Computers and Geotechnics 64 (2015) 32–47

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Computers and Geotechnics

journal homepage: www.elsevier .com/ locate/compgeo

Stress-accurate Mixed FEM for soil failure under shallow foundationsinvolving strain localization in plasticity

http://dx.doi.org/10.1016/j.compgeo.2014.10.0040266-352X/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: lbenedetti@cimne.upc.edu (L. Benedetti).

Lorenzo Benedetti ⇑, Miguel Cervera, Michele ChiumentiInternational Center for Numerical Methods in Engineering (CIMNE), Technical University of Catalonia (UPC), Edificio C1, Campus Norte, Jordi Girona 1-3, 08034 Barcelona, Spain

a r t i c l e i n f o a b s t r a c t

Article history:Received 29 July 2014Received in revised form 17 September2014Accepted 7 October 2014Available online 18 November 2014

Keywords:Strain softeningStrain localizationLocal plasticity modelsMixed finite elementsSlip linesShear bandsShallow foundationsSoil failureVon MisesDrucker–Prager

The development of slip lines, due to strain localization, is a common cause for failure of soil in many cir-cumstances investigated in geotechnical engineering. Through the use of numerical methods – like finiteelements – many practitioners are able to take into account complex geometrical and physical conditionsin their analyses. However, when dealing with shear bands, standard finite elements display lack of pre-cision, mesh dependency and locking. This paper introduces a (stabilized) mixed finite element formula-tion with continuous linear strain and displacement interpolations. Von Mises and Drucker–Prager localplasticity models with strain softening are considered as constitutive law. This innovative formulationsucceeds in overcoming the limitations of the standard formulation and provides accurate results withinthe vicinity of the shear bands, specifically without suffering from mesh dependency. Finally, 2D and 3Dnumerical examples demonstrate the accuracy and robustness in the computation of localization bands,without the introduction of additional tracking techniques as usually required by other methods.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

The stability analysis of a slope, both in the small scale of a roadembankment and in the larger one of a mountain slope, is a veryfrequent example of geotechnical engineering. The prevention offailure shear bands is a fundamental requirement to ensure thesafety of a volume of soil. In the geotechnical practice, standarddesign procedures require the computation of a safety factor. Thisis usually done by comparing the value of the acting forces to thevalue of the resisting ones through simplified methods. The firstrecorded case of stability analysis was performed by Hultin andPettersson in 1916 (documented only in 1955), for the StigbergQuay in Gothenburg (Sweden), where the slip surface was takento be circular and the sliding mass was divided into slices. In thesame period, the first major result was the Bishop method, pro-posed by Prof. A. Bishop as an extension of the ‘‘Swedish Slip CircleMethod’’ [1]. Although these methods are very useful as prelimin-ary evaluation tool, the validity of the approach is strongly limitedwhen simplified assumptions on soil mechanical constitutive law,geometry and slip lines shape are required a priori. The

introduction of the Finite Elements Method represented a soundalternative to tackle detailed problems of geotechnical nature,thanks to their potential versatility and vast application. FEMmakes possible the study of materials failure and its complex cou-pling with environmental actions such as seepage flow.

In the last three decades, the scientific community invested aconsiderable effort seeking a consistent description of failuremodes through the use of numerical methods. A slip line is a phys-ical discontinuity created by a localization of strains, as it isdepicted in part b of Fig. 1 reported from Cervera et al. [2]. Froma mathematical stand point, the numerical discontinuity in thefield variables can be treated in various ways. In the approachadopted in this work the strain localization is assumed to occurin a band of finite width where the displacements are continuousand the strains are discontinuous but bounded [3]. Actually, thisis a regularization of the discontinuity over a finite length, as it ispossible to see in part a of Fig. 1.

It is well known that this kind of ‘‘smeared’’ approach posessome challenges. The standard irreducible formulation of FEM isknown to be heavily affected by spurious mesh dependence whensoftening behavior occurs and, consequently, slip lines evolution isbiased by the orientation of the mesh [4]. Moreover, in the case ofisochoric behavior, unbounded pressure oscillations arise and the

Fig. 1. Localized failure: strong (right) and smeared (left) discontinuities.

L. Benedetti et al. / Computers and Geotechnics 64 (2015) 32–47 33

consequent locking of the stresses pollutes the numerical solution.Both problems can be shown not to be related to the mathematicalstatement of the continuous problem but instead to its discrete(FEM) counterpart [5,6].

Mixed formulations in terms of both the pressure and the flowvelocity are classical in the numerical solution of Darcy’s equation[7–12], where the focus is placed in achieving enhanced accuracyin the velocity. The mathematical structure of Darcy’s and Cauchy’sproblems is analogous, with the pressure and velocity fields in thefirst one corresponding to the displacement and stress fields in thesecond one. Therefore, similar mixed methods can be applied toboth problems.

In the last decade, the use of mixed finite elements for thedescription of failure mechanics has proved to be extremely useful.Initially, a stabilized displacement–pressure (u=p) formulation wasintroduced to address the problem of incompressibility in elasto-plasticity [13]. Later, it was shown that a continuum isotropic dam-age constitutive law can be fitted in such formulation [14].Recently, Badia and Codina [7], for the Stokes–Darcy problem,and then Cervera et al. [15], for the linear and nonlinear mechani-cal problem, discussed the local convergence properties of mixedformulations. From these, it follows that the reliability in the pre-diction of strain concentration bands depends directly on the capa-bility of the method to converge to a meaningful solution. In nearlysingular situations, such as when a slip line forms, the u=p formu-lation presents satisfactory global convergence in the interpolatedvariables, but it lacks of local convergence in the stress field,although a large number of tests showed a well behaved solutionin many cases [16].

In order to achieve local convergence of stresses, and, in turn,objectivity of results with respect of the mesh alignment, a stabi-lized mixed strain–displacement (e=u) formulation was developed

by Cervera et al. [17,18] and applied to problems involving soften-ing isotropic damage materials. In these references, it is shown thatthe enhancement of accuracy attained by the use of mixed strain–displacement (e=u) formulation overcomes the spurious mesh-biasdependency observed when using the standard irreducible FEMformulation.

In this work, the strain–displacement mixed formulation isextended for the purpose of solving problem involving compress-ible and incompressible plasticity. The effectiveness of the formu-lation, outperforming both the standard irreducible and the mixeddisplacement–pressure (u=p) approaches, is demonstrated inexamples involving failure and strain concentration bands.

The paper is organized as follows. First, the mixed finite ele-ment method is derived and the mathematical basis are presented.Then, the Drucker–Prager constitutive model is introduced as apressure-dependent generalization of the incompressible VonMises model. Finally, numerical examples are reported in orderto demonstrate the robustness and the accuracy of the proposedmixed finite elements.

2. e� u mixed finite elements

2.1. Strong form

Consider a body occupying the space domain X, its boundarybeing @X. The field of total strain has be compatible with the dis-placement field, so that

�eþ $su ¼ 0 ð1Þ

where u is the field of displacements and e is the field of infinites-imal strains. The equilibrium of the body in a (quasi-) staticmechanical problem is described by the following equation:

34 L. Benedetti et al. / Computers and Geotechnics 64 (2015) 32–47

$ � rþ f ¼ 0 ð2Þ

where r is the Cauchy stress tensor and f are the external forcesapplied to the body. The symbol $ � ð�Þ refers to the divergence oper-ator whereas $sð�Þ is used to denote the symmetric gradient. Insmall strain plasticity the strain tensor is decomposed additively as

e ¼ ee þ ep ð3Þ

with ee the elastic strain tensor and ep the plastic strain tensor. Theconstitutive equation can be written as

r ¼ C : ee ¼ C : e� ep� �

ð4Þ

where C is the fourth order elastic constitutive tensor. Now, substi-tuting (4) in (2), the problem reads

� eþ $su ¼ 0$ � C : e� ep

� �� �þ f ¼ 0

ð5Þ

In order to obtain a symmetric system, the first equation is pre-multiplied by the elastic constitutive tensor C:

� C : eþ C : $su ¼ 0$ � C : e� ep

� �� �þ f ¼ 0

ð6Þ

Hence, (6) is the final system of partial differential equations instrong form in terms of the total strains e and displacements u forthe mechanical problem involving plasticity. The mixed problem issolved for both unknown fields u; e½ � introducing appropriateboundary conditions and evolution laws for the plastic strain field[19]. For the sake of shortness and recalling (4), it can be written:

� C : eþ C : $su ¼ 0$ � rþ f ¼ 0

ð7Þ

2.2. Weak form

The weak form of the set of equations presented in (6) is:

�Z

Xc : C : eþ

ZXc : C : $su ¼ 0 8c 2 GZ

Xv : $ � rð Þ þ

ZX

v : f ¼ 0 8v 2 V

ð8Þ

The functional space V represents the set of test functions v for thedisplacement field u, whereas G is the set of test function tensorsfor the strain e. Integrating by parts the second equation, it can bewritten:

�Z

Xc : C : eþ

ZXc : C : $su ¼ 0 8c 2 GZ

X$sv : r ¼ F vð Þ 8v 2 V

ð9Þ

where the boundary terms accounting for stresses on the boundaryand body forces f are collected in the term

F vð Þ ¼Z@X

v : r � nð Þ þZ

Xv : f ð10Þ

in which n represents the outward normal vector with respect tothe boundary @X. From the mathematical requirements of the prob-lem in (9), V will be in the space of square integrable functions vwhich are at least square integrable and have square integrable firstderivative, whereas G will belong to the set of square integrablesymmetric tensors c.

2.3. Discrete Galerkin formulation

The discretized version of the continuous weak form is obtainedconsidering a finite set of interpolating functions for both the

solution and the test function. For this reason the discretefunctional spaces are a subset of their continuous version:

Gh � G # L2ðXÞdim�dim and Vh � V # H1ðXÞdim ð11Þ

where dim is the number of the dimensions of the domain of theproblem. Now, the strain tensor e and the displacement field uare approximated as

e! eh ¼Xnpts

i¼1

cðiÞh eðiÞh ch 2 Gh

u! uh ¼Xnpts

i¼1

v ðiÞh uðiÞh vh 2 Vh

ð12Þ

The system of Eq. (9), in its discrete form, reads

�Z

Xch : C : eh þ

ZXch : C : $suh ¼ 0 8ch 2 GhZ

X$svh : r ¼ F vhð Þ 8vh 2 Vh

ð13Þ

In the following, we will introduce equal interpolation finite ele-ment spaces for displacements and strains. Particularly interestingwill be the case of linear and bilinear interpolations, i.e. P1P1 andQ1Q1 elements. However, it is well known that the stability of a dis-crete mixed formulation depends from the choice of the finite ele-ment spaces Gh and Vh as stated by the Inf-Sup condition [20].Using equal order of interpolation does not satisfy the previous con-dition; consequently, a Variational Multiscale Stabilization proce-dure is now introduced.

2.4. Variational Multiscale Stabilization

The Variational Multiscale Stabilization was developed in firstinstance by Hughes et al. [21] and then generalized by Codina[22]. This technique modifies appropriately the variational formof the problem in order to provide the required numerical stability.The corresponding modified Inf-Sup condition is milder than theoriginal one and it holds for most common equal order finite ele-ment spaces [23].

The stabilization procedure supposes that the solution of vari-ables ðe;uÞ is given by a resolvable scale ðeh;uhÞ, calculated onthe FEM mesh, and an irresolvable one ð~e; ~uÞ, called subscalesolution:

e ¼ eh þ ~e

u ¼ uh þ ~uð14Þ

The subscale variables and their test functions pertain to theirrespective functional spaces ~G for the strain subscale and ~V forthe displacement subscale. This initial hypothesis allows us to con-sider extended solution spaces given by G � Gh � ~G andV � Vh � ~V. The subscale part ð~e; ~uÞ can be thought as a high fre-quency solution that cannot be captured with the coarse FEM mesh.

The plastic strains ep are computed by the return mapping algo-rithm, given the stress tensor r ¼ C : e� ep

� �as input data. Since

the total strain field e has both coarse and subscale contribution,then also the plastic strain tensor ep could present a correspondingsubscale part. However, since the subscale contribution is assumedto be small, the plastic strain will be approximated as:

ep ¼ ep rð Þ � ep rhð Þ ð15Þ

with

rh ¼ C : eh � ep rhð Þ� �

ð16Þ

Within this enhanced functional setting, the set of equations can bewritten as:

L. Benedetti et al. / Computers and Geotechnics 64 (2015) 32–47 35

�Z

Xch : C : eh þ ~eð Þ þ

ZXch : C : $s uh þ ~uð Þ ¼ 0 8ch 2 GhZ

X$svh : C : eh þ ~e� ep

� �� �¼ FðvhÞ 8vh 2 Vh

�Z

X

~c : C : eh þ ~eð Þ þZ

X

~c : C : $s uh þ ~uð Þ ¼ 0 8~c 2 ~GZX

~v : $ � C : eh þ ~e� ep� �� �� �

þZ

X

~v : f ¼ 0 8~v 2 ~V

ð17Þ

Rewriting the second group of equations, tested against the sub-scale test functions, and assuming that the subscale ð~e; ~uÞ vanisheson the boundary, it follows

�Z

X

~c : C : ~eþZ

X

~c : C : $s ~u ¼Z

X

~c : C : eh � $suh½ � ~c 2 ~GZX

$s ~v : C : ~e ¼ �Z

X

~v : $ � rh þ f½ � ~v 2 ~V

ð18Þ

The last system of equations shows that the solution of the subscalevariables depends on the residuals of the strong form of the equa-tions upon substitution of the FEM solution. Defining R1;h and R2;h

as the residuals of the equations defined as:

R1;h ¼ �C : eh þ C : $suh

R2;h ¼ $ � rh þ fð19Þ

Eq. (18) represent the projection of the residuals on the subscalegrid. They can be rewritten as:eP1 �C : ~eþ C : $s ~uð Þ ¼ eP1 C : eh � C : $suhð Þ ¼ �eP1 R1;h

� �eP2 $s ~v : C : ~eð Þ ¼ �eP2 $ � rh þ fð Þ ¼ �eP2 R2;h

� � ð20Þ

Following the work of Codina [22], it is possible to approximate thesubscale variables within each element as:

~e ¼ se C�1 : eP1 R1;h� �

~u ¼ sueP2 R2;h� � ð21Þ

where su and se are the stabilization parameters that, for this prob-lem, will be computed as:

su ¼ cuhL0

land se ¼ ce

hL0

ð22Þ

In the last expression, cu and ce are arbitrary positive numbers; l isa mechanical parameter of the problem, usually chosen as twice theshear modulus of the material G; h is the representative size of thefinite element mesh and L0 is a characteristic length of the problem.To complete the stabilization method, an appropriate projectionoperator has to be selected in order to be able to compute the sub-scale variables.

2.4.1. ASGSIn the Algebraic Subgrid Scale Stabilization method [7], the pro-

jection operator is taken as the identity, that is:

eP ¼ I )~e ¼ se �eh þ $suhð Þ~u ¼ su $ � rh þ fð Þ

ð23Þ

Back-substituting in the system of equations tested against thefinite element functions and rearranging:

� 1� seð ÞZ

Xch : C : eh þ 1� seð Þ

ZXch : C : $suh

þ su

ZXch : C : $s $ � rh þ fð Þ ¼ 0 8ch 2 GhZ

X$svh : C : 1� seð Þeh þ se$

suh � ep� �� �

¼ FðvhÞ 8vh 2 Vh

ð24Þ

Now, integrating again by parts in the first equation and takingch ¼ 0 on @X, the final system of equations reads:

� 1� seð ÞZ

Xch : C : eh � $suhð Þ

� su

ZX

$ � C : chð Þ½ � � $ � rh þ f½ � ¼ 0 8ch 2 GhZX

$svh : C : 1� seð Þeh þ se$suh � ep

� �¼ FðvhÞ 8vh 2 Vh ð25Þ

The first term in the first equation represents a projection (smooth-ing) of the strain field obtained by differentiation of the discrete dis-placement field. The second additional term is given by thedisplacement subscale that, in turn, depends on the residual ofthe strong form of the equilibrium equation. The second equationis related to the balance of momentum. Defining the stabilized totalstrain field as:

estab ¼ 1� seð Þeh þ se$suh ð26Þ

the system of Eq. (25) reads:

� 1� seð ÞZ

Xch : C : eh � $suhð Þ

� su

ZX

$ � C : chð Þ½ � � $ � rh þ f½ � ¼ 0 8ch 2 GhZX

$svh : C : estab � ep� �

¼ FðvhÞ 8vh 2 Vh ð27Þ

2.4.2. OSGSIn the Orthogonal Subgrid Scale Stabilization [7], the projection

operator selected to solve the unresolvable scale variables is theorthogonal projectoreP?h Xð Þ ¼ I Xð Þ � Ph Xð Þ ð28Þ

where Ph represents the projection over the finite element mesh. Itrepresents the L2 projection of X, or least square fitting, on the finiteelement space [13]. It is performed taking advantage of the orthog-onality conditionZ

XXP � Xð Þ : gh ¼ 0 8gh 2 Vh or Gh ð29Þ

where XP is the projected value of X on the mesh nodes. Substitut-ing in (21), the subscale variables ~u and ~e can be approximated as:

~e ¼ se C�1 : R1;h � Ph R1;h� �� �

~u ¼ su R2;h � Ph R2;h� �� � ð30Þ

with the residuals R1;h; R2;h defined in (19). First of all, asPh ehð Þ ¼ eh, the strain subscale is given by

~e ¼ se �eh þ $suhð Þ � Ph �eh þ $suhð Þ½ � ¼ se $suh � Ph $suhð Þ½ � ð31Þ

Now, comparing the Eqs. (13) and (29), the following substitution isdone:Z

Xch : C : Ph $suhð Þ ¼

ZXch : C : eh ð32Þ

For the displacement subscale, assuming that Ph fð Þ ¼ f , it can bewritten:

~u ¼ su $ � rh � Ph $ � rhð Þ½ � ð33Þ

Back-substituting in the set of equations of the problem, it reads:

� 1� seð ÞZ

Xch : C : eh � $suhð Þþ

þ su

ZXch : C : rs $ � rh � Ph $ � rhð Þ½ � ¼ 0 8ch 2 GhZ

X$svh : C : 1� seð Þeh þ se$

suh � ep� �

¼ FðvhÞ 8vh 2 Vh

ð34Þ

36 L. Benedetti et al. / Computers and Geotechnics 64 (2015) 32–47

Integrating by parts the second equation and rearranging, the finalset of equations is:

� 1� seð ÞZ

Xch : C : eh � $suhð Þþ

� su

ZX

$ � C : chð Þ½ � � $ � rh � Ph $ � rhð Þ½ � ¼ 0 8ch 2 GhZX

$svh : C : estab � ep� �

¼ FðvhÞ 8vh 2 Vh

ð35Þ

The set of equations with OSGS stabilization resembles the one forthe ASGS, except for the second term in the first equation. In orderto compute the projection of stresses at each time step, we canrecall expression (29), and writeZ

XPr � $ � rhð Þ : gh ¼ 0 8gh 2 Gh ð36Þ

and, with the additional projection equation, it reads:

� 1� seð ÞZ

Xch : C : eh � $suhð Þ

� su

ZX

$ � C : chð Þ½ � � $ � rh �Pr½ � ¼ 0 8ch 2 GhZX

$svh : C : estab � ep� �

¼ FðvhÞ 8vh 2 VhZX

Pr � $ � rhð Þ : gh ¼ 0 8gh 2 Gh

ð37Þ

The OSGS scheme is less diffusive than the ASGS scheme [24]. How-ever, this comes at the price of solving an additional equation: inthe implementation details it is shown how this problem can becircumvented.

3. Implementation details

In the presented formulation, the presence of the non-linearplastic strains ep ¼ ep rð Þ requires an iterative procedure to dealwith the nonlinearity of the system. Iterative solution schemes,such as Picard or Newton–Raphson methods, need to be intro-duced. Constitutive laws involving plasticity are usually writtenin terms of rate equations and, consequently, the matrices involvedin the resulting algebraic set of equations are tangent. Hence, theuse of the Newton–Raphson scheme will be considered in thefollowing.

Consider the nonlinear multidimensional-multivariableproblem

F Xð Þ ¼ 0 ð38Þ

where X ¼ e;u½ �T is the unknown vector. Such problem can besolved starting from a Taylor approximation around the solutionpoint at iteration iþ 1 in a particular time step nþ 1:

F iþ1nþ1 � F i

nþ1 þ J inþ1 dXiþ1 ð39Þ

where the Jacobian matrix J is defined as

J ¼ @F@X

ð40Þ

Assuming that F iþ1nþ1 ¼ 0, an iterative correction is computed as

dXiþ1 ¼ � J inþ1

h i�1F i

nþ1 ð41Þ

and the solution vector is updated as

Xiþ1 ¼ Xi þ dXiþ1 ð42Þ

The Jacobian matrix can be found by differentiating the set of equa-tion with respect to the unknowns variables X ¼ e;u½ �T at iteration i.

The advantage of such method is a quadratic convergence rate inthe iteration at each time step.

3.1. ASGS implementation

In the case of the ASGS scheme, differentiating the system ofequations at iteration i of time step nþ 1, the Jacobian matrix pre-sents the mathematical structure:

J inþ1 ¼

Ms Gs

Ds Ks

� �i

nþ1

ð43Þ

where M is a mass-like projection matrix, G is a gradient matrix, Dis a divergence matrix and K is the stiffness matrix. The subscript srefers to the fact that those matrices incorporate stabilizationterms. Differentiating (25), within the hypothesis introduced inEq. (15) that the plastic strain depends only on eh, the previousmatrices read:

Ms ¼ � 1� seð ÞZ

XNT

eCNe � su

ZX

CBBTCnþ1

ep ð44Þ

Gs ¼ 1� seð ÞZ

XNT

eCB ð45Þ

Ds ¼Z

XBT

Cnþ1ep � seC

h iNu ð46Þ

Ks ¼ se

ZX

BTCB ð47Þ

where Ne and Nu are the matrices of shape functions of the respec-tive strain and displacement fields and B is the matrix of the gradi-ent of those shape functions. The resulting algebraic system ofequations is, in general, not symmetric. Note that disregarding theterms due to plasticity, the system matrix is symmetric and it coin-cides with the one presented in Cervera et al. [17,18]. Details on thedifferentiation of the plastic strain tensor ep with respect to theproblem unknown eh are given in Appendix A.

3.2. OSS implementation

The OSS implementation is identical to the ASGS implementa-tion, except for the additional projection of the nodal stresses.The projection equation gives some additional terms in the Jaco-bian matrix when differentiating (37):

�Ms Gs DT

P

Ds Ks 0DP 0 MP

264375

i

nþ1

deh

duh

dPh

264375

iþ1

nþ1

¼R1;h

R2;h

R3;h

264375

i

nþ1

ð48Þ

where deh; duh; dPhð Þ are the iterative corrections for eh;uh;Phð Þ inthe Newton–Raphson scheme. The added projection matrices arecomputed as:

MP ¼ �Z

XNT

e Ne ð49Þ

DP ¼Z

XBT Ne ð50Þ

Alternatively to this procedure, a staggered scheme can be devised.First, the projection of the stresses Ph is computed at the beginningof the time step. Then, Ph is used for the solution of ðeh;uhÞ. Withthis substitution, the matrix depicted in (48) can be formally con-densed [13] and it becomes:

� Ms � DTPM�1

P DP Gs

Ds Ks

" #i

nþ1

deh

duh

� �iþ1

nþ1

¼R1;h

R2;h

� �i

nþ1

ð51Þ

L. Benedetti et al. / Computers and Geotechnics 64 (2015) 32–47 37

This scheme is preferred with respect to the monolithic one due tothe reduced computation time required.

4. Drucker–Prager plasticity model

The Drucker–Prager model is a pressure dependent plasticitymodel frequently used in geomechanics. It has a singular point incorrespondence of the maximum allowed mean stress. In the fol-lowing sections, this particular model is introduced and detailson the return mapping are given.

4.1. Definition of the space of admissible stresses

The Drucker–Prager plasticity model may be constructed as alinear combination of a J2 Von Mises plasticity model and a PurePressure plasticity model. The Von Mises yield criterion states thata material reaches the elastic limit when the equivalent octahedral

p

√3J2

pmin

φ

Fig. 2. Drucker–Prager elastic domain in the ðp; J2Þ plane.

Fig. 3. Drucker–Prager elastic domain in the p

stress is equal to a known uniaxial maximum admissiblethreshold:

f r; qð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi3J2ðrÞ

p� rd qð Þ ¼ 0 ð52Þ

where q is a stress-like hardening/softening variable. The value rd qð Þrepresents a limit in the admissible stress with respect to the sec-ond invariant of the deviatoric tensor J2ðrÞ. The Pure Pressure yieldcriterion relates the hydrostatic pressure with a maximum admissi-ble threshold:

f r; qð Þ ¼ 13

I1ðrÞ � rp qð Þ ¼ p� rp qð Þ ¼ 0 ð53Þ

where the value rp qð Þ represents a limit in the admissible pressure.In the Drucker–Prager model, the angle of friction / is introduced torelate the admissible deviatoric stresses to the pressure as:

f r; qð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi3J2ðrÞ

p� rd qð Þ

h iþ a

13

I1ðrÞ � rp qð Þ� �

tanð/Þ ¼ 0 ð54Þ

Plotting this yielding surface on the ðp; J2Þ plane, the result is a linewith a slope equal to tanð/Þ (Fig. 2). In the principal stress Haig–Westergaard space, the Drucker–Prager yield surface appears as asymmetric cone with the axis coinciding with the hydrostatic pres-sure and a circular trace on the octahedral plane (Fig. 3). The param-eter a ¼ 1 controls the sign of the pressure part and theorientation of the admissible plane of stresses. This means thatthe material may fail due to high tension states (a ¼ 1) or due tohigh compression states (a ¼ �1). The point ðpmin;0Þ in Fig. 2 repre-sents the vertex of the cone, the minimum allowed mean stressstate. In geotechnical engineering, the value of a ¼ 1 is usuallyassumed. Taking advantage of some trigonometric identities, it ispossible to rewrite the surface of failure explicitly as:

f r; qð Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi3J2ðrÞ

p� rd qð Þ

þ a 1� qð Þ 1

3I1ðrÞ � rp qð Þ

� �¼ 0

ð55Þ

rincipal stress Haig–Westergaard space.

38 L. Benedetti et al. / Computers and Geotechnics 64 (2015) 32–47

where q ¼ 1=ð1þ tanð/ÞÞ. In this work, the pressure threshold istaken as rp ¼ 0 to allow a direct comparison between J2 incompress-ible plasticity and Drucker–Prager plasticity. On the other side, thedeviatoric stress threshold reads:

rd qð Þ ¼ ry � qðnÞ ð56Þ

where qðnÞ is the hardening/softening function and n is an internalstrain-like parameter. The function qðnÞ controls the value of theintersection between the yielding surface and the deviatoric axisin Fig. 2. In the linear softening case, the function rd nð Þ is:

rd nð Þ ¼ry 1� HS

ryn

for 0 6 n 6 ry

HS

0 for ry

HS6 n 61

8<: ð57Þ

whereas, in the case of exponential softening, rd nð Þ assumes theform:

rd nð Þ ¼ ry exp�2HS

ryn

� �for 0 6 n 61 ð58Þ

Rewriting the invariants J2ðrÞ ¼ 12 devrk k and I1ðrÞ ¼ 1

3 trr, the fail-ure criteria takes the form:

f r; qð Þ ¼ qffiffiffi32

rdevrk k � ry � q nð Þ

� � !þ a 1� qð Þ1

3trr ¼ 0 ð59Þ

4.2. Return mapping algorithm

Assuming associative plasticity and the existence of a plasticpotential that coincides with the definition of the admissible stresssurface f r; qð Þ, the evolution equations for the plastic variablesread:

_ep ¼ _c@f r; qð Þ@r

_n ¼ _c@f r; qð Þ@q

ð60Þ

where _c is the plastic multiplier or plastic consistency parameter.Substituting the definition of the failure surface and differentiating:

_ep ¼ _c@rf r; qð Þ ¼ _c qffiffiffi32

rdevrdevrk k þ

a 1� qð Þ3

1

" #_n ¼ _c@qf r; qð Þ ¼ _cq

ð61Þ

Additionally, the Karush–Kuhn–Tucker and consistency conditionshold:

cP0; f r;qð Þ60; cf r;qð Þ¼0 ð62Þ

if f r; qð Þ ¼ 0 ) _c P 0; _f r; qð Þ 6 0 and _c _f r; qð Þ ¼ 0

ð63Þ

Given the last set of conditions, _c is computed as [19]:

_c ¼ @rf : C : _eh i@f@r : C : @f

@rþ@f@q

dqdn

@f@q

ð64Þ

The time derivative of the evolution equations of the plastic vari-ables can be approximated introducing a Backward–Euler schemewith time steps of length Dt, considering the ½tn; tnþ1� span. Then,the discrete-in-time version of (61) reads:

_ep �eðnþ1Þ

p � eðnÞp

Dt¼ cðnþ1Þ � cðnÞ

Dtq

ffiffiffi32

rdevrðnþ1Þ

devrðnþ1Þk k þa 1� qð Þ

31

" #

_n � nðnþ1Þ � nðnÞ

Dt¼ q

cðnþ1Þ � cðnÞ

Dtð65Þ

The trial state is defined at step nþ 1 with the plasticity variablesfrozen at step n:

eðnþ1Þp;trial ¼ eðnÞp

nðnþ1Þtrial ¼ nðnÞ

ð66Þ

Therefore, the trial stresses are:

rðnþ1Þtrial ¼ C : eðnþ1Þ � eðnÞp

qðnþ1Þ

trial ¼ qðnÞð67Þ

Plasticity occurs if f ðnþ1Þtrial P 0. The trial yielding function is:

f ðnþ1Þtrial ¼ q

ffiffiffi32

rdevrðnþ1Þ

trial

� rdy � qðnþ1Þ

trial

!þ a 1� qð Þ 1

3trrðnþ1Þ

trial

� �ð68Þ

The change of plastic multiplier Dcðnþ1Þ ¼ cðnþ1Þ � cðnÞ is computedwith the discrete counterpart of (64) as:

Dcðnþ1Þ ¼ f ðnþ1Þtrial

1� qð Þ2K þ 3Gq2 þ q2dqdn

ðnþ1Þ ð69Þ

where K is the bulk modulus and G is the shear modulus of thematerial. Notice that q nðnþ1Þ

implicitly depends on the value of

Dcðnþ1Þ as shown in (65).

4.3. Constitutive elasto-plastic tangent operator

The constitutive elasto-plastic tangent fourth order tensor canbe written as a function of _c. Defining:

D ¼ @f@r

: C :@f@rþ @f@q

dqdn

@f@q

ð70Þ

On one hand, the constitutive elastoplastic tensor in continuousform is [25]:

Cep ¼ C�C : @f

@r

C : @f

@r

D

ð71Þ

On the other hand, considering the discrete Backward Euler timeintegration, the algorithmic consistent constitutive elasto-plastictensor can be computed as:

Cðnþ1Þep ¼ drðnþ1Þ

deðnþ1Þ ð72Þ

Carrying out the differentiation, it yields:

Cðnþ1Þep ¼C�

q2Gffiffi32

qnðnþ1Þ

d;trialþa 1�qð ÞK1h i

q2Gffiffi32

qnðnþ1Þ

d;trialþa 1�qð ÞK1h i

Dðnþ1Þ

�Dcðnþ1Þð2GÞ2qffiffiffi32

rI� 1

311� �

�nðnþ1Þd;trialnðnþ1Þ

d;trial

h idevrðnþ1Þ

trial

ð73Þ

where Dðnþ1Þ is the discrete counterpart of (70):

Dðnþ1Þ ¼ 1� qð Þ2K þ q23Gh i

� q3dq nðnÞ þ qDcðnþ1Þ

dnð74Þ

and nðnþ1Þd;trial is the unit vector in the deviatoric stress direction:

nðnþ1Þd;trial ¼

devrðnþ1Þtrial

devrðnþ1Þtrial

ð75Þ

l

l/5

Fig. 5. Geometry for Prandtl’s punch problem.

-30

-25

]

Strain-Displacement formulationPressure-Displacement formulation

L. Benedetti et al. / Computers and Geotechnics 64 (2015) 32–47 39

4.4. Apex return mapping

The Drucker–Prager model presents a singular point in theyielding surface: the apex of the cone. For the return mapping inthose cases when this point is involved, an ‘‘ad hoc’’ procedure isnecessary. In the literature, deBorst [26] and Peric and de SouzaNeto [27] proposed some general methods to tackle this problem.In this case, a particular return mapping algorithm is devised inorder to have a scalar condition on the components of Cauchystress tensor.

Consider the yielding surface function in Eq. (59). The minimumvalue of the admissible pressure defines the apex of the cone inFig. 4 and its value is:

pmin ¼q ry � q� �

a 1� qð Þ ð76Þ

The part located outside the admissible stress space can be dividedin two zones by considering the orthogonal line to the yielding sur-face passing through the apex (Fig. 4). The standard return mapping,described in the previous section, is used in the cases where thetrial stress state falls in the ‘‘Zone 1’’ domain. When the trial stressis in the complementary ‘‘Zone 2’’ of the cone, the differentiation ofthe yielding surface cannot be performed since the normal vector tothe yielding surface does not have a unique definition. However, afamily of sub-differentials of the yielding surface exists and thereturn mapping can be performed, for example, by consideringthe principal components pressure p and deviatoric stress devr tosatisfy some particular conditions.

In order to find the condition to discriminate the two situations,consider the return mapping for the deviatoric components, i.e.along the vertical axis of Fig. 4. Once the variation of the plasticmultiplier is known, the deviatoric components of the stress tensorare updated with the new plastic strains as:

devrðnþ1Þ ¼ devrðnþ1Þtrial

� Dcðnþ1Þqffiffiffi32

r2G ð77Þ

As the norms are positive definite, it follows that:

Dcðnþ1Þ6

devrðnþ1Þtrial

q

ffiffi32

q2G

ð78Þ

If this condition is verified, then the return mapping is madethrough the standard procedure described in the previous section.Otherwise, the return mapping will be made to the apex of theDrucker–Prager cone.

The stress at the apex point is:

f ðr; qÞ ¼ 0 and p ¼ pmin ) devrk k ¼ 0 ð79Þ

Calling ntrial ¼ napex the unit vector that points from rtrial to thevertex of the cone ðpmin;0Þ, the plastic flow is:

Fig. 4. Drucker–Prager domain in the ðJ2; pÞ plane with return mapping zones.

_ep ¼ _cnapex ð80Þ

and in discrete form

eðnþ1Þp ¼ eðnÞp þ Dcðnþ1Þnðnþ1Þ

apex ð81Þ

The trial stress is

rðnþ1Þtrial ¼ devrðnþ1Þ

trial þ pðnþ1Þtrial 1 ð82Þ

and the stress after the return mapping reads:

rðnþ1Þ ¼ pmin1 ð83Þ

Therefore:

eðnþ1Þp ¼ eðnÞp þ a

pðnþ1Þtrial � pmin

3K1þ devrðnþ1Þ

trial

2G

" #¼ eðnÞp þ Deðnþ1Þ

p ð84Þ

Notice that the value of pmin depends on the value of the isotropichardening q ¼ qðnÞ. Consequently, an iterative procedure is neces-sary in order to evaluate correctly the plastic multiplier.

4.5. Apex consistent elasto-plastic tangent operator

In the case of return mapping to the apex, the consistent consti-tutive tensor is the null fourth tensor. This means that once thestress state arrives at the vertex of the cone, it will remain at theapex unless unloading or neutral loading occurs.

-20

-15

-10

-5

0-0.2-0.15-0.1-0.05 0

Ver

tical

Rea

ctio

n [k

N

Vertical Displacement [m]

Fig. 6. Prandtl’s punch problem with J2 plasticity: vertical reaction force vs.imposed vertical displacement of the foundation.

40 L. Benedetti et al. / Computers and Geotechnics 64 (2015) 32–47

4.6. Softening behavior

In a softening process, the energy dissipated by inelastic behav-ior is linked with the fracture energy Gf [28], defined by unit sur-face. When using a plastic model defined in terms of stress andstrain, the dissipated plastic energy Wp is defined by unit volume.

Fig. 7. Results for Prandtl’s punch problem: u=p (left column) and e=u (right column). Cvectors of principal strains (third row).

In the discrete FE setting, these two definitions are related througha characteristic length lch, connected to the mesh resolution:

Wp ¼Gf

lchð85Þ

In the plastic model, the rate of plastic work is computed as:

ontours of: total displacements (first row), equivalent plastic strain (second row),

Fig. 8. Results for Prandtl’s punch problem in the case of standard irreducible finite element formulation.

L. Benedetti et al. / Computers and Geotechnics 64 (2015) 32–47 41

_Wp ¼ r : _ep ¼ �r _�ep ¼ a r nð Þ _n ð86Þ

where �r is the equivalent Drucker–Prager stress:

�r¼qffiffiffi32

rdevrk kþ 1�qð Þa1

3trr tan /ð Þ¼q ry�q

� �¼ r nð Þ ð87Þ

and _�ep is the rate of equivalent plastic strain:

_�ep ¼ _ep

¼ _c qffiffiffi32

rþ a 1� qð Þ

" #ð88Þ

and, finally, a is a scaling factor depending on the friction angle,

a ¼ffiffiffi32

rþ 1� q

qð89Þ

In both the linear and exponential softening cases, where r nð Þ isdefined respectively by (57) and (58), the total plastic work is calcu-lated then as:

Wp ¼Z t¼1

t¼0

_Wpdt ¼Z n¼1

n¼0a r nð Þ _n ¼ a

ry2

2HSð90Þ

and this represent the area underlying the r–n curve. Now, compar-ing expressions (85) and (90), the parameter HS can be computedas:

HS ¼ ary

2

2Gflch ¼ HSlch ð91Þ

The parameter HS depends only on material properties, whereas lch

depends on the resolution of the discretization. As pointed out byCervera et al. [18], the size of the strain concentration band dependson the finite element technology. For instance, irreducible finite ele-ments provide a concentration band within a single element span,due to the discontinuous strain field. On the contrary, in the e� umixed FE formulation, with inter-elemental continuous strain, theslip line spans two elements. The characteristic length lch is takenaccordingly.

4.7. Plastic dissipation rate

The condition of positive rate of dissipation

_Wp ¼ r : _ep P 0 8 _e ð92Þ

has to hold in both classical and apex return mappings in order tohave a thermodynamically consistent model. In the first case, since0 6 q � 1 and the initial stress threshold ry > 0, it holds:

_Wp ¼ qry _c P 0 8 _e ð93Þ

In the return of the apex case, a continuous expression is not avail-able, but, using (83) and (84) the incremental dissipation takes theform:

D _Wðnþ1Þp ¼ 1

3Kpðnþ1Þ

min 1 : pðnþ1Þmin 1

h iP 0 ð94Þ

which is positive by construction.

5. Numerical examples

The objective of the following numerical examples is to high-light the benefits of a stress-accurate finite element method, suchas the proposed e=u mixed FEM, in order to capture softeningbehavior and failure due to the formation of strain localizationlines. In all the examples, the convergence tolerance used for theiterative Newton–Raphson procedure is 10�5. Computations havebeen realized using an enhanced version of COMET-Coupledmechanical and thermal analysis [29], developed by the authorsat the International Center of Numerical Methods in Engineering(CIMNE) in Barcelona, Spain. The geometrical models have beencreated using GiD, a pre and post-processing software, also devel-oped by CIMNE.

5.1. Prandtl’s punch problem with J2 plasticity

In this first example, the relative performance of the displace-ment–pressure formulation u=p and the strain–displacement for-mulation e=u in the 2D Prandtl’s punch test is assessed.Incompressible J2 plasticity (/ ¼ 0�) is assumed. The problem con-sists of a foundation loading a semi-infinite soil domain. A portionof 10 by 5 meters of soil is modeled, with a 2 meters wide loadingzone. Due to symmetry conditions, only one half of the domain hasbeen meshed. The geometry of the problem is shown in Fig. 5. Theload is given by an imposed vertical displacement of 0.2 m in thedownward direction. Young’s modulus is 10 MPa and Poisson’sratio is 0.4. The maximum tensile strength is 10 kPa, whereas afracture energy of 200 J/m2 is considered for the strain softeningcase. All cases are run with 400 time steps and an unstructuredmesh of 4340 triangular P1-P1 elements (typical size of h ¼ 0:25).

In Fig. 7(a) and (b), the norm of displacement field obtainedwith both formulations at the end of the loading is shown. Theresults computed with the two formulations are very similar. Thisis due to the fact that both formulations have the same order ofconvergence rate in the displacement field. The equivalent plasticstrain is presented in Fig. 7(d) and (e). It can be seen that, even if

42 L. Benedetti et al. / Computers and Geotechnics 64 (2015) 32–47

the displacements do not present substantial difference, the plasticstrain field differs for the two formulations, not in the path of theslip line but rather in the quality of the description of the shearband. In fact, the e=u solution presents a continuous distributionof strains whereas the u=p formulation yields a element-wise con-stant but inter-element discontinuous field. Principal strain vectorsare shown in Fig. 7(g) and (h). Here, the largest differencesbetween the solution of the two formulations can be observed. Inthe u=p formulation, the strain tensor is computed summing thevolumetric part of the deformation, computed starting from thepressure field, and the deviatoric one, given by differentiation ofthe displacement field. Clearly, the latter one is mesh dependent

Fig. 9. Prandtl’s punch problem with Drucker–Prager plasticity: displacem

across the slip line and this fact biases the orientation of the prin-cipal axes of strain. Although the overall behavior is correct and thesolution is the expected one, some sharp changes in the directionof vectors are observed locally in the u=p solution. Contrariwise,in the e=u solution strain is a continuous variable throughout thedomain. This was noted already for the irreducible formulationagainst the mixed one by Cervera et al. [18]. This discrepancyexplains the slightly difference in post peak behaviors of the u=pand e=u formulations presented in the reaction-displacement plotin Fig. 6.

For the sake of comparison, Fig. 8(a) and (c) shows displace-ment and plastic strain contours for the same problem, obtained

ent contour maps for 0, 15 and 30 degrees of internal friction angle.

(a) φ = 0◦

(c) φ = 15◦

(d) φ = 30◦

Fig. 10. Prandtl’s punch problem with Drucker–Prager plasticity: equivalent plastic strain for 0, 15 and 30 degrees of internal friction angle.

L. Benedetti et al. / Computers and Geotechnics 64 (2015) 32–47 43

using standard irreducible formulation. The solution is stronglymesh dependent. This is due to two factors. On one hand, the meanstress (the pressure) is completely locked because of the isochoricnature of the plastic flow. On the other hand, the deviatoric part ofthe strain field fails to converge to the correct, mesh independent,solution of the problem.

5.2. Prandtl’s punch problem with Drucker Prager plasticity

With the geometry and the mechanical properties of the pre-vious example, the Prandtl’s punch problem is now analyzedconsidering the Drucker–Prager plasticity model. The dimen-

sions of the meshed domain have been doubled in order toallow the complete formation of the slip lines and, therefore,to avoid interaction with the boundaries. Again, due to symme-try, the modeled domain is half of the total, 10 by 10 m with a1 m wide footing. The footing applies a downward imposed dis-placement of 0.2 m. The mesh is a structured quadrilateral gridwith one hundred Q1Q1 elements per side (typical size ofh ¼ 0:1). The solution is computed for the cases of / = 0�, 15�,30� of internal friction angle. The e=u formulation is used inall three cases.

In Fig. 9, the comparison in the displacement field shows con-siderable differences in the volume of domain subjected to the

Fig. 11. Prandtl’s punch problem with Drucker–Prager plasticity: J2 strain for 0, 15 and 30 degrees of internal friction angle.

44 L. Benedetti et al. / Computers and Geotechnics 64 (2015) 32–47

effect of the footing. As the angle of friction increases, the affectedzone becomes larger, both in the horizontal and vertical directions.In the J2 case, the failure mechanism follows Prandtl’s theoreticalresult for incompressible soils, in which a triangular elastic domainof material, with 45� slip lines, creates a single fan-shaped partsliding along a circular localization zone. Instead, when the angleof friction increases, the shape of the triangular elastic portionunder the footing changes and additional fans sliding along the slipline are created. In the equivalent plastic strain plots (Fig. 10), thechange in the failure mechanism can be clearly appreciated. Theratio of deviatoric and volumetric strain is bigger for higher anglesof friction (Figs. 11 and 12). Fig. 13 shows the resulting reaction

force under the footing plotted against the vertical displacementfor each one of the considered friction angle. The graph confirmsthat materials with larger internal friction angle have higher yield-ing stresses when pressure increases.

5.3. Rigid footing on a 3D cube

The final example is a 3D cube of soil with side l ¼ 2 mwith an imposed vertical displacement d ¼ l=10 ¼ 0:2 m in afootprint of l=2� l=2 ¼ 1� 1 m2 (Fig. 14(a)). The properties ofthe material are the same as in the previous examples: Young’smodulus of 10 MPa, Poisson ratio of 0.4, yielding stress 10 kPa

Fig. 12. Prandtl’s punch problem with Drucker–Prager plasticity: volumetric strain for 0, 15 and 30 degrees of internal friction angle.

L. Benedetti et al. / Computers and Geotechnics 64 (2015) 32–47 45

and a fracture energy of 200 J/m2. The plasticity model is VonMises (/ ¼ 0). The vertical displacement of 0.2 is applied in100 time steps. The cube is supported on the three faces thatare not adjacent to the footing. The mesh is a structured hexa-hedral grid with 20 elements per side (h ¼ 0:1). This examplewas solved with the u=p formulation and P1P1 tetrahedral ele-ments in [13].

The computed results show how the cube deforms under theimposed displacement of the rigid foundation (Fig. 14(b)): atetrahedral wedge detaches from the corner of the cube slidingin diagonal along the slip line. Failure is symmetric as expected

by the geometry of the problem. Fig. 14(c) depicts the computeddisplacement field, showing an almost rigid motion of the wedgeonce the failure mechanism is fully developed. Fig. 15 confirmsthat, at the end of the loading process, the reaction force is lessthan one tenth of the peak value. Fig. 14(d) shows the localizationof the plastic strains. From the latter two plots it is possible to seethat both displacement and plastic strain are continuous across theslip line. The resulting shear band width is, at most, two elementswide. Finally in Fig. 15, the vertical reaction force is plotted againstthe imposed vertical displacement, showing the full developmentof the resulting softening branch.

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0-0.2-0.15-0.1-0.05 0

Ver

tical

Rea

ctio

n [k

N]

Vertical Displacement [m]

Angle of friction = 0Angle of friction = 15Angle of friction = 30

Fig. 13. Vertical reaction force vs. vertical displacement for the Prandtl’s punchproblem with Drucker–Prager plasticity.

l/2

l

(a) (b)

(c) (d)

Fig. 14. Results for a rigid footing on a 3D cube: (a) geometry of the problem, (b) final deformed mesh, (c) contours of total displacements, (d) equivalent plastic strain.

-60

-50

-40

-30

-20

-10

0-0.2-0.15-0.1-0.05 0

Ver

tical

reac

tion

[kN

]

Vertical displacement [m]

Strain-Displacement formulation

Fig. 15. Rigid footing on a 3D cube: vertical reaction vs. imposed displacement.

46 L. Benedetti et al. / Computers and Geotechnics 64 (2015) 32–47

L. Benedetti et al. / Computers and Geotechnics 64 (2015) 32–47 47

6. Conclusion

A mixed strain–displacement finite element formulation hasbeen developed and applied to model the failure due to plasticstrain localization of different geotechnical examples. The problemof computation of slip lines has been addressed in pressure depen-dent as well as incompressible plasticity. The enhanced accuracy inthe strain field that this formulation provides, compared to thepreviously used displacement/pressure formulation, allows a moredetailed description of the failure mechanism, peak load and post-peak behavior. Various examples of 2D and 3D geotechnical prob-lems were solved using triangular, quadrilateral and hexahedralmeshes, showing no mesh dependency, theoretical consistencyand remarkable robustness.

Acknowledgments

Financial support from the EC 7th Framework Programmeunder the MuMoLaDe project – Multi-scale Modelling of Land-slides and Debris Flows – within the framework of Marie CurieITN (Initial Training Networks) and the Spanish Ministry of Econ-omy and Competitivity under the EACY project – Enhanced accu-racy computational and experimental framework for strainlocalization and failure mechanisms – within the ‘‘Excellency’’ Pro-gram for Knowledge Generation is gratefully acknowledged.

Appendix A. Differentiation of the plastic strain tensor p

In previous sections, when the linearization of the discreteweak problem was performed, the derivative of the plastic straintensor was introduced. The plastic strain eðnþ1Þ

p depends directlyon the trial stress via the return mapping.

Consider the derivative with respect to the nodal strains eh.Using the chain rule, it reads:

@eðnþ1Þp

@eh¼ @eðnþ1Þ

p

@rðnþ1Þtrial

@rðnþ1Þtrial

@ehð95Þ

The first term on the right hand side represents the variation of theplastic strain with respect to the trial stress at the actual time step.From the discrete evolution equations presented in (65), the plasticstrain tensor is updated as:

eðnþ1Þp ¼ eðnÞp þ Dcðnþ1Þ @rf rðnþ1Þ; qðnþ1Þ� �

ð96Þ

where @rf ¼ nðnþ1Þ is the normal to the yield surface. It was shownin Eq. (61) that, for Drucker–Prager plasticity model, the vectornðnþ1Þ is

nðnþ1Þ ¼ @rf r; qð Þ ¼ qffiffiffi32

rdevrdevrk k þ

a 1� qð Þ3

1 ð97Þ

Being eðnÞp a constant value, the derivative of eðnþ1Þp with respect to

the trial stresses is:

@eðnþ1Þp

@rðnþ1Þtrial

¼ @Dcðnþ1Þ

@rðnþ1Þtrial

nðnþ1Þ þ Dcðnþ1Þ @nðnþ1Þ

@rðnþ1Þtrial

ð98Þ

Recall that the tangent consistent elasto-plastic tensor is defined as:

Cðnþ1Þep ¼ @r

ðnþ1Þ

@eðnþ1Þ ¼ C� C@Dcðnþ1Þ

@eðnþ1Þ nðnþ1Þ � Dcðnþ1ÞC@nðnþ1Þ

@eðnþ1Þ ð99Þ

Comparing (98) and (99), it is possible to write:

@eðnþ1Þp

@rðnþ1Þtrial

¼ C�1 C� Cðnþ1Þep

h iC�1 ð100Þ

The second term on the right hand side of (95) is the derivative ofrðnþ1Þ

trial with respect to eðnþ1Þ:

rðnþ1Þtrial ¼ C : eðnþ1Þ � eðnÞp

) @rðnþ1Þ

trial

@eðnþ1Þ ¼ C ð101Þ

Finally, the derivative expressed in (95) can be computed as:

@eðnþ1Þp

@eh¼ @eðnþ1Þ

p

@rðnþ1Þtrial

@rðnþ1Þtrial

@eh¼ C�1 C� Cðnþ1Þ

ep

h ið102Þ

which provides the algebraic system matrices in (44)–(47).

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