three-dimensional meshfree-enriched finite element formulation for micromechanical hyperelastic...

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng (2012)Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.4306

Three-dimensional meshfree-enriched finite element formulationfor micromechanical hyperelastic modeling of particulate

rubber composites

C. T. Wu1,*,† and M. Koishi2

1Livermore Software Technology Corporation, Livermore, CA 94550, USA2CAE Laboratory, The Yokohama Rubber Co. Ltd., Hiratsuka, Japan

SUMMARY

A three-dimensional microstructure-based finite element framework is presented for modeling themechanical response of rubber composites in the microscopic level. This framework introduces a novelfinite element formulation, the meshfree-enriched FEM, to overcome the volumetric locking and pressureoscillation problems that normally arise in the numerical simulation of rubber composites using conventionaldisplacement-based FEM. The three-dimensional meshfree-enriched FEM is composed of five-noded tetra-hedral elements with a volume-weighted smoothing of deformation gradient between neighboring elements.The L2-orthogonality property of the smoothing operator enables the employed Hu–Washizu–de Veubekefunctional to be degenerated to an assumed strain method, which leads to a displacement-based formulationthat is easily incorporated with the periodic boundary conditions imposed on the unit cell. Two numericalexamples are analyzed to demonstrate the effectiveness of the proposed approach. Copyright © 2012 JohnWiley & Sons, Ltd.

Received 2 November 2011; Revised 17 January 2012; Accepted 7 February 2012

KEY WORDS: micromechanics; composites; finite element method; nonlinear; near-incompressible

1. INTRODUCTION

Styrene Butadiene rubber (SBR), because of its good aging stability, good abrasion, and water andchemical resistance, finds widespread use in automobile tires. However, such rubber without theincorporation of any reinforcing fillers often performs poorly in terms of mechanical strength. Animprovement of the mechanical strength in SBR can be of great benefit for the automobile industry.The improvement can be achieved by optimizing the microstructure of the rubber with the additionof either particulate or wire/fiber as fillers into the rubber matrix. For example, adjusting the volumefraction of the carbon black fillers and their morphology allows the improvements of loading factorand wear life in tires. The numerical modeling of microscopic mechanical response of rubber com-posite has typically been conducted by assuming a particle or wire of simple geometry in a unit cellmodel. Two-dimensional representative volume elements (RVE) have been successfully employedto model the microscopic mechanical response of rubber composite including particle debonding[1, 2]. Although simplifications in unit cell models may aid in computation, they fail to capture thecomplex morphology, size, and spatial distribution of the reinforcement. It follows that an accuratesimulation of material can really only be obtained by incorporating actual three-dimensional (3D)microstructures as a basis for the model [3].

*Correspondence to: Cheng-Tang Wu, Livermore Software Technology Corporation (LSTC), 7374 Las Positas Road,Livermore, CA 94551, USA.

†E-mail: [email protected]

Copyright © 2012 John Wiley & Sons, Ltd.

C. T. WU AND M. KOISHI

On the other hand, the rubber composites in 3D microscopic level often experience excessivestrains, which could be in the order of several hundred percent and are difficult to be simulatedusing the standard FEM. Furthermore, the characteristic of incompressibility in rubber has pre-sented another difficulty in the numerical simulation. The so-called volumetric locking is due tothe over-constrained nature in the low-order displacement-based FEM and special numerical for-mulation has to be employed. Many approaches have been developed to overcome this difficulty.Among some popular methods for hyperelasticity materials are the methods of mixed formula-tion [4] and enhanced assumed strain formulation [5]. The u/p mixed formulation requires a stableinf–sup pair of spaces for the displacement and pressure, and usually a high-order finite element isrecommended. For mixed formulation with continuous pressure approach, additional attention hasto be paid to enforce the periodic boundary condition at the pressure nodes on RVE. To counteractthe lack of inf–sup stability, low-order pairs are usually supplemented by stabilization or post-processing procedures [6] that remove spurious pressure modes. A 3D stabilized Petrov–Galerkinformulation using equal-order interpolations for displacement and pressure was adopted by Matoušand Geubelle [7] to model large deformation and particle debonding of rubber composites involvinguniaxial loading of unit cell in meso-scale. The basic idea of enhanced strain formulation consistsin enriching the space of discrete strains by means of suitable local modes or bubble functions.However, those local modes or bubble functions may not be derived from admissible displace-ments, which lead to a nonconforming approximation and require consistency error estimates for aproof of robust convergence of the formulation. Furthermore, it was soon discovered that classicalenhanced strain formulation suffered from undesirable nonphysical instabilities, especially whenapplied to strong compression tests [8]. In addition, classical enhanced strain formulation experi-ences a decline in accuracy when the initial element is distorted [9], thus not suitable for the 3Dmicrostructure analyses.

Alternatively, several nontraditional numerical methods such as meshfree methods [10, 11] andgeneralized FEMs [12, 13] have been applied to relieve the volumetric locking problem. Wanget al. [14] first applied a displacement-based Galerkin meshfree method using an interface-enrichedreproducing kernel particle approximation [11] to analyze the magnetostrictive particle-reinforcedrubber composites. On the other hand, the compatibility across material interface can also be weaklyenforced by the introduction of a meshfree discontinuous Galerkin approach [15] that requires noadditional unknowns. Because conventional meshfree approximation requires special considerationsto impose the boundary conditions, Wu and Koishi [2] proposed a convex generalized mesh-free approximation to simplify the treatment of periodic boundary conditions on RVE. However,the inf–sup stability of their formulations was not addressed. Recently, Ortiz et al. [16, 17] pre-sented a maximum-entropy meshfree method in the framework of mixed formulation to resolve thevolumetric locking problem. Their numerical inf–sup tests have demonstrated the stability of theirmethod for the linear analysis of incompressible and near-incompressible problems. Dolbow andDevan [18] presented a geometrically nonlinear assumed strain method for the nonlinear analysisof hyperelastic materials involving displacement discontinuity where the strain is enriched basedon the generalized finite element approach. A similar idea of using the generalized finite elementapproach was presented by Srinivasan et al. [19] in the framework of mixed FEM for nonlinearanalysis of rubber composites. Recently, the iso-geometric discretization based on nonuniformrational B-splines (NURBS) [20] has presented a promising alternative to solve the incompress-ible or near-incompressible problems. The high continuity of the NURBS interpolation allows usto solve the incompressible elasticity as an elliptic fourth-order problem in terms of a scalar streamfunction whose curl gives the displacement field [21]. A nonlinear F-bar projection method [22]using the higher-order NURBS interpolation was also proposed for the nonlinear analysis of near-incompressible elasticity. More recently, a meshfree-enriched FEM (ME-FEM) was proposed byWu and Hu [23] to overcome the volumetric locking problem. The meshfree-enriched linear ele-ment is established by introducing a first-order convex meshfree approximation [2, 24] into a linearelement with an enriched meshfree node. Additional strain smoothing procedure [23] is devel-oped in cooperation with the meshfree-enriched finite element interpolation to acquire the discretedivergence-free property for the nearly incompressible analysis of the linear elasticity problem.An equivalent mixed formulation was also derived in [23] for the stability study of triangular and

Copyright © 2012 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2012)DOI: 10.1002/nme

THREE-DIMENSIONAL MESHFREE-ENRICHED FINITE ELEMENT FORMULATION

tetrahedral elements. Their numerical inf–sup [25] study indicates the pair of spaces in displacementand pressure fields is inf–sup stable.

The purpose of this paper is to present a three-dimensional microstructure-based finite elementframework using the nonlinear meshfree-enriched finite element formulation to study the mechani-cal response of rubber composites in the microscopic level. The remainder of the paper is outlined asfollows: In Section 2, we define a linear bimaterial problem and its variational equation. In addition,an overview on the linear ME-FEM for the volumetric locking-free analysis is given. In Section 3,we introduce a 3D nonlinear meshfree-enriched finite element formulation for rubber compositesin the hyperelasticity analysis. A mixed three-field Hu–Washizu–de Veubeke variational principleis utilized to formulate the nonlinear bimaterial problem and an equivalent assumed strain methodis derived. In Section 4, we define the RVE in rubber composites to be modeled. Furthermore,the enforcement of periodic boundary condition in RVE is described and the final discrete equa-tion is derived. Two numerical examples including one multiparticle unit cell model are studied inSection 5. Final remarks are given in Section 6.

2. GENERAL FRAMEWORK IN BIMATERIAL FINITE ELEMENT ANALYSIS

2.1. Bimaterial linear elasticity problem and its variational equation

The single-phase reinforced rubber composite is a type of bimaterial solid and its elasticity can beconsidered as a boundary value problem, which is modeled by the following second-order ellipticequations containing discontinuous coefficients [26].

�r � .C.x/ ru .x//D f .x/ x 2�n� (1)

uD g.x/ x 2 @�D (2)

C .x/ru.x/ � n0 D t.x/ x 2 @�N, (3)

where u.x/ is the displacement, f 2 L2 .�/ is the prescribed body force over a convex polygon��R3. Let @�D and @�N be two open subsets of boundary @� such that @�D @�D [ @�N and@�D \ @�N D 0. g.x/ is the prescribed displacement applied on the Dirichlet boundary @�D andt 2 L2 .@�N/ is the prescribed traction applied on the Neumann boundary @�N with n0 denoting theoutward unit normal to the boundary @�N. The elastic body is composed of two perfectly boundedmaterials with zero-thickness interface � . Without loss of generality, we assume the interface � isa smoothed and closed surface that divides the global domain � into two regions: �C representingthe base matrix and�� denoting the reinforcement such that their union gives the global domain�,N�D�C[�� and � D @�C\ @�� as depicted in Figure 1. The symbol n in Figure 1 denotes theoutward unit normal vector on � . The equilibrium configuration of the elastic body is characterized

Figure 1. Graphical depiction for bimaterial elasticity problem.



by the continuity of displacement and continuity of normal stress across the material interface � .This leads to the following homogenous Dirichlet jump condition and Neumann jump condition onthe interface �

Œu�D 0 x 2 � (4)

ŒC .x/ru.x/ � n�D 0 x 2 � (5)

We also define the jump operator Œ�� by

Œq�.x/D qC.x/� q�.x/ (6)

in which + and – denote the two sides of the interface � with the jump of quantity q across theinterface. The body force and material constants can exhibit discontinuities across the interface � ,but have smooth restrictions f C,CC to �C and f �,C� to ��. They are given by

f D

²f Cin�C

f �in��and C D

²CCin�C

C�in��(7)

The infinitesimal strain tensor �.u/ is defined by

� .u/DrsuD1

2

�ruC .ru/T

�(8)

CC and C� 2 L1 .�/ are elasticity tensors with major and minor symmetries and are correspond-ing to the base material in �C and reinforcement in ��, respectively. In the case of linear isotropicelasticity, we take CC and C� to be constants. The Cauchy stress tensor � and strain tensor " havethe following relationship:²

�C D CC � �D 2�C�C �C t r.�/I in�C

�� D C� � �D 2��C �� t r.�/I in��, (9)

where the positive constants �C,�� and �C, �� are Lamé constants.The variational form of this problem is to find the displacement u 2 V g D

®u 2H 1 .�/ W u D g

on @�Dº such that for all v 2 V

a .u, v/D l.v/, (10)

where the functional space V D H 10 .�/ consists of functions in Sobolev space H 1 .�/, which

vanish on the boundary @� and is defined by

V .�/D®v W v 2H 1, vD 0 on @�D

¯. (11)

The bilinear form a .�, �/ and linear functional l.�/ are obtained by multiplying the test functionv 2 V to both sides of Equation (1) and integrating over the regions �C and �� separately usingGreen’s formula.R�C f

C.x/ � vd�CR�� f

�.x/ � vd�D�R�C r �

�CC .x/ � rsu.x/

�� vd�

�R�� r � .C

� .x/ � rsu.x// � vd�DR�C

�.u/ �CC � �.v/d�

�R�CC .x/ � rsu.x/ � n

C � v d�

CR��.u/ �C� � �.v/d��

R� C

�.x/ � rsu.x/ � n� � v d� �

R@�N

.t � v/d@�

.

(12)Using the fact that nC D�n�, we can rewrite the above equation to becomeR

�C fC.x/ � vd� C

R�� f

�.x/ � vd�DR�C

�.u/ �CC��.v/d�

CR��.u/ �C��.v/d��

R@�N

.t � v/d@��R�ŒC .x/ � rsu.x/ � n� � vd�

.

(13)



Applying the homogenous Neumann jump condition to Equation (13) yields

Z�C

�.u/ �CC��.v/d�C

Z��

�.u/ �C� � �.v/d��

Z�Cf C � vd��

Z��f � � vd�

�

Z@�N

t � vd@�D a .u, v/� l.v/D 0

, (14)

where

a .u, v/DZ�C

� .u/ �CC � �.v/d�C

Z��

�.u/ �C� � �.v/d� (15)

l.v/D

Z�Cf C � vd�C

Z��f � � vd�C

Z@�N

t � vd@�. (16)

It is noted that the elasticity tensors CC and C� are symmetric, and homogenous Neumann jumpcondition is enforced in the variational level. Obviously, the bilinear form a .�, �/ in Equation (15)is symmetric, bounded, and coercive by Friedrichs’ inequality. The existence and uniqueness of theproblem is ensured by the Lax–Milgram theorem [27].

2.2. Three-dimensional meshfree-enriched linear finite element formulation

To overcome the volumetric locking and pressure oscillation problems in nearly incompress-ible analysis of rubber composites using conventional finite element formulation, we employ theME-FEM. Because our focus is on the nonlinear analysis, this section only provides an overviewon the construction of meshfree-enriched finite element interpolations in linear tetrahedral elementsand their strain smoothing in the linear analysis. A detailed description of linear ME-FEM and itscorresponding mathematical proofs for the uniqueness of the solution and discrete divergence-freeproperty in the incompressible limit can be found in [23]. Let us consider a locally quasi-uniformtriangulation Qh of the polygonal domain �, where Qh consists of simplexes and is denoted byQh D [eTe. Each simplex Te contains four corner nodes xi , 1 6 i 6 4 and one enriched meshfreenode x5. Let F e 2 P1

�NTe�

be the affine transformation that maps the reference triangle NTe onto tothe tetrahedron Te 2Qh as depicted in Figure 2 and defined by

Figure 2. Isoparametric mapping in the five-noded meshfree-enriched tetrahedral finite element.



F e W NTe! Te, x D F e./D�F 1e ,F 2e ,F 3e

�D

5XiD1

xii . , �, &/ ,5XiD1

yii . , �, &/ ,5XiD1

´ii . , �, &/

!8 2 NTe

,

(17)where x D Œx,y, ´�T , D Œ , �, &�T and P1

�NTe�D span ¹i , i D 1, � � � 5º denotes that the space

contains a set of basis functions in NTe that reproduces first-order polynomials.In Figure 2, the reference element NTe is an equilateral tetrahedron, with dark circles denoting

the finite element node and open circles denoting the enriched meshfree node. The location of theenriched meshfree node in reference element NTe is given by

.5, �5, &5/D

4XiD1

i=4,4XiD1

�i=4,4XiD1

&i=4

!, (18)

which is the centroid of the reference element. i D .i , �i , &i /, i D 1, 2 , 3, 4, 5 are nodalco-coordinates of the reference element. The shape functions §i , i D 1, 2, 3, 4, 5 of the referenceelement are constructed using a meshfree convex approximation. In this study, we employ the gen-eralized meshfree approximation (GMF) method [24] to obtain the meshfree convex approximation.The convex GMF approximation is constructed using the inverse tangent basis function. The cubicspline window function is chosen to be the weight function in GMF method. In this study, eachnode in Figure 2 is assigned to a weight function with same sphere support in the reference elementNTe. The element mapping in the ME-FEM has been proven [23] to be bijective. In other words, the

determinant of the Jacobian matrix computed using Equation (18) in the element mapping is posi-tive everywhere in the element. A detail derivation of GMF method and its mathematical propertiesis given in [24].

Giving the five-noded ME-FEM shape functions, we define the following approximation spacefor the displacement field:

V h .�/D°uh W uh 2

�H 10

�3,uh jTe D Nu

h ıF �1e , Nuh 2P1�NTe�

for allTe 2Qh

±, (19)

which consists of functions in Sobolev space H 1 .�/ that vanish on the boundary. Because theshape functions constructed using the GMF method are convex, they reduce to the standard lin-ear finite element shape functions along the element edge. Compared with the standard Lagrangianfinite element, the ME-FEM shape functions are not mutually .�, �/

V h-orthonormal.

To provide a volumetric locking-free analysis using meshfree-enriched tetrahedral elements, avolume-weighted strain smoothing scheme is introduced for the near-incompressible analysis of thelinear elastic problem. The strain smoothing scheme is briefly described in the following:

Let xji , 1 6 i 6 4 be the four vertices of a tetrahedral element Tj 2 Qh. xj5 is the centroid

of the tetrahedral element Tj . We connect xj5 to the four vertices of the tetrahedron by straightlines to divide the tetrahedron into four subtetrahedrons Sl ,Sm,Sn, and So. Each subtetrahedronSm shares the element face m of the tetrahedron and carries one integration point. Because eachsubtetrahedron occupies the same volume, all four integration points are assigned to the same weightfor the numerical integration. For simplicity, we choose the integration point to locate at the centroidof the element face in this study.

The smoothing domain �m corresponding to the edge m for the adjacent elements is definedby �m D [S2SmS as depicted in Figure 3 and the smoothed strain is given in terms of itscomponents by

Q�h D Qruh D1

Vm

Z�m

ruh$m.x/d�, (20)

where Vm is the volume of the smoothing domain �m, $m.x/ is the characteristic or smoothingfunction of the smoothing domain �m defined by

$m.x/D

²1, ifx 2�m0, else

. (21)



Figure 3. Strain smoothing in ME-FEM tetrahedral linear elements.

It is worth mentioning that an eigenvalue analysis verifies that the ME-FEM tetrahedral elements donot contain spurious zero energy modes and nonphysical volumetric locking modes [23].

3. 3D NONLINEAR MESHFREE-ENRICHED FINITE ELEMENT FORMULATION FORBIMATERIAL MODELING

This section devotes to the application of ME-FEM tetrahedral elements to the 3D bimaterial modelin the nonlinear hyperelasticity analysis. Hyperelasticity is a typical path-independent material,where the internal energy can be expressed in terms of deformation gradient. For the nonlinear anal-ysis, the smoothing of strain for the linear ME-FEM introduced in the previous section is generalizedto the smoothing of deformation gradient and performed in each material domain

��C and��

�. The

smoothed deformation gradient is given in the following form:

NFhD…h

�F�uh��D

1

Vm

Z�m

F�uh�$m .X/d� (22)

where …h denotes the smoothing operator of deformation gradient. Adopting Einstein’s repeatedsubscripts convention, Equation (22) can be written by

NF hij�Xgk

�D

1

Vm

Z�m

�F hij

�Xgk

��$m

�Xgk

�d�

D1

Vm

Z�m

@uhi@XjC ıij

!$m

�Xgk

�d�

D1

Vm

Z�m

@uhi@Xj

!$m

�Xgk

�d�C ıij

D1

Vm

Z�m

5XID1

�@‰I

@XjuiI

$m

�Xgk

�d�C ıij � Nh

hij

�Xgk

�C ıij

(23)

and

Nhhij�Xgk

�D

1

Vm

Z�m

5XID1

�@I

@XjuiI

$m

�Xgk

�d�, (24)

where Vm is the volume of the smoothing domain �m, Xgk , k D 1, 2, 3, 4 is the integration pointof the tetrahedral element Tj as depicted in Section 2.2 of the linear formulation. Each tetrahedralelement Tj contains four smoothing domains. For each smoothing domain �mwhose element Tjshares a common face of a neighboring element Tn, a simple volume-weighted integration rule is



employed for the evaluation of deformation gradient as illustrated in Figure 3. Note that the diver-gence theorem utilized in the original strain smoothing technique [28] for meshfree method was notapplied to the evaluation of smoothed deformation gradient in the present method. Accordingly, thesmoothed deformation gradient of Equation (22) can be further expressed by

NFhD

1

2Vm

�F j

�Xgk

�det

�Jj1

�CF n

�Xgk

�det

�J n2��

(25)

Vm D�det

�J j�C det .J n/

�= 2, (26)

where F j�Xgk

�denotes the deformation gradient evaluated at integration point Xgk of the tetra-

hedral element Tj and J j is its Jacobin. J n is the Jacobin of neighboring element Tn evaluated atthe same integration point.To introduce the smoothed deformation gradient into Galerkin approximation in the nonlinear anal-ysis, the following mixed three-field Hu–Washizu–de Veubeke energy functional [29] is consideredfor the derivation of nonlinear finite element formulation for bimaterial hyperelastic materials.

UHW�u, NF , N�

�D

Z�C

W C�NF�d�C

Z��

W ��NF�d�C

Z�CN� W�r0u� NF

�d�

C

Z��N� W�r0u� NF

�d��

Z�Cf C0 � ud��

Z��f �0 � ud��

Z@�N

t0 � ud� ,

(27)where the displacements u, smoothed deformation gradient NF and smoothed first Piola–Kirchhoffstresses N� are independently varied. The symbol r0 denotes the gradient operator with respect tothe original configuration. W ˙ D W ˙

�NF�

represent the assumed strain energy density functionsin base matrix and reinforcement respectively. f ˙0 are the body force density of base matrix andreinforcement measured in the original configuration and t0 is the prescribed traction on the naturalboundary @�N defined in the original configuration. The smoothed first Piola–Kirchhoff stresses N�is related to the smoothed deformation gradient NF by the following constitutive law:

N�˙ D@W ˙

@ NF. (28)

The energy functional in Equation (27) is taken as a basis for the numerical discretization. Becausethe material frame indifference restricts the dependence of strain energy density function W ˙ onsmoothed deformation gradient NF , it is advantageous to express the variational equation in terms ofthe total Lagrangian formulation. The discretization of undeformed domain � using the ME-FEM

elements gives the following discretized variational formulation: Find�uh, NF

h, N�h

�2 V g�„h�‚h

such thatZ�

N�h W r0ıuhd��

Z�Cf C0 � ıu

hd��

Z��f �0 � ıu

hd��

Z@�N

t0 � ıuhd� D 0 8 ıuh 2 V h

(29)Z�

ı N�h W�r0u

h � NFh�d�D 0 8 ı N�h 2‚h (30)

Z�C

@W C

@ NFh� N�h

!W ı NF

hd�C

Z��

@W �

@ NFh� N�h

!W ı NF

hd�D 0 8 ı NF

h2„h, (31)

where V h .�/ denotes the approximation space of displacement defined in Equation (19). Becausethe discrete smoothed deformation gradient is defined locally on each smoothing domain �m andno continuity conditions are applied at the boundaries of �m, the approximation space of smootheddeformation gradient can be defined by

„h .�/D°NFhW NF

h2 L2 .�/ , NF

hcontains piecewise constants8 �m 2Qh

±(32)



where Qh is a triangulation of the polygonal domain �. The space ™h 2 L2 of smoothed firstPiola–Kirchhoff stresses also contains piecewise constants in �. Apparently, Equation (31) can beenforced strongly using the constitutive relationship in Equation (28). Equation (30) can be furtherexpressed using Equation (22) to yieldZ

�

ı N�h�r0u

h � NFh�d�D

nmXmD1

ı N�hZ�m

�r0u

h � NFh�d�

D

nmXmD1

ı N�h�Z

�m

r0uhd�� NF

hVm

D

nmXmD1

ı N�h�Z

�m

r0uhd��

Z�m

r0uhd�

D 0

. (33)

In other words, Equation (30) also can be enforced automatically. The index nm in Equation (33)denotes the total number of smoothing domain �m in � (� D [m�m/. Equation (33) impliesan orthogonal condition between the stress field N�h and the difference of the deformation gradi-

ent r0uh field and the smoothed strain field NFh. This is equivalent to the assumed strain variation

principle of Simo and Hughes [30].Using the orthogonal condition in Equation (33), we arrive at the following modified variational

equation depending only on displacement and smoothed deformation gradient fields

ıU hHWmod

�uh, NF

h�D

Z�

N�h W r0ıuhd�� ıWext

�uh�

D

Z�

N�h W ı NFhd�� ıWext

�uh� , (34)

where ıWext�uh�

designates the external work given by

ıWext

�uh�D

Z�Cf C0 � ıu

hd�C

Z��f �0 � ıu

hd�C

Z@�N

t0 � ıuhd� . (35)

The two-field variation problem can be condensed to the following primal problem by the definitionof Equation (22) to yield: Find uh 2 V g such that

ı NU hHWmod

�uh�D

Z�C

ıW C�…h

�F h

�uh��

d�C

Z��

ıW ��…h

�F h

�uh��

d�

� ıWext

�uh�8uh 2 V h (36)

4. MICROSTRUCTURED-BASED MODELING

4.1. Representative volume element for rubber composite

Numerical homogenization techniques allow for rubber composites to be approximated by macro-scopic finite element models, thus simplifying numerical analyses of structures directly made fromthe heterogeneous materials. The main idea of the numerical homogenization techniques is to find aglobally homogenous medium close to the original composite, where the strain energy stored in bothsystems is approximately the same. The common approach to model the macroscopic properties ofthe rubber composites is to create the RVE that can capture the major features of the underly-ing microstructure. The RVE represents an idealization of the actual microstructure of the rubbercomposites and the incorporation of three-dimensional microstructures of the rubber composites iscrucial for obtaining an accurate predication of material behavior. Figure 4 shows two typical finiteelement RVE models of wire-reinforced and particle-reinforced rubber composites in tires.

After the finite element RVE model is created, a numerical homogenization procedure is carriedout on the RVE. Effective macroscopic properties for the rubber composite are obtained based on



Figure 4. Two examples of finite element RVE model in tire.

the stress–strain behavior of the RVE. The stress–strain behavior of RVE is characterized by solv-ing the boundary value problem defined on an RVE. Different types of boundary conditions canbe applied to the RVE. Among them the periodicity conditions have been shown [31, 32] to rendera more reasonable estimation of the stress–strain constitutive than other boundary conditions suchas uniform displacement and uniform traction boundary conditions. In addition, periodic boundarycondition is known to be the most efficient approach in terms of convergence when the RVE sizeincreases. In this paper, the periodic boundary condition is considered.

The enforcement of a periodic boundary condition on RVE requires a periodic mesh. This impliesthat the same nodal distribution has to be imposed on any pair of two opposing faces on the RVEunless special approximation is adopted [33]. Furthermore, conforming mesh is considered for thediscretization of rubber composite across the material interface in the current approach. Mesh gen-eration for 3D finite element RVE model that contains periodic and conforming mesh is not aneasy task when the model involves complex geometry of reinforcements. In this study, we useTSV-Pre software [34] to create the periodic and conforming mesh for finite element RVE model.The generated finite element RVE model is composed of linear tetrahedral elements, which areknown to suffer from volumetric locking problem for nearly incompressible materials. To overcomethe volumetric locking and pressure oscillation problems using linear tetrahedral elements in thehyperelastic analysis, the 3D nonlinear meshfree-enriched finite element formulation introduced inthe previous section is incorporated in the study. The material interface between reinforcement andrubber matrix is assumed to be perfectly bounded. Because the enriched meshfree node is addedat the element center, the mesh along the material interface remains unchanged and the enforce-ment of a periodic boundary condition can be imposed as in the standard displacement-based finiteelement formulation.

4.2. Enforcement of periodic boundary condition and the discrete equations

The periodic boundary conditions in a three-dimensional unit cell are described by [35]

u .X�/D u .XC/ on @� (37)



� .X�/ � n� D � .XC/ � nC on @�, (38)

where n� denotes the outward unit normal to the boundaries @�� and @�C as shown in Figure 5.d D d1Ei 1Cd2Ei 2Cd3Ei 3 shown in Figure 5 denotes position vector of the RVE. Equations (37) and(38) imply that the opposing points X� on RVE opposing faces @��

�@ N�D @ N�� [ @ N�C

�have

equal displacements u and their normal stresses � are also equal but opposite in sign.Because RVE in equilibrium state considers no external work, the variational boundary valueproblem defined in Equation (36) can be rewritten to find uh 2 V p such that

ı NU hHWmod

�uh�D

Z�C

ıW C�…h

�F h

�uh��

d�C

Z��

ıW ��…h

�F h

�uh��

d�8ıuh 2 V h

(39)

V P .�/D°vh W vh 2H 1, vh .X�/D v

h .XC/8 pairs ¹X�,XCº±

, (40)

where the discrete form of periodic displacement boundary condition in Equation (37) is enforcedas an essential boundary condition. After a standard derivation as in Equations (12)–(14), the strongform of periodic traction boundary condition in Equation (38) can be merged as a natural bound-ary condition, which is satisfied automatically by the variation process in the displacement-basedGalerkin formulation. In other words, additional enforcement of periodic traction boundary condi-tion is neither necessary nor correct [36]. The discrete form of Equation (37) can be considered asthe dependent multiple constraint equations. Common approaches to solve the multiple constraintequations include Lagrange multipliers, penalty method and transformation method [37]. In thisstudy, we follow the two-dimensional transformation method in [2] to impose the periodic displace-ment condition in the 3D case. Let the discrete form of Equation (37) be rearranged and written inmatrix form as

Œ˛�®Up

¯D ¹0º , (41)

where [˛] is the matrix containing the coefficients of the constraint equations and ¹U pº is the cor-responding periodic degrees of freedom. Gaussian elimination can be applied to the [˛] matrix onthe left-hand side of Equation (41) such that Equation (41) can be further partitioned as

Œ˛e ˛c�

²Ue

Uc

³D ¹0º , (42)

where ¹U cº is the displacement vector with a collection of independent nodes, which includethe nodes on retained entity @�C and a set of corner nodes ZI D ¹nI , I D 1, � � � 8n7º. ¹U eº isthe displacement vector containing the nodes that need to be eliminated. The dependent nodesare the nodes on tied entity @�� and corner node n7. Œ˛e ˛c� is the corresponding coefficientmatrix, where the submatrix [˛e] is square and nonsingular.

Figure 5. Three-dimensional unit cell.



A standard linearization of Equation (39) with introducing the approximations of displacement inEquation (19) and smoothed deformation gradient in Equation (23) to the linearized equation yieldsthe following incremental matrix equation:

ı ¹UrºT�QT�T NKvnC1 � QT � .� ¹Urº/

vC1nC1 D�ı ¹Urº

T�QT�Tf intv

I ,nC1 , (43)

which is used to solve the periodic boundary value problem in quasi-static analysis and to obtainthe displacement vector ¹U rº of the retained nodes. The square matrix

�QT�

in Equation (43) is theglobal transformation matrix. The definition of ¹U rº and

�QT�

are given in Appendix A. The tangentstiffness matrix NKvnC1 .�¹U º/

vC1nC1 contains the material and geometric stiffness matrices that evalu-

ated at the v-th iteration during the (nC 1) time incremental step. f intv

I ,nC1 is the nodal internal forcevector. They are given by

NKDIJ

nmCXlD1

°BT

I

hCC�Fh�C S

�Fh�iBJVl

±C

nm�XlD1

°BT

I

hNC��Fh�C S

�Fh�iBJVl

±(44)

C˙

ijkl

�Fh�D@2W ˙

�Fh�

@F ij @F kl(45)

S ijkl

�Fh�D ıikF

h

mj ml

�Fh�

(46)

f intI D

nmCXlD1

°BT

I �Fh�Vl

±C

nm�XlD1

°BT

I �Fh�Vl

±, (47)

where NBI is the smoothed gradient matrix with its components .1=Vm/R�m

�@I=@Xj

�$m.X/d�,

j D 1, 2, 3 expressed in the standard gradient matrix format. The indexes nmC and nm� denotethe total number of smoothing domains in �C and ��, respectively. Because the smoothed gradi-ent matrix BI is defined in the original configuration, the smoothing procedure only needs to beperformed at once. We further assume a quadratic form in volumetric strain energy density functionQW�NJ�

given by

QW�NJ�Dk

2

�NJ � 1

�2, (48)

where k is the bulk modulus and NJ is determinant of the smoothed deformation gradients definedby NJ D det

�NF�. Equation (48) leads to a discrete hydrostatic pressure expressed in terms of a total

form by

ph D@ QW

@ NJ� k

�NJ h � 1

�D�k

�1

Vm

Z�m

J h$m .X/d�� 1

. (49)

Although the pressure does not directly involve in the nonlinear computation, the stability of thenonlinear formulation is still subject to the inf–sup pair requirements for displacement field andthe implicit pressure field induced by Equation (49) as in the linear analysis. The failure to satisfyinf–sup condition indicates the pair of spaces in displacement and pressure is not stable. As a result,numerical solutions may exhibit locking and possible pressure oscillation. This stability requirementis examined in the following numerical examples by evaluating the load–displacement response andchecking the pressure field in the nonlinear analysis. After the displacements of retained nodes aresolved, the displacements of dependent nodes ¹U eº are obtained using Equation (A.1).



5. RESULTS AND DISCUSSION

In this section, we analyze two numerical examples to study the performance of the proposed methodfor the micromechanical hyperelastic modeling of particulate rubber composites. As a comparison,we also provide the results using: (1) standard displacement-based meshfree method (meshfree-S)[2] and (2) standard displacement-based meshfree method with a least squares procedure for pres-sure smoothing [38] (meshfree-PS). Both meshfree methods (1) and (2) utilize the same discretiza-tion in ME-FEM with additional decomposition to split apart each five-noded element into fourcongruent tetrahedral elements. In other words, we use the same total DOFs in three numericalmethods for the comparison. The GMF method is employed for the construction of meshfree con-vex approximation in the standard displacement-based meshfree method. In meshfree methods (1)and (2), the tetrahedral elements are adopted as the integration cells and one-point quadrature rule isused for the numerical integration. A normalized nodal support size of 1.2 is considered for all mesh-free methods including the proposed method. A standard Newton–Raphson method is employed tosolve the nonlinear Equation (41). If the number of iterations reaches the maximum number ofallowed equilibrium iterations, the current time step is reset and the iteration with a reduction oftime increment by a scale factor of 0.5 is redone. This execution repeats until the convergence isachieved. If the number of resets in single time step reaches its maximum of 15, the executionis terminated.

5.1. Compression test of a cubic cell with single rubber material

The first example is a standard benchmark problem used to examine whether the numerical methodis likely to provide a volumetric locking-free and pressure oscillation-free solution in the nonlinearanalysis of hyperelastic materials. The model shown in Figure 6 consists of a 100 �m� 100 �m�100 �m cubic cell subjected to a fixed end and compressed by a rigid, frictionless, and flat plate witha prescribed displacement. The material property of the rubber matrix is modeled by Yeoh’s cubicenergy density function defined in Equation (50) with coefficients C10 D 0.333 MPa, C20 D 0.0MPa, C30 D 5.05E � 4 MPa and bulk modulus k D 333.0 MPa.

W .I1/D C10 .I1 � 3/CC20 .I1 � 3/ 2CC30 .I1 � 3/3 , (50)

where I1 is the first invariant of the smoothed Green deformation tensor Gij D F kiF kj . Threeregularly refined finite element discretizations shown in Figure 7 are used to study the convergenceproperty of the proposed method in the nonlinear analysis.

The load–displacement curves generated by the proposed method using three different meshesare shown in Figure 8. The results suggest the convergence of the proposed method in the non-linear analysis as the mesh is refined. Figure 9 compares the load–displacement response of theproposed method, meshfree-PS and meshfree-S using the most refined mesh. The results show that

Figure 6. Compression of unit cell.



(a) (b) (c)

Figure 7. Discretization of unit cell (a) 6000 elements (b) 20,250 elements and (c) 48,000 elements.

Figure 8. Convergence study of nonlinear behavior using the proposed method.

Figure 9. Comparison of displacement–force predictions from various numerical methods.

two meshfree methods behave stiffer than the proposed method. The locking behavior of standarddisplacement-based meshfree method can be reduced when the smoothing for pressure is consid-ered. The locking behavior of the two meshfree methods can be further illustrated in their pressuredistribution plots. Figure 10(a) shows a severe pressure oscillation in a side view of the solutionobtained from the displacement-based meshfree method. Using the pressure smoothing procedure,



(a)

(b)

(c)

Figure 10. Comparison of pressure contours in 48,000 elements model in intermediate stage: (a)displacement-based meshfree Galerkin method (b) displacement-based meshfree Galerkin method with

pressure smoothing, and (c) proposed method.

the pressure oscillation in displacement-based meshfree method is improved but is still visible asdisplayed in Figure 10(b). Figure 10(c) demonstrates that the pressure oscillation is effectively elim-inated by the proposed method. Figure 11 shows the progressive deformation using the proposedmethod with 48,000 elements model. The pressure field in the final stage remains at a stable andnonoscillating level.

5.2. Representative volume element of rubber composite with multiple spherical fillers

In this example, an RVE model with a length of 100 �m per side as shown in Figure 12(a) is consid-ered as an idealization of a microstructure consisting of several nonaggregated carbon black fillerssurrounded by rubber matrix. The nonaggregated carbon black fillers are modeled by multiple spher-ical fillers as shown in Figure 12(b). The RVE size was chosen to be at least twice the diameter of the



Figure 11. History of deformation plots in 48,000 element model using the proposed method.

(a) (b)

Figure 12. Finite element model of particle-reinforced rubber composite (a) unit cell and (b) particlereinforcement.

Figure 13. Comparison of displacement–force predictions from various numerical methods.



(a)

(b)

(c)

Figure 14. Comparison of pressure contours: (a) displacement-based meshfree Galerkin method, (b)displacement-based meshfree Galerkin method with pressure smoothing, and (c) presented method.

spherical filler for the accuracy consideration [39]. The unit cell is subjected to a macroscopic ten-sile loading with prescribed displacements. The corner node n1 of the unit cell shown in Figure 5 isselected to be fixed to prevent the rigid body motion. The material property of rubber matrix is mod-eled by Yeoh’s cubic energy density function in Equation (50) with coefficients CC10 D 0.333MPa,CC20 D 0.0MPa, CC30 D 5.05E � 4MPa and bulk modulus kC D 333.0MPa. The material prop-erty of carbon black fillers is modeled by the same material but with higher material modulus.



Figure 15. History of unit cell deformation plots using the proposed method.

They are given by C�10 D 333.0MPa, C�20 D 0.0MPa, C�30 D 5.05E � 1 MPa and bulk modulusk� D 333, 000. MPa. The FEM model contains 59,628 meshfree-enriched finite elements (46,489elements in base matrix and 13,139 elements in fillers). The volume fraction of spherical fillers isabout 29.26%.

Figure 13 shows the comparison of load–displacement response in three methods. The resultobtained from the standard displacement-based meshfree method apparently exhibits volumetriclocking. This volumetric locking starts as early as in the linear range. Although the locking can bereduced with the use of relatively large normalized support, the computation becomes expensiveand is unstable in the nonlinear range. With additional pressure smoothing procedure, the standarddisplacement-based meshfree method improves the solution in the load–displacement curve evenwith a small support size. On the other hand, the proposed method presents the softest result amongthree numerical methods, which is consistent with the results in example 1.

Superior performance of the proposed method over two other methods is presented in the pres-sure solution. The displacement-based meshfree method without pressure smoothing procedureproduces severe pressure oscillation as plotted in Figure 14(a). It is important to note that, unlikethe volumetric locking phenomenon, pressure oscillation cannot be eliminated simply by increas-ing the nodal support size. Although the pressure oscillation is less pronounced using the meshfreemethod with pressure smoothing procedure, pressure oscillation pattern is still visible as displayed inFigure 14(b). The existence of volumetric locking and pressure oscillation in two meshfree solutionsimplies the failures of two meshfree methods to deliver the stable solutions in the hyperelastic anal-ysis. Figure 14(c) depicts a very smooth pressure distribution using the proposed method. This resultsuggests a satisfaction of inf–sup stability in the ME-FEM as reported in the linear analysis [23].The progressive deformation of the RVE shown in Figure 15 has demonstrated the capability of theproposed method in the large strain analysis of rubber composite in the microscopic level.

6. CONCLUSION

Recent advances in rubber composite technology allow for the design and fabrication of microscaledstructures to serve as enhancements in macroscale materials for improving their mechanical per-formance. Via micromechanical modeling techniques, the finite element model of a representativevolume element is commonly used to parametrically study the mechanical properties of the com-posite microstructures. However, there still exist great challenges for the conventional numericalsimulation of microstructures in rubber composites when the large strain, incompressibility and the



requirement of consistent periodic boundary condition are simultaneously presented in the model.This paper attempts to provide an effective way in the finite element framework to overcome thosenumerical challenges.

The proposed three-dimensional nonlinear meshfree-enriched finite element formulation inheritsnice numerical properties such as free of volumetric locking and free of pressure oscillation from itslinear version [23] for the hyperelastic modeling of rubber-like materials. Different from the mixedformulation with continuous pressure approach, the proposed formulation is a displacement-basedapproach in which the periodic traction boundary condition is merged naturally in the variationallevel and periodic displacement boundary condition can be imposed on the RVE as an essentialboundary condition without special treatments. Two numerical examples are studied and the resultsare validated by comparison with other numerical methods. The numerical results have demon-strated that the proposed method is capable of delivering volumetric locking-free and pressureoscillation-free solutions in the large strain analysis of rubber composites subjected to periodicboundary conditions.

At present, a decoupled microscale and macroscale modeling method [40] is pursued to extendthe presented approach to the two-scale analysis of rubber composites in tires. The resulting multi-scale constitutive model will be addressed in a separate paper. To better understand the mechanismsthat give rise to the macroscopic behavior of rubber composites in tires, the considerations ofvisco-hyperelastic material, wire-reinforced composite, and material interface debonding in theformulation are under investigation and will be discussed in the near future.

APPENDIX A

The vector ¹U eº can be further expressed using Equation (42) to yield

¹Ueº D � Œ˛e��1 Œ˛c� ¹Ucº D Œˇ� ¹Ucº , (A.1)

where

Œˇ�D� Œ˛e��1 Œ˛c� . (A.2)

Matrix [ˇ] describes the relationship of dependent nodes ¹U eº and independent nodes ¹U cº.Consequently, the constraint equations associated with the periodic boundary conditions can bewritten as ®

Up

¯D

²Ue

Uc

³D�NT�¹Ucº , (A.3)

where�NT�

is the local transformation matrix involves the periodic boundary conditions and isdefined as �

NT�D

ˇ

I

�(A.4)

and I denotes the unit tensor.The global displacements now become

¹U º D

8<:Ue

Uc

Ui

9=;D � QT � ¹Urº (A.5)

with

¹Urº D

²Uc

Ui

³(A.6)

and �QT�D

NT 00 I

�, (A.7)



where ¹U rº is the retained nodes that include the independent nodes ¹U cº in the constraint equa-tions and the rest of nodes ¹U iº in the unit cell that do not involve the constraint equations.

�QT�

is the global transformation matrix that contains the local transformation matrix in Equation (A.4)and a unit tensor. The global transformation matrix carries the information of retained nodes andeliminated nodes that can be used to condense the structural equations for the solving of periodicboundary value problem.

ACKNOWLEDGEMENTS

The authors would like to thank Dr. John O. Hallquist of LSTC for his support to this research. The supportfrom Yokohama Rubber Co, Ltd, Japan is also gratefully acknowledged.

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