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A smoothing technique based beta finiteelement method ( FEM) for crystal plasticity modeling

ARTICLE in COMPUTERS & STRUCTURES JANUARY 2016Impact Factor: 2.13 DOI: 10.1016/j.compstruc.2015.09.007

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A smoothing technique based beta nite element method ( bFEM)for crystal plasticity modeling

W. Zeng a ,b , , G.R. Liub , D. Lic, X.W. Dong b ,da CEAS-Biomedical Engineering (BME), University of Cincinnati, 2901 Woodside Dr., Cincinnati, OH 45221, USAb CEAS-School of Aerospace Systems, University of Cincinnati, 2851 Woodside Dr., Cincinnati, OH 45221, USAc Transportation and Vehicle Engineering School, Shandong University of Technology, Zibo 255049, Chinad College of Mechanical and Electronic Engineering, China University of Petroleum, 66 Changjiang Rd, Huangdao District, Qingdao 257061, China

a r t i c l e i n f o

Article history:Received 6 July 2015Accepted 22 September 2015

Keywords:Strain smoothing techniquesBeta nite element method ( bFEM)Large strain plasticitySingle crystalStrain localizationPolycrystalline deformation

a b s t r a c t

This paper presents a novel class of smoothing techniques based beta nite element method ( bFEM) formodeling of crystalline materials. The method is rst examined by a simple standard patch test andapplied in elastic problems. It is then implemented to model the anisotropic plastic deformation of rate-independent single crystals and bi-crystal. Several representative examples are studied to demon-strate the capability of proposed method with the integration algorithm for capturing the strain localiza-tion and dealing with plastic incompressibility. It is also performed to simulate the mechanical behaviorof polycrystalline aggregates through modeling the synthetic microstructure constructed by Voronoitessellation technique.

2015 Elsevier Ltd. All rights reserved.

1. Introduction

The response of crystalline materials is essentially anisotropicat mesoscale. This means the instantaneous elasticplastic defor-mations in crystals depends on the directions of external loadingand/or constraints. This crystalline anisotropy is physically due tothe direction-dependent elastic properties and the orientationdependentmovement of dislocations in micromechanical for a par-ticular crystalline structure, even if the internal defects are notconsidered. The continuum slip theory of crystals can provide aframework of nite plasticity with physically well-dened rootsin the dislocation mechanics of crystalline materials [1] . Earlyapproaches to describe the anisotropic crystal plasticity at macro-scopic level have been explored by the pioneering researchers suchas Sachs [2] , Taylor [3] , Schmid and Boas [4] , Taylor [5] , Bishop andHill [6] , and Kster [7] .

The plastic sliding on certain set(s) of crystallographic systemswas revealed as the essential reason causing inelastic behavior of ductile monocrystals (single crystals). The quantitative under-standing of elasticplastic deformation of metal crystals was

proposed in description of small strain [8] and in forms of largestrains [911] . Geometrically, the continuum slip framework of nite crystal elastoplasticity is the postulating of the multiplicativeslip of local deformation gradient into an elastic component due tothe stretching and rotation of the crystal lattice, and a plasticcomponent which arises from the microscopic sliding alongcrystallographic planes. Intensive research has been conductedon theconstitutive and numerical aspects of monocrystal modelingduring the past three decades. The crystal plasticity nite elementmethod (CPFEM) is regarded as one of the most powerful numeri-cal tool to simulate the crystalline behaviors. We refer to the workcompleted by Mandel [12] , Peirce et al. [13] , Asaro [14] , Asaro andNeedleman [15] , Cuitio and Ortiz [16] , Havner [17] , Rashid andNemat-Nasser [18] , Borja and Wren [19] , Miehe [20] , Steinmannand Stein [21] , etc. Some critical analysis of the predictions fromthese models and new explorations can be found in literatures(e.g., [2228] . Generally, CPFEM is a variational approach in theformof nite element approximations for crystal mechanical mod-eling, which is based on certain crystal plasticity constitutive laws.CPFEM models are used in many applications because of their abil-ity to predict the anisotropic inelastic behavior and localizationprocesses with various advanced and physically based designmethod at microscopic crystallographic sliding level.

Another advantage of CPFEM models is their capability tosimulate crystal mechanical responses for polycrystals with agreat quantity of individual grains under complex geometric

http://dx.doi.org/10.1016/j.compstruc.2015.09.007

0045-7949/ 2015 Elsevier Ltd. All rights reserved.

Corresponding author at: CEAS-Biomedical Engineering (BME), University of Cincinnati, 2901 Woodside Dr., Cincinnati, OH 45221, USA and CEAS-School of Aerospace Systems, University of Cincinnati, 2851 Woodside Dr., Cincinnati, OH45221, USA. Tel.: +1 513 556 2661.

E-mail address: [email protected] (W. Zeng).

Computers and Structures 162 (2016) 4867

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conguration and complicated internal and/or external boundaryconditions. Polycrystals can be regarded as assemblies of largenumbers of monocrystals (grains), each of which can deform bycrystallographic slip with varying orientations. As such, the actualsolution of the equilibrium of a deforming polycrystal is that of ahighly complex elasticplastic boundary value problem for acrowd of anisotropic, continuous and fully contiguous crystallites[29] . Earlier approximate polycrystal models adopted assumptionsof averaging the crystal interactions to describe macroscopicbehavior. Forexample, the Taylor model [5] assumed all monocrys-tal grains within the aggregate experience the same state of defor-mation (strain), which assures compatibility conditions betweenthe grains, but violates equilibrium conditions across the grainboundaries. The response represents an upper bound on the aver-aged stress in the polycrystalline aggregate. While the Sachs model[2] was based on the presumption that each grain is subject to thesame stress state, which satises the equilibrium condition acrossthe grain boundaries but violates the compatibility conditions. Tosatisfy both compatibility and equilibrium conditions across thegrain boundaries, some hybrid models were proposed, for example,constrained hybrid model [30] , relaxed constraints model [31] , andself-consistent methods [3237] . To link the micromechanicalbehavior of grains to the response of a polycrystalline aggregate,we can nd works provided by Refs. [11,14,38,39] , and so on. Ontopics related to kinematics, homogenization and multiscalemethods of crystal plasticity modeling, one can refer to reviewsin Ref. [40] . So far, many important aspects of mutiscale analysesfor polycrystals can be revealed by numerous crystal plasticitymodels under the general framework of nite element modeling.

In view of the plastic incompressibility of monocrystals, anappropriate FE technique which can deal with unstable/volumetriclocking phenomena is very necessary. It is a well-known fact thatthe performance of virtual work-based standard rst-order niteelements becomes extremely poor to tackle large strain problemsof nearly incompressible solid as unstable/spurious volumetriclocking phenomena [41] . (Bi) linear plane elements and trilinearsolid elements in this case are ineffective to grant the completenullity of the volumetric strain. Then the near zero or verysmall values of displacements will be obtained because of theoverestimation of stiffness values involved with the volumetricdeformation [42] . Besides, for modeling polycrystal, the mesh dis-cretization for a polycrystal domain with a large collection of com-pleted internal grain boundaries needs to consider the facility orfeasibility. The T-mesh (using triangular for 2D and tetrahedralfor 3D, or T-element) are the best option (sometimes the onlyone) to discretize the complex domain with high quality. However,the standard FEM models with T-elements are also suffered withshear and volumetric locking issues, and they often exhibit overlystiff behavior leading to poor results, compared to the other meshtypes (e.g., quadrilateral mesh for 2D or hexahedral mesh for 3D).

So it is usually not recommended using T-mesh in most commer-cial FEM software packages. The second-order T-elements can beadopted to remedy the locking phenomenon but will be ineffectivefor extremely large strain problems due to intermediate nodes.Reduced, selective reduced integration (SRI) techniques [43,44]and B-bar [45] method might be effective approaches in alleviatingspecically the volumetric locking for quadrilateral or hexahedralelements, but they are not compatible with T-elements [46] . Inaddition, the application of SRI and B-bar cases is in some extentlimited to materials whose property matrix is able to be decom-posed into volumetric and deviatoric parts.

In the past fewyears, Liuand his group proposed applications of strain smoothing technique of meshfree methods to the nite ele-ment method, and named the resulting technique smoothed nite

element methods (S-FEM) [4753] . The essential idea in the S-FEMis to utilize a standardrst-order niteelement mesh (inparticular

T-mesh) to build numerical models with good performance [54] .This is implemented by modifying/constructing the compatiblestrain eld in a Galerkin weak form model to deliver some goodproperties. Other than element based implementation in thestandard FEM, the S-FEM techniques implement and evaluate theweak form based on smoothing domains. Such an implementationin S-FEM can be located within the elements but more oftenbeyond the elements which bring in the information from theneighboring elements. Based on this, a variety of S-FEM modelswere proposed: the cell-based smoothed FEM (CS-FEM) [50,53] ,node-based smoothed FEM (NS-FEM) [52] , edge based smoothedFEM (ES-FEM) [49] and face-based smoothed FEM (FS-FEM) [55] ,etc. Owing to the strain smoothing technique on smoothingdomains, the overestimation of stiffness values of the standardFEM can be reduced or alleviated and hence the accuracy of bothprimal and dual quantities can be improved signicantly [56] .Moreover, S-FEM does not require the shape function derivativesand S-FEM models developed in elasticity are insensitive to meshdistortion because of absence of isoparametric mapping [57,58] .Due to its versatility, the class of S-FEMs was successfully appliedin various types of solid mechanics area and has been becoming asimple and effective numerical tool for practical problems.

Among these S-FEMs, the NS-FEM possesses some properties insolid mechanics such as: (1) it can extremely soften the over-stiffness of the standard FEM model and can deliver upper boundsolution with respect to the exact solution [59] ; (2) it is effectivein overcoming volumetric locking for nearly incompressible hyper-elastic materials [60] ; (3) it provides very accurate and oftensuper-convergent properties of stress solutions; (4) it can adoptT-elements for discretization of background grid (NS-FEM-T3 orNS-FEM-T4); (5) it is spatially stable but may behave temporallyinstable with non-zero-energy spurious modes [61] and (6) theproperties of NS-FEM for plastic materials have not been analyzedyet. Meanwhile, the ES-FEMshows some valuable properties for 2Dsolid mechanics analyses as following: (1) the ES-FEM possesses aclose-to-exact stiffness, which offers solution with properties of super-convergence and accuracy; (2) ES-FEM usually produces alower bound to the exact solution in strain energy, still with thefeature of overestimation of stiffness; (3) the ES-FEM models areoften temporally stable and always stiffer than NS-FEM, partiallydue to the number of edges being always larger than the numberof nodes, for a problem domain with background T-mesh; (4) itsperformance revealed some deciencies in alleviating the volumet-ric locking for extreme large strain occurrence [54] ; and (5) itworks effectively with T-mesh that can be generated automaticallyfor complicated geometries.

In this work, a novel ultra-accurate beta nite element method(bFEM) using triangular base mesh is rst proposed and thenapplied in crystal plasticity modeling. The essential idea of bFEMis to construct the smoothing domains via mixed smoothing

techniques of edge-based and node-based, in which the parameterb 2 0; 1 tunes the portion of area of the edge-based and node-based smoothing domains. Since the contributions from both theNS-FEM and ES-FEM are included though the introduced parame-ter b, a continuous function of strain energy is then able to beestablished. The bFEM can be regarded as a utilization of the over-estimation property of ES-FEM using T-elements and the uniqueunder-estimation property of NS-FEM method, and hence can betuned to have good features of both. Based on the fact that boththe NS-FEM and ES-FEM with T-elements are spatially stable [54] ,our bFEM will be stable and ensures the convergence. Besides, thescheme ensures the variational consistence and the compatibilityof the displacement eld, by which guarantees reproducing lineareld exactly [62,63] . In this paper, the proposed method is rst

used to nd the exact solution (at least nearly exact or calledclose-to-exact) in strain energy for two given plane elastic

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problems. After veried by the linear problems, the method is alsoapplied to nonlinear crystal plasticity problems with proper consti-tutive integration scheme at large strain. In such cases, the exactsolutions are usually difcult to obtain, but the capability of thismethod will be embodied in exploring to handle plastic incom-pressibility of single crystals as well as to capture strain localiza-tion phenomena. The work comprises the formulation andnumerical implementation of a hyperelastic-based multiplicativeplasticity constitutive model based on b FEM scheme to treat therate independent anisotropic crystal plasticity with large strain.The mathematical framework for kinematics starts fromthe multi-plicative decomposition of deformation gradient by introducingthe isoclinic intermediate conguration. A multisurface plasticow rule and the classic isotropic Taylor hardening law areadopted for the plane double slip model. For the stress update pro-cedure, the multivector return-mapping algorithm with exponen-tial map-based integration [20,64] is incorporated in hyperelasticlaw.

2. Incremental equations for nite element approximation

Let us consider a solid initially occupying an initial congura-tion B 0 X , which is bounded by smooth boundary @ B 0 and closureB 0 : B [ @ B 0 . Assume t n ; t n 1 be the standard time interval forincremental equations, then we can consider a process of incremental loading whereby the displacement eld of particlesmapping over B 0 changes from u n at time t n to u n 1 u n d attime t n 1 t n Dt . With the eld a n at time t n and the body forcesand the prescribed surface tractions at t n 1 , the weak form of equilibrium for standard FEM at time t n 1 according to the virtualprinciple can be written as the following incremental form

Z B 0 Pn 1 : r 0 g dX Z B 0 f bn 1 g dX Z @ B 0 f t n 1 g dC 0 1where f bn 1 and f t n 1 are the body forces and the prescribed tractionvector, respectively, g denes an admissible virtual displacementeld satisfying the homogeneous form of essential boundary condi-tions, r 0 denotes the material gradient, and Pn 1 represents the rstPiolaKirchhoff stress eld at time t n 1 , which can be determined bya general form using certain algorithm of stress updating method[65] , that is

Pn 1 P F n 1 ; a n 2

where a n stands for the set of internal variables and the deforma-tion gradient at t n 1 is given as

F n 1 r 0 u n 1 3

Substitute Eqs. (2) and (3) into (1) , we rewrite the equation into

Z B 0 P F n 1 ; an : r 0 g f bn 1 gh idX Z @ B 0 f t n 1 g dC 0 4which denes a set of non-linear equations that can be solved toobtain the updated deformation mapping u n 1 . If the NewtonRaphson iteration method is employed, the linearized problem forthe incremental displacements follows the form [16] :

Z B 0 r 0 g : Kn 1 : r 0d dX r 0 5where r is the residual force term, and

^Kn 1 denotes the consistenttangents.

3. The idea of present b FEM

3.1. Local gradient smoothing operation

The strain smoothing technique was applied in the Galerkinmesh-free methods, which uses the moving least-squares (MLS)and reproducing kernel approximations [66] . The so-called weak-

ened weak (W2) formulation based on the G space theory [63]was subsequently developed by extending the gradient smoothingtechnique to a class of discontinuous shape functions. The strain isexpressed as the divergence of a spatial average of the standard(compatible) strain eld, i.e. symmetric gradient of the displace-ment eld [67] . The strain smoothing operation is carried out overthe so-called local smoothing domain which can be constructedwithin elements (e.g., CS-FEM) but more often beyond theelements (e.g., ES-FEM, NS-FEM and FS-FEM). The smoothed straineld ~e k , for computation of stiffness matrix, will be in generallycomputed by a weighted average of the standard strain elde h x . For example, if we divide the problem domain into a fewnon-overlapping and non-gap representative smoothing domainsXskk 1 ; 2 ; . . . with boundary C

skk 1 ; 2 ; . . . , the smoothing of

the strain eld (smoothing of the gradient of the displacementeld) at a point in a given smoothed domain Xsk can be carriedout using

~e k x C Z Xh e h x Uk x x C dX 6where Uk x x C is a distribution function or a smoothing functionthat satises at least unity property such as

Uk x x C P 0 and Z Xsk Uk x x C dX 1 7The most frequently adopted smoothing function is the

Heaviside-type piecewise constant function dened in the follow-ing form

Uk x x C 1= Ask; x 2 Xsk

0; x R Xsk( 8where Ask R Xsk d X is the area of the smoothing domain X

sk . Substi-

tuting Eq. (8) into Eq. (6) and introducing the divergence theorem,the smoothed strains have the form

~e k 1 Ask Z Xsk r S d hdX 1 Ask Z Csk n sk x d h x dC 9

where C sk is the boundary of the smoothing domain Xsk , and n sk x

denotes the outward normal matrix on the boundary Csk given as

n sk x nskx 0 n

sky

0 nsky n

skx

" #T

10

where n skx and nsky are theunit outward normal components in x-axis

and y-axis, respectively.

3.2. Brieng of edge-based strain smoothing

The various S-FEM models evaluate the weak form based on dif-ferent smoothing domains created from the entities of the elementmesh such as cells/elements, or nodes, or edges, or faces [68] . Inthe ES-FEM the smoothing domains are constructed in associationwith edges of the three-node triangle elements as illustrated inFig. 1 . These smoothing domains are generated based on the edgesof the elements such that X S N edk1 X

sk and X

si \ X

s j for i j, in

which N ed is the number of edges of all elements in the entire prob-lem domain. For triangular elements, the smoothing domain Xsk

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associated with the edge kk l for interior edges and k m forboundary edges) is created by connecting two endpoints of theedge to centroids of corresponding adjacent element(s) as sketchedin Fig. 1 .

If a mesh of linear triangular elements (ES-FEM-T3) is used, thesmoothed strain in Eq. (9) for smoothing domain Xsk can beobtained by following matrix form

~e k XI 2S sk ~BI d I 11 where d I is the vector of the associated nodal displacements, S sk isthe set of supporting nodes for the smoothing domain Xsk , i.e., theset of all nodes of the elements which share the common edge k.For ES-FEM using triangle elements with sample smoothingdomains as shown in Fig. 1 , S sk is the set of nodes f A; B; C g for bound-ary edge AC , and f D; E ; F ; Gg for the interior edge DF . ~BI is the

smoothed straindisplacement matrix evaluated by

~BI 1 Ask Z Csk n sk x NI x dC

~bIx 0 ~bIy0 ~bIy ~bIx" #

T

12

with

~bIh 1 Ask Z Csk N I x nskh x dC; h x; y 13

From the above equation, we know that only the values of shape functions N I (not the derivatives) are involved on the bound-ary of the smoothing domain Csk . If a linearly compatible displace-ment eld is utilized along Csk, then a single Gaussian point issufcient for numerical integration along each segment Csk;t of the boundary Csk . It is now possible to obtain the form by Gaussquadrature

~bIh 1

Ask X

nsC

t 1

N I x GP t nskh

;t l

sk;t ; h x; y 14

G

A

B

C

O

D E

F

P

Q

smoothing domainfor boundary edges

smoothing domainfor interior edges

interior edge ( )l DF

(segments: , , , ) sl DP PF FQ QD

(4-node domain ) sl DPFQ

boundary edge (CA)m

(segments: , , ) sm AO OC CA

(triangle domain ) sm AOC

: centroids of triangles: field nodes

Fig. 1. Division of a problem domain into triangular elements and edge-based smoothing domains. For example, the smoothing domain Xsm for boundary edge m is a triangle

AOC , and the smoothing domain Xsl for interior edge l is four-sided convex polygon DPFQ .

: centroids of triangles

: field nodes

node q

(boundary of smoothing

domain)

sq

(smoothing domain) sq

: mid-edge-points

D

C

A

B

E

Fig. 2. Division of a problem domain into triangular elements and node-based smoothing domains. For example, the smoothing domain Xsq for node q is a polygon with 2 n eqsides (where n eq is the number of elements surrounding node q).

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~ As b2 As_

and As 1 b2 As_

; b 2 0; 1 24

For 1-Dand 3-Dproblems, Eq. (24) will be expressed in forms as

~ As b As_

and As 1 b As_

; b 2 0; 1 25

~ As b3 As_

and As 1 b3 As_

; b 2 0; 1 26

where As_

denotes the length of the smoothing domain for 1-D prob-lems, and the volume of the smoothing domain for 3-D problems.

In a bFEM scheme, the smoothed straindisplacement matrix ~BI for Xsk can be expressed as

~BI 1 AskX

nek

j1

13

b2 Ae j Be j 27

The smoothed straindisplacement matrix BI for Xsq can berewritten as

BI 1 Asq X

neq

l1

13

1 b2 Ael Bel 28

Then the smoothed stiffness matrix ~KkIJ or smoothed tangent

stiffness matrix ~KkTIJ for smoothing domain Xsk will be dened as

~KkIJ

Z Xsk

~BT I D~B J dX b2 ~BT

I D~B J As

k 29

or

~KkTIJ Z Xsk ~G T I ~c ~G J dX b2 ~G T I ~c ~G J Ask 30 where D stands for the elasticity tensor for linear elastic material, ~cis the smoothed elastoplastic consistent tangent.

For node-based smoothing domain X sq , the smoothed stiffness

matrix KqIJ or smoothed tangent stiffness matrix KqTIJ would be

obtained by a similar fashion as follows

KqIJ Z Xsq BT I DB J dX 1 b2 BT I DB J Asq 31 or

KqTIJ Z Xsq G T I c G J dX 1 b2 G T I c G J Asq 32 The global stiffness matrix K

_or global tangent stiffness matrix

KT _

for bFEM can be assembled from the ~KkIJ and KqIJ as follows

K_

XN e

k1

~KkIJ XN n

q1

KqIJ 33

or

KT _

XN e

k1

~KkTIJ

XN n

q1

KqTIJ 34

where N e and N n denote the number of total edges and total nodes.The properties of NS-FEM and ES-FEM (such as displacement

compatibility, variational consistence, solution continuity, and soon) have been analyzed or discussed [54,62,63] . Now the bFEM isequipped with a continuous scalar factor b which is regarded asa knob controlling the contributions from the NS-FEM and ES-FEM. If the factor b varies from 0 to 1, one may get a continuoussolution function from the solution of the NS-FEM to that of ES-FEM. Some basic properties can be studied by analysis of energy.Here we can take elastic problems as an example. If we denotethe strain eld computed by ES-FEM and NS-FEM as ~e and erespectively, the potential energy functional of bFEM for elasticproblems can be determined by the virtual work principle usedalso for standard FEM, which reads as

P_

g ; b P_

int g P_

ext g

b2 ~P int g 1 b2 P int g P_

ext g 35

For isotropic linear elastic material, Eq. (35) can be written as

P_

g ; b b2 Z B 0 12 ~e Tg D~eg dX 1 b2 Z B 0 12 e Tg De g dXZ B 0 f bn 1 g dX Z @ B 0 f t n 1 g dC 36

where g denes an admissible virtual displacement eld satisfy-ing the given essential boundary conditions, the given continuous

scalar factor b 2 0 ; 1 ; D stands for the elasticity tensor of isotropiclinear elastic material.

D C

A

B E

F H

G

node-basedsmoothing domain

: centroids of triangles: field nodes

: domain-dividing-points on edges

edge-basedsmoothing domain

F AG H

1l 2l 3l

sq

sq

sk

sk

Fig. 3. Division of representative elements into smoothing domains using bFEM-T3: the node-base smoothing domains are shown by red dotted lines and the edge-basedsmoothing domains indicated with green dashed lines. (For interpretation of the references to colour in this gure legend, the reader is referred to the web version of thisarticle.)

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By Eqs. (35) and (36) , we can nd some important properties asfollows:

(1) P_

int g P int g , if the factor is set as b 0. The bFEMscheme is essentially the same as the NS-FEM, which leadsto an underestimation of stiffness values;

(2) P_

int g ~P int g , if the factor is set as b 1. The bFEMscheme becomes the scheme of ES-FEM and the stiffness willbe overestimated, though its accuracy will be better thanstandard FEM;

(3) If the scaling factor b changes from 0.0 to 1.0, the property of underestimation of stiffness will become overestimation,continuously;

(4) The solution of bFEM shall be within the narrow intervalwhich bounds the exact solution, owing to the fact that ES-FEM generates the nearly exact solution from the lowerbound and NS-FEM produces the unique upper bound solu-tion (noted that this interval will be narrower than aFEM, asthe solution of ES-FEM is closer to exact solution from thelower bound than standard FEM);

(5) It is possible to nd the exact (or close-to-exact) solution(s)of strain energy during the procedure of tuning of factorbecause of the solution continuity property [69] ;

(6) Even with a small portion of ES-FEM (choosing a small valueof b, the constructed stiffness matrix exhibits the propertiesof overestimation, which can alleviate the temporal instabil-ity brought by the pure NS-FEM.

4. Material model for large strain single crystal plasticity

The kinematics and mathematical continuum descriptions of elasticplastic deformations in crystals have been developedby Hill [8] in the small-strain format and by Lee [70] , Rice[9] , Hill and Rice [71] , Hill and Havner [72] , and Asaro [11] inthe context of nite strains. The kinematical basis for thefollowing is taken from [11,73,74] . The precise mathematicaltheory and numerical implementation in the next sectionfollows the framework in [16,41,75] . The mechanical responseof inelastic deformation of crystalline is dominated by crystallo-graphic slip, in which the other mechanisms such as the slidingeffect of grain boundaries, twinning or diffusion will not bediscussed here.

4.1. Basic kinematics

The fundamental kinematics basis of the continuum slip theoryis to geometrically describe the plastic deformation as shearing of given crystallographic slip systems. This macroscopically descrip-tion is micromechanically owing to the movement of dislocationsof the idealized crystal lattice as graphically depicted in Fig. 4 . Dur-ing the process of ow, the crystal lattice undergoes rigid rotationand stretching, which is able be recovered by complete unloadingof the material. Although these two deformation modes arisesimultaneously, they can be multiplicatively decomposed [70]locally in mathematical models by introducing the intermediateconguration as indicated in Fig. 4 , i.e., considering the multiplica-tive elastoplasticity decomposition [7577] , the deformation gra-dient is decomposed into elastic and plastic parts

F @ x@ X

F e F p 37

where the elastic part F e describes the distortion and rigid bodymotions of the lattice, and F p denes as the cumulative effect of dis-location motion, i.e., a local plastic intermediate (or unrotated) con-guration which is supposed to be obtained by the evolutionconstitutive equation _ F p @ F p=@ t with the initial condition F p

t t t 0 1 at the reference conguration [12] .In Fig. 5 , a pair of orthonormal slip system vectors (initial slip

direction vector s a0 and initial slip plane normal vector na0 dened

the ath slip system in initial (or undeformed) conguration; theunit slip system vectors s a and n a remain orthonormal sinces a

0 n a0 s a n a 0.The associated plastic velocity gradients in the intermediate

conguration can be represented as the sum of plastic slip rates_ca for all active crystallographic slip systems by [9,11]

L p _ F p F p 1 XN as

a1

_ca s a n a 38

where N as is the number of active slip systems, ca is the plasticincrement within the slip system.

We now can dene the rotation tensor R e and the right stretchtensor U e through the polar decomposition of F e as

F e R eU e 39

In addition, the unit vectors s a and n a will be dened by

p F

0 n

= e F F F

0 s

n

p F

e F

s

n

s

0 ( ) X

( ) X

( ) X

Fig. 4. Basic mechanism of elastoplastic deformation of crystalline solid by crystallographic slip: multiplicative elastoplasticity decomposition of deformation gradient, F F e F p (involving the initial conguration B 0 X , intermediate conguration BX, and deformed conguration B X .

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s a R es a0 and na R en a0 40

4.2. Constitutive relations

Here we present the constitutive relations and ow rule of ageneral large strain, rate-independent, multisurface elastoplasticmodel of crystals. Themechanisms of inelastic deformationof crys-

tals resulting from shear deformations occurring on one or moreslip systems have been mentioned previously, but no descriptionwas given yet to reckon the stresses required to initiate and sustainthe deformation. A crystal deforms plastically only when the stresscomponent on a slip plane and in the slip directionreaches the crit-ical resolved shearstress. The resolved shear stress on a system a isgiven as

sa R eT s R e : s a n a 41

where s denotes the Kirchhoff stress, which is symmetric and has arelation with the Cauchy stress tensor via Jacobi

s J r 42

Compared to plastic behavior, the elastic distortions of crystal

lattices are generally quite small. For numerical convenience, ahyperelastic law is adopted here in constitutive relation to describethe reversibility behavior of crystalline. Assume the strain energydensity function arisen by elastic contribution is denoted byW eF e, the Kirchhoff stress s in Eq. (41) can be evaluated as

s @ W e

@ F e F eT 43

The evaluation of yield functions for rate-independent elasto-plastic model of f.c.c crystals here is determined by the relation-ship in terms of the resolved shear stress and the criticalresolved shear stress. For the a th slip system, it will be

f a sa s tr ; sa y sa s tr sacr ; a 1; 2; . . . ; N as 44

where s acr is the critical resolved shear stress for the a th slip system,s a s tr is the resolved shear stress related to the trial state of stresss tr in the general return-mapping procedure. Note that f a will beanisotropic functions of s a . The plastic slip for an associated slipsystem may commence when f a > 0, that is, the state of resolvedshear stress exceeds the corresponding anisotropic yield surface[31] . The set of systems for which s a s acr is called as the set of potentially active or critical systems [13] .

It is worthwhile to note that the critical resolved shear stresssa s tr depends on both the strain and the history of strain if hard-ening behavior is involved. After rst introduced in Taylors dislo-cation model [3] , a number of ow rules have been developed todescribe the hardening behavior of crystals for both rate-independent [35,71,7884] . The general form of the constitutivelaws for slip (shear) on a th slip system can be expressed as the fol-lowing evolution equation [13]

_s acr XN as

b1

hab _cb ; a 1; 2; . . . ; N as 45

where hab are slip-plane plastic hardening moduli that characterizethe work hardening rate of the crystal, the sum ranges over all acti-vated slip systems. The diagonal components haa b a representself-hardening on a slip system and off diagonal terms h ab b adenote latent hardening, viz., hardening of one slip system due toslip on another [91] . In this study, the classical commonly used Tay-lor isotropic hardening rule is employed where the self and latenthardening are considered equal. Besides, the resolved critical stress

is assumed to be a function of a single internal variable c, or so-called the Taylor cumulative shear strain on all slip system [92] , i.e.

c XN as

a1 Z t

0_caj jdt 46

Then the evolution equation for the plastic deformation gradi-ent (Eq. (38) ) follows the multisurface plastic ow rule as

_ F p

XN as

a1

_ca s a n a

" # F p 47

where _ca should ensure the following KuhnTucker conditions[93]

f a 6 0; _ca P 0 and f a _ca 0; a 1; 2; . . . ; N as 48

5. The integration algorithm and implementation aspects

In order to illustrate the materialmodel outlined above, we nowconsider the algorithmic counterpart of the single crystal plasticitymodels by constructing multisurface-type stress update algo-rithms. For real crystal plastic deformation, it may comprise anumber of slip systems, e.g., face centered cubic (f.c.c.) crystalshave 12 slip systems. In most situations, the slip initiation andshear band formation is observed in crystals undergoing multi-slip, often with a double mode of primary-conjugate slip [13] . Forthe simplicity of implementation, a general integration algorithmfor planar primary-conjugate double slip crystal model will be dis-cussed in this section.

To solve the elastoplastic form of crystal plasticity theory in thepreceding section, the numerical integration is required for thenonlinear algebraic system of constitutive equations. A few differ-ent implicit and explicit integration schemes have been developedfor both rate-independent [8,71,7884] and rate-dependent for-mulations [15,16,18,8590] . Here we adopt the standard New-tonRaphson scheme to solve the associated system of equations.

5.1. A planar double slip model

The plane deformation of f.c.c. crystals under certain crystallo-graphic orientations and boundary conditions can be characterizedby a planar double slip model with two effective slip systems. Eachslip systemcan be viewed as a pair of two mirroredslip systems forconvenience. In Fig. 5 , h0 represent the initial orientation of slipsystems 1 and / 0 is the angle between system 1 and 2. The mir-rored slip systems 3 and 4 have the relations with systems 1 and2 as

hs3 ; n3 i h s1 ; n1 i ; hs4 ; n4 i h s2 ; n2 i 49

Noted that the compatible active slip sets will be h1 i ; h2 i ; h3 i ; h4 ifor the set has one systemor h1 ; 2 i ; h2 ; 3 i ; h3 ; 4 i ; h4 ; 1 i for the set hastwo systems.

To verify our numerical results in the next section with somereferences, the same compressible hyperelastic Neo-Hookean typemodel proposed in [64] is utilized here to produce a relatively sim-ple format of return-mapping equations. In the present case, thestored energy function is given as

W e 12

j ln 2 J e 12 l F eiso : F

eiso 3 50

where j and l stand for the bulk modulus and shear modulus,respectively. Here, the equation introduced the tensor F eiso , the iso-choric component of the elastic left CauchyGreen strain tensor,which reads as

F eiso J e 1=3 F e with J e det F e 51

Considering Eq. (43) , the Kirchhoff stress then can then beobtained as

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s j ln J e I l dev F eiso : F eiso 52

Since the slip system tensors are deviatoric intrinsically, theinner products by hydrostatic components of Kirchhoff stress willbe eliminated. Then if we substitute Eq. (52) into Eq. (41) , theresolved shear stress will have the form as

s a l R eT F eiso F eiso

T R e : s a n a l C eiso : s a n a 53

with the isochoric right CauchyGreen strain tensor

C eiso R

eT F eiso F eiso

T R e F eisoT F e

iso 54

Then a simple constitutive formulation of the Schmid stressesbased on the NeoHookean type model can be inferred as

s a l s a n a 55 where the pairs of Eulerian vectors s a ; n a of the slip systems arecalculated via mapping of the orthonormal slip system vectorss a ; n a by the unimodular part F eiso of the elastic deformationgradient,

s a F eisosa and n a F eison

a 56

5.2. Stress-updating algorithm for crystal plasticity

The elastic predictor/return-mapping algorithm for integrationof the general constitutive equations is used for stress-updatingprocedure. For the corresponding exponential map algorithm forisotropic elastoplastic response in computational multiplicativeelasto-plasticity, we refer here to available literatures such as[9498] .

If the pseudo-time interval t n ; t n 1 is considered, all variables attime t n are assumed to be known. The incremental deformationgradient would be also known and computed as

F inc I r nD d 57

The elastic deformation gradient at trial state follows the incre-mental procedure, which can be evaluated as the following manner

F etr n 1 F inc F

en 58

where F en is the elastic deformation gradient at time t n . The associ-ated trial unimodular part F eiso is then to be evaluated by the incre-mental form of Eq. (51) and the trial Schmid stresses are easilywritten as

s a tr n 1 l s a tr n a tr 59

where

s a tr F etr iso sa and n a tr F etr iso n

a 60

Once the trial stresses are obtained, we need to check that thetrial state is within the elastic domain or lies on the yield surfaceif the return-mapping algorithm applied, viz., check the yieldingfunction

f a tr sa tr n 1 sacr ctr n 1 l s a tr n a tr s cr ctr n 1 ; with a 1; 2; . . . ; N as 61

If f a tr 6 0, it locates in the elastic state, then the incrementalplastic multipliers will be set to be zero and stress state will beset to trial state directly; Otherwise, the plastic multipliers foractive systems should be non-negative, and it can be expressedby accumulative form of the corrector dcan 1 in the kth NewtonRaphson scheme as

Dc

an 1;k :

Dc

an 1;k 1 dc

an 1 with Xa ;b2 Adc

an 1 J

ab

f a

62

where A is an active working set and the components of the Jaco-bian matrix J of return mapping system can be computed as

Jab l s a n a n a s a : F etr iso De : s a n a dscr

dc 63

where D e represents the derivative of the exponential map atP a2 A Dca s a n a .

When the plastic return-mapping is applied for active workingset, the corresponding yield function involving Dc yields

f a D c l s a D c n a D c scr cn 1 D c with a 2 A 64

To nd the pairs of Eulerian vectors s a ; n a in Eq. (64) , Eq. (59)can be employed, and the corresponding term F eisoDc by theexponential map-based update algorithm leads to

F eisoD c F

etr iso

P eiso; with P

eiso exp Xa2 A D ca s a n a " # 65

For a generic unsymmetric argument, the computation of thetensor exponential function (or exponential map) can be obtainedby truncating the innite series with rst nmax terms [20]

P eiso exp N X

nmax

n0

1n!

Nn 1 N 12!

N2 13!

N3 66

where the tensor argument N P a2 A Dca s a n a , and maximumpower nmax is determined by a tolerance check.

The updated elastic deformation gradient and Kirchhoff stressat the end of the interval t n ; t n 1 will be obtained via Eqs. (51)and (52) , i.e., we compute

F en 1 J

1=3 F eiso 67

and

s n 1 j ln J en 1 I l dev F eiso F

eiso

T 68

The tedious derivation of elastoplastic consistent tangent c forthe stress updating algorithm above, will not be presented hereand we adopt the similar expression of algorithmic moduliemployed by [64] , that is

c j 1 ln J e ~1 ~1 2j ln J e ~I

23 l tr

F eiso F eiso

T h i ~I13

~1 ~1 23 dev s ~1 ~1 dev s Xa2 A Xb2 A J

ab 1l dev s a n a n a s a

l dev s b n b n b s b 69 In Appendix A , a step-by-step algorithm procedure for imple-menting the presented stress-updating algorithm above issummarized.

6. Numerical examples and discussion

In this section, a simple standard patch test is illustrated in rstexample for linear elastic material. To study the convergence rateof the presented bFEM, the cantilever beam under a tip load underplane stress conditions is carried out. In the thirdexample, an elas-tic innite plate with a circular hole under plane strain conditionsis tested for simple volumetric locking. The numerical procedureproposed for single crystal plasticity is carried out in two examples(4 and 5) with the planar slip single crystal model in the context of rate-independent localization computations. The proposed method

and algorithms are also applied to model the mechanical behaviorof bicrystal and polycrystalline aggregate in the last two examples.

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6.1. A standard patch test

In order to check the convergence of presented b FEM, we per-form a standard patch test as shown in Fig. 6 . A simple squaredomain for the model is discretized using a patch of irregular tri-angular elements, which is displayed by the red mesh. To pass thepatch test, the computed displacements of all the interior nodesshould follow exactly (to machine precision) the same linear func-tion of the displacements imposed along the edges of the domain.The parameter material properties are set as Young modulusE 3 10 7 Pa and Poisson ratio m 0:30. The linear displacementeld is assigned as

u 18

x and v 18

y 70

The displacement error norm dened as follow is utilized toexamine the numerical convergence rate

ed Pndof i1 ui u

hi

Pndof i1 uij j

100 % 71

where ui and uhi denote the exact and numerical solution of displacements, respectively.

In Fig. 6 , the deformed mesh is plotted by blue dash-dot lines.The numerical results show that the simple model with irregular

elements can pass the designed patch test within machine preci-sion for any value of b 2 0; 1 in Table 1 . Hence, the displacementcompatibility is guaranteed and the convergence of numericalsolutions (toward exact results) will be conrmed.

6.2. Cantilever beam under a tip load: Study of accuracy and solutionbounds

In this example, a rectangular cantilever linear elastic beamwith length L and height H is studied here. The beam is xed alongthe left side edge and subjected to a parabolic traction P at free endas shown in Fig. 7 (a). The beam is assumed to be a plane stressproblem with unit thickness. The analytical solution of displace-ments can be found in Ref. [99] , which reads as follows

u x P 6EI

6L 3 x xy 2 m y y2 H 2

4 !" #u y

P 6EI

4 5m H 2 x4

3L x x2 3m y2 L x " # 72

The corresponding stresses can be expressed as

r xx x; y P L x y

I ; r yy x; y 0; s xy x; y

P 2I

y2 D2

4 ! 73 where I is the moment of inertia for the beam and can be written asI H 3 =12 for this problem. The related geometry/loading parame-ters and material properties are given as: L 2 :4 m, H 0:6 m,P 5000 N, Youngs modulus E 3 10 7 Pa and Poisson ratiov 0:3.

In Fig. 7 (b), a sample mesh using 512 triangular elements (or256 quadrilateral elements with same number of nodes) is illus-trated. To check the accuracy of bFEM, the displacement values

along the neutral axis obtained by different methods are comparedin Fig. 8 . The bound properties of strainenergy are investigatedandcompared in Fig. 9 . From these numerical results, it reveals severalfacts as: (1) compared to analytical solution, the FEM-T3, FEM-Q4,ES-FEM produce stiffer solutions of deformation and show theoverestimation property of stiffness, which can evaluate the exactsolution from the lower-bound of deformation or energy; (2) theES-FEM solution is most accurate one among these methods fromthe lower-bound, and it behaviors even (slightly) more accuratethan FEM-Q4; (3) the NS-FEM generates overly-soft solutiondue to the underestimation behavior, which brings on the uniqueproperty of the upper-bound; (4) bFEM can achieve the super-accurateor close-to-exact solution when it adopts the proper valueof adjustable parameter b . For example, b 2 0:8 ; 0 :95 for this prob-

lem works well compared to analytical solution; and (5) thenumerical results of bFEM is within the interval bounded by thesolutions of NS-FEM and ES-FEM.

6.3. Innite plate with a circular hole: test for accuracy and volumetric locking

Fig. 10 illustrates a plate with a central circular hole subjectedto a unidirectional tensile stress of p 1 :16 10 6 N/m 2 at innityin the x direction. Since the stress concentration around the hole ishighly localized and decays very rapidly, essentially disappearingwhen the distance to the center is greater than 5 a , only a niteplate with L 5a is necessary to be modeled. Here one quarter(the upper right quadrant) of the plate is chosen and discretized

into T3 or Q4 elements, owing to the symmetry of problem. Thesymmetry boundary conditions are imposed along the left and

n 1s 2 s 1

n 2

0

0

x

y

Fig. 5. Schematic drawing of a planar double-slip crystal model.

0 1 2 3 40

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Fig. 6. A patch test for bFEM using triangular mesh.

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bottom edges and inner edge of the hole is traction free. Planestrain condition is considered and the geometrical parametersare assumed as a 0:1 m and L 0:5 m. The exact solution for dis-placement components is given as [99]

u x pa8l 2

ar

r a 1 j cos h 2

ar

1 a2

r 2 cos3 h u y

pa8l 2

ar

r a 1 j sin h 2

ar

1 a2

r 2 sin3 h 74

where the shear modulus l E =21 m and bulk modulusj 3 4m for plane strain conditions, r ; h are the polar coordi-nates with counterclockwise measured h. The exact solution forthe stress is given as [99]

r 11 p 1 a2

r 232

cos2 h cos4 h 3a4

2r 4 cos4 h

r 22 p a2

r 21

2 cos 2 h cos4 h

3a4

2r 4 cos4 h

s12 p a2r 2 12 sin 2 h sin4 h 3a4

2r 4 sin4 h

75

In order to investigate the strain energy, the energy curvesagainst to mesh index are compared and plotted in Fig. 11 . Herethe parameters are set to be: Youngs modulus E 9 10 7 Pa, Pois-son ratio v 0:4, and b 0:8. The gure shows that the values of strain energy obtained using NS-FEM are always larger than thereference exact energy and those computed from other methods.On the other side, the curves computed using FEM-T3, FEM-Q4,ES-FEM and bFEM are lower than the exact one. Moreover, the pre-sented bFEM provides the most accurate results which can beregarded as the close-to-exact solution among these methods.

These phenomena also indicate the reference/exact solution isalways bounded from both side: by NS-FEM from upper-bound

of energy or deformation and by FEM-T3, FEM-Q4 or ES-FEM fromthe lower-bound. Tuning the value of the scaling factor b in bFEM,the solutions will be shifted from above to below of the exact oneand the features of both NS-FEM and ES-FEM will be inheritedwhen the factor b 2 0; 1 is selected.

For the volumetric locking issue in the nearly incompressibleelastic materials under plane strain condition, we can test the prob-lem via setting the Poissons ratio as values being close to 0.5, i.e.,m 0:4=0:49 =0 :499 =0:4999 =0:49999 =0 :499999 =0 :4999999. Table 2and Fig. 12 shows the displacement error norms at differentPoissons ratios for FEM-T3, FEM-Q4, ES-FEM, and bFEM withb 0 (i.e., NS-FEM) and b 1 m. Obviously both the standardFEM using T3 and Q4 suffer from the volumetric locking. Whilethe bFEM, which inherits the property of NS-FEM, is immune fromvolumetric locking and it is suitable for treating near incompress-ible situations. The case b 0 for bFEM also veried the property:the NS-FEM is effective in overcoming volumetric locking.

Table 1

Displacement error norm.

b 0.0000 0.2000 0.3689 a 0.6171 a 0.8000 0.9000 1.0000ed 5.0021e 14 4.6966e 14 3.2898e 14 2.9913e 14 2.3874e 14 2.5579e 14 2.7924e 14

a Random generated number.

(a)

(b)

x

y

L

H

P

Fig. 7. Computation model of cantilever beam: (a) sketch of geometry and loading;(b) domain discretization using 512 triangular (or 256 quadrilateral) elements.

0 0.5 1 1.5 2 2.50.05

0.04

0.03

0.02

0.01

0

0.01

x (y=0)

V e r t

i c a

l d i s p

l a c e m e n

t

Analytical solu.T3Q4NSFEMESFEMFEM

Fig. 8. Vertical displacement at central line ( y 0 using the mesh with 85 nodes.

8x2 16x4 24x6 32x860

80

100

120

140

160

180

Mesh index N N

S t r a

i n e n e r g y

Exact energyT3Q4NSFEMESFEM FEM

Fig. 9. Solution bounds of energy for the problem of cantilever beam.

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6.4. Planar tension of single crystal with symmetric localization

In this example, we consider the localization of a rectangularsingle crystal strip under plane strain conditions, which has beenstudied using special elements such as Q 1E 4 enhanced incompat-ible elements [100] or F -bar elements [42] in some classicalRefs. [20,41,64] . Here the problem is treated within the presentedframework of rate-independent single-crystal plasticity underquasi-static conditions. The geometry of the strip in Fig. 13 is char-acterized by the relation (width/length): W =H = 60 mm/20 mm.The material parameters for calculation are set to be: Young

modulus E 55 :4911 GPa, Poisson ratio m 0:315; and the harden-ing function s yc y0 y0 ;1 1 exp c=c0 hc, with the param-eters c0 0:0929 ; h 0:0010 GPa, ow stress y0 0:0600 GPa and y0 ;1 0:0480 GPa. The initial crystallographic orientation of the

rst slip plane is assumed to be h0 30 :0 w.r.t the horizontaldirection and the angle between the second slip system and therst one is / 0 120 :0 , i.e., the crystal lattice is oriented symmet-rically with respect to the axis of tension. The specimen will bedeformed up to a prescribed elongation of DD 5 :0 mm at bothends in the horizontal direction. By exploiting the symmetry alongthe centerlines of the specimen, only one quadrant of the specimensubjected to appropriate boundary conditions is modeled andanalyzed.

To implement the simulation by bFEM, the domain is dis-cretized with a base mesh of constant strain triangle elements(CST or T3), and then followed by the construction of strainsmoothing domains for b FEM as illustrated in Fig. 3 . Without lossof generality, here the specimen utilizes free unstructured meshwith 4 709 triangle elements for coarse mesh model ( Fig. 14

x

y r

a

p p

L

L

y

x

Fig. 10. An innite plate with a circular hole and its quarter model.

8x8 16x16 24x24 32x321610

1615

1620

1625

1630

1635

1640

1645

Mesh index N N

S t r a

i n e n e r g y

Exact energyT3Q4NSFEMESFEM FEM

Fig. 11. Solution bounds of energy for the problem of innite plate with a circularhole.

Table 2

Displacement error norm.

Mesh Possions ratio FEM-T3 FEM-Q4 ES-FEM bFEM ( b 0) bFEM ( b 1 m)

16 16 0.4 1.08677 0.29847 0.14065 0.90085 0.8517316 16 0.49 3.96202 1.80973 0.43391 0.85934 0.8493316 16 0.499 7.34553 7.67250 1.89630 0.85573 0.8545016 16 0.4999 8.35639 15.12191 4.59207 0.85556 0.8554316 16 0.49999 8.49039 17.45636 7.37184 0.85554 0.8555316 16 0.499999 8.50438 17.75544 8.23398 0.85554 0.8555416 16 0.4999999 8.50579 17.78638 8.34642 0.85554 0.85554

0.4 0.49 0 .499 0.4999 0 .49999 0 .499999 0 .49999990

2

4

6

8

10

12

14

16

18

Possons Ratio

D i s p

l a c e m e n

t e r r o r n o r m

( % )

T3Q4ESFEMFEM ( =0)

FEM ( =0.5 )

Fig. 12. Displacement error norms vs. different Poissons ratios.

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(a)) and 4 2426 elements for ne mesh model ( Fig. 14 (b)). Forcrystal plasticity problems, the parameter b 2 0 :7 ; 0 :9 is recom-mended to use and b 0:8 is adopted in this example. The perfor-mance of proposed formulations and algorithms implementedusing bFEM in this example is compared with standard FEM basedon the same initial ne mesh plotted in Fig. 14 (b). In order totrigger the localization of the geometrically perfect specimen, a

material imperfection is assumed in the center of the specimenas shown in Fig. 13 . Here the ow stress values are reduced by10% for those elements attached directly at the center node.Fig. 14 (c), (d) an