn,en,ee) iter=iter+1 goto 10
endif
sigp(1)=dlam*en(1) sigp(2)=dlam*en(2) sigp(4)=dlam*en(4) sigp(3)=dlam*en(3) return end
c–––––––––––––––––––––––-c
c––––––––––––––––––––––––c subroutine vmfunc(s,y0,yi,h,q,cp1,func,iend,cp2,yc,qc,cp) c––––––––––––––––––––––––c c Function : von Mises Yield Function c Non-linear Hardening/Softening
c––––––––––––––––––––––––c
implicit real*8 (a-h,o-z) integer*4 iend dimension s(4) trs1=s(1)+s(2)+s(4) trs2=s(1)*s(1) +s(2)*s(2) +s(4)*s(4) +2.d0*s(3)*s(3) c––Quadratic Hardening/Softening
if(iend.eq.1)then
yn=0.5d0*cp*q*q+h*q+y0 c––Exponential SIMO Hardening/Softening elseif(iend.eq.2)then yn=y0+(yi-y0)*(1.d0-dexp(-cp1*q))+h*q c––Modified Exponential SI MO Hardenin g/Softening
CODRI.F
197elseif(iend.eq.3.and.q.lt.qc)then
yn=y0+(yi-y0)*(1.d0-dexp(-cp1*q))+h*q elseif(iend.eq.3.and.q.ge.qc) then yn=yc*dexp(cp2*(q-qc)) else write(*,*)'Hardening/Softe ning not available',iend
stop
endif
func=dsqrt(.5d0*trs2-trs1*trs1/6.d0)-yn/dsqrt(3.d0) return end c––––––––––––––––––––––––c subroutine vmgrad (s,rn)
c––––––––––––––––––––––––c
c Function:Gradient of associated von Mises plastic potential c––––––––––––––––––––––––c implicit real*8 (a-h,o-z) dimension s(4), rn(4) trs1=s(1)+s(2)+s(4) trs2=s(1)**2+s(2)**2+s(4)**2+2.d0*s(3)**2
trd2=trs2-trs1**2/3.d0
den=dsqrt(2.d0/3.d0*trd2) rn(1)=(-trs1/3.d0+s(1))/den rn(2)=(-trs1/3.d0+s(2))/den rn(4)=(-trs1/3.d0+s(4))/den rn(3)= s(3)/den
c––Engineering strain
rn(3)=2.d0*rn(3) return end c––––––––––––––––––––––––c subroutine vmdir(rn,en,ee) c––––––––––––––––––––––––c
c Function:Direction of the von Mises plastic stress
c––––––––––––––––––––––––c implicit real*8 (a-h,o-z) dimension rn(4), en(4), ee(4,4) c en(1)=ee(1,1)*rn(1)+ee(1,2)*rn(2)+ee(1,4)*rn(4)
en(2)=ee(2,1)*rn(1)+ee(2,2)*rn(2)+ee(2,4)*rn(4)
en(4)=ee(4,1)*rn(1)+ee(4,2)*rn(2)+ee(4,4)*rn(4) c––Engineering strain en(3)=ee(3,3)*rn(3) return end c––––––––––––––––––––––––c
subroutine vmdlmt(st,rn,en,q,h,y0,yi,cp1,func,dlmt,iend,cp2,yc,
+ qc,cp) c––––––––––––––––––––––––c c Function :Consistenze condition c Non-linear Hardening/Softening c––––––––––––––––––––––––c
implicit real*8 (a-h,o-z)
integer*4 irhd dimension st(4), rn(4), en(4)
trne=rn(1)*en(1)+rn(2)*en(2)+rn(4)*en(4)+ + rn(3)*en(3) trm2=rn(1)*rn(1)+rn(2)*rn(2)+rn(4)*rn(4)+
+ rn(3)*rn(3)/2.d0
c––Quadratic Hardening/Softening if(iend.eq.1)then yn=0.5d0*cp*q*q+h*q+y0 dfdq=(-cp*q-h)/dsqrt(3.d0) c––Exponential SIMO Hardening/Softening elseif(iend.eq.2)then
yn=y0+(yi-y0)*(1.d0-dexp(-cp1*q))+h*q
dfdq=-( (yi-y0)*cp1*dexp(-cp1*q)+h )/dsqrt(3.d0) c––Modified Exponential SI MO Hardenin g/Softening
elseif(iend.eq.3.and.q.lt.qc) then yn=y0+(yi-y0)*(1.d0-dexp(-cp1*q))+h*q dfdq=-( (yi-y0)*cp1*dexp(-cp1*q)+h )/dsqrt(3.d0)
elseif(iend.eq.3.and.q.ge.qc) then
yn=yc*dexp(cp2*(q-qc)) dfdq=-(yc*(dexp(cp2*(q-qc)))*cp2)/dsqrt(3.d0) else write(*,*)'Hardening/Softe ning not available',iend
stop endif
dq=dsqrt(2.d0/3.d0*trm2)
dlmt=func/(trne-dfdq*dq) return end c––––––––––––––––––––––––c subroutine vmdlam (st,rn,sn,en,dq,q,h,y0,yi,cp1,dlmt,dlam,
CODRI.F
198+ iend,cp2,yc,qc,cp)
c––––––––––––––––––––––––c c Function:Consistence condition c Non-linear Hardening/Softening c––––––––––––––––––––––––c implicit real*8 (a-h,o-z) integer*4 iend
dimension st(4),rn(4),sn(4),en(4), sigp(4)
tre1=en(1) +en(2) +en(4) tre2=en(1)*en(1)+en(2)*en(2)+en(4)*en(4)+ + en(3)*en(3)+en(3)*en(3) trs1=st(1) +st(2) +st(4) trs2=st(1)*st(1)+st(2)*st(2)+st(4)*st(4)+
+ st(3)*st(3)+st(3)*st(3)
trse=st(1)*en(1)+st(2)*en(2)+st(4)*en(4)+ + st(3)*en(3)+st(3)*en(3) trm2=rn(1)*rn(1)+rn(2)*rn(2)+rn(4)*rn(4)+ + rn(3)*rn(3)/2.d0 dq=dsqrt(2.d0/3.d0*trm2) w0=(0.5d0*trs2-trs1*trs1/6.d0)
w1=(-trse+trs1*tre1/3.d0)
w2=(0.5d0*tre2-tre1*tre1/6.d0) c––Newton-Raphson iteration dlam=dlmt iter=0 10 iter=iter+1
qn1=q+dlam*dq
c––Quadratic Hardening/Softening if(iend.eq.1)then yn1=(0.5d0*cp*qn1*qn1+h*qn1+y0)/dsqrt(3.d0) dyn=(cp*qn1*dq+h*dq)/dsqrt(3.d0) c––Exponential SIMO Hardening/Softening elseif(iend.eq.2)then
yn1=(y0+(yi-y0)*(1.d0-dexp(-cp1*qn1))+h*qn1)/dsqrt(3.d0)
dyn=( (yi-y0)*cp1*dq*dexp(-cp1*qn1)+h*dq )/dsqrt(3.d0) c––Modified Exponential SI MO Hardenin g/Softening
elseif(iend.eq.3.and.qn1.lt.qc)then yn1=(y0+(yi-y0)*(1.d0-dexp(-cp1*qn1))+h*qn1)/dsqrt(3.d0) dyn=( (yi-y0)*cp1*dq*dexp(-cp1*qn1)+h*dq )/dsqrt(3.d0)
elseif(iend.eq.3.and.qn1.ge.qc) then
yn1=yc*dexp(cp2*(qn1-qc))/dsqrt(3.d0) dyn=yc*dexp(cp2*(qn1-qc))*cp2*dq/dsqrt(3.d0) else write(*,*)'Hardening/Softe ning not available',iend
stop endif
fn1=dsqrt(dlam*dlam*w2+dlam*w1+w0)-yn1
dfn= (dlam*2.d0*w2+w1)/2.d0/dsqrt(dlam*dlam*w2+dlam*w1+w0)-dyn dlam=dlam-fn1/dfn if(dabs(fn1/dfn).gt.1.d-10.and.iter.le.50) goto 10 if(dlam.lt.0.d0) write(*,*)' dlam < null !! ',dlam dq=dq*dlam
sn(1)=st(1)-dlam*en(1)
sn(2)=st(2)-dlam*en(2) sn(4)=st(4)-dlam*en(4) sn(3)=st(3)-dlam*en(3) return end
c–––––––––––––––––––––––-c
c DRUCKER-PRAGER PLASTICITY MODEL c c (two parameters) c c–––––––––––––––––––––––-c c Variables c c–––––––––––––––––––––––-c c c
c inc – number of increment c
c est – indicative of state c c = 1 plane strains c c = 2 plane stresses c c = 3 axisymmetric c c con – control type c
c = 1 strains c
c = 2 mixed c c = 3 stresses c c mat – material proprieties c c e – Elastic module c c poi – Poisson's ratio c c alfa – yield function parameter c
c beta – plastic potential parameter c
c cp – Hardening/Softening parameter of quadratic function c c cp1 – Hardening/Softening parameter of Simo's function c c cp2 – Hard./Soft.parameter of Modified Simo's function c
c h – Hard./Soft.parameter of Simo and Mod Simo's function c c yi – Hard./Soft.parameter of Simo and Mod Simo's function c
CODRI.F
199c iend – indicator for the hardening/softening function c
c = 1 Hard./Soft. Quadratic c c = 2 Hard./Soft. exponential SIMO c c = 3 Hard./Soft. modified exponential SIMO c c y0 – Hard./Soft.parameter of Q-S-MS function c c ymax – Hard./Soft.paramet er of Q-S-MS function c
c qmax – Hard./Soft.paramet er of Q-S-MS function c
c yc – Hard./Soft.parameter of Q-S-MS function c
c qc – Hard./Soft.parameter of Q-S-MS function c
c yr – Hard./Soft.parameter of Q-S-MS function c
c qr – Hard./Soft.parameter of Q-S-MS function c
c ee – Elastic matrix c c dd – Inverse elastic matrix c
c func – yield function c
c ds – Imposed stress increment (input) c
c de – Imposed strain increment (input) c
c dsm – Calculated stress increment (output) c c dem – Calculated stra in increment (output) c
c epsi – strain increments c
c sigi – Elastic stress increment (elastic predictor) c
c sigini – Auxiliar y stress state c
c sig – Calculated stress c
c rm – Gradient function of plastic potential c c rn – Gradient function of yield function c c en – plastic corrector direction c c dsigp – variation of plastic corrector c
c sigp – plastic corrector at anterior iteration c
c sigpnew – plastioc corrector: actual iteration c c norma1 – norm of dsigp c c norma2 – norm of sigpnew c c itr – number of iteration (stress control) c c maxitr – maxim number of iterations (stress control) c c maxitc – maxim number of ite rations (palstic corrector) c
c errrel – maxim relative error between two iterations c c minmin – tolerance to the zero c c c c–––––––––––––––––––––––-c c–––––––––––––––––––––––-c subroutine linealdp(dsm,dem,ds,de,sigini,mat,inc,con,est,q)
c–––––––––––––––––––––––-c
c Calculation of stress and strain increments c c for No-Associated Plastic Potential c c–––––––––––––––––––––––-c implicit none real*8 de,ds,dem,dsm, mat,ee,dd,dq,norma,norma1,norma2, * sigp,sigpnew,dsigp,epsi,sigi,sig,e,sigini,ymax,qmax,
* poi,alpha,beta,func,yi,q,dlam,cp1,yc,qc,yr,cp,
* minmin,errrel,cp2,h,y0,qr integer*4 inc,con,est,itr,maxitr,maxitc,i,j,iend dimension de(3),ds(3),dem(3),dsm(3),mat(16), * ee(4,4),dd(4,4),epsi(4),sigi(4),sig(4), * sigp(4),sigpnew(4),dsigp(4),sigini(4)
c––Tolerances
minmin=1./10.**14 errrel=0.0001 maxitr=100 maxitc=200 c––Material and model data
e = mat(1)
poi = mat(2) y0 = mat(3) alpha= mat(4) beta = mat(5) cp1 = mat(6) h = mat(7)
yi = mat(8)
iend = mat(9) cp2 = mat(10) yc = mat(11) qc = mat(12) ymax = mat(13)
qmax = mat(14)
yr = mat(15) qr = mat(16) c––Elastic Matrix call elast(mat,ee,dd) c––Plastic corrector reset sigp(1) = 0.d0
sigp(2) = 0.d0
sigp(3) = 0.d0 sigp(4) = 0.d0 if(inc.eq.2)q=0 c––START TO THE ITERATION (fo r mixed and stress control)
do itr=1,maxitr
CODRI.F
200call predel(sigi,epsi,sig,sigini,sigp,ds,de,ee,dd,poi,con,est)
c––Flow Condition call vcfunc(sig,alpha,y0,yi,h,q,cp1,func,iend,cp2,yc,qc,cp) c––Calculated stress in elastic regime if(func.le.minmin)exit c––Plastic Corrector call vcresp(epsi,sig,ee,q,dlam,y0,yi,alpha,beta,h,cp1,iend,
+ sigpnew,dq,cp2,yc,qc,cp)
c–- Final stress update sig(1)=sig(1)-sigpnew(1) sig(2)=sig(2)-sigpnew(2) sig(3)=sig(3)-sigpnew(3) sig(4)=sig(4)-sigpnew(4)
c Exit condition for strain control
if(con.eq.1) exit c Exit condition for stress and mixed control do i=1,4 dsigp(i)=sigpnew(i)-sigp(i) end do norma1=norma(dsigp)
norma2=norma(sigpnew)
do i=1,4 sigp(i)=sigpnew(i) end do if(dabs(norma1/norma 2).lt.errrel) exit
c––END OF ITERACION (for stress and mixed control)
end do
if(itr.ge.maxitr) write(*,*)' The model does not converge', . ' using the stress control' c––State variable update q=q+dq c––Real stress increments call incten(dsm,ds,sig,sigini,poi,con,est)
c––Real strain increments
dem(1)=epsi(1) dem(2)=epsi(2) dem(3)=epsi(4) end subroutine c–––––––––––––––––––––––-c
subroutine vcresp(epsi,st,ee,q,dlam,y0,yi,alpha,beta,h,cp1,iend,
+ sigp,dq,cp2,yc,qc,cp) c–––––––––––––––––––––––-c c Function :No-associated Drucker Prager. c Plastic corrector for plastic potential function c Non-linear Hardening/Softening c –––––––––––––––––––––––c
implicit real*8 (a-h,o-z)
integer*4 iend dimension epsi(4),ee(4,4),sigp(4) dimension sn(4),st(4),em(4),rm(4) c––Plastic potential gradient call vcgrad(st,rm,beta)
c––Plastic poten tial direction
call vcsdir(rm,em,ee)
c––Initial value for the plastic multiplier call vcdlmt(st,rm,em,q,h,y0,yi,alpha,cp1,func,dlma,iend, + cp2,yc,qc,cp) c––Development of non-linear consistence condition
iter=1
dlma=0.d0 10 call vcdlam(st,rm,sn,em,dq,q,h,y0,yi,alpha,cp1,dlma,dlam,iend, + cp2,yc,qc,cp) if(dabs((dlam-dlma)/dlam).gt.1.d-9.and.iter.le.50)then dlma=dlam call vcgrad(sn,rm,beta)
call vcsdir(rm,em,ee)
iter=iter+1 goto 10 endif sigp(1)=dlam*em(1) sigp(2)=dlam*em(2)
sigp(4)=dlam*em(4)
sigp(3)=dlam*em(3) return end c––––––––––––––––––––––––c subroutine vcfunc(s,alpha,y0,yi,h,q,cp1,func,iend,cp2,yc,qc,cp) c––––––––––––––––––––––––c
c Function : Drucker-Prager Yield Function (two parameters)
c Non-linear Hardening/Softening c––––––––––––––––––––––––c implicit real*8 (a-h,o-z) integer*4 iend dimension s(4)
CODRI.F
201trs1=s(1) +s(2) +s(4)
trs2=s(1)**2+s(2)**2+s( 4)**2+s(3)**2+s(3)**2
trd2=trs2-trs1*trs1/3.d0 c––Quadratic Hardening/Softening if(iend.eq.1)then yn=0.5d0*cp*q*q+h*q+y0 c––Exponential SIMO Hardening/Softening
elseif(iend.eq.2)then
yn=y0+(yi-y0)*(1.d0-dexp(-cp1*q))+h*q c––Modified Exponential SI MO Hardenin g/Softening
elseif(iend.eq.3.and.q.lt.qc)then yn=y0+(yi-y0)*(1.d0-dexp(-cp1*q))+h*q elseif(iend.eq.3.and.q.ge.qc) then
yn=yc*dexp(cp2*(q-qc))
else write(*,*)'Hardening/Softe ning not available',iend
stop endif func=alpha*trs1+dsqrt(.5d0*trd2)-yn/dsqrt(3.d0) return
end
c––––––––––––––––––––––––c subroutine vcgrad(s,rm,beta) c––––––––––––––––––––––––c c Function:Gradient of Drucker-Prager plastic potential c––––––––––––––––––––––––c
implicit real*8 (a-h,o-z)
dimension s(4),rm(4) trs1=s(1) +s(2) +s(4) trs2=s(1)**2+s(2)**2+s( 4)**2+s(3)**2+s(3)**2
trd2=trs2-trs1*trs1/3.d0 den=2.d0*dsqrt(.5d0*trd2) rm(1)=beta+(s(1)-trs1/3.d0)/den
rm(2)=beta+(s(2)-trs1/3.d0)/den
rm(4)=beta+(s(4)-trs1/3.d0)/den rm(3)=1.d0*(s(3) )/den c––Engineering strain rm(3)=2.d0*rm(3) return
end
c––––––––––––––––––––––––c subroutine vcsdir(rm,em,ee) c––––––––––––––––––––––––c c Function:Direction of the Drucker-Prager plastic stress c––––––––––––––––––––––––c implicit real*8 (a-h,o-z)
dimension rm(4),em(4),ee(4,4)
em(1)=ee(1,1)*rm(1)+ee(1, 2)*rm(2)+ee(1,4)*rm(4)
em(2)=ee(2,1)*rm(1)+ee(2, 2)*rm(2)+ee(2,4)*rm(4)
em(4)=ee(4,1)*rm(1)+ee(4, 2)*rm(2)+ee(4,4)*rm(4)
c––Engineering strain em(3)=ee(3,3)*rm(3)
return
end c––––––––––––––––––––––––c subroutine vcdlmt(st,rm,em,q,h,y0,yi,alpha,cp1,func,dlmt,iend, + cp2,yc,qc,cp) c––––––––––––––––––––––––c
c Function :Consistenze condition
c Non-linear Hardening/Softening c––––––––––––––––––––––––c implicit real*8 (a-h,o-z) integer*4 irhd dimension st(4),rm (4),rn(4),em(4)
call vcgrad(st,rn,alpha)
trne=rn(1)*em(1)+rn(2)*em(2)+rn(4)*em(4)+
+ rn(3)*em(3) trm2=rm(1)*rm(1)+rm(2)*rm(2)+rm(4)*rm(4)+ + rm(3)*rm(3)/2.d0 c––Quadratic Hardening/Softening if(iend.eq.1)then
yn=0.5d0*cp*q*q+h*q+y0
dfdq=(-cp*q-h)/dsqrt(3.d0) c––Exponential SIMO Hardening/Softening elseif(iend.eq.2)then yn=y0+(yi-y0)*(1.d0-dexp(-cp1*q))+h*q dfdq=-( (yi-y0)*cp1*dexp(-cp1*q)+h )/dsqrt(3.d0) c––Modified Exponential SI MO Hardenin g/Softening
elseif(iend.eq.3.and.q.lt.qc)then yn=y0+(yi-y0)*(1.d0-dexp(-cp1*q))+h*q dfdq=-( (yi-y0)*cp1*dexp(-cp1*q)+h )/dsqrt(3.d0) elseif(iend.eq.3.and.q.ge.qc) then yn=yc*dexp(cp2*(q-qc)) dfdq=-(yc*(dexp(cp2*(q-qc)))*cp2)/dsqrt(3.d0)
CODRI.F
202else
write(*,*)'Hardening/Softe ning not available',iend
stop endif dq=dsqrt(2.d0/3.d0*trm2) dlmt=func/(trne-dfdq*dq) return
end
c––––––––––––––––––––––––c subroutine vcdlam(st,rm,sn,em,dq,q,h,y0,yi,alpha,cp1,dlmt,dlam, + iend,cp2,yc,qc,cp) c––––––––––––––––––––––––c c Function:Consistence condition
c Non-linear Hardening/Softening
c––––––––––––––––––––––––c implicit real*8 (a-h,o-z) integer*4 iend dimension st(4),rm(4), sn(4),em(4), sigp(4)
tre1=em(1) +em(2) +em(4) tre2=em(1)*em(1)+em(2)*em(2)+em(4)*em(4)+
+ em(3)*em(3)+em(3)*em(3)
trs1=st(1) +st(2) +st(4) trs2=st(1)*st(1)+st(2)*st(2)+st(4)*st(4)+ + st(3)*st(3)+st(3)*st(3) trse=st(1)*em(1)+st(2)*em(2)+st(4)*em(4)+ + st(3)*em(3)+st(3)*em(3)
trm2=rm(1)*rm(1)+rm(2)*rm(2)+rm(4)*rm(4)+
+ rm(3)*rm(3)/2.d0 dq=dsqrt(2.d0/3.d0*trm2) w0=(0.5d0*trs2-trs1*trs1/6.d0) w1=(-trse+trs1*tre1/3.d0) w2=(0.5d0*tre2-tre1*tre1/6.d0) c––Newton-Raphson iteration
dlam=dlmt
iter=0 11 iter=iter+1 qn1=q+dlam*dq c––Quadratic Hardening/Softening if(iend.eq.1)then
yn1=(0.5d0*cp*qn1*qn1+h*qn1+y0)/dsqrt(3.d0)
dyn=(cp*qn1*dq+h*dq)/dsqrt(3.d0) c––Exponential SIMO Hardening/Softening elseif(iend.eq.2)then yn1=(y0+(yi-y0)*(1.d0-dexp(-cp1*qn1))+h*qn1)/dsqrt(3.d0) dyn=( (yi-y0)*cp1*dq*dexp(-cp1*qn1)+h*dq )/dsqrt(3.d0) c––Modified Exponential SI MO Hardenin g/Softening
elseif(iend.eq.3.and.qn1.lt.qc)then yn1=(y0+(yi-y0)*(1.d0-dexp(-cp1*qn1))+h*qn1)/dsqrt(3.d0) dyn=( (yi-y0)*cp1*dq*dexp(-cp1*qn1)+h*dq )/dsqrt(3.d0) elseif(iend.eq.3.and.qn1.ge.qc) then yn1=yc*dexp(cp2*(qn1-qc))/dsqrt(3.d0) dyn=yc*dexp(cp2*(qn1-qc))*cp2*dq/dsqrt(3.d0)
else
write(*,*)'Hardening/Softe ning not available',iend
stop endif fn1=alpha*(trs1-dlam*tre1)+dsqrt(dlam*dlam*w2+dlam*w1+w0) + -yn1
dfn=-alpha*tre1
+ +(dlam*2.d0*w2+w1)/2.d0/dsqrt(dlam*dlam*w2+dlam*w1+w0) + -dyn dlam=dlam-fn1/dfn if(dabs(fn1/dfn).gt.1.d-10.and.iter.le.200) goto 11 if(dlam.lt.0.d0) write(*,*)' dlam < null !! ',dlam dq=dq*dlam
sn(1)=st(1)-dlam*em(1)
sn(2)=st(2)-dlam*em(2) sn(4)=st(4)-dlam*em(4) sn(3)=st(3)-dlam*em(3) call vcfunc(sn,alpha,y0,yi,h,(q+dq),cp1,func,iend,cp2,yc,qc,cp) if(dabs(func).gt.1.d-4)then
write(*,*)' false dlam, func= ',func
endif return end c–––––––––––––––––––––––-c c DRUCKER-PRAGER PLASTICITY MODEL c c (three parameters) c
c–––––––––––––––––––––––-c
c Variables c c–––––––––––––––––––––––-c c c c inc – number of increment c c est – indicative of state c
CODRI.F
203c = 1 plane strains c
c = 2 plane stresses c c = 3 axisymmetric c c con – control type c c = 1 strains c c = 2 mixed c c = 3 stresses c
c mat – material proprieties c
c e – Elastic module c c poi – Poisson's ratio c c alfa – yield function parameter c c z – yield function parameter c beta – plastic potential parameter c
c cp – Hardening/Softening parameter of quadratic function c
c cp1 – Hardening/Softening parameter of Simo's function c c cp2 – Hard./Soft.parameter of Modified Simo's function c
c h – Hard./Soft.parameter of Simo and Mod Simo's function c c yi – Hard./Soft.parameter of Simo and Mod Simo's function c c iend – indicator for the hardening/softening function c c = 1 Hard./Soft. Quadratic c
c = 2 Hard./Soft. exponential SIMO c
c = 3 Hard./Soft. modified exponential SIMO c c y0 – Hard./Soft.parameter of Q-S-MS function c c ymax – Hard./Soft.paramet er of Q-S-MS function c
c qmax – Hard./Soft.paramet er of Q-S-MS function c
c yc – Hard./Soft.parameter of Q-S-MS function c
c qc – Hard./Soft.parameter of Q-S-MS function c
c yr – Hard./Soft.parameter of Q-S-MS function c
c qr – Hard./Soft.parameter of Q-S-MS function c
c ee – Elastic matrix c c dd – Inverse elastic matrix c c func – yield function c c ds – Imposed stress increment (input) c
c de – Imposed strain increment (input) c
c dsm – Calculated stress increment (output) c c dem – Calculated stra in increment (output) c
c epsi – strain increments c
c sigi – Elastic stress increment (elastic predictor) c c sigini – Auxiliar y stress state c
c sig – Calculated stress c c rm – Gradient function of plastic potential c c rn – Gradient function of yield function c c en – plastic corrector direction c c dsigp – variation of plastic corrector c c sigp – plastic corrector at anterior iteration c c sigpnew – plastioc corrector: actual iteration c
c norma1 – norm of dsigp c
c norma2 – norm of sigpnew c c itr – number of iteration (stress control) c c maxitr – maxim number of iterations (stress control) c c maxitc – maxim number of ite rations (palstic corrector) c
c errrel – maxim relative error between two iterations c
c minmin – tolerance to the zero c
c c c–––––––––––––––––––––––-c c–––––––––––––––––––––––-c subroutine dp3(dsm,dem,ds,de,sigini,mat,inc,con,est,q) c–––––––––––––––––––––––-c
c Calculation of stress and strain increments c
c for No-Associated Plastic Potential c c–––––––––––––––––––––––-c implicit none real*8 de,ds,dem,dsm, mat,ee,dd,dq,norma,norma1,norma2, * sigp,sigpnew,dsigp,epsi,sigi,sig,e,sigini,ymax,qmax, * poi,alpha,beta,func,yi,q,dlam,cp1,yc,qc,yr,cp,z,
* minmin,errrel,cp2,h,y0,qr,rm,t
integer*4 inc,con,est,itr,maxitr,maxitc,i,j,iend dimension de(3),ds(3),dem(3),dsm(3),mat(17),rm(4), * ee(4,4),dd(4,4),epsi(4),sigi(4),sig(4), * sigp(4),sigpnew(4),dsigp(4),sigini(4) c––Tolerances
minmin=1./10.d0**14
errrel=0.0001 maxitr=100 maxitc=200 c––Material and model data e = mat(1) poi = mat(2)
y0 = mat(3)
alpha= mat(4) beta = mat(5) cp1 = mat(6) h = mat(7) yi = mat(8)
CODRI.F
204iend = mat(9)
cp2 = mat(10) yc = mat(11) qc = mat(12) ymax = mat(13) qmax = mat(14) yr = mat(15)
qr = mat(16)
z = mat(17) c––Elastic Matrix call elast(mat,ee,dd) c––Plastic corrector reset sigp(1) = 0.d0
sigp(2) = 0.d0
sigp(3) = 0.d0 sigp(4) = 0.d0 if(inc.eq.2)q=0 c––START TO THE ITERATION (fo r mixed and stress control)
do itr=1,maxitr call predel(sigi,epsi,sig,sigini,sigp,ds,de,ee,dd,poi,con,est)
c––Flow Condition
call vcfunc3(sig,alpha,y0,yi,h,q ,cp1,func,iend,cp2,yc,qc,cp,z
+ ,t) c––Calculated stress in elastic regime if(func.le.minmin)exit c––Plastic Corrector
call vcresp3(func,epsi,sig,ee,q,dlam,y0,yi,alpha,beta,h,cp1,iend,
+ sigpnew,dq,cp2,yc,qc,cp,z,rm) c–- Final stress update sig(1)=sig(1)-sigpnew(1) sig(2)=sig(2)-sigpnew(2) sig(3)=sig(3)-sigpnew(3) sig(4)=sig(4)-sigpnew(4)
c Exit condition for strain control
if(con.eq.1) exit c Exit condition for stress and mixed control do i=1,4 dsigp(i)=sigpnew(i)-sigp(i) end do
norma1=norma(dsigp)
norma2=norma(sigpnew) do i=1,4 sigp(i)=sigpnew(i) end do if(dabs(norma1/norma 2).lt.errrel) exit
c––END OF ITERACION (for stress and mixed control)
end do
if(itr.ge.maxitr) write(*,*)' The model does not converge', . ' using the stress control' c––State variable update q=q+dq c––Real stress increments
call incten(dsm,ds,sig,sigini,poi,con,est)
c––Real strain increments dem(1)=epsi(1) dem(2)=epsi(2) dem(3)=epsi(4) end subroutine
c–––––––––––––––––––––––-c
subroutine vcresp3(func,epsi,st,ee,q,dlam,y0,yi,alpha,beta,h,cp1, + iend,sigp,dq,cp2,yc,qc,cp,z,rm) c–––––––––––––––––––––––-c c Function :No-associated Drucker Prager. c Plastic corrector for plastic potential function c Non-linear Hardening/Softening
c –––––––––––––––––––––––c
implicit real*8 (a-h,o-z) integer*4 iend dimension epsi(4),ee(4,4),sigp(4) dimension sn(4),st(4),em(4),rm(4) c––Plastic potential gradient
call vcgrad3(st,rm,beta,z,dtrd2s1,dtrd3s1,dqjs1,dts1,dpds1,
+ dfds1) c––Plastic poten tial direction
call vcsdir3(rm,em,ee) c––Initial value for the plastic multiplier call vcdlmt3(st,rm,em,q,h,y0,yi,alpha,cp1,func,dlma,iend, + cp2,yc,qc,cp,z,yn,dfdq,dq)
c––Development of non-linear consistence condition
10 call vcdlam3(st,rm,sn,em,dq,q,h,y0,yi,alpha,cp1,dlma,dlam,iend, + cp2,yc,qc,cp,z, + fn1,tl,pl,yn1,dfn) if(dabs((dlam-dlma)/dlam).gt.1.d-10.and.iter.le.50)then dlma=dlam
CODRI.F
205call vcgrad3(sn,rm,beta,z,dtrd2s1,dtrd3s1,dqjs1,dts1,dpds1,
+ dfds1) iter=iter+1 goto 10 endif sigp(1)=dlam*em(1) sigp(2)=dlam*em(2)
sigp(4)=dlam*em(4)
sigp(3)=dlam*em(3) return end c––––––––––––––––––––––––c subroutine vcfunc3(s,alpha,y0,yi,h,q,cp1,func,iend,cp2,yc,qc,cp,z
+ ,t)
c––––––––––––––––––––––––c c Function : Drucker-Prager Yiel d Function (three parameters)
c Non-linear Hardening/Softening c––––––––––––––––––––––––c implicit real*8 (a-h,o-z) integer*4 iend
dimension s(4)
trs1=s(1) +s(2) +s(4) trs2=s(1)*s(2)+s(2)*s(4)+s(4)*s(1)-s(3)*s(3) trs3=s(1)*s(2)*s(4)-s(3)*s(3)*s(4) trd2=1/3.d0*(trs1*trs1-3*trs2) trd3=1/27.d0*(2*trs1*trs1*trs1-9*trs1*trs2+27*trs3)
p=-trs1/3.d0
qj=dsqrt(trd2*3.d0) t=1/2.d0*qj*(1+1/z-(1-1/z)* (1/qj)**3*(27/2.d0*trd3))
c––Quadratic Hardening/Softening if(iend.eq.1)then yn=0.5d0*cp*q*q+h*q+y0 c––Exponential SIMO Hardening/Softening
elseif(iend.eq.2)then
yn=y0+(yi-y0)*(1.d0-dexp(-cp1*q))+h*q c––Modified Exponential SI MO Hardenin g/Softening
elseif(iend.eq.3.and.q.lt.qc)then yn=y0+(yi-y0)*(1.d0-dexp(-cp1*q))+h*q elseif(iend.eq.3.and.q.ge.qc) then
yn=yc*dexp(cp2*(q-qc))
else write(*,*)'Hardening/Softe ning not available',iend
stop endif func=t-alpha*p-yn return
end
c––––––––––––––––––––––––c subroutine vcgrad3(s,rm,beta,z,dtrd2s1,dtrd3s1,dqjs1,dts1,dpds1, + dfds1) c––––––––––––––––––––––––c c Function:Gradient of Drucker-Prager plastic potential
c––––––––––––––––––––––––c
implicit real*8 (a-h,o-z) dimension s(4),rm(4) trs1=s(1) +s(2) +s(4) trs2=s(1)*s(2)+s(2)*s(4)+s(4)*s(1)-s(3)*s(3) trs3=s(1)*s(2)*s(4)-s(3)*s(3)*s(4)
trd2=1/3.d0*(trs1*trs1-3*trs2)
trd3=1/27.d0*(2*trs1*trs1*trs1-9*trs1*trs2+27*trs3) p=-trs1/3.d0 qj=dsqrt(trd2*3.d0) t=1/2.d0*qj*(1+1/z-(1-1/z)* (1/qj)**3*(27/2.d0*trd3))
dtrd2s1=1/3.d0*(2*trs1-3*(s(2)+s(4))) dtrd3s1=1/27.d0*(6*trs1*trs1-9*trs2-9*trs1*(s(2)+s(4))+27*
+ (s(2)*s(4)))
dqjs1=1/2.d0/(dsqrt(trd2*3.d0))*3.d0*dtrd2s1 dts1=1/2.d0*(dqjs1*(1+1/z-(1-1/z)*(1/qj)**3*(27/2.d0*trd3))+qj* + (-(1-1/z)*(-3*(1/qj)**4* dqjs1*(27/2. d0*trd3)+
+ (1/qj)**3*27/2.d0*dtrd3s1))) dpds1=-1/3.d0
dfds1=dts1-beta*dpds1
rm(1)=dfds1 dtrd2s2=1/3.d0*(2*trs1-3*(s(1)+s(4))) dtrd3s2=1/27.d0*(6*trs1*trs1-9*trs2-9*trs1*(s(1)+s(4))+27* + (s(1)*s(4))) dqjs2=1/2.d0/(dsqrt(trd2*3.d0))*3.d0*dtrd2s2 dts2=1/2.d0*(dqjs2*(1+1/z-(1-1/z)*(1/qj)**3*(27/2.d0*trd3))+qj*
+ (-(1-1/z)*(-3*(1/qj)**4* dqjs2*(27/2. d0*trd3)+
+ (1/qj)**3*27/2.d0*dtrd3s2)))
dpds2=-1/3.d0 dfds2=dts2-beta*dpds2 rm(2)=dfds2 dtrd2s4=1/3.d0*(2*trs1-3*(s(2)+s(1)))
CODRI.F
206dtrd3s4=1/27.d0*(6*trs1*trs1-9*trs2-9*trs1*(s(2)+s(1))+27*
+ (s(2)*s(1))) dqjs4=1/2.d0/(dsqrt(trd2*3.d0))*3.d0*dtrd2s4 dts4=1/2.d0*(dqjs4*(1+1/z-(1-1/z)*(1/qj)**3*(27/2.d0*trd3))+qj* + (-(1-1/z)*(-3*(1/qj)**4* dqjs4*(27/2. d0*trd3)+
+ (1/qj)**3*27/2.d0*dtrd3s4))) dpds4=-1/3.d0
dfds4=dts4-beta*dpds4
rm(4)=dfds4 dtrd2s3=1/3.d0*(-3*(-2.d0*s(3))) dtrd3s3=1/27.d0*(-9*trs1*(-2.d0*s(3))+27*(-s(4)*2.d0*s(3))) dqjs3=1/2.d0/(dsqrt(trd2*3.d0))*3.d0*dtrd2s3 dts3=1/2.d0*(dqjs3*(1+1/z-(1-1/z)*(1/qj)**3*(27/2.d0*trd3))+qj*
+ (-(1-1/z)*(-3*(1/qj)**4* dqjs3*(27/2. d0*trd3)+
+ (1/qj)**3*27/2.d0*dtrd3s3)))
dpds3=0.d0 dfds3=dts3-beta*dpds3 rm(3)=dfds3 c––Engineering strain rm(3)=2.d0*rm(3)
return
end c––––––––––––––––––––––––c subroutine vcsdir3(rm,em,ee) c––––––––––––––––––––––––c c Function:Direction of the Drucker-Prager plastic stress
c––––––––––––––––––––––––c
implicit real*8 (a-h,o-z) dimension rm(4),em(4),ee(4,4) em(1)=ee(1,1)*rm(1)+ee(1, 2)*rm(2)+ee(1,4)*rm(4)
em(2)=ee(2,1)*rm(1)+ee(2, 2)*rm(2)+ee(2,4)*rm(4)
em(4)=ee(4,1)*rm(1)+ee(4, 2)*rm(2)+ee(4,4)*rm(4)
c––Engineering strain
em(3)=ee(3,3)*rm(3)
return end c––––––––––––––––––––––––c subroutine vcdlmt3(st,rm,em,q,h,y0,yi,alpha,cp1,func,dlmt,iend, + cp2,yc,qc,cp,z,yn,dfdq,dq)
c––––––––––––––––––––––––c
c Function :Consistenze condition c Non-linear Hardening/Softening c––––––––––––––––––––––––c implicit real*8 (a-h,o-z) integer*4 irhd dimension st(4),rm (4),rn(4),em(4)
call vcgrad3(st,rn,alpha,z,dtrd2s1,dtrd3s1,dqjs1,dts1,dpds1, + dfds1) trm2=rm(1)*rm(1)+rm(2)*rm(2)+rm(4)*rm(4)+rn(3)*em(3) trne=rn(1)*em(1)+rn(2)*em(2)+r n(4)*em(4)+rm(3)*rm(3)/2.d0
c––Quadratic Hardening/Softening if(iend.eq.1)then
yn=0.5d0*cp*q*q+h*q+y0
dfdq=(-cp*q-h) c––Exponential SIMO Hardening/Softening elseif(iend.eq.2)then yn=y0+(yi-y0)*(1.d0-dexp(-cp1*q))+h*q dfdq=-( (yi-y0)*cp1*dexp(-cp1*q)+h )
c––Modified Exponential SI MO Hardenin g/Softening
elseif(iend.eq.3.and.q.lt.qc)then
yn=y0+(yi-y0)*(1.d0-dexp(-cp1*q))+h*q dfdq=-( (yi-y0)*cp1*dexp(-cp1*q)+h ) elseif(iend.eq.3.and.q.ge.qc) then yn=yc*dexp(cp2*(q-qc)) dfdq=-(yc*(dexp(cp2*(q-qc)))*cp2)
else
write(*,*)'Hardening/Softe ning not available',iend
stop endif dq=dsqrt(2.d0/3.d0*trm2) dlmt=func/(trne-dfdq*dq)
return
end c––––––––––––––––––––––––c subroutine vcdlam3(st,rm,sn,em,dq,q,h,y0,yi,alpha,cp1,dlmt,dlam, + iend,cp2,yc,qc,cp,z, + fn1,tl,pl,yn1,dfn) c––––––––––––––––––––––––c
c Function:Consistence condition
c Non-linear Hardening/Softening c––––––––––––––––––––––––c implicit real*8 (a-h,o-z) integer*4 iend dimension st(4),rm(4), sn(4),em(4), sigp(4)
CODRI.F
207trs1=st(1) +st(2) +st(4)
trs2=st(1)*st(2)+st(2)*st(4) +st(4)*st(1)-st(3)*st(3)
trs3=st(1)*st(2)*st(4)-st(3)*st(3)*st(4) trd2=1/3.d0*(trs1*trs1-3*trs2) trd3=1/27.d0*(2*trs1*trs1*trs1-9*trs1*trs2+27*trs3) p=-trs1/3.d0 qj=dsqrt(trd2*3.d0)
t=1/2.d0*qj*(1+1/z-(1-1/z)* (1/qj)**3*(27/2.d0*trd3))
tre1=em(1) +em(2) +em(4)
tre2=em(1)*em(2)+em(2)*em(4)+em(4)*em(1)-em(3)*em(3) tre3=em(1)*em(2)*em(4)-em(3)*em(3)*em(4) trde2=1/3.d0*(tre1*tre1-3*tre2) trde3=1/27.d0*(2*tre1*tre1*t re1-9*tre1*tre2+27*tre3)
pe=-tre1/3.d0 qje=dsqrt(trde2*3.d0) te=1/2.d0*qje*(1+1/z-(1-1/z)*(1/qje)**3*(27/2.d0*trde3)) trm2=rm(1)*rm(1)+rm(2)*rm(2)+rm(4)*rm(4)+rm(3)*rm(3)/2.d0 dq=dsqrt(2.d0/3.d0*trm2) c––Newton-Raphson iteration dlam=dlmt
iter=0
11 iter=iter+1 qn1=q+dlam*dq c––Quadratic Hardening/Softening if(iend.eq.1)then yn1=(0.5d0*cp*qn1*qn1+h*qn1+y0)
dyn=(cp*qn1*dq+h*dq)
c––Exponential SIMO Hardening/Softening elseif(iend.eq.2)then yn1=(y0+(yi-y0)*(1.d0-dexp(-cp1*qn1))+h*qn1) dyn=( (yi-y0)*cp1*dq*dexp(-cp1*qn1)+h*dq ) c––Modified Exponential SI MO Hardenin g/Softening
elseif(iend.eq.3.and.qn1.lt.qc)then
yn1=(y0+(yi-y0)*(1.d0-dexp(-cp1*qn1))+h*qn1)
dyn=( (yi-y0)*cp1*dq*dexp(-cp1*qn1)+h*dq ) elseif(iend.eq.3.and.qn1.ge.qc) then yn1=yc*dexp(cp2*(qn1-qc)) dyn=yc*dexp(cp2*(qn1-qc))*cp2*dq else
write(*,*)'Hardening/Softe ning not available',iend
stop
endif trs1l=st(1)+st(2)+st(4)-dlam*(em(1)+em(2)+em(4)) trs2l=trs2+dlam**2*tre2-dlam*(st(1)*em(2)+em(1)*st(2)+ + st(2)*em(4)+em(2)*st(4)+ st(4)*em(1)+em(4)*st(1))
+ +2.d0*dlam*st(3)*em(3)
trs3l=trs3-dlam*(st(1)*st(2)*em(4)+st(1)*em(2)*st(4)+
+ st(2)*em(1)*st(4))+dlam**2*(st(1)*em(2)*em(4)+ + st(2)*em(4)*em(1)+st(4)* em(1)*em(2))-dlam**3*tre3
+ +st(3)*st(3)*dlam*em(4)-dlam**2.d0*em(3)*em(3)*st(4) + +2.d0*st(3)*dlam*em(3)*st(4)-2.d0*st(3)*dlam*em(3)*dlam*em(4) trd2l=1/3.d0*(trs1l*trs1l-3*trs2l)
trd3l=1/27.d0*(2*trs1l*trs1l*trs1l-9*trs1l*trs2l+27*trs3l)
pl=-trs1l/3.d0 qjl=dsqrt(trd2l*3.d0) tl=1/2.d0*qjl*(1+1/z-(1-1/z)*(1/qjl)**3*(27/2.d0*trd3l)) dtrs1l=-(em(1)+em(2)+em(4)) dtrs2l=2*dlam*tre2-(st(1)*em(2)+ em(1)*st(2)+st(2 )*em(4)+em(2)*
+ st(4)+st(4)*em(1)+em(4)* st(1))+2.d0*st(3)*em(3)
dtrs3l=-(st(1)*st(2)*em(4)+st(1)*e m(2)*st(4)+st(2)*em(1)*st(4))+
+ 2*dlam*(st(1)*em(2)*em(4)+st(2)*em(4)*em(1)+st(4)*em(1)* + em(2))-3*dlam**2*tre3 + +st(3)*st(3)*em(4)-2*dlam*em(3)*em(3)*st(4) + +2.d0*st(3)*em(3)*st(4)-4.d0*st(3)*em(3)*dlam*em(4) dtrd2l=1/3.d0*(2*trs1l*dtrs1l-3*dtrs2l)
dtrd3l=1/27.d0*(6*trs1l*trs1l*dtrs1l-9*dtrs1l*trs2l-9*trs1l*dtrs2l
+ +27*dtrs3l) dqjl=1/2.d0/qjl*3.d0*dtrd2l dtl=1/2.d0*((dqjl*(1+1/z-(1-1/z)*(1/qjl)**3*(27/2.d0*trd3l))+qjl* + (-(1-1/z))*((-3)*1/qjl**4*dqjl*(27/2.d0*trd3l)+ + (1/qjl)**3*27/2.d0*dtrd3l)))
dpl=-1/3.d0*dtrs1l
fn1=tl-alpha*pl-yn1 dfn=dtl-alpha*dpl-dyn dlam=dlam-fn1/dfn if(dabs(fn1/dfn).gt.1.d-10.and.iter.le.200) goto 11 if(dlam.lt.0.d0) write(*,*)' dlam < null !! ',dlam dq=dq*dlam
sn(1)=st(1)-dlam*em(1)
sn(2)=st(2)-dlam*em(2) sn(4)=st(4)-dlam*em(4) sn(3)=st(3)-dlam*em(3) return end
CODRI.F
208c–––––––––––––––––––––––-c
c BRESLER-PISTER PLASTICITY MODEL c c (three parameters) c c–––––––––––––––––––––––-c c Variables c c–––––––––––––––––––––––-c c c
c inc – number of increment c
c est – indicative of state c c = 1 plane strains c c = 2 plane stresses c c = 3 axisymmetric c c con – control type c
c = 1 strains c
c = 2 mixed c c = 3 stresses c c mat – material proprieties c c e – Elastic module c c poi – Poisson's ratio c c b – yield function parameter c
c c – yield function parameter c
c bb – plastic potential parameter c c cc – plastic potential parameter c c cp – Hardening/Softening parameter of quadratic function c c cp1 – Hardening/Softening parameter of Simo's function c c cp2 – Hard./Soft.parameter of Modified Simo's function c
c h – Hard./Soft.parameter of Simo and Mod Simo's function c c yi – Hard./Soft.parameter of Simo and Mod Simo's function c c iend – indicator for the hardening/softening function c c = 1 Hard./Soft. Quadratic c c = 2 Hard./Soft. exponential SIMO c c = 3 Hard./Soft. modified exponential SIMO c c a0 – Hard./Soft.parameter of Q-S-MS function c
c amax – Hard./Soft.paramet er of Q-S-MS function c
c qmax – Hard./Soft.paramet er of Q-S-MS function c
c ac – Hard./Soft.parameter of Q-S-MS function c
c qc – Hard./Soft.parameter of Q-S-MS function c
c ar – Hard./Soft.parameter of Q-S-MS function c
c qr – Hard./Soft.parameter of Q-S-MS function c
c ee – Elastic matrix c c dd – Inverse elastic matrix c c func – yield function c c ds – Imposed stress increment (input) c
c de – Imposed strain increment (input) c
c dsm – Calculated stress increment (output) c c dem – Calculated stra in increment (output) c
c epsi – strain increments c
c sigi – Elastic stress increment (elastic predictor) c c sigini – Auxiliar y stress state c
c sig – Calculated stress c c rm – Gradient function of plastic potential c c rn – Gradient function of yield function c
c en – plastic corrector direction c
c dsigp – variation of plastic corrector c c sigp – plastic corrector at anterior iteration c c sigpnew – plastioc corrector: actual iteration c c norma1 – norm of dsigp c c norma2 – norm of sigpnew c
c itr – number of iteration (stress control) c
c maxitr – maxim number of iterations (stress control) c c maxitc – maxim number of ite rations (palstic corrector) c
c errrel – maxim relative error between two iterations c c minmin – tolerance to the zero c c c c–––––––––––––––––––––––-c
c–––––––––––––––––––––––-c
subroutine bp(dsm,dem,ds,de,sigini,mat,inc,con,est,q) c–––––––––––––––––––––––-c c Calculation of stress and strain increments c c for No-Associated Plastic Potential c c–––––––––––––––––––––––-c
implicit none
real*8 de,ds,dem,dsm, mat,ee,dd,dq,norma,norma1,norma2, * sigp,sigpnew,dsigp,epsi,sigi,sig,e,sigini,amax,qmax, * poi,b,c,bb,cc,func,yi,q,dlam,cp1,ac,qc,ar,cp, * minmin,errrel,cp2,h,a0,qr,rm integer*4 inc,con,est,itr,maxitr,maxitc,i,j,iend dimension de(3),ds(3),dem(3),dsm(3),mat(17),rm(4),
* ee(4,4),dd(4,4),epsi(4),sigi(4),sig(4),
* sigp(4),sigpnew(4),dsigp(4),sigini(4) c––Tolerances minmin=1./10.d0**14 errrel=0.0001 maxitr=100
CODRI.F
209maxitc=200
c––Material and model data e = mat(1) poi = mat(2) a0 = mat(3) b = mat(4) !failure function c = mat(5) !failure function
bb = mat(6) !plastic potential
cc = mat(7) !plastic potential cp1 = mat(8) h = mat(9) yi = mat(10) iend = mat(11)
cp2 = mat(12)
ac = mat(13) qc = mat(14) amax = mat(15) qmax = mat(16) qr = mat(17) c––Elastic Matrix
call elast(mat,ee,dd)
c––Plastic corrector reset sigp(1) = 0.d0 sigp(2) = 0.d0 sigp(3) = 0.d0 sigp(4) = 0.d0
if(inc.eq.2)q=0
c––START TO THE ITERATION (fo r mixed and stress control)
do itr=1,maxitr call predel(sigi,epsi,sig,sigini,sigp,ds,de,ee,dd,poi,con,est) c––Flow Condition call bpfunc(sig,b,c,a0,yi,h,q,cp1,func,iend,cp2,ac,qc,cp) c––Calculated stress in elastic regime
if(func.le.minmin)exit
c––Plastic Corrector call bpresp(func,epsi,sig,ee,q,dlam,a0,yi,b,c,bb,cc,h,cp1,iend, + sigpnew,dq,cp2,ac,qc,cp,rm) c–- Final stress update sig(1)=sig(1)-sigpnew(1)
sig(2)=sig(2)-sigpnew(2)
sig(3)=sig(3)-sigpnew(3) sig(4)=sig(4)-sigpnew(4) c Exit condition for strain control if(con.eq.1) exit c Exit condition for stress and mixed control do i=1,4
dsigp(i)=sigpnew(i)-sigp(i)
end do norma1=norma(dsigp) norma2=norma(sigpnew) do i=1,4 sigp(i)=sigpnew(i)
end do
if(dabs(norma1/norma 2).lt.errrel) exit
c––END OF ITERACION (for stress and mixed control) end do if(itr.ge.maxitr) write(*,*)' The model does not converge', . ' using the stress control'
c––State variable update
q=q+dq c––Real stress increments call incten(dsm,ds,sig,sigini,poi,con,est) c––Real strain increments dem(1)=epsi(1) dem(2)=epsi(2)
dem(3)=epsi(4)
end subroutine c–––––––––––––––––––––––-c subroutine bpresp(func,epsi,st,ee,q,dlam,a0,yi,b,c,bb,cc,h,cp1, + iend,sigp,dq,cp2,ac,qc,cp,rm) c–––––––––––––––––––––––-c
c Function :No-associat ed Bresler Pister.
c Plastic corrector for plastic potential function
c Non-linear Hardening/Softening c –––––––––––––––––––––––c implicit real*8 (a-h,o-z) integer*4 iend dimension epsi(4),ee(4,4),sigp(4)
dimension sn(4),st(4),em(4),rm(4)
c––Plastic potential gradient call bpgrad(st,rm,bb,cc,dtrd2s1,dtrj2s1,dfds1 ) c––Plastic poten tial direction
call bpsdir(rm,em,ee) c––Initial value for the plastic multiplier
CODRI.F
210call bpdlmt(st,rm,em,q,h,a0,yi,b,c,cp1,func,dlma,iend,
+ cp2,ac,qc,cp,an,dfdq,dq) c––Development of non-linear consistence condition 10 call bpdlam(st,rm,sn,em,dq,q,h,a0,yi,b,c,cp1,dlma,dlam,iend, + cp2,ac,qc,cp,fn1,an1,dfn) if(dabs((dlam-dlma)/dlam).gt.1.d-10.and.iter.le.50)then dlma=dlam
call bpgrad(sn,rm,bb,cc,dtrd2s1,dtrj2s1,dfds1)
iter=iter+1 goto 10 endif sigp(1)=dlam*em(1) sigp(2)=dlam*em(2)
sigp(4)=dlam*em(4)
sigp(3)=dlam*em(3) return end c––––––––––––––––––––––––c subroutine bpfunc(s,b,c,a0,yi,h,q,cp1,func,iend,cp2,ac,qc,cp) c––––––––––––––––––––––––c
c Function : Bresler Pister Yield Function (three parameters)
c Non-linear Hardening/Softening c––––––––––––––––––––––––c implicit real*8 (a-h,o-z) integer*4 iend dimension s(4)
trs1=s(1) +s(2) +s(4)
trs2=s(1)*s(2)+s(2)*s(4)+s(4)*s(1)-s(3)*s(3) trd2=1/3.d0*(trs1*trs1-3*trs2) trj2=dsqrt(trd2) c––Quadratic Hardening/Softening if(iend.eq.1)then an=0.5d0*cp*q*q+h*q+a0
c––Exponential SIMO Hardening/Softening
elseif(iend.eq.2)then an=a0+(yi-a0)*(1.d0-dexp(-cp1*q))+h*q c––Modified Exponential SI MO Hardenin g/Softening
elseif(iend.eq.3.and.q.lt.qc)then an=a0+(yi-a0)*(1.d0-dexp(-cp1*q))+h*q
elseif(iend.eq.3.and.q.ge.qc) then
an=ac*dexp(cp2*(q-qc)) else write(*,*)'Hardening/Softe ning not available',iend
stop endif func=-an+b*trs1-c*trs1**2.d0+trj2
return
end c––––––––––––––––––––––––c subroutine bpgrad(s,rm,bb,cc,dtrd2s1,dtrj2s1,dfds1) c––––––––––––––––––––––––c c Function:Gradient of Bresler-Pister plastic potential
c––––––––––––––––––––––––c
implicit real*8 (a-h,o-z) dimension s(4),rm(4) trs1=s(1) +s(2) +s(4) trs2=s(1)*s(2)+s(2)*s(4)+s(4)*s(1)-s(3)*s(3) trd2=1/3.d0*(trs1*trs1-3*trs2)
trj2=dsqrt(trd2)
dtrd2s1=1/3.d0*(2*trs1-3*(s(2)+s(4))) dtrj2s1=1/2.d0/dsqrt(trd2)*dtrd2s1 dfds1=bb-cc*2.d0*trs1+dtrj2s1 rm(1)=dfds1 dtrd2s2=1/3.d0*(2*trs1-3*(s(1)+s(4))) dtrj2s2=1/2.d0/dsqrt(trd2)*dtrd2s2
dfds2=+bb-cc*2.d0*trs1+dtrj2s2
rm(2)=dfds2 dtrd2s4=1/3.d0*(2*trs1-3*(s(1)+s(2))) dtrj2s4=1/2.d0/dsqrt(trd2)*dtrd2s4 dfds4=+bb-cc*2.d0*trs1+dtrj2s4 rm(4)=dfds4
dtrd2s3=1/3.d0*(-3*(-2.d0*s(3)))
dtrj2s3=1/2.d0/dsqrt(trd2)*dtrd2s3 dfds3=+dtrj2s3 rm(3)=dfds3 c––Engineering strain rm(3)=2.d0*rm(3) return
end
c––––––––––––––––––––––––c subroutine bpsdir(rm,em,ee) c––––––––––––––––––––––––c c Function:Direction of the Bresler-Pister plastic stress
c––––––––––––––––––––––––c
CODRI.F
211implicit real*8 (a-h,o-z)
dimension rm(4),em(4),ee(4,4) em(1)=ee(1,1)*rm(1)+ee(1, 2)*rm(2)+ee(1,4)*rm(4)
em(2)=ee(2,1)*rm(1)+ee(2, 2)*rm(2)+ee(2,4)*rm(4)
em(4)=ee(4,1)*rm(1)+ee(4, 2)*rm(2)+ee(4,4)*rm(4)
c––Engineering strain em(3)=ee(3,3)*rm(3)
return
end c––––––––––––––––––––––––c subroutine bpdlmt(st,rm,em,q,h,a0,yi,b,c,cp1,func,dlmt,iend, + cp2,ac,qc,cp,an,dfdq,dq) c––––––––––––––––––––––––c
c Function :Consistenze condition
c Non-linear Hardening/Softening c––––––––––––––––––––––––c implicit real*8 (a-h,o-z) integer*4 irhd dimension st(4),rm (4),rn(4),em(4)
call bpgrad(st,rn,b,c,dtrd2s1,dtrj2s1,dfds1)
trm2=rm(1)*rm(1)+rm(2)*rm(2)+rm(4)*rm(4)+rn(3)*em(3)
trne=rn(1)*em(1)+rn(2)*em(2)+r n(4)*em(4)+rm(3)*rm(3)/2.d0
c––Quadratic Hardening/Softening if(iend.eq.1)then an=0.5d0*cp*q*q+h*q+a0 dfdq=(-cp*q-h)
c––Exponential SIMO Hardening/Softening
elseif(iend.eq.2)then an=a0+(yi-a0)*(1.d0-dexp(-cp1*q))+h*q dfdq=-( (yi-a0)*cp1*dexp(-cp1*q)+h ) c––Modified Exponential SI MO Hardenin g/Softening
elseif(iend.eq.3.and.q.lt.qc)then an=a0+(yi-a0)*(1.d0-dexp(-cp1*q))+h*q
dfdq=-( (yi-a0)*cp1*dexp(-cp1*q)+h )
elseif(iend.eq.3.and.q.ge.qc) then an=ac*dexp(cp2*(q-qc)) dfdq=-(ac*(dexp(cp2*(q-qc)))*cp2) else write(*,*)'Hardening/Softe ning not available',iend
stop endif dq=dsqrt(2.d0/3.d0*trm2) dlmt=func/(trne-dfdq*dq) return end c––––––––––––––––––––––––c
subroutine bpdlam(st,rm,sn,em,dq,q,h,a0,yi,b,c,cp1,dlmt,dlam,
+ iend,cp2,ac,qc,cp,fn1,an1,dfn) c––––––––––––––––––––––––c c Function:Consistence condition c Non-linear Hardening/Softening c––––––––––––––––––––––––c
implicit real*8 (a-h,o-z)
integer*4 iend dimension st(4),rm(4), sn(4),em(4), sigp(4)
trs1=st(1) +st(2) +st(4) trs2=st(1)*st(2)+st(2)*st(4) +st(4)*st(1)-st(3)*st(3)
trd2=1/3.d0*(trs1*trs1-3*trs2)
trj2=dsqrt(trd2)
tre1=em(1) +em(2) +em(4) tre2=em(1)*em(2)+em(2)*em(4)+em(4)*em(1)-em(3)*em(3) trde2=1/3.d0*(tre1*tre1-3*tre2) trje2=dsqrt(trde2) trm2=rm(1)*rm(1)+rm(2)*rm(2)+rm(4)*rm(4)+rm(3)*rm(3)/2.d0 dq=dsqrt(2.d0/3.d0*trm2)
c––Newton-Raphson iteration
dlam=dlmt iter=0 11 iter=iter+1 qn1=q+dlam*dq c––Quadratic Hardening/Softening
if(iend.eq.1)then
an1=(0.5d0*cp*qn1*qn1+h*qn1+a0) dan=(cp*qn1*dq+h*dq) c––Exponential SIMO Hardening/Softening elseif(iend.eq.2)then an1=(a0+(yi-a0)*(1.d0-dexp(-cp1*qn1))+h*qn1) dan=( (yi-a0)*cp1*dq*dexp(-cp1*qn1)+h*dq )
c––Modified Exponential SI MO Hardenin g/Softening
elseif(iend.eq.3.and.qn1.lt.qc)then
an1=(a0+(yi-a0)*(1.d0-dexp(-cp1*qn1))+h*qn1) dan=( (yi-a0)*cp1*dq*dexp(-cp1*qn1)+h*dq ) elseif(iend.eq.3.and.qn1.ge.qc) then an1=ac*dexp(cp2*(qn1-qc))
CODRI.F
212dan=ac*dexp(cp2*(qn1-qc))*cp2*dq
else write(*,*)'Hardening/Softe ning not available',iend
stop endif trs1l=st(1)+st(2)+st(4)-dlam*(em(1)+em(2)+em(4)) trs2l=trs2+dlam**2*tre2-dlam*(st(1)*em(2)+em(1)*st(2)+
+ st(2)*em(4)+em(2)*st(4)+ st(4)*em(1)+em(4)*st(1))
+ +2.d0*dlam*st(3)*em(3)
trd2l=1/3.d0*(trs1l*trs1l-3*trs2l) trj2l=dsqrt(trd2l) dtrs1l=-(em(1)+em(2)+em(4)) dtrs2l=+2.d0*dlam*tre2-(st (1)*em(2)+em(1)*st(2)+
+ st(2)*em(4)+em(2)*st(4)+ st(4)*em(1)+em(4)*st(1))
+ +2.d0*st(3)*em(3) dtrd2l=1/3.d0*(2.d0*trs1l*dtrs1l-3.d0*dtrs2l) dtrj2l=1/2.d0/dsqrt(trd2l)*dtrd2l fn1=-an1+b*trs1l-c*trs1l**2.d0+trj2l dfn=-dan+b*dtrs1l-c*2.d0*trs1l*dtrs1l+dtrj2l dlam=dlam-fn1/dfn
if(dabs(fn1/dfn).gt.1.d-10.and.iter.le.200) goto 11
if(dlam.lt.0.d0) write(*,*)' dlam < null !! ',dlam dq=dq*dlam sn(1)=st(1)-dlam*em(1) sn(2)=st(2)-dlam*em(2) sn(4)=st(4)-dlam*em(4)
sn(3)=st(3)-dlam*em(3)
return end
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