Ahadi Implicit Integration of Plasticity Models for Granular Materials

8/2/2019 Ahadi Implicit Integration of Plasticity Models for Granular Materials

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Implicit integration of plasticity models for granular materials

Aylin Ahadi a,*, Steen Krenk b

a Division of Mechanics, Lund University, Box 118, S-221 00 Lund, Swedenb Department of Mechanical Engineering, Technical University of Denmark, DK-2800 Lyngby, Denmark

Received 9 July 2002; received in revised form 14 April 2003; accepted 23 April 2003

Abstract

A stress integration algorithm for granular materials based on fully implicit integration with explicit updating is

presented. In the implicit method the solution makes use of the gradient to the potential surface at the final stress state

which is unknown. The final stress and hardening parameters are determined solving the non-linear equations itera-

tively so that the stress increment fulfills the consistency condition. The integration algorithm is applicable for models

depending on all the three stress invariants and it is applied to a characteristic state model for granular material. Since

tensile stresses are not supported the functions and their derivatives are not representative outside the compressive

octant of the principal stress space. The elastic predictor is therefore preconditioned in order to ensure that the first

predictor is within the valid region. Capability and robustness of the integration algorithm are illustrated by simulating

both drained and undrained triaxial tests on sand. The algorithm is developed in a standard format which can be

implemented in several general purpose finite element codes. It has been implemented as an ABAQUS subroutine, and atraditional geotechnical problem of a flexible strip footing resting on a surface of sand is investigated in order to

demonstrate the global accuracy and stability of the numerical solution.

2003 Elsevier B.V. All rights reserved.

Keywords: Integration algorithm; Granular materials; FE implementation; Footing analysis; Large strains

1. Introduction

Modeling the behaviour of granular materials under various loading conditions is technically important

and theoretically challenging. Recent advances in computational techniques have made it possible to solve

advanced geotechnical engineering problems numerically using the finite element method. The availabilityof powerful computers enables engineers to perform a three-dimensional finite element analysis of large

scale boundary value problems using realistic constitutive models. The overall accuracy of the analysis is

directly affected by the precision of the numerical algorithm used to integrate the constitutive equations.

This creates a need of developing accurate and robust constitutive drivers that can easily be implemented in

finite element codes.

* Corresponding author.

E-mail address: [email protected] (A. Ahadi).

0045-7825/03/$ - see front matter 2003 Elsevier B.V. All rights reserved.

doi:10.1016/S0045-7825(03)00354-2

Comput. Methods Appl. Mech. Engrg. 192 (2003) 34713488

www.elsevier.com/locate/cma
http://mail%20to:%[email protected]/http://mail%20to:%[email protected]/


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In this paper a stress integration algorithm based on fully implicit integration with explicit updating is

presented. The algorithm is applied to a non-associated plasticity model for granular materials developed in

[1]. The model is a three-dimensional generalization of the CamClay model, introducing dilation before

failure, dependence on the third stress invariant and a consistent limitation to compressive stresses. TheCamClay model based on the critical state concept developed in [2] is perhaps the most widely used model

today for geotechnical analyses. It is formulated in the two-dimensional stress space with mean stress p and

maximum shear stress 12q and includes isotropic hardening only.

The CamClay model has been implemented using a return mapping algorithm in [3] and later using

implicit integration algorithms in [46]. An extension of the CamClay model including kinematic

hardening has also been integrated using an implicit formulation [7]. The yield surface and plastic po-

tential function in the model used here are represented by functions including the third stress invariant I3,

to accurately describe the behaviour in triaxial as well as plane strain conditions. The elasticity in the

present model has the same simple form as in the original CamClay with constant shear modulus and

bulk modulus increasing linearly with the mean stress. In this model a characteristic state separating

contractive and dilative behaviour is distinguished from the ultimate state, corresponding to perfectly

plastic behaviour. As a result, one of the major shortcoming in the CamClay model is overcome, and

the dilative behaviour of granular material is modelled with very good accuracy as demonstrated in [8].

This is achieved by introducing a hardening law depending on both the shear and the volumetric strain

increments.

Fully implicit algorithms have been widely used in finite element formulations, since they have shown

good robustness and efficiency for simple elasto-plastic material models. Implicit integration of the con-

stitutive equations has also been used for more complex elasto-plastic and viscoplastic constitutive relations

[9]. Plasticity formulations for granular materials including the third stress invariant I3 are typically highly

non-linear. In recent years implicit solution strategies for models depending on the third stress invariant

have been developed, for associative isotropic elasto-plastic and viscoplastic models [10], for associative

elasto-plastic models with kinematic hardening [11] and for models of general isotropic elasto-plastic

geomaterials [12]. To reduce the number of equations in the implicit scheme fully implicit integration al-gorithms with explicit updating have been developed, e.g. for J2 plasticity in [13].

The third stress invariant I3 typically results in a high degree of non-linearity and complex numerical

algorithms. The aim here is to develop an implicit integration procedure for granular materials as simple

as the implicit integration algorithms developed for the CamClay model, and at the same time more

general and suitable for more versatile constitutive models with a high degree of non-linearity both in the

elastic as well as plastic components. The solution of the non-linear constitutive equations is carried out

with the backward Euler difference scheme slightly modified since the hardening parameter can be de-

termined explicitly at each intermediate state. The elastic predictor is preconditioned in order to ensure

that it is inside the valid stress region, which for cohesionless granular materials is the compressive

octant. The proposed numerical integration algorithm does not depend on the particular set of con-

stitutive expressions. It is suitable for granular material and can be extended to other both associated andnon-associated plasticity based material models without any conceptual changes. The algorithm can

therefore be expressed in a standard format which can be implemented in several general purpose finite

element codes.

The capability, accuracy and robustness of the numerical algorithm is tested at the local Gauss point

level, as well as at the global level using an implementation as a plastic material model in the finite element

code ABAQUS. The performance of the integration procedure is illustrated by simulating a triaxial test on

single element of sand. In addition a boundary value problem of traditional flexible strip footing resting on

a surface of sand is investigated in order to demonstrate the global accuracy and stability of the numerical

solution. General purpose finite element codes often contain a finite strain implementation of plasticity

models, and this feature is also illustrated.

3472 A. Ahadi, S. Krenk / Comput. Methods Appl. Mech. Engrg. 192 (2003) 34713488


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2. Constitutive equations for infinitesimal plasticity

Assuming a small deformation during a generic increment of loading, the infinitesimal strain tensor is

decomposed into elastic and plastic parts,de dee dep: 1

The elastic response of the material is characterized by the generalized Hooke s law which relates stress and

elastic strain increments linearly through the elastic constitutive tensor Ce. This relation expressed in a form

relating the component of the elastic strain increment to the component of stress increment, takes the

following form:

dee C1e dr; 2where C1e is the elastic tangent flexibility matrix.

In non-associated plasticity theory, the plastic strain increment is proportional to the gradient of the

plastic potential. This is known as a flow rule where the direction of dep is the gradient of the plastic

potential function and its magnitude is given by the plastic multiplier dv,

dep dv ogorT

: 3

The plastic multiplier dv is determined by a plastic hardening via the consistency relation, according to

which a stress point remains on the yield surface f during plastic loading,

df ofor

dr Hdv 0: 4

H is the hardening parameter describing the evolution of the plastic variables, and it typically consists of

two parts,

H ofoa

oa

ov H1H2: 5

The first factor H1 of=oa describes the changes in shape, size and position of the yield surface, i.e. itsdependence on the hardening parameters a, while the second factor H2 oa=ov describes the evolution ofthe hardening parameters. In the case of multiple hardening parameters a a1; a2; . . .T the factor H1 is arow vector, while the factor H2 is a column vector.

For a plastic work hardening material there is only one hardening parameter and the second part H2 can

conveniently be written as H2 og=orMar, where the matrix M defines a weighting between hydrostaticand deviatoric plastic work. In the following the hardening is assumed to depend only on the current stress

state, and M is a constant matrix.

The elasto-plastic stiffness matrix Cep is needed for use in the global, non-linear equation system forpredicting the size and direction of the next strain increment. It is determined from (1) by inserting the

relations between the stress increment dr and the elastic and plastic strain increments respectively from (2)

and (3),

Cep Ce Ce og=orTof=orCe

H of=orCe og=orT ; 6

where the plastic multiplier dv has been eliminated by use of the consistency relation (4).

In the following we will present an algorithm for computation of stresses and hardening parameters

consistent with the predicted strain increment done by a backward difference scheme.

A. Ahadi, S. Krenk / Comput. Methods Appl. Mech. Engrg. 192 (2003) 34713488 3473


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3. Fully implicit integration with explicit updating

The solution of the global, non-linear FE-equations gives an estimate of the total strain for each stress

point. The computation of corresponding stresses and hardening parameters that fulfill the yield conditionis then performed by integration of the constitutive equations. The precision of the numerical algorithm

used to integrate these equations has a direct impact on the overall accuracy of the global solution. For

elasto-plastic materials the constitutive equation are non-linear and must thus be solved using iterative

techniques. While the equilibrium must be checked on the global level, the integration of the stress in-

crement Drin for a given strain increment Dein can be performed at each Gauss point independently. The

integration procedure described in the following concerns determination of the stress increment at a single

point for a given total strain increment and the iteration index i is generally omitted. The subscript (n 1)denotes the last established equilibrium state, and subscript n denote the final, still unknown state.

The purpose of the integration scheme is to determine the stress changes Dr and hardening parameters

corresponding to a total change of displacement De within the load increment. The total strain increment is

decomposed into elastic and plastic parts,

De Dee Dep: 7The plastic part of the strain increment is estimated from the flow rule

Dep Dv og

or: 8

Depending on how the gradient og=or of the plastic potential function is computed the two families ofnumerical algorithms are obtained. In the generalized trapezoidal rule the gradient og=or is represented asan average over the increment, while in the generalized mid-point rule the gradient is evaluated at a rep-

resentative stress state. Use of the previous equilibrium stress rn1 result in the forward Euler scheme, whileused of the final stress rn gives the backward Euler scheme. These integration rules have been evaluated in

[14] and the generalized mid-point rule was found to be superior. However, the backward Euler scheme wasfound to be numerically stable for larger strain increments, which is desirable to use in FE-computations.

The backward Euler difference scheme is also considerably simpler to implement, and therefore used in this

paper. Fig. 1 illustrates the iteration strategy based on the backward difference, where Dr and r refers to the

current iteration, which converges to Drn, determining the stress state rn.

Fig. 1. Geometric interpretation of stress point integration algorithm.



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3.1. Integration scheme

The integration scheme determines the stress changes Dr and hardening parameters corresponding to a

total change of displacement De within the load increment. In the implicit backward Euler method thesolution makes use of the final stress state rn; an. The final stress rn and hardening parameters an aredetermined by solving the non-linear equations iteratively so that the stress increment fulfills the consis-

tency condition. The current estimate state r; a is determined in each iteration step relative to the lastequilibrium state rn1; an1.

The total strain increment De en en1 is the sum of the elastic and plastic parts, see (7). The elasticpart is written as Dee eern eern1, while for the plastic part use of a non-associated flow rule impliesDe

p Dvogrn=orT. Insertion into (7) yields

en en1 eern eern1 Dv ogrnorT

: 9

The stresses rn and the plastic multiplier Dv are unknown, while the prescribed total strain en and terms

related to the previous equilibrium state (n 1) are known.The computed stress and hardening parameters at final state (rn; an) must fulfill the consistency relation

frn; an 0: 10The process of plastic loading is generally associated with hardening, and the hardening parameters a must

be determined to satisfy

an an1 H2rnDv; 11where H2 oa=ov describes the hardening parameters introduced in (5).

3.2. Newton iteration

The non-linear equation system comprising (9)(11) can be solved using NewtonRaphson iteration

scheme. In the following a slightly modified version will be employed, as it is possible to eliminate the

hardening equation (11) by calculating it exactly for each intermediate state.

The incremental form of the constitutive relation (9) is obtained by making a first order Taylor ex-

pansion around the current state (r; a). The elastic strain in the last equilibrium state eern1 is constantduring iterations and does not contribute. e is the current strain estimate obtained in the previous iteration.

ei1n en1 ein en1

o

ore

e Dv ogorT

dr

i ogorT

dvi;

f

r

i1n ; a

i1n

f

r

in; a

in

of

ordr

i

of

oada

i;

ai1n an1 ain an1 Dv

oH2

ordr

i H2 dvi:

8>>>>>>>>>>>:

12

The subincrement of the hardening parameters dai ai1n ain can be calculated explicitly for each inter-mediate state from the last equation in (12)

dai Dvi oH2

ordr

i H2 dvi: 13

Insertion of (13) into the second equation in (12) allows us to reduce the number of equations in the

iteration scheme. Introducing strain residual dei ei1n ein and the residual of the yield function frin; aingives the following non-linear equation system to be solved:



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C1e Dvo2g

orTor

og

orT

of

or DvH1oH2

or H1H2

2

664

3

775i

dr

dv

!i

de

f

!i

; 14

where the elastic tangent flexibility matrix C1e oee=or has been used.For a finite plastic step the iteration matrix in (14), subsequently called A, is generally non-symmetric,

even for associated plasticity models. It has a similar form as the algorithmic elasto-plastic stiffness matrix,

see e.g. [15], where the use of finite increments leads to non-symmetric global stiffness matrix.

The equation system (14) is solved iteratively for dr; dvi, and the increments (Dr;Dv) are updated bysubincrements until the residuals are smaller than the prescribed tolerances,

Dri1 Dri dri; Dvi1 Dvi dvi: 15

Note, that while the final solution is independent of the individual subincrement, the iterative scheme re-

quires yield function and gradients of the plastic potential function to be defined also at the intermediate

states used for updates. After solving (14) the total increment of the hardening parameter Da is updatedexplicitly,

Dan H2Dv: 16It is important to realize that the subsequent update ofDa by use of (16) is necessary in order to ensure that

linearization of (11) does not produce any residual.

The iterations defined by (14) are carried out when the stress point turns out to be in plastic loading.

Thus an iteration procedure must start with an elastic predictor step r in order to determine, whether thereis plastic loading or elastic unloading. In case of plastic loading the predictor leads to a stress state outside

the current yield surface, as indicated in Fig. 2 and the iteration matrix A in (14) is computed for (r; an1).The NewtonRaphson procedure for calculation of the plastic corrector corresponding to the elastic

predictor implicitly assumes the existence of the yield function and the gradients of the yield function andthe plastic potential at the current state. It is therefore essential that the first estimated stress state r

corresponds to meaningful directions and magnitudes of the iterative corrections. Yield functions and

plastic potentials depending on the third deviatoric stress invariant J3 usually have equipotential surfaces

consisting of several sheets, and therefore the gradients may point towards a secondary potential surface.

This creates a need for bringing the elastic predictor into the valid domain. A robust way of doing that is

considered in the following section.

Fig. 2. Geometric interpretation of the first iterative step.



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3.3. First preconditioning step

In most friction material models stiffness and gradients cannot be evaluated in the tensile stress region.

Therefore, there is a need for an efficient way to ensure that the first elastic predictor lies within the validdomain. In case of the elastic predictor falling into the tensile region, the algorithm should be able to pull it

inside the compression octant of the principal stress space bounded by coordinate planes r1 0, r2 0 andr3 0. For any stress state (r1;r2;r3) a pressure pc can be defined via the equation

r1 pcr2 pcr3 pc 0 17such that the translated stress state (r1 pc;r2 pc;r3 pc) is located on one of the coordinate planes ofthe stress space coordinate system. In terms of the mean stress p, the second and the third deviatoric in-

variants J2 and J3, this equation takes the following form:

J3 p pcJ2 p pc3 0: 18This is a cubic equation in pc and the relevant roots are found by introducing the Lode angle h as

cos3h 3ffiffiffi

3p2

J3

J3=22

: 19

Substitution of J3 from (19) into (18) yields a new cubic equation

4cos3 w 3cosw cos3h; cosw ffiffiffi

3p

2

p pcffiffiffiffiJ2

p : 20

The relevant root of (20) is obtained by setting cos 3h cos3/ and using the trigonometric identity4cos3 w 3cos w cos3w,

pc 2ffiffi3p

ffiffiffiffiJ2p

cos h p; 06 h6p=3; 21where h is determined from (19). Now, the condition for determining whether a stress point lies within the

compressive octant is the following:

pc6 0 ) inside the compressive octant;> 0 ) outside the compressive octant:

&22

In case of the predictor stress point being outside the compressive octant, it can be moved inside the

compressive octant by a correction consisting of a hydrostatic translation and a reduction of the magnitude

of the deviatoric component. For a stress state r with mean stress p this operation can be written as

r r pc1 /r p1 1 /r pc /p1; 23where 1 is the second order unit tensor, and / < 1 is a scalar multiplier. The hydrostatic translation

pc1

brings the stress on to one of the coordinate planes, while application of the factor / to the deviatoriccomponent (r p1) defines a contraction towards the hydrostatic axis. In the examples the value / 0:01has been used.

4. Specific model formulation

The integration procedure described in the previous section is applied to a non-associated plasticity

model for granular materials based on the concept of a characteristic state where the incremental dilation

vanishes, [1]. The model is a three-dimensional generalization of the classical CamClay model, which is



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based on the classical critical state theory developed in [2] and formulated in the two-dimensional stress

space with mean stress p and maximum shear stress 12q. The elasticity in the present model has the same

simple form as in the original CamClay model with constant shear modulus and bulk modulus increasing

linearly with the mean stress. In the present model the characteristic state which separates the contractivefrom dilative behaviour is distinguished from the ultimate state which corresponds to perfectly plastic

behaviour. In the classical critical theory these two states coincide into a single critical state and as a result

of this the transition to dilative behaviour before failure of granular materials cannot be modelled. The

present more general but yet simple model has only six material parameters which can be determined from

data of a single standard triaxial test according to the calibration procedure developed in [8]. A brief de-

scription of the model is given below.

4.1. Description of non-associated plasticity model

The same generic format is used for both yield and plastic potential surface families, having different

shapes controlled by shape functions. The yield criterion for a plastic model defines whether plasticity is

activated or not. In terms of the mean pressure pand the third stress invariant I3 the isotropic yield function

in the model is expressed as

fr I3 p3gfp: 24The yield function grows in self-similar way. The shape parameter gf changes the deviatoric contour

continuously from triangular to circular when taking values between 0 and 1. The shape parameter is

expressed in terms of the mean pressure as

gfp p=pfm: 25The size of the yield function is controlled by parameter pf, which is the only hardening parameter in the

model, and the exponent m, assumed to be a material constant. The plastic potential is assumed to be

associated in the deviatoric plane, while the volumetric part is non-associated, leading to the similar formatas for the yield surface,

gr I3 p3ggp: 26The shape function, which is derived from an approximate friction hypothesis in [16], has the following

form:

gg 1 c2gp; cgp 1 p=pgn; 27

where the exponent n is assumed to be a material constant.

Use of associated deviatoric flow leads to identical deviator contours for the yield and flow potential

functions, i.e. the shape function gf and gg are equal. For a point of yielding this implies that gfgg

I3=p3 from which pg may be explicitly calculated. Since pg is not an independent model parameter the sizeparameter of the yield function pf is the only hardening parameter in the model, i.e. a pf. The yieldsurface and plastic potential function are illustrated in Fig. 3.

In elastic and elasto-plastic states it is assumed that the specific volume depends linearly on the logarithm

of the mean pressure p,

deev j

pdp; dev k

pdp: 28

The two non-dimensional flexibility parameters j and k are the inclination of the evlnp line in the elastic

and the elasto-plastic state, respectively. The elastic constitutive matrix in six-component format is



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Ce G

a b b

b a b

b b a

1

1

1

26666664

37777775

; 29

where a p=jG 4=3 and b p=jG 2=3. The shear modulus G is assumed to be constant.The direction of the plastic strain increment dep is the gradient of the plastic potential function and its

magnitude is given by the plastic multiplier dv

dep dv ogor

: 30

The change of the yield function per unit change of the plastic multiplier v is determined by the hardening

parameter H of=ov. The hardening of the yield function in the present model is controlled by the sizeparameter pf, and thus

H ofopf

opf

ov H1H2: 31

By differentiating the yields function (24), the factor H1 becomes

H1 of

opf mp2

gm

1

=m

f : 32The hardening of the loading surface depends on both plastic volumetric and deviatoric strain increments in

order to model dilatancy before failure of a normally consolidated material. Thus, the hardening rule in the

model is a weighted sum of the volumetric and deviatoric parts of the plastic work,

dpf 1k j pde

pv

wsT dep; 33where s is the deviatoric part of the stress, and e is the deviatoric part of the strain. w is a small non-

dimensional weight parameter, assumed to be a material constant. The value of w follows from the

introduction of the ultimate state line, which defines a stress state of ideal plasticity. The value of w is

Fig. 3. (a) Yield surface, (b) plastic potential surface.



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determined from the difference in inclination between the characteristic state line Mc and ultimate state

line Mu. In the CamClay model the two lines are lumped into a common critical state, i.e. w 0 and thehardening rule in CamClay model is obtained from (33) by setting w 0. The disadvantage of this isthat the hardening stops when de

p

v 0 and the material can never pass the characteristic line and hencethe an important characteristic of granular material as transition from compaction to dilation cannot berepresented.

Differentiating relation (33) after introducing the plastic strain increments from (3), the hardening factor

H2 is obtained in the form

H2 opfov

1k j p

og

op

w og

oss

1k j

og

orMr; 34

where the stress r has been decomposed into hydrostatic pressure p and deviatoric stress s. The matrix M

defines the general work hardening model and for the present model it is

M 131 w11T wI 35

with I the identity matrix. M is a constant matrix and by setting w 0 the volumetric hardening model ofthe critical state theory, is obtained.

4.2. Gradients of the yield and plastic potential functions

The gradients of the yield function (24) and plastic potential function (26) can be written in the similar

form

of

or oI3

or op

3gfor

oI3or

hfp21; 36

og

or oI3

or o

p3gg

or oI3

or hgp21; 37where the non-dimensional factors hf and hg have been introduced as

hf 13p2

o

opp3gf 1

1

3m

gf; 38

hg 13p2

o

opp3gg 1 cg 1

1

2

3n

cg

: 39

It is important to notice that in the differentiation gf and gg are the functions defined as gfp p=pfmand gg 1 c2gp, with cgp 1 p=pgn, while in the final results they are determined directly from theconditions f

r

0 and g

r

0, respectively.

The second derivative of the plastic potential g needed for computations is convenient written as

o2g

orTor o

2I3

orTor o

2p3ggorTor

o2I3

orTor 1

3h00gp11

T; 40

where h00g is defined as

h00g 1

3p

o2

op2p3gg 21 cg 1

1

3n 1 n 1

2

3n

cg

41

and as in the previous the differentiation is done for the constant cg 1 p=pgn, while in the final form itis calculated directly from gr 0 as c2g 1 I3=p3.



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The third stress invariant is defined as

I3 r11r22r33 2r23r31r12 r11r223 r22r231 r33r212: 42

Written in component form the first derivative of I3 becomes

oI3

orT

r22r33 r223r33r11 r231r11r22 r212

2r31r12 r11r232r12r23 r22r322r23r32 r33r12

26666664

37777775

43

and the second derivative is

o2

I3orTor

0 r33 r22 2r23 0 0r33 0 r11 0 2r31 0r22 r11 0 0 0 2r122r23 0 0 2r11 2r12 2r310 2r31 0 2r12 2r22 2r230 0 2r12 2r31 2r23 2r33

2

6666664

3

7777775: 44

Having computed the gradients of f and g the derivative of H2 with respect to r is obtained by differen-

tiation of (34),

oH2

orT 1k j M

og

orT

o

2g

orTorMrT

: 45

4.3. Integration algorithm

The constitutive calculations are performed using the implicit integration algorithm formulated in

Section 3. For numerical computation it is convenient to express the stress and the strain increment tensors

used in the stressstrain relations in six-component format as follows:

r r11;r22;r33; r23;r13;r12; e e11; e22; e33; 2e23; 2e13; 2e12: 46The shear strain increments are multiplied with a factor two in order to obtain tensor consistency and as in

most finite element codes tension is assumed to be positive. The model operates with the traditional split of

stresses and strains into hydrostatic and deviatoric parts

p 131Tr; s r p1; 47

ev

1Te; e

e

1

3

ev1;

48

where p is the hydrostatic pressure, s is the deviatoric stress tensor, ev is volumetric strain and e is deviatoricstrain tensor.

Integration of the stresses and the hardening parameter for a given strain increment requires evaluation

of the iteration matrix A. The current values of all terms in (14) must be computed. In addition values of H2and oH2=or are needed for updating the hardening parameter pf. These computation implicitly assume A tobe well defined for every iteration, even for the first. In the current model the yield function and plastic

potential function are third degree polynomials if the stress, which leads to regions where the gradients do

not represent the assumed slope towards the yield and potential surfaces, yielding invalid directions and

magnitude of the plastic strain increment. This problem is most likely to occur near the tensile region where

the bounding triangle narrows in the solution space. Therefore the elastic predictor should be scaled



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rationally, so that it lies within the circumscribing triangle. The elastic predictor is transformed as described

in Section 3.3. The Newton iteration are carried out and the increments of stress and plastic multiplier are

updated by subincrements. The hardening parameter pf is updated explicitly at each iteration. The sub-

sequent update of pf is necessary in order to ensure that linearization of (11) do not produce any residual.

The iterations stop when the residuals of kdpfk=kdpf0k and kdek=kde0k fulfill the required convergencetolerances p and e respectively. The value of p and e used in the calculation is 10

8. This general, yetsimple, integration algorithm is summarized in Table 1.

5. Numerical examples

The accuracy, stability and convergence properties of the numerical algorithm are tested at both local

and global level. A single element test was carried out under both drained and undrained conditions and the

boundary value problem chosen simulates a classical geotechnical bearing capacity problem of a strip

footing resting on surface of sand. The model parameters were determined using test data from a single

triaxial test on sand, [17], and calibration procedures developed in [8]. The following material parameters

were used in the simulations G 11:3 MPa, k 0:0142, j 0:00755, n 0:959, m 0:600 and w 0:251.

5.1. Triaxial test on single element

To investigate the local stress integration algorithm a triaxial test on single element has been simulated.The test starts at initial hydrostatic pressure p0 0:2 MPa and the element was compressed 5% of its initialheight in vertical direction. Both triaxial drained and undrained tests have been considered here. It is as-

sumed that the sand remains homogeneous and that no strain localization occurs during the tests.

The increase of the solution accuracy with the increased step number is ensured by the algorithmic

consistency. The algorithm is therefore tested for three different size of steps and the number of steps is

varied between 15, 30 and 60. The simulation of undrained test carried out with step number 60 is des-

ignated as the exact integration of the constitutive equations. Then the same simulation are carried out for

step numbers 30 and 15. The comparison of these simulation are seen in Fig. 4. The algorithm captures the

stressstrain responses with very good accuracy. The responses are very similar and solutions with number

of steps greater that 30 are practically identical to the exact solution.

Table 1

Integration algorithm

Inputs: en1, rn1, pn1f , en, Dv 0elastic predictor: r

rn

1

dr

e

en

en

1;rn

1

pc 2=ffiffiffi

3p J2 cos h p1st octant? if pc > 0 then r

1 /r pc /p1iterations i 1; 2; . . . ; imax

A Ar;pf;Dv, f fr;pfdr

dv

! A1 def

!

Dv Dv dvr r drpf pn1f H2rDvde en een1;rn1; r;Dvdpf pf pI3=p31=m

current stress r rstop iteration when kdpfk=kdpf0k < p and kdek=kde0k < e

rn r



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The drained triaxial test have been simulated for the same three different numbers of steps 60, 30 and 15.

The comparison of these results are presented in Fig. 5. The stressstrain curves are practically identical to

each other, demonstrating the good performance of the algorithm. The volumetric curves are very similar

i.e. the algorithm captures the volumetric responses with very good accuracy. For the solution with only 15

steps the loss of accuracy is relatively small considering the large steps, and solutions with more than 30

steps are very close to the exact solution. These results demonstrate the robustness and accuracy of the

integration algorithm. Table 2 illustrates the behaviour of the integration algorithm for local iterations for

drained triaxial test. The residuals for four typical load steps are presented and the number of iterationsneeded to meet the convergence tolerance of 108 is between 4 and 5 per load step.

Fig. 4. Stressstrain curves for different numbers of steps, undrained triaxial test.

Fig. 5. Stressstrain and volumetric curves for different numbers of steps, drained triaxial test.

Table 2

Normalized residual strain norm kdek=kde0k for drained triaxial testIteration Step 10 Step 20 Step 40 Step 50

1 1.0000e)00 1.0000e)00 1.0000e)00 1.0000e)00

2 1.9974e)02 1.0568e)02 8.7621e)02 3.0766e)02

3 3.3896e)04 8.9362e)05 5.5725e)03 2.9590e)04

4 4.4331e)08 8.9827e)09 3.0052e)05 8.7546e)08

5 8.4318e)10



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The computational algorithm was also checked with respect to the stress paths including load reversals.

Results of a typical load reversal path oabcd is seen in Fig. 6. The unloadingreloading behaviour is

assumed to be elastic and the loading curve oac is the same as that obtained without load reversals, which

shows the accuracy of the integration algorithm.

5.2. Footing analysis

The performance of the numerical algorithm at the global level is investigated by multi-element test. A

traditional geotechnical problem of a flexible strip footing resting on a surface of sand has been simulated.

The computations are performed using ABAQUS finite element code, which provides a facility for im-

plementing user defined material behaviour in FORTRAN subroutines. The constitutive model described

in Section 4.1 is programmed in the user subroutine UMAT. This subroutine is called by ABAQUS at eachelement integration point, for each increment, and during each load step. The main functions of the

subroutine are to integrate stresses and solution dependent state variables, and to provide the Jacobian

matrix oDr=oDe used in overall Newton iteration. For simplicity in this version of the algorithm we use thecontinuum tangent stiffness defined in (6), instead of the asymptotic tangent stiffness matrix.

The number of solution dependent state variables and the required material parameters are introduced in

an input file and the subroutine is linked with the ABAQUS-solver. The hardening parameter pf is the only

state variable in the present UMAT and the material used in this simulation is the same as the triaxial test

simulation with material parameters given in Section 5.

The finite element mesh of width 2 m and depth 1 m is shown in Fig. 7. A strip footing may be considered

as a plane strain problem, but the analysis is made using the three-dimensional finite element procedure

described in Section 3. A plane strain condition is applied by restraining the degree of freedom normal tothe vertical plane. Due to symmetry of geometry and loading only half of the footing is modelled. The mesh

consist of 342 nodes and 144 eight-noded brick elements. Half of the footing with 0.475 m width spans six

elements in the upper left corner of the mesh. As well as choosing values of the material parameters, the

simulation requires definition of realistic initial conditions prior the application of the footing load. The

initial condition in terms of stresses were generated in preliminary step in which the unit weight c 0:2MN/m3 was applied, the stresses at the Gauss points were computed and then displacements were the reset

to zero. In addition a load of q 0:1 MN/m2 was applied on the ground surface, and then the uniformlydistributed footing load was applied in increments. The simulation failed to converge at footing load of 2.65

MPa. The analytically computed ultimate load according to Therzaghi s theory has been calculated to 2.735

MPa. The numerically computed limit solution is in good agreement with this value.

Fig. 6. Stressstrain curve with load reversals.



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The numerical performance of the algorithm at both local and global levels is illustrated in Tables 3 and

4. Table 3 shows the normalized strain norm of the strain subincrement at the local level when using the

integration algorithm of Table 1. The quadratic convergence of the local integration algorithm is seen

clearly. The global convergence of the equilibrium iterations is illustrated in Table 4. Only the tangent

stiffness is transferred from the material subroutine UMAT to the main program, and it is seen that the

convergence is fast, but not quadratic. The convergence criteria for the nodal residual force is specified as a

tolerance 104 times the average nodal force, given in the last row of the table, and this is combined with aconvergence tolerance on the last displacement subincrement du of 103 times the displacement increment

Du.Results of the FE simulation corresponding to the computed limit load are summarized in Figs. 8 and 9

in which contour plots of the stress r22 in vertical direction and mean stress p are reported.

Table 3

Normalized residual strain norm kdek=kde0k for top center element below strip footingIteration Step 1 Step 30 Step 50

1 1.0000e)00 1.0000e)00 1.0000e)00

2 2.2024e)02 1.0533e)02 3.9310e)03

3 3.4069e)05 9.2453e)05 1.0898e)05

4 8.2574e)

10 7.1230e)

09 8.2481e)

11

Table 4

Residual nodal force in strip footing analysis

Iteration Step 1 Step 30 Step 50

1 4.038e)04 3.097e)05 7.214e)04

2 3.185e)05 3.547e)06 3.126e)05

3 2.073e)06 1.859e)07 1.946e)06

4 1.822e)07 5.770e)07

Mean force 6.462e)03 6.998e)03 7.558e)03

Fig. 7. FE-mesh and deformed mesh of the footing problem.



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5.3. Large strain analysis

The effects of using large strains can be included in the analysis in a simple way. The ABAQUS finite

element code provides a parameter NLGEOM accounting for geometric non-linearities during the load

step. When activating this parameter the elements are formulated in the current configuration using current

nodal position. The calculated stresses are the Cauchy stresses. NLGEOM is included in the input file and

no modifications of the implemented integration algorithm are needed. The footing problem in the previous

example was simulated using large strains. The initial conditions prior the application of the footing loadwere applied in a preliminary small strain step in the same way as described in the previous example. The

uniformly distributed footing load was then applied including the effects of large strains.

In Figs. 10 and 11 results from the large strain simulation are compared to the small strain solution. As it

is seen in Fig. 10a, in the region near the footing center there is no significant difference in the contact stress

beneath the footing between the two simulation, while in the region near the edge the large strain solution

predicts somewhat lower contact stress. The distribution of the vertical stress along the symmetry axis is

plotted in Fig. 10b. There is an obvious differences between the two solutions. At the same depth the small

strain solution predicts higher stress in vertical direction compared to the large strain solution. The com-

puted loaddisplacement curve of the center of the footing is seen in Fig. 11. As expected the use of large

strains results in smaller vertical displacement compared to the small strain solution.

Fig. 8. Stress r22 in vertical direction at the end of the simulation.

Fig. 9. Mean stress at the end of the simulation.



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6. Conclusions

A fully implicit stress integration algorithm with explicit updating has been presented in this paper. The

final stresses and hardening parameters are determined solving the non-linear equations iteratively so that

the stress increment fulfills the consistency condition. The number of equation to be solved was reduced

since the hardening parameter can be updated explicitly. The integration algorithm was applied to a

characteristic state model for granular materials developed in [1], but it can be applied to other both as-

sociated and non-associated plasticity based material models without any conceptual changes.

The good accuracy and robustness of the numerical algorithm has been demonstrated at the local Gausspoint level, as well as at the global level. Numerical results from triaxial tests on sand illustrate the good

performance of the integration procedure. The global accuracy and stability was demonstrated by per-

forming three-dimensional simulations of geotechnical engineering problems. The boundary value problem

of traditional geotechnical flexible strip footing resting on a surface of sand was investigated and the nu-

merical results show very good performance. The algorithm is developed in a standard format which en-

ables implementation into multipurpose finite element codes, and the present analyses were made using an

implementation of the material model in the ABAQUS code. This code has a facility for using updated

geometry, simulating a sequence of incremental steps. The computations were performed using both small

and finite strains, and comparison of the contact stress, vertical distribution of the stress beneath the

footing and loaddisplacement curves demonstrate a visible but moderate effect of finite strains.

Fig. 10. (a) Contact stress distribution beneath the footing, (b) vertical stress distribution along the symmetry axis.

Fig. 11. Loaddisplacement curve of the footing center.



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