A three-invariant cap model with isotropic–kinematic ... · A double-surface plasticity model was...

Available online at www.sciencedirect.com

International Journal of Solids and Structures 45 (2008) 631–656

www.elsevier.com/locate/ijsolstr

A three-invariant cap model with isotropic–kinematichardening rule and associated plasticity for granular materials

H. DorMohammadi, A.R. Khoei *

Center of Excellence in Structural and Earthquake Engineering, Department of Civil Engineering,

Sharif University of Technology, P.O. Box 11365-9313, Tehran, Iran

Received 16 October 2006; received in revised form 24 August 2007Available online 8 September 2007

Abstract

In this paper, a three-invariant cap model is developed for the isotropic–kinematic hardening and associated plasticityof granular materials. The model is based on the concepts of elasticity and plasticity theories together with an associatedflow rule and a work hardening law for plastic deformations of granulars. The hardening rule is defined by its decompo-sition into the isotropic and kinematic material functions. The constitutive elasto-plastic matrix and its components arederived by using the definition of yield surface, material functions and non-linear elastic behavior, as function of hardeningparameters. The model assessment and procedure for determination of material parameters are described. Finally, theapplicability of proposed plasticity model is demonstrated in numerical simulation of several triaxial and confining pres-sure tests on different granular materials, including: wheat, rape, synthetic granulate and sand.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Three-invariant plasticity; Cap-plasticity surface; Isotropic–kinematic hardening; Granular materials

1. Introduction

Modeling the behavior of granular materials under various loading conditions poses several important andchallenging problems. It is a characteristic of granular materials that the relative motion between the grainsleads to irrecoverable deformation, and the use of plasticity theory is necessary as a theoretical frameworkfor the mechanical behavior of granular media. The characterization of the stress–strain and failure behaviorof granular materials is so complex due to particulate nature. It can be particularly observed in triaxial teststhat starting from different confining pressure and initial void ratio, such as: hardening, densification, dilation,or combinations of these behaviors. Thus, it is of great interest to be able to accurately predict the mechanicalbehavior of granular materials by an appropriate constitutive model. This study focuses on the frictionaleffects of granular materials based on a combined isotropic–kinematic hardening rule and associated plasticity.

0020-7683/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijsolstr.2007.08.019

* Corresponding author. Tel.: +98 21 6600 5818; fax: +98 21 6601 4828.E-mail address: [email protected] (A.R. Khoei).

mailto:[email protected]

632 H. DorMohammadi, A.R. Khoei / International Journal of Solids and Structures 45 (2008) 631–656

There are a number of plasticity models available in the literature for constitutive relations of granularmaterials, such as: sand, concrete and rock. A review of various plasticity models for frictional and granularmaterials can be seen in Desai and Siriwardane (1984), Chen and Baladi (1985) and Lewis and Schrefler(1998). Basically, there are two different approaches in constitutive modeling of granular materials; ‘the micro-mechanical model’ and ‘the macromechanical model’. The micromechanical models are considered as idealapproaches, and cannot be directly applied to analyze general boundary value problems (Jenkins and Strack,1993; McDowell and Bolton, 1998; Chang and Hicher, 2005). Therefore, the continuum approaches whichconsist of implementation of macroscopic constitutive models based on plasticity theories have been proposedfor granular materials.

The crucial factor that governs the behavior of granular soils is stress-dilatancy, the soil’s ability to increasein volume due to shear stresses and geometrical effects. A plasticity theory was developed by Dorris andNemat-Nasser (1982) for finite deformation of granular materials, which accounts for the true stress triaxial-ity, pressure sensitivity and dilatancy. They captured the effect of stress triaxiality by including the third devi-atoric stress invariant in the yield function and the flow potential. A simple constitutive model was developedby Wan and Guo (1998) within the plasticity theory by introducing a modified stress-dilatancy law. They mod-eled the dependencies of granular soil behavior on void ratio and stress by a modified stress-dilatancy law. Byusing a void ratio-dependent factor that measures the deviation of the current void ratio from the critical one,Rowe’s stress dilatancy equation was modified. This modification was related to a new energy dissipationequation for a granular assembly which sustain the kinematical constraints under the action of stresses.

The experimental results of Watson and Wert (1993) and Brown and Abou-Chedid (1994) demonstratedthat a ‘two-mechanism-model’, such as: Drucker–Prager or Mohr–Coulomb and elliptical cap models canbe utilized to construct the suitable phenomenological constitutive models which capture the major featuresof the response of geological and frictional materials. These models consist of two yield surfaces; a ‘distortionsurface’ and a ‘consolidation’ or ‘cap’ surface, which has an elliptical shape. The distortion surface controlsthe ultimate shear strength of material and the cap-surface captures the hardening behavior of material undercompression. Gudehus (1996) presented a constitutive model by incorporation of density and pressure depen-dencies for granular materials within hypoplasticity formulation. A cone-cap model based on a density-depen-dent Drucker–Prager yield surface and a non-centered ellipse was developed by Brandt and Nilsson (1999).A double-surface plasticity model was proposed by Lewis and Khoei (2001) for the non-linear behavior ofporous materials in the concept of the generalized plasticity formulation.

The non-linear behavior of frictional and granular materials is adequately described by double-surface plas-ticity models. However, it suffers from a serious deficiency when the stress-point reaches in the intersection ofthese two different yield functions. In the flow theory of plasticity, the transition from an elastic state to an elas-to-plastic state appears more or less abruptly. For double-surface plasticity models, it is very difficult to define thelocation of yield surface and special treatment must be performed to avoid numerical difficulties in the intersec-tion of these two surfaces (Khoei et al., 2004). Lade and Kim (1995) proposed a single hardening plasticity modelfor frictional materials, which does not have such a drawback. Their model was based on a single, isotropic yieldsurface shaped as an asymmetric teardrop with the pointed apex at the origin of the principal stress space. Theyield surface was expressed in terms of stress invariants to describe the locus at which the total plastic work isconstant. The total plastic work due to shear and volumetric strains was served as the hardening parameter,and used to define the location and shape of the yield surface. The non-associated flow rule was derived froma potential function which describes a three-dimensional surface shaped with an asymmetric cross-section.

A non-associated plasticity theory was developed by Krenk (2000) and Krenk and Ahadi (2000) for gran-ular materials based on the concept of a characteristic stress state of vanishing incremental dilation. The the-ory was presented based on a common format for the yield and flow potential surfaces, representing thesurfaces in terms of stress invariants and a single shape function. The flow potential surface was determinedby an approximate friction hypothesis, and the plastic work hardening was introduced in a linear invariantform that permits dilation before the ultimate state, by including the work associated with shape change inaddition to the traditional contribution from volume change. Bigoni and Piccolroaz (2004) proposed ayield/damage function for modeling the inelastic behavior of a broad class of pressure-sensitive, frictional,ductile and brittle-cohesive materials. The yield function was based on a single, convex and smooth surfacein stress space approaching as limit situations well-known criteria and the extreme limits of convexity in

H. DorMohammadi, A.R. Khoei / International Journal of Solids and Structures 45 (2008) 631–656 633

the deviatoric plane. A three-invariant isotropic/kinematic hardening cap plasticity model was presented byFoster et al. (2005) for geomaterials based on an associated flow rule. Their yield surface was an exponentialshear failure function at its core with triangularity in p-plane that captures the pressure-dependence of theshear strength of geomaterials.

Recently, a single cap plasticity with an isotropic hardening rule and associated plasticity was developed byKhoei and Azami (2005), which could generate some common elliptical yield surface and double-surface plas-ticity models, as special cases. In the present study, the cap-plasticity model developed by Khoei and Azami(2005), Khoei and DorMohammadi (2007) and Khoei et al. (2007) is extended to three-invariant cap plasticitywith isotropic–kinematic hardening rule and associated plasticity for granular materials. The paper focuses ondifferent aspects of material behavior, including: isotropic and kinematic hardening and dilation. A general-ized framework for the cap plasticity is presented based on three invariants of stress states, which describes theisotropic and kinematic hardening behavior of material. The constitutive elasto-plastic matrix and its compo-nents are extracted. The model assessment and parameter determination are described. Finally, the applica-bility of the model is demonstrated in several numerical examples.

2. Three-invariant isotropic–kinematic cap plasticity

In order to present a constitutive plasticity model for granular media, several requirements must be takeninto account (Collins, 2005). Firstly, the soils and granular materials are multiple phase materials, consistingof solid grains together with voids. It is therefore necessary that the model considers the effects of porosity andvoids ratio during loading conditions. Secondly, the elasto-plastic behavior depends on the current stress stateand the pressure history, particularly the pre-consolidation pressure, which is the fundamental hardeningparameter in granular materials. Thus, the material elasto-plasticity matrix must be defined as a functionof hardening parameter. Thirdly, granular materials undergo significant plastic volume changes, and presentthe dilation and contraction depending on the current voids ratio and pressure. The dilation can be caused dueto the fact that individual grains must ride up over each other in order to accommodate shear strains. Thus,the model must capture both the dilation and compaction behavior of material.

Basically, there are various procedures in the literature for the derivation of plasticity models. Someresearchers used the concept of characteristic stress state by using surfaces that can be characterized by their‘traces’ in the deviatoric plane and in planes containing a ‘radial’ centre-line (Krenk, 2000). Other researchersapplied the shear failure function which captures the pressure-dependence of the shear strength (Foster et al.,2005). The current research is based on the second procedure to derive the constitutive plasticity model fol-lowing the works of senior author given in Khoei et al. (2003) and Khoei and Bakhshiani (2004). In these ref-erences, an endochronic plasticity theory was developed based on coupling between deviatoric and hydrostaticbehavior for pressure-sensitive materials. Although the concept of yield surface has not been explicitlyassumed in endochronic theory, it was presented by Khoei and Bakhshiani (2004) that the single-cap plasticitycan be derived as a special case of their endochronic model.

2.1. The yield function

The constitutive model is developed here for both isotropic and kinematic hardening behaviors based onthree invariants of stress states, J1, J2D and J3D. As the first step in derivation of plasticity model, a shear-fail-ure function is defined as

F f ¼ f 2d �

fd

fh

� �2

J 21 ð1Þ

where J1 is the first invariant of stress tensor, and fh and fd are the hydrostatic and deviatoric material func-tions, respectively, and will be defined later as functions of hardening parameters. The function Ff captures thepressure-dependence of the shear strength of material, which increases with more compressive mean stresses,as shown in Fig. 1. It will be shown later that different combinations of fh and fd lead to different shapes of theyield surface. The initial yield surface f0 is offset from the pressure-dependence function Ff, as shown in Fig. 2.

fF

1J−

2f dF f=

1 hJ f= −

Fig. 1. The shear-failure function Ff.

fF

fF

0f

1J−

Fig. 2. The shear-failure function Ff and the initial yield surface f0.


Considering the shear-failure function Ff, the yield function can be written in (J1,J2D) stress space as

F 1 ¼2

3J 2D � F f ¼ 0 ð2Þ

or

F 1 ¼2

3J 2D þ

fd

fh

� �2

J 21 � f 2

d ¼ 0 ð3Þ

where J2D is the second invariant of deviatoric stress tensor. In Fig. 3, the yield function F1 is presented inðffiffiffiffiffiffiffiffiJ 2D

p; J 1Þ stress space. In above relation, we define the coefficients /h and /d in order to indicate the effect

of material functions fh and fd in the yield function. The yield function (3) can be therefore rewritten as

F 2 ¼2

3J 2D þ

/dfd

/hfh

� �2

J 21 � ð/dfdÞ2 ¼ 0 ð4Þ

Finally, applying the effect of third invariant of deviatoric stress tensor J3D into Eq. (4), i.e. the triangularity ofdeviatoric trace along the hydrostatic axis, the yield function can be written in its final representation as

F ðr; a; jÞ ¼ wJ 2=33D þ

2

3J 2D þ

/dfd

/hfh

� �2

J 21 � ð/dfdÞ2 ¼ 0 ð5Þ

1F

2DJ

1J−

Fig. 3. The cap-surface function F1 in ðffiffiffiffiffiffiffiffiJ 2D

p; J 1Þ stress space.


where w is a constant, and a and j are the kinematic and isotropic hardening parameters, respectively. It mustbe mentioned that the material functions fh and fd can be decomposed into two parts, the isotropic and kine-matic parts, which control the shape of yield surface (5). Fig. 4 presents the 3D representation of yield surface(5) for the isotropic and kinematic hardening behavior of material.

The isotropic and kinematic hardening parameters j and a evolve with plastic deformation. The evolutionof j is related to the mean stress, and more directly to the volumetric plastic strain ep

v, i.e. j ¼ epv, while the

evolution of a is related to the deviatoric plastic strain ep. As can be expected, the kinematic hardening param-eter a = {a1 a2 a3}T can be decomposed in two directions J1 and J2D in meridian plane, which contains twoparts as follows

Fig. 4.materi

a ¼ a1 exp a2ððepÞT : epÞa3

� �mþ a4ððepÞT : epÞa3

� �ep ð6Þ

where the first term controls the movement of yield surface in J1-axis and the second term controls the move-ment of yield surface in perpendicular direction to J1-axis. In relation (6), m is the unit vector, defined asm = {1 1 1}T, and a1, a2, a3 and a4 are the material parameters. The three components of kinematic hardening

Trace of 3D representation of yield surface (5) in principal stress space for the isotropic and kinematic hardening behavior ofal.


parameter a, i.e. a1, a2 and a3, determine the values of the yield surface movements in the directions of prin-cipal stresses r1, r2 and r3, respectively.

2.2. The isotropic and kinematic material functions

As mentioned earlier, the material functions fh and fd control the size and movement of the yield surface,and are functions of hardening parameters. Then, these functions can be decomposed into two parts, i.e. theisotropic and kinematic parts, as

fh ¼ fhisotropicþ fhkinematic

ð7Þfd ¼ fdisotropic

þ fdkinematicð8Þ

The isotropic part of material functions fh and fd are the exponential increasing functions of the isotropichardening parameter j ¼ ep

v, defined as

fhisotropic¼ ðb1 þ b2 expðb3e

pvÞÞdðep

vÞ ð9Þfdisotropic

¼ ðc1 þ c2 expðc3epvÞÞdðep

vÞ ð10Þ

where b1, b2, b3, c1, c2 and c3 are the material parameters and dðepvÞ is defined as

dðepvÞ ¼

1 if epv 6¼ 0

0 if epv ¼ 0

�ð11Þ

In order to determine the kinematic parts of material functions (7) and (8), consider two different stress spacesri and Ni; the former is located in the center of the yield surface before kinematic hardening and the later isplaced in the center of the yield surface after kinematic hardening (Fig. 5). The distance of centers of two coor-dinate systems can be defined by ai, in which the relationship between the principal stresses in two stress spacesis defined as

Ni ¼ ri þ ai ð12Þ

2σ

1σ

3σ2Ξ

1Ξ

3Ξ

α

Fig. 5. The stress spaces before kinematic hardening ri and after kinematic hardening Ni.


Defining the parameters J1a and J2Da similar to the definitions of the invariants of stress and deviatoric stresstensors as

J 1a ¼ a1 þ a2 þ a3 ð13Þ

J 2Da ¼1

6ða1 � a2Þ2 þ ða2 � a3Þ2 þ ða3 � a1Þ2h i

ð14Þ

Considering the definition of the principal stresses in two stress spaces, defined by Eq. (12), the three-invariantsingle plasticity (5) can be written in new stress space as

wJ 2=3IIID þ

2

3J IID þ

/dfd

/hfh

� �2

J 2I � /2

df 2d ¼ 0 ð15Þ

where JI is the first invariant of stress tensor and JIID and JIIID are the second and third invariants of devi-atoric stress tensors in the second stress space, defined as

J I ¼ J 1 þ J 1a ð16ÞJ IID ¼ J 2D þ J 2Da þ Jra ð17Þ

where

Jra ¼2ffiffiffi3p

ffiffiffiffiffiffiffiffiJ 2D

pcos x a1 �

1

2a2 �

1

2a3

� �þ

ffiffiffi3p

2sin xða2 � a3Þ

!ð18Þ

where

x ¼ 1

3cos�1 3

ffiffiffi3p

2

J 3D

J 3=22D

!0� 6 x 6 60� ð19Þ

Substituting relations (16) and (17) into the yield surface (15) in the absence of third invariants of deviatoricstress with respect to Eq. (5), results in

/dfd

/hfh

� �2

¼ � 2

3

ðJ 2Da þ JraÞð2J 1aJ 1 þ J 2

1aÞð20Þ

According to Eq. (20), the kinematic parts of material functions fh and fd can be therefore written as

fhkinematic¼ 1

/h

ð2J 1aJ 1 þ J 21aÞ

1=2 ð21Þ

fdkinematic¼ 1

/d

� 2

3ðJ 2Da þ JraÞ

� �1=2

ð22Þ

Finally, the material functions fh and fd can be defined by substituting Eqs. (9), (10), (21) and (22) into (7) and(8) as

fh ¼ ðb1 þ b2 expðb3epvÞÞdðep

vÞ þ1

/hð2J 1aJ 1 þ J 2

1aÞ1=2 ð23Þ

fd ¼ ðc1 þ c2 expðc3epvÞÞdðep

vÞ þ1

/d

� 2

3ðJ 2Da þ JraÞ

� �1=2

ð24Þ

3. Model assessment

In order to demonstrate the performance of the proposed plasticity model in prediction of granular mate-rial behavior, the experimental tests must be performed to determine and calibrate the parameters of materialfunctions fh and fd, defined by (23) and (24), in the yield surface (5). These two material functions control thesize and movement of the yield surface, and are decomposed into the isotropic and kinematic parts, given by


(7) and (8), as functions of the hardening parameters j and a, or directly the plastic volumetric strain epv and

the deviatoric plastic strain ep. It must be noted that the kinematic hardening parameter a indicates the move-ment of the yield surface in the direction of J1-axis and perpendicular direction to J1-axis.

It is worth mentioning that different values of material functions fh and fd result in different aspects of theyield surface (5). Consider that the first two terms of Eq. (5) are zero, it leads to

/dfd

/hfh

� �2

J 21 � /2

df 2d ¼ 0 ð25Þ

The above equation generally yields to three roots, the points of intersection of yield surface with J1-axis, i.e.J1 = ± /h fh and one more from fd = 0.0, which has been defined in Eq. (24). If fd = 0.0 does not lead to anyvalue for J1, the yield surface of Eq. (5) yields to two roots for J1, i.e. J1 = ± /hfh, which results in an ellipticalshape in meridian plane. The 3D representation of the yield surface (5) is shown in Fig. 6 in principal stressspace for different values of isotropic hardening, where the intersection point of yield surface with J1-axis are

-20

-10

0

10

20

-20

-10

0

10

20

-20

-10

0

10

20

σ

σ

σX Y

Z

Fig. 6. Trace of 3D elliptical yield function in principal stress space for the isotropic hardening behavior.


�/hfh and +/hfh. This representation clearly shows how the yield surface grows with densification, eventuallybecoming independent of the hydrostatic stress J1 at full dense material, where the von-Mises yield surface isgenerated. This yield surface is very similar to the elliptical yield functions developed by authors for porousmetals based on an extension of von-Mises’s concept (Doraivelu et al., 1984; Oliver et al., 1996).

If fd = 0.0 leads to the value of J1 between �/hfh and +/hfh, the cone-cap yield surface will be producedfrom Eq. (5) based on the intersection points of J1 = � /hfh and the value obtained from fd = 0.0 (Fig. 3). The3D representation of Fig. 3 is shown in Fig. 7 for different values of isotropic hardening. As can be observedfrom this figure the yield surface grows with densification and reduces to the Drucker–Prager yield functionfor full dense bodies. This yield surface is very similar to the double-surface cap models, i.e. a combination ofMohr–Coulomb or Drucker–Prager and elliptical surfaces, which has been extensively used by authors todemonstrate the behavior of granular and powder materials (Brandt and Nilsson, 1999; Lewis and Khoei,2001; Khoei et al., 2004). It is worth mentioning that as the double-surface plasticity consists of two differentyield functions, special treatment should be made to avoid numerical difficulties in the intersection of these twosurfaces (Khoei et al., 2004), however – the single yield surface (5) does not have such a drawback. Fig. 8 pre-sents the effect of parameter w in the yield surface (5) that causes triangularity of deviatoric trace along the

-10-5

0

5

10

15

20

25

-10-5

05

1015

2025

-10-5

05

1015

2025

σ

σ

σX Y

Z

Fig. 7. Trace of 3D cone-cap yield function in principal stress space for the isotropic hardening behavior.

-10

-5

0

5

10

15

20

25

-10-5

05

1015

2025

-10-5

05

1015

2025

σ

σ

σX Y

Z

Fig. 8. 3D irregular hexagonal pyramid of the Mohr–Coulomb and cone-cap yield function in principal stress space for w = 0.1.


hydrostatic axis. This yield surface is similar to the irregular hexagonal pyramid of the Mohr–Coulomb andcone-cap yield surface employed by researchers for description of soil and geomaterial behavior (Lade andKim, 1995). It must be noted that the single-surface plasticity (5) is not only for the isotropic compression partof the triaxial tests, and is capable to model the complete triaxial and confining pressure tests.

4. Parameter determination

The important issue in prediction of granular material behavior is the identification of parameters of theproposed plasticity model. The calibration procedure for three-invariant isotropic–kinematic cap plasticityis carried out based on a series of isostatic and triaxial tests. An organized approach to determine modelparameters is to utilize an optimization routine. Mathematically, an objective function and a search strategyare necessary for the optimization. The objective function, which represents the constitutive model, capturesthe material behavior and can be used in a simultaneous optimization against a series of experimental data.The simplest search strategy is based on the direct search approach, which is proved to be reliable and its rel-ative simplicity make it quite easy to program into the code.


For the proposed constitutive model, the total number of 10 material constants need to be determined forthe material functions fh and fd. The parameters of isotropic parts of fh and fd (i.e. b1, b2, b3 and c1, c2, c3) arefirstly evaluated using the confining pressure test, where the values of J2D and J3D in the yield surface (5) arezero. The parameters of the kinematic parts of fh and fd (i.e. a1, a2, a3 and a4) are then estimated performingthe LSM method on the data obtained by a series of triaxial tests. These parameters control different aspects ofpredicted stress–strain curves obtained numerically by fitting the stress path to the triaxial and confining pres-sure tests, including: the slope, transition, expansion and contraction.

a1 controls the slope of ‘stress–axial strain’ and ‘radial strain–axial strain’ diagrams.a2 controls the slope of ‘radial strain–axial strain’ diagram after dilation.a3 controls the transition of ‘radial strain–axial strain’ diagram before dilation.a4 controls the transition of ‘stress–axial strain’ and ‘radial strain–axial strain’ diagrams, and the slope of

‘stress–axial strain’ curve.b1 controls the transition, expansion and contraction of ‘radial strain–axial strain’ diagram, and the

slope, expansion and contraction of ‘stress–axial strain’ curve.b2 controls the transition, expansion and contraction of ‘stress–axial strain’ and ‘radial strain–axial

strain’ diagrams, and the slope of ‘radial strain–axial strain’ curve.b3 controls the slope and transition of ‘radial strain–axial strain’ curve, and the slope of ‘stress–axial

strain’ diagram.c1 controls the slope, expansion and contraction of ‘stress–axial strain’ and ‘radial strain–axial strain’

diagrams.c2 controls the slope, transition, expansion and contraction of ‘stress–axial strain’ diagram, and the

slope of ‘radial strain–axial strain’ diagram.c3 controls the slope of ‘stress–axial strain’ curve, and the slope and transition of ‘radial strain–axial

strain’ diagram after dilation.

The procedure of parameter determination is performed as follows;

Step 1: Based on the results obtained from the confining pressure test, the values of J1 are evaluated usingthe yield surface (5) where the values of J2D and J3D are zero. From the values of volumetric strain ev,the elastic and plastic volumetric strains, ee

v and epv, are estimated. The parameters b1, b2 and b3 in the

isotropic part of fh are computed. The parameters c1, c2 and c3 in the isotropic part of fd are thencalculated by a least square method on the data obtained from the confining pressure test.

Step 2: Applying the results of triaxial tests and the isotropic parameters of fh and fd obtained from Step 1,the kinematic parameters of fh and fd, i.e. a1, a2 and a3, in the first term of relation (6) are first esti-mated. Parameter a4 in the second term of relation (6) is then obtained by performing the leastsquare method on the data obtained from the triaxial tests.

5. Numerical simulation results

5.1. Triaxial tests on dry silo materials

The first three examples refer to set of triaxial tests on wheat, rape and synthetic granulate materials per-formed by Kolymbas and Wu (1990). The device was a triaxial apparatus specially designed to test dry silomaterials. The triaxial apparatus was designed in the Institute of Soil Mechanics and Rock Mechanics ofthe Karlsruhe University. The axial force was measured beneath the pressure chamber by a load cell with aprecision of ±30 N. As the axial force was measured beneath the pressure chamber, the measurement wasnot influenced by the friction between the loading piston and the sealing or by the confining pressure. Thelateral strain of the sample was measured directly by means of three collars which contact the sample inthe upper, middle and lower parts, respectively. The axial and lateral deformations were measured duringhydrostatic loading by the displacement transducer mounted between the end plates and the three collars.


The specimens are prepared by pluviation. The specimens are cylindrical with the initial diameter of 100.0 mmand length of 100.0 mm. The axial load is exerted by moving the loading piston. The apparatus allows a max-imum axial load of 100 kN. The maximum design confining pressure is 1400 kPa.

5.1.1. Compaction of wheat

Based on the procedure described in preceding section, the material parameters of the yield surface cali-brated for the wheat material are as follows

Fig. 9.results

a1 ¼ �5:0e� 11 a2 ¼ �32:0 a3 ¼ �32:0 a4 ¼ �5:0

b1 ¼ 1500:3 b2 ¼ 600:8 b3 ¼ 600:5

c1 ¼ 3000:0 c2 ¼ 500:0 c3 ¼ �83:5

Axial Strain (%)

Rad

ial S

trai

n (%

)

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Three-invariant isotropic-kinematiccap model

Experiment (Kolymbas and Wu,1990)

Axial Strain (%)

Rad

ial S

trai

n (%

)

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

a

b

Radial strain versus axial strain in isostatic compression test for wheat; (a) a comparison between the numerical and experimental, (b) schematically isotropic hardening behavior.


In Fig. 9(a), the radial strain versus axial strain is presented for the hydrostatic pressure test. It shows agood agreement between the model and experimental results. In Fig. 9(b), the relevant yield surfaces corre-sponding to the isotropic hardening behavior has been shown schematically. This representation clearly showshow the yield surface grows isotropically by increasing the hydrostatic pressure. In this figure, the inner circleshows the initial yield surface and the outer one presents the yield surface at the specified point.

In Figs. 10 and 11, the variations of volumetric strain and stress ratio (axial stress/radial stress) are depictedwith the axial strain at confining pressures 50 kPa and 400 kPa, respectively, corresponding to complete triax-ial tests. In Figs. 10(b) and 11(b), the relevant yield surfaces are plotted corresponding to the kinematic hard-ening behavior. In these figures, the isotropic hardening behavior can be clearly observed in triaxial tests dueto different confining pressures. Obviously, the initial yield surface at the beginning of triaxial test with con-fining pressure 400 kPa is bigger than that presented at the confining pressure 50 kPa. Thus, both isotropic andkinematic hardening behavior can be observed in this example. As can be seen, the proposed model capturesthe behavior of wheat in complete triaxial experiment.

Axial Strain (%)0 5 10 15

-4

-3

-2

-1

0

1

2

3

4



Axial Strain (%)

Vol

umet

ric S

trai

n (%

)S

tres

s R

atio

Vol

umet

ric S

trai

n (%

)S

tres

s R

atio

0 5 10 15-4

-3

-2

-1

0

1

2

3

4

a

b

Fig. 10. Stress ratio and volumetric strain versus axial strain in triaxial test for wheat (confining pressure = 50 kPa); (a) a comparisonbetween the numerical and experimental results, (b) schematically isotropic–kinematic hardening behavior.

Axial Strain (%)

Vol

umet

ric S

trai

n (%

)S

tres

s R

atio

0 5 10 15-4

-3

-2

-1

0

1

2

3

4



Axial Strain (%)

Vol

umet

ric S

trai

n (%

)S

tres

s R

atio

0 5 10 15-4

-3

-2

-1

0

1

2

3

4

a

b

Fig. 11. Stress ratio and volumetric strain versus axial strain in triaxial test for wheat (confining pressure = 400 kPa); (a) a comparisonbetween the numerical and experimental results, (b) schematically isotropic–kinematic hardening behavior.


5.1.2. Compaction of rapeIn the compaction of rape material, the material parameters calibrated for the yield surface are as follows

a1 ¼ �5:0e� 4 a2 ¼ �9:0 a3 ¼ 2:0 a4 ¼ �1:0

b1 ¼ 900:0 b2 ¼ 170:5 b3 ¼ 400:0

c1 ¼ 1500:5 c2 ¼ 600:2 c3 ¼ �5:0

In Fig. 12(a), the variation of radial strain with axial strain is presented for the isostatic test. The isotropicbehavior of material during the expansion of the yield surfaces is demonstrated in Fig. 12(b) by increasingthe hydrostatic pressure. In Figs. 13(a) and 14(a), the variations of volumetric strain and stress ratio (axialstress/radial stress) with axial strain are depicted at confining pressures 200 kPa and 400 kPa, respectively, cor-responding to complete triaxial tests. Also plotted in Figs. 13(b) and 14(b) are the relevant yield surfaces cor-responding to the isotropic and kinematic hardening behavior.

Axial Strain (%)

Rad

ial S

trai

n (%

)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5



Axial Strain (%)

Rad

ial S

trai

n (%

)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

a

b

Fig. 12. Radial strain versus axial strain in isostatic compression test for rape; (a) a comparison between the numerical and experimentalresults, (b) schematically isotropic hardening behavior.


5.1.3. Compaction of synthetic granulate

The material parameters corresponding to the yield function (4) calibrated for the synthetic granulate mate-rial are as follows

a1 ¼ �0:8e� 6 a2 ¼ 15:0 a3 ¼ �14:0 a4 ¼ �3:0

b1 ¼ �100:0 b2 ¼ �70:7 b3 ¼ 150:0

c1 ¼ 30:0 c2 ¼ 910:0 c3 ¼ �0:95

In Fig. 15(a), the variation of radial strain with axial strain is presented for the isostatic test. The isotropichardening behavior of granular material is illustrated in Fig. 15(b) due to expansion of the yield surfacesby increasing the hydrostatic pressure. Clearly, the effect of confining pressure on isotropic hardening of mate-rial can be observed in this figure. Figs. 16 and 17 correspond to the complete triaxial compression tests. Thevariations of the volumetric strain and stress ratio (axial stress/radial stress) are presented with the axial strain

Axial Strain (%)0 5 10

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

Three-invariantisotropic-kinematiccapmodel

Experiment(Kolymbasand Wu,1 990)

Axial Strain (%)

Vol

umet

ric S

trai

n (%

)S

tres

s R

atio

Vol

umet

ric S

trai

n (%

)S

tres

s R

atio

0 5 10-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

a

b

Fig. 13. Stress ratio and volumetric strain versus axial strain in triaxial test for rape (confining pressure = 200 kPa); (a) a comparisonbetween the numerical and experimental results, (b) schematically isotropic–kinematic hardening behavior.


in Figs. 16(a) and 17(a). Figs. 16(b) and 17(b) present the relevant yield surfaces corresponding to the isotropicand kinematic hardening behavior in complete triaxial tests. Both isotropic and kinematic hardening behaviorwith dilatancy can be observed in this example.

5.2. Triaxial tests on loose and dense sands

The next two examples refer to set of triaxial tests on loose and dense sands performed by Krenk and Ahadi(2000). The two materials are loose Baskarp sand with an initial specific pore volume e = 0.85 (Borup andHedegaard, 1995), and dense Lund sand with initial specific pore volume e = 0.55 (Ibsen and Jakobsen,1996). In both tests, the initial isotropic confining pressure was p0 = 0.64 MPa. The shear modulus of thedense sand is nearly four times larger, and the elastic and elasto-plastic flexibilities are two to three times smal-ler. Clearly, this gives smaller strains in the dense sand. The height of the test specimens was equal to the diam-eter, thus preventing localized failure near the peak load (Krenk and Ahadi, 2000).

Axial Strain (%)

Vol

umet

ric S

trai

n (%

)S

tres

s R

atio

Axial Strain (%)

Vol

umet

ric S

trai

n (%

)S

tres

s R

atio

0 5 10 15-4

-3

-2

-1

0

1

2

3

4



0 5 10 15-4

-3

-2

-1

0

1

2

3

4

a

b

Fig. 14. Stress ratio and volumetric strain versus axial strain in triaxial test for rape (confining pressure = 400 kPa); (a) a comparisonbetween the numerical and experimental results, (b) schematically isotropic–kinematic hardening behavior.


The material parameters calibrated for the loose Baskarp sand are as follows

a1 ¼ 1:0 e� 6 a2 ¼ 0:05 a3 ¼ 20:0 a4 ¼ �0:06

b1 ¼ 12:3 b2 ¼ �3150:0 b3 ¼ 1500:0

c1 ¼ �10:0 c2 ¼ 2650:0 c3 ¼ �2:6

The results for loose sand are shown in Figs. 18 and 19. In Fig. 18(a), the normalized stress, i.e. the axialstress–radial stress/radial stress is presented with axial strain. The relevant yield surfaces corresponding tothe kinematic hardening behavior are shown in Fig. 18(b). In Fig. 19, the variation of volumetric strain withaxial strain is presented together with the relevant yield surfaces corresponding to the kinematic hardeningbehavior. It must be noted that the cap plasticity model presented here is based on an associated flow rule,and the associated flow rule cannot address the contraction response of the loose sand once the stress-point

Axial Strain (%)

Rad

ial S

trai

n (%

)

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

3.5



Axial Strain (%)

Rad

ial S

trai

n (%

)

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

3.5

a

b

Fig. 15. Radial strain versus axial strain in isostatic compression test for synthetic granulate; (a) a comparison between the numerical andexperimental results, (b) schematically isotropic hardening behavior.


locates on the wing part of the yield surface. In this case, a non-associated flow rule is necessary. Thus, it isassumed that the stress-point locates on the cap part of the yield surface for the loose Baskarp sand.

The results for the dense sand are shown in Figs. 20 and 21 with the corresponding material parametersgiven as follows

a1 ¼ 0:9e� 6 a2 ¼ 15:0 a3 ¼ �4:0 a4 ¼ �1:0

b1 ¼ �125:0 b2 ¼ �60:1 b3 ¼ 260:0

c1 ¼ 35:0 c2 ¼ 103:0 c3 ¼ �44:9

The variation of normalized stress, i.e. the axial stress–radial stress/radial stress, is presented with axial strainin Fig. 20(a). Also plotted in Fig. 21(a) is the variation of volumetric strain with axial strain. In Figs. 20(b) and21(b), the relevant yield surfaces corresponding to the kinematic hardening behavior are shown.

It is of interest to be highlighted from the results of experimental and numerical simulation that the exam-ples of loose and dense sands have been used here to demonstrate the capability of proposed model even in the

Axial Strain (%)

Axial Strain (%)

Vol

ume

Str

ain

(%)

Str

ess

Rat

io

0 5 10 15

-1

0

1

2


Experiment (Kolymbas and Wu,1990)V

olum

e S

trai

n (%

)S

tres

s R

atio

0 5 10 15

-1

0

1

2

3

a

b

Fig. 16. Stress ratio and volumetric strain versus axial strain in triaxial test for synthetic granulate (confining pressure = 100 kPa); (a) acomparison between the numerical and experimental results, (b) Schematically isotropic–kinematic hardening behavior.


case of non-associated material, however – it must be mentioned that an associated plasticity with combinedisotropic–kinematic hardening cannot be replaced by a non-associated plasticity model. In addition, it can beobserved from the curves that the dilation happens in axial strain of 7% for the loose sand and axial strain of2% for the dense sand, so it results in different material parameters for these two sands. These two examplesdemonstrate the capability of the proposed plasticity model to represent the combined isotropic and kinematichardening behavior together with the dilatancy in standard triaxial tests for the loose and dense sands.

6. Conclusion

In the present paper, a three-invariant cap plasticity model was developed based on an isotropic–kinematichardening rule and associated plasticity for granular materials. Two material functions were introduced tocontrol the size and movement of the yield surface, which are decomposed into the isotropic and kinematicparts as functions of the hardening parameters. The kinematic hardening parameter was defined to indicate

Axial Strain (%)

Axial Strain (%)

Vol

ume

Str

ain

(%)

Str

ess

Rat

io

0 5 10 15

-1

0

1

2


Experiment (Kolymbas and Wu,1990)V

olum

e S

trai

n (%

)S

tres

s R

atio

0 5 10 15

-1

0

1

2

3

a

b

Fig. 17. Stress ratio and volumetric strain versus axial strain in triaxial test for synthetic granulate (confining pressure = 200 kPa); (a) acomparison between the numerical and experimental results, (b) schematically isotropic–kinematic hardening behavior.


the movement of the yield surface in the direction of J1-axis and in the direction of perpendicular to J1-axis.The constitutive elasto-plastic matrix and its components were derived by using the definition of yield surface,material functions and non-linear elastic behavior, as function of hardening parameters. The calibration pro-cedure for three-invariant isotropic–kinematic cap plasticity was demonstrated based on a series of standardisostatic and triaxial tests. Finally, the applicability of the proposed model was illustrated in modeling ofexperiments on several granular materials, including: wheat, rape, synthetic granulate, loose sand and densesand. Different aspects of material behavior, including: isotropic and kinematic hardening and dilation wereinvestigated. The variation of the radial strain with axial strain was presented for the hydrostatic pressure test.It was shown how the yield surface grows isotropically by increasing the hydrostatic pressure. Furthermore,the variations of axial stress and volumetric strain were obtained with the axial strain at different confiningpressures corresponding to the set of triaxial tests to represent the isotropic and kinematic hardening behavior

Axial Strain (%)

Axial Strain (%)

Str

ess

Diff

eren

ce (

%)

0 5 10 150

10

20

30

40

50

60

70

80

90


Experiment (Borup and Hedegaard,1995)

Str

ess

Diff

eren

ce (

%)

0 5 10 150

10

20

30

40

50

60

70

80

90

a

b

Fig. 18. Normalized stress versus axial strain in triaxial test for loose Baskarp sand; (a) a comparison between the numerical andexperimental results, (b) schematically isotropic–kinematic hardening behavior.


with dilatancy in granular materials. Remarkable agreements were observed between experimental and numer-ical results. In a later work, we will show how an extension and modification of proposed plasticity model canbe effectively used to capture the softening and cyclic response of granular materials.

Appendix A. Computation of material property matrix

The object of the mathematical theory of plasticity is to provide a theoretical description of the relationshipbetween stress and strain or more commonly, between increments of stress and increments of strain using theassumption that the material behaves plastically only after a certain limiting value has been exceeded. Theelasto-plastic constitutive relation in its incremental form can be presented by dr = Depde, with dr denotingthe incremental stress vector, de the incremental strain vector and Dep the constitutive elasto-plastic matrix.The yield surface F(r,a,j) = 0 determine the stress level at which the plastic deformation begins. The materialproperty matrix Dep is defined as

Axial Strain (%)

Axial Strain (%)

Vol

ume

Str

ain

(%)

0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3


Experiment (Borup and Hedegaard,1995)

Vol

ume

Str

ain

(%)

0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

a

b

Fig. 19. Volumetric strain versus axial strain in triaxial test for loose Baskarp sand; (a) a comparison between the numerical andexperimental results, (b) schematically isotropic–kinematic hardening behavior.


Dep ¼ De �nTDT

e Deng

Hþ nTDeng

ðA:1Þ

where n = oF/or and ng = oQ/or are the normal vector to the yield and potential plastic surfaces, respectively.In present study, an associated flow rule is used so F = Q. In Eq. (A.1), H is the hardening plastic modulusdefined as

H ¼ � oFol

dldk

ðA:2Þ

where dk is the plastic multiplier and l ¼ ðep; epvÞ.

In order to derive the constitutive elasto-plastic matrix and its components, we need to calculate De, n, ng

and H in Eq. (A.1). The normal vector to the yield surface is determined by

Axial Strain (%)

Axial Strain (%)

Str

ess

Diff

eren

ce (

%)

0 2 4 6 8 100

10

20

30

40

50

60

70

80

90


Experiment (Ibsen and Jakobsen,1996)

Str

ess

Diff

eren

ce (

%)

0 2 4 6 8 100

10

20

30

40

50

60

70

80

90

a

b

Fig. 20. Normalized stress versus axial strain in triaxial test for dense Lund sand; (a) a comparison between the numerical andexperimental results, (b) schematically isotropic–kinematic hardening behavior.


n ¼ oFor¼ oF

oJ 1

oJ 1

orþ oF

oJ 2D

oJ 2D

orþ oF

oJ 2=33D

oJ 2=33D

orðA:3Þ

where

oFoJ 1

¼ 2J 1

/dfd

/hfh

� �2

þ /2df 2

d

/3hf 3

h

2J 21J 1affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�ð2=3Þð2J 1aJ 1 þ J 21aÞ

q ðA:4Þ

oFoJ 2D

¼ 2

3þ 2J 2

1

fd/2d

/2hf 2

h

ofd

oJ 2D

� �� 2/2

dfd

ofd

oJ 2D

� �ðA:5Þ

oF

oJ 2=33D

¼ wþ 2J 21

fd/2d

/2hf 2

h

ofd

oJ 2=33D

!� 2/2

dfd

ofd

oJ 2=33D

!ðA:6Þ

Axial Strain (%)

Axial Strain (%)

Vol

ume

Str

ain

(%)

0 2 4 6 8-2

-1.5

-1

-0.5

0

0.5

1

1.5


Experiment (Ibsen and Jakobsen,1996)

Vol

ume

Str

ain

(%)

0 2 4 6 8-2

-1.5

-1

-0.5

0

0.5

1

1.5

a

b

Fig. 21. Volumetric strain versus axial strain in triaxial test for dense Lund sand; (a) a comparison between the numerical andexperimental results, (b) schematically isotropic–kinematic hardening behavior.


where

ofd

oJ 2D

¼ ða1 � a2Þðcos x�ffiffiffi3p

sin xÞ4/d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3J 2DðJ 2Da þ JraÞ

p þ 3J 3D

8/dJ 22D

ða1 � a2Þð� sin x�ffiffiffi3p

cos xÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðJ 2Da þ JraÞð1� 27

4J 2

3D=J 32DÞ

q ðA:7Þ

ofd

oJ 2=33D

¼ 3J 1=33D

8/d

ða1 � a2Þð� sin x�ffiffiffi3p

cos xÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðJ 2Da þ JraÞ 1� 27

4J 2

3D=J 32D

� q ðA:8Þ

According to the definition of stress invariants, i.e. J1 = rii, J 2D ¼ 12sijsij and J3D = det(sij) in which the devi-

atoric stress tensor defined as sij ¼ rij � 13dijrmm, the vectors oJ1/or, oJ2D/or and oJ 2=3

3D =or, defined in (A.3),can be calculated as


oJ 1

or¼ f1; 1; 0; 1gT ðA:9Þ

oJ 2D

or¼ fsx; sy ; 2sxy ; szgT ðA:10Þ

oJ 2=33D

or¼ 2

3J�1=3

3D s2x þ s2

xy ; s2y þ s2

xy ; 2sxyðsx þ syÞ; s2z

n oT

ðA:11Þ

The hardening plastic modulus H can be determined by substituting the yield surface (5) into definition (A.2)as (Khoei, 2005)

H ¼ � oFofh

ofh

oep

dep

dkþ oF

ofd

ofd

oep

dep

dk

� �þ oF

ofh

ofh

oepv

depv

dkþ oF

ofd

ofd

oepv

depv

dk

� �� ðA:12Þ

where

oFofh

¼ �2J 21

/2df 2

d

/2hf 3

h

!ðA:13Þ

oFofd

¼ 2J 21

/2dfd

/2hf 2

h

!� 2/2

dfd ðA:14Þ

ofh

oepv¼ ðb2b3 expðb3e

pvÞÞdðep

vÞ ðA:15Þ

ofd

oepv¼ ðc2c3 expðc3e

pvÞÞdðep

vÞ ðA:16Þ

ofh

oep¼ � 1

/h

ðJ 1 þ J 1aÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ð2J 1aJ 1 þ J 2

1aÞq 2

oa1

oepþ oa2

oep

� �ðA:17Þ

ofd

oep¼ ða2 � a1Þ

3/d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiJ 2Da þ Jra

p oa2

oep� oa1

oep

� �þ

ffiffiffiffiffiffiffiffiJ 2D

pðcos x�

ffiffiffi3p

sin xÞ2/d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ðJ 2Da þ JraÞ

p oa1

oep� oa2

oep

� �ðA:18Þ

depv

dk¼ oF

oJ 1

ðA:19Þ

dep

dk¼ oF

or� 1

3

oFoJ 1

m ðA:20Þ

where m is the unit vector m = {1 1 1}T.

References

Bigoni, D., Piccolroaz, A., 2004. Yield criteria for quasibrittle and frictional materials. Int. J. Solids Struct. 41, 2855–2878.Borup, M., Hedegaard, J., 1995. Data Report 9403, Baskarp Sand No. 15, Geotechnical Engineering Group, Aalborg University, Denmark.Brandt, J., Nilsson, L., 1999. A constitutive model for compaction of granular media, with account for deformation induced anisotropy.

Mech. Cohes. Frict. Mater. 4, 391–418.Brown, S., Abou-Chedid, G., 1994. Yield behavior of metal powder assemblages. J. Mech. Phys. Solids. 42, 383–398.Chang, C.S., Hicher, P.Y., 2005. An elasto-plastic model for granular materials with microstructural consideration. Int. J. Solids Struct.

42, 4258–4277.Chen, W.F., Baladi, G.Y., 1985. Soil Plasticity, Theory and Implementation. Elsevier, Amsterdam.Collins, I.F., 2005. Elastic/plastic models for soils and sands. Int. J. Mech. Sci. 47, 493–508.Desai, C.S., Siriwardane, H.J., 1984. Constitutive Laws for Engineering Materials, with Emphasis on Geologic Materials. Prentice-Hall,

New Jersey.Doraivelu, S.M., Gegel, H.L., Gunasekera, J.S., Malas, J.C., Morgan, J.T., Thomas, J.F., 1984. A new yield function for compressible P/

M materials. Int. J. Mech. Sci. 26, 527–535.Dorris, J.F., Nemat-Nasser, S., 1982. A plasticity model for flow of granular materials under triaxial stress states. Int. J. Solids Struct. 18,

497–531.Foster, C.D., Regueiro, R.A., Fossum, A.F., Borja, R.I., 2005. Implicit numerical integration of a three-invariant, isotropic/kinematic

hardening cap plasticity model for geomaterials. Comput. Methods Appl. Mech. Eng. 194, 5109–5138.


Gudehus, G., 1996. A comprehensive constitutive equation for granular materials. Soils Foundations 36, 1–12.Ibsen, L.B., Jakobsen, F.R., 1996. Data Report, Lund Sand No. 0, Part 1 and Part 2, Geotechnical Engineering Group, Aalborg

University, Denmark.Jenkins, J.T., Strack, O., 1993. Mean-field inelastic behavior of random array of identical spheres. Mech. Mater. 16, 25–33.Khoei, A.R., 2005. Computational Plasticity in Powder Forming Processes. Elsevier, UK.Khoei, A.R., Azami, A.R., 2005. A single cone-cap plasticity with an isotropic hardening rule for powder materials. Int. J. Mech. Sci. 47,

94–109.Khoei, A.R., Azami, A.R., Haeri, S.M., 2004. Implementation of plasticity based models in dynamic analysis of earth and rockfill dams; a

comparison of Pastor-Zienkiewicz and cap models. Comput. Geotech. 31, 385–410.Khoei, A.R., Bakhshiani, A., 2004. A hypoelasto-plastic finite strain simulation of powder compaction processes with density dependent

endochronic model. Int. J. Solids Struct. 41, 6081–6110.Khoei, A.R., Bakhshiani, A., Mofid, M., 2003. An implicit algorithm for hypoelasto-plastic and hypoelasto-viscoplastic endochronic in

finite strain isotropic–kinematic hardening model. Int. J. Solids Struct. 40, 3393–3423.Khoei, A.R., DorMohammadi, H., 2007. A three-invariant cap plasticity with isotropic–kinematic hardening rule for powder materials:

model assessment and parameter calibration. Comp. Mater. Sci., in press.Khoei, A.R., DorMohammadi, H., Azami, A.R., 2007. A three-invariant cap plasticity model with kinematic hardening rule for powder

materials. J. Mater. Proc. Tech. 187, 680–684.Krenk, S., 2000. Characteristic state plasticity for granular materials, Part I: basic theory. Int. J. Solids Struct. 37, 6343–6360.Krenk, S., Ahadi, A., 2000. Characteristic state plasticity for granular materials, Part II: model calibration and results. Int. J. Solids Struct.

37, 6361–6380.Kolymbas, D., Wu, W., 1990. Recent results of triaxial tests with granular materials. Powder Tech. 60, 99–119.Lade, P.V., Kim, M.K., 1995. Single hardening constitutive model for soil, rock and concrete. Int. J. Solids Struct. 32, 1963–1978.Lewis, R.W., Khoei, A.R., 2001. A plasticity model for metal powder forming processes. Int. J. Plasticity 17, 1652–1692.Lewis, R.W., Schrefler, B.A., 1998. The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous

Media. Wiley, England.McDowell, G.R., Bolton, M.D., 1998. On the micromechanics of crushable aggregates. Geotechnique 48, 667–679.Oliver, J., Oller, S., Cante, J.C., 1996. A plasticity model for simulation of industrial powder compaction processes. Int. J. Solids Struct.

33, 3161–3178.Wan, R.G., Guo, P.J., 1998. A simple constitutive model for granular soils: modified stress-dilatancy approach. Comput. Geotech. 22,

109–133.Watson, T.J., Wert, J.A., 1993. On the development of constitutive relations for metallic powders. Metall. Trans. 24, 1993–2071.

A three-invariant cap model with isotropic–kinematic ... · A double-surface plasticity model was...

Documents

Transcript of A three-invariant cap model with isotropic–kinematic ... · A double-surface plasticity model was...