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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci

A new multiscale XFEM for the elastic properties evaluation of heterogeneous materials R.U. Patil, B.K. Mishra, I.V. Singh

MARK

⁎

Department of Mechanical and Industrial Engineering, Indian Institute of Technology Roorkee, 247667, Uttarakhand, India

A R T I C L E I N F O

A B S T R A C T

Keywords: Micro-element Macro-element MsXFEM Elastic properties Heterogeneous material RVE

In the present work, extended ﬁnite element method (XFEM) in conjunction with multiscale ﬁnite element method (MsFEM) is employed to model heterogeneous materials i.e. matrix having particles and/or voids. This approach, named as multiscale extended ﬁnite element method (MsXFEM), is used to determine the eﬀective elastic properties of heterogeneous materials. The elastic properties are calculated by analyzing a representative volume element (RVE) under periodic boundary conditions. Three cases of particles and voids in the matrix i.e. matrix with hard particles, matrix with voids, and matrix with voids and hard particles both, are considered for analysis. The particles and voids are randomly distributed in the matrix. A strain energy based homogenization is carried out numerically for reinforcement volume fractions up to 50%. RVEs containing large number of discontinuities are analyzed by the proposed MsXFEM. The evaluated eﬀective properties of the heterogeneous material are compared with the results available in the literature. These simulations show that the use of MsXFEM leads to a signiﬁcant reduction in the CPU time as compared with the standard XFEM.

1. Introduction In the past few decades, many metal matrix composites have been developed to meet rising industrial demands. Particle reinforced and short ﬁber reinforced composites have advantage in terms of easy manufacturing and good mechanical properties. The particle-reinforced composites can be easily prepared by molding, casting or sintering to produce complex shaped components. These spatially distributed reinforcing particles provide more balanced properties and lead to improved through-the-thickness stiﬀness/strength. Components with complicated shapes can be created easily. Moreover, porous solids, particularly composite foams, are potential materials for lightweight structural applications involving energy absorption, noise attenuation, blast protection, etc. Hence, composites with spatially distributed particles or voids ﬁnd many applications in nuclear, aerospace and automobile industries. The determination of macroscopic equivalent properties of these heterogeneous composites is essential for designing structures or components made from such heterogeneous composites. Several analytical [1], semi-analytical [2–7] and numerical techniques have been used to evaluate the eﬀective homogeneous properties of composite materials. The ﬁnite element method (FEM) [8–10] is widely used to model the microstructure and also to evaluate equivalent properties. Kari et al. [8] and Singh et al. [9] evaluated the eﬀective elastic material properties of randomly distributed spherical

⁎

particle reinforced composites using periodic and uniform boundary conditions. Brassart et al. [10] predicted the eﬀective elasto-plastic response of composite materials using representative volume element (RVE) and FEM. FEM requires that the boundary of the reinforcement particles coincide with the edges of the ﬁnite elements, i.e. a conformal mesh is needed. Hence, modeling and simulation of an RVE with large number of heterogeneities (particles and voids) using FEM becomes quite cumbersome. Furthermore, the unknown ﬁeld variables in an element are interpolated by suitable functions while keeping the material properties (such as elastic modulus, conductivity and permeability) constant over each element. In these situations, accurate solution is obtained only if the mesh size is kept smaller than the size of heterogeneities (particles and voids). This generates a large number of degrees of freedom for the problems where the material such as a particulate composite consists of many voids and particles with a large variation in their size. To extract the equivalent properties, RVE containing a large number of heterogeneities needs to be modelled [8,11]. Moreover, stochastic homogenization involves computational analysis of a large number of randomly generated conﬁgurations of the composite medium [12,13]. In stochastic analysis, a problem with diﬀerent geometric conﬁgurations is solved for many possible combinations. The modeling of a large number of randomly dispersed heterogeneities/ discontinuities by FEM with conformal meshing is quite

Corresponding author. E-mail address: [email protected] (I.V. Singh).

http://dx.doi.org/10.1016/j.ijmecsci.2017.01.028 Received 13 September 2016; Received in revised form 9 December 2016; Accepted 17 January 2017 Available online 20 January 2017 0020-7403/ © 2017 Elsevier Ltd. All rights reserved.

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response, which results in a large reduction in the degrees of freedom, CPU time and memory required. The CPU time is further reduced by the use of XFEM [12,19] in each macro element to build the multiscale base functions in the presence of heterogeneities/discontinuities. In the present work, it is assumed that the discontinuities are randomly distributed in the RVE in such a way that they neither intersect each other nor the boundary of the RVE. This assumption avoids a need of complex level set function. Moreover, if the boundary of the macro element is intersected by the interface/discontinuity, it may cause loss of accuracy in the evaluation of basis function due to non-smoothness of the solution [38]. Since, the boundary of the RVE is matching with the boundary of the macro element hence the interaction of the discontinuities with the boundary of the RVE is avoided. This work is organized as: an introduction to the problem is provided in the Section 1. A detailed formulation of the MsXFEM is presented in Section 2. A detailed procedure to generate the 2-D RVE consisting of randomly dispersed circular shaped particles and/or voids of varying radii is discussed in Section 3. The periodic boundary conditions are also explained in this section. The RVE is then analyzed by MsXFEM to evaluate the elastic moduli of the heterogeneous material for three cases i.e. matrix with particles, matrix with voids and matrix with particles and voids both. In Section 4, the elastic properties obtained by MsXFEM are compared with results obtained from various previous models [2,3,7] for volume fraction ranging from 5% to 50%. Finally, the concluding remarks are given in Section 5.

cumbersome and time-consuming task. To overcome the problems associated with FEM, the extended ﬁnite element method (XFEM) [14–17] and level set method (LSM) [18] have been developed to model the geometric discontinuities independent of the mesh. In XFEM, the modeling of arbitrary discontinuities is performed by enriching the ﬁeld function approximation [18]. Many researchers [12,13,19] have used the XFEM for simulation of heterogeneous materials to obtain the eﬀective properties. The inclusions of various shapes, sizes, and orientations were modelled. They [12,19] solved these problems by XFEM and found a signiﬁcant reduction in the computational time in comparison to FEM. In the past two decades, various multiscale methods were developed to simulate the mechanical behavior of heterogeneous materials at diﬀerent length scales. The main advantage of the multiscale homogenization methods is to provide an accurate global response at low computational cost. Hou and Wu [20] developed a multiscale FEM for solving an elliptic problem of the composite materials and ﬂow through porous media. The method is designed to capture the large scale nature of the solution without resolving all the micro-scale heterogeneities. Kouznetsova et al. [21] presented the ﬁrst-order multiscale homogenization scheme for heterogeneous mechanical problems. The secondorder computational homogenization scheme [22–24] were developed to evaluate constitutive behavior of heterogeneous materials. Multiscale methods were also used to handle fracture and damage problems. A hierarchical variational multiscale framework was developed by Mergheim [25] to model propagating discontinuities. A multiscale projection method, developed by Loehnert and Belytschko [26], was used to simulate the eﬀects of both macro-cracks and micro-cracks. Belytschko et al. [27] presented a coarse-graining multiscale method for fracture mechanics problems. Verhoosel et al. [28] proposed a multiscale framework to incorporate microstructural information in a macroscale discrete fracture model. The computational homogenization algorithm was presented for both adhesive and cohesive failure at the macroscale. The multiscale homogenization was further extended by Nguyen et al. [29,30] for cohesive crack modeling of heterogeneous quasi-brittle materials using new evolutionary boundary condition for microscopic samples. Rate dependent fracture in heterogeneous quasibrittle materials was also presented under dynamic loading [31]. Some researchers utilized XFEM [32,33] at microscale for modeling the discontinuities without conformal meshing in multiscale environment. The mechanical behavior of glass ﬁber reinforced composite was obtained by Kastner et al. [33] using XFEM based multiscale approach. Hettich et al. [32] coupled the XFEM and LSM with variational multiscale approach [34] to simulate the linear and nonlinear mechanical behavior of the heterogeneous composite. Recently, Zhang et al. [35–37] developed a new multiscale ﬁnite element method (MsFEM) based on multiscale basis function and FEM. They used MsFEM for solving elasto-plastic and dynamic problems of heterogeneous materials. The main idea behind the method is to numerically construct the multiscale base functions, which can eﬃciently capture the micro-scale features in each macro element by locally solving the Dirichlet boundary value problems. Thus, the important micro-scale information is obtained for a macro-scale simulation using multiscale basis functions, which generate an equivalent stiﬀness matrix for the macro-element. In the present work, the merits of XFEM [14] and LSM [18] are used to avoid conformal meshing whereas the MsFEM [35–37] is used to reduce the computational time. The XFEM coupled with the MsFEM, is named as multiscale extended ﬁnite element method (MsXFEM). The equivalent properties of heterogeneous material containing large number of discontinuities (particles and/or voids) have been accurately and eﬃciently evaluated by MsXFEM using the concept of RVE. In MsXFEM, the macroscopic behavior of the material has been modelled by MsFEM at macro scale whereas XFEM has been employed to model the discontinuities at a micro level. The coarse-scale mesh is ﬁnally solved for external boundary conditions to obtain the overall global

2. Implementation of MsXFEM The standard FEM and MsXFEM can be diﬀerentiated on the basis of construction of base functions. Consider a heterogeneous structure containing large number of discontinuities (particles and voids) as shown in Fig. 1(a). The analysis of such domain directly by FEM or XFEM requires huge number of degrees of freedoms and memory space, as the element size required must be less than the size of discontinuities. In this work, multiscale extended ﬁnite element method (MsXFEM) is developed for simulation of heterogeneous material. The key steps of MsXFEM are as follows, 1. A physical domain is converted into ﬁnite element model using macro elemental mesh as shown in Fig. 1(b), which is ﬁnally solved for macro-scale displacements. 2. Each macro-element is further subdivided into micro-mesh, called micro-scale ﬁnite element model as shown in Fig. 1(c) with microelements. The size of micro-elements is kept suﬃciently smaller than the size of discontinuities.

īt

(a)

N

(b)

īu (c)

Fig. 1. Schematic diagram of heterogeneous structure for MsXFEM; (a) heterogeneous structure with boundary conditions; (b) ﬁnite element model with macro-mesh; (c) discretization of macro-element into micro-mesh.

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enrichment can be written as [14,18],

3. The discontinuities/defects are analyzed at micro-scale by developing a micro-scale model using XFEM and level set to avoid conformal micro-mesh. 4. The micro-scale model is solved using special displacement boundary conditions to numerically construct the basis functions (N ). The basis functions act as a via medium to share information from microscale to macro-scale and vice-versa. 5. Once the basis functions are created, the micro-scale information is gathered into macro-scale through the equivalent stiﬀness matrices evaluated for each macro element. 6. The macro-scale stiﬀness matrices of all elements are assembled and then solved for macroscopic displacement as per actual external boundary conditions. 7. Once the macroscopic displacements are calculated, the microscopic displacements are obtained using the basis functions.

∑ ψI (x )uI + ∑

u(x ) =

I∈n

J ∈ nc

ψJ (x )χ[ϕ(x )] aJ

(2)

where, ψI is the ﬁnite element shape function associated with node I satisfying the partition of unity criterion, χ[ϕ(x )] is the enrichment function, aJ is the vector of additional unknowns due to enrichment, n is the set of all the nodes in the domain Ωm and nc is the set of nodes whose support contains the evaluation point x and are intersect by the discontinuity. In XFEM, a void is modelled with enrichment function V (x ) for nodes, which interact with the boundary of the void. If a node lies inside the void then V (x ) = 0 and if a node is outside the void then V (x ) = 1. The particle interface modeling is carried out with enrichment function χ[ϕ(x )], deﬁned as χ[ϕ(x )] = ϕ(x ) where ϕ(x ) is the level set function. The circular discontinuity in a 2-D space is deﬁned by the level set function ϕ(x , y ) = x 2 + y 2 − r , where ϕ = 0 is on the circle.

Since the problem domain is solved only for the macro degrees of freedom, the computation time reduces signiﬁcantly.

2.3. MsXFEM approximation 2.1. Equilibrium equation, boundary conditions and constitutive relations In multiscale formulation, the micro-scale and macro-scale displacements are bridged using multiscale basis functions. The micro-scale displacement ﬁelds within a particular macro-element occupying region Ωm in Fig. 2 can be expressed as,

Consider a 2-D heterogeneous domain (Ω ) containing randomly distributed discontinuities such as particles and/or voids as shown in Fig. 1(a). The equilibrium and boundary conditions for this problem are given as,

∇. σ + b = 0 in Ω

(1a)

σ . nˆ = t on Γt

(1b)

Γu

(1c)

u=u

on

4

4

4

vj = ∑i =1 Niyx (xj , yj ) ui + ∑i =1 Niyy(xj , yj ) vi

3 e4

1

4

(3b)

e3

u = N uE

e2

where, u is the nodal displacement vector of the micro-scale mesh in a particular macro-element, uE is the nodal displacement vector at the macro-scale level and N is the multiscale base functions matrix. The expanded form of terms u , uE and N are as follows,

e e1

4

where uj and vj are the micro-scale displacement components along x and y-directions respectively; akx and aky are the additional displacement degrees of freedom at micro-scale due to enrichment along x and ydirections respectively; ui and vi are the macro-scale displacement components at a particular node of the macro-element (occupying region Ωm ) along x and y-directions respectively; Nixx (xj , yj ) and Niyy(xj , yj ) are the basis functions of a macro-mesh node i in x and y -directions respectively; Nixy(xj , yj ) is an additional coupling term relating the displacement ﬁeld in the x direction within the microelement to the unit displacement of macro-mesh node i in the y -direction. Niyx (xj , yj ) is deﬁned in a similar way as Nixy(xj , yj ).Naixx (xk , yk ), Naiyy(xk , yk ), Naixy(xk , yk ) and Naiyx (xk , yk ) are the basis functions associated with additional displacement degrees of freedom. i =1–4 are the nodes of macro-element occupying a region Ωm ; j=1, 2, …, n are the total number of micro-mesh nodes in a macro-element; k=1, 2, …, m are the total number of enriched nodes in the macroelement. Eqs. (3a) and (3b) can be expressed in a vector form as,

The schematic of MsXFEM is depicted in Fig. 2, where the microscale mesh for the evaluation of reference solution by XFEM and the macro-scale mesh for the computation by MsXFEM are shown. The micro-scale displacement u(x ), within a particular macro element Ωm containing discontinuities is approximated using XFEM. Hence, the XFEM displacement approximation in the generalized form for the domain containing particles and voids with the partition of unity 4

4

aky = ∑i =1 Naiyx (xk , yk ) ui + ∑i =1 Naiyy(xk , yk ) vi

2.2. XFEM approximation

Particle

4

(3a)

akx = ∑i =1 Naixx (xk , yk ) ui + ∑i =1 Naixy(xk , yk ) vi

where, σ is the Cauchy stress tensor, u is the displacement ﬁeld, b is the vector of body force per unit volume and nˆ is the unit outward normal. The displacements are prescribed on displacement boundary, Γu and tractions are prescribed on traction boundary, Γt . The strain-displacement relations are given as, ε = ε(u) = ∇s u . The constitutive relation for the elastic material is given by Hooke's law σ = D ε where, D is the Hooke's tensor.

Void

4

uj = ∑i =1 Nixx (xj , yj ) ui + ∑i =1 Nixy(xj , yj ) vi

Macro-element with micro mesh

2

Micro-element

u = {u1 v1 u 2 v2 … un vn : a1x a1y a 2x a 2y … amx amy}T

(5)

uE = {u1 v1 u 2 v2 u3 v3 u4 v4}T

(6)

N = {R1T R2T …

y

(4)

RnT : RaT1 RaT2 …

T

T Ram }

(7)

where,

⎡ N1xx (xj , y ) N1xy(xj , y ) … N4xy(xj , y )⎤ j j j ⎥ Rj = ⎢ ⎢⎣ N1yx (xj , yj ) N1yy(xj , yj ) … N4yy(xj , yj )⎥⎦

x Fig. 2. Schematic description of the MsXFEM.

279

(8a)

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⎡ Na1xx (xk , y ) Na1xy(xk , y ) … Na 4xy(xk , y )⎤ k k k ⎥ Rak = ⎢ ⎢⎣ Na1yx (xk , yk ) Na1yy(xk , yk ) … Na 4yy(xk , yk )⎥⎦

p

q

KE = ∑ f =1 Kf + ∑e =1 Ke where Kf = GfT Kf Gf

(8b)

Ke = GeT KeGe

(13)

Standard ﬁnite element method is implemented and all the macro scale stiﬀness matrices are assembled to get the global stiﬀness matrix (K ). The macroscopic displacement ﬁeld (U ) can be obtained by solving the following equation for externally applied boundary condition,

2.4. Construction of base functions Consider a macro-element as shown in Fig. 2, which occupies a subregion Ωm (Ωm ⊂Ω ). The equilibrium equation i.e. Eqs. (1a)–(1c) is solved for this macro element along with the speciﬁed boundary conditions [35–37]. This solution provides micro-scale displacement (u ) at each node of the micro-mesh within this macro-element. The macro-scale displacement (uE ) over this element is known from the speciﬁed boundary condition. As u and uE are known, the basis function matrix (N ) for the speciﬁed boundary condition is numerically constructed from Eq. (4) (for more details on the construction of basis functions, refer Refs. [35–37]). The choice of boundary conditions for the construction of the base functions has got a major inﬂuence on the accuracy of the multiscale solution. In this work, periodic boundary conditions are used to construct the basis function for the heterogeneous material. The multi-constraints of periodic boundary conditions are implemented by using Lagrangian multiplier method [39]. The numerically obtained base functions take care of the extra degrees of freedom due to enrichment at micro-scale.

KU = Fext

(14)

where, Fext is a vector of externally applied forces. 2.6. Strain and stress at micro-scale The numerically evaluated base functions are further used to evaluate the strain and stress in the micro-scale element. The macroscopic displacements obtained from Eq. (14), are used to obtain the nodal displacements of the micro scale element e . Thus, the strain and stress of the micro scale enriched (e.g. e) and non-enriched (e.g. f ) elements are given as,

εe = Beue = BeGeuE

(15a)

εf = Bf u f = Bf Gf uE

(15b)

σe = Dε e e = DB e eGeuE

(16a)

σf = Df εf = Df Bf Gf uE

(16b)

2.5. Equivalent stiﬀness matrix of macro element Once the base functions are obtained, the equivalent stiﬀness matrix of a macro-element can be derived. Consider an arbitrary microelement e which is intersected by a macro-scale discontinuity as shown in Fig. 2. The elemental strain energy of the micro-element is given by

Πe =

1 T ue Keue 2

where

Ke =

∫Ω (BeT DB e et ) dΩe

This is called downscaling to evaluate strain and stress at micro scale and can be used further to calculate the total strain energy of the deformed structure. A detailed multiscale framework to obtain the microscopic displacement, strain and stress is shown in Fig. 3.

(9)

e

3. Evaluation of equivalent property

where, ue is the displacement vector of element e containing the standard and the extra degrees of freedom due to enrichment, Ke is the element stiﬀness matrix of the enriched element, Be is the enriched strain displacement transformation matrix, De stress strain transformation matrix and t is the element thickness. Using Eqs. (4), (8a) and (8b), we have

Accurate evaluation of the equivalent properties of heterogeneous composite material requires a large number of discontinuities to be modelled in the selected domain of the composite. This selected domain is known as the representative volume element (RVE). Thus, this section describes a methodology for RVE generation along with a procedure for the evaluation of equivalent properties. The periodic boundary conditions used to obtain the elastic properties are also explained. In addition, diﬀerent analytical homogenization techniques are also reviewed brieﬂy.

ue = GeuE T

where Ge = [ Re1 Re2 Re3 Re 4 Rea1 Rea2 Rea3 Rea 4 ]

(10)

where, Ge is a transition matrix obtained from the basis function matrix which denotes the mapping relations between the displacement vectors of micro-element nodes and macro-element nodes. The strain energy of the micro-element Πe may be written as,

1 Πe = uET GeT KeGeuE 2

3.1. RVE generation Physical and geometrical properties of a composite material are evaluated through using a suitable RVE. An RVE is a model of a material microstructure, which is used to obtain the response of the homogenized macroscopic continuum at a macroscopic material point. The proper choice of RVE largely determines the accuracy of the modeling of a heterogeneous material. The actual choice of the RVE size is a rather delicate task. The RVE should be large enough to represent the microstructure, without introducing non-existing properties (e.g. undesired anisotropy) and at the same time, it should be small enough to allow eﬃcient computational modeling [40]. Hill [41] argued that an RVE is well deﬁned if it reﬂects the material microstructure and if the responses to periodic boundary conditions converge to constant values. From convergence study [9,11], it has been noticed that approximately 200–300 particles/pores are required in a RVE for accurate evaluation of equivalent properties of a non-uniform and non-regular microstructure. To verify this, the eﬀective elastic moduli of a particulate composite containing a large number of hard particles and voids (30% by volume) is evaluated using MsXFEM under periodic boundary conditions. The elastic moduli of matrix and hard particles used for the analysis are

(11)

Adding up the strain energy of all the micro-scale elements, the total strain energy of the macro-element is obtained as, p

ΠE =

∑ Πf f =1

q

+

∑ Πe = e =1

⎞ ⎛ p q ⎞ 1 T⎜ 1 ⎛ uE ⎜ ∑ GfT Kf Gf ⎟⎟uE + uET ⎜⎜∑ GeT KeGe⎟⎟uE 2 ⎝ f =1 2 ⎝ e =1 ⎠ ⎠ (12)

where, p and q respectively are the total number of standard nonenriched (e.g. f shown in Fig. 2) and enriched micro-scale elements (e.g. e shown in Fig. 2) within the micro-mesh; Kf and Gf are the standard ﬁnite element stiﬀness and transition matrices for nonenriched element respectively. In transition matrix Gf , only the ﬁrst four terms are used from the transition matrix Ge to obtain potential energy of non-enriched micro-element. Thus, the total strain energy of the macro-element can be written with the help of macro-scale displacement uE . Thus, the equivalent stiﬀness matrix of the macroelement can be given as, 280

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Table 3 Eﬀect of number of voids on the elastic moduli (30% volume fraction).

discretize structure into macromesh

discretize macro-element with micro-mesh calculate stiffness matrix using Eq. (9)

loop over microelements

assemble micro-scale stiffness solve equilibrium Eq. (1) for specified boundary condition

Number of particles

Mean shear modulus (GPa)

Standard deviation

Mean Young's modulus (GPa)

Standard deviation

20 20 20 20 20 20 20

20 45 64 125 180 245 320

14.28 14.06 13.95 13.88 13.84 13.83 13.82

0.71 0.62 0.40 0.29 0.24 0.19 0.18

37.52 37.24 37.19 37.09 37.02 36.91 36.91

1.16 0.96 0.64 0.37 0.36 0.34 0.34

loop over macroelements

52.6

calculate stiffness matrix using Eq. (13) obtain macro-element stiffness matrix using Eq. (13)

loop over microelements

assemble the macroscopic stiffness to obtain global

140.0

Effective shear modulus Effective Young's modulus

using Eq. (4) Effective shear modulus (GPa)

construct basis function

Number of trials

52.4 139.5

52.2 139.0

Effective Young's modulus (GPa)

START

52.0

solve Eq. (14) to calculate global displacement

0

100

200

300

138.5

Number of particles

calculate and using Eq. (15)-(16), which are further processed for calculation of equivalent properties

Fig. 4. Eﬀective elastic moduli with number of hard particles for 30% volume fraction.

14.4 37.5

Effective shear modulus Effective Young's modulus Effective shear modulus (GPa)

Fig. 3. Flowchart showing multiscale framework. Table 1 Elastic properties of the constituent materials. Material

Young's Modulus (GPa)

Shear modulus (GPa)

Poisson's ratio

Matrix (Zr-2.5 Nb) Hard Particles (SiC)

97 450

36.16 192.30

0.341 0.17

14.2 37.2

14.0 36.9

Effective Young's modulus (GPa)

END

13.8 Table 2 Eﬀect of number of hard particles on the elastic moduli (30% volume fraction). Number of trials

Number of particles

Mean shear modulus (GPa)

Standard deviation

Mean Young's modulus (GPa)

Standard deviation

20 20 20 20 20 20 20

20 45 64 125 180 245 320

52.65 52.23 52.05 52.00 51.98 51.98 51.98

0.38 0.24 0.14 0.09 0.08 0.07 0.07

139.76 139.04 138.76 138.66 138.63 138.62 138.62

0.63 0.50 0.33 0.24 0.18 0.17 0.16

0

100

200 Number of particles

300

36.6

Fig. 5. Eﬀective elastic moduli with number of voids for 30% volume fraction.

Tables 2 and 3 respectively. It is observed that the number of particles do not inﬂuence the mean eﬀective elastic properties much although the standard deviation of the eﬀective properties decreases with the increase in the number of particles. Figs. 4 and 5 show the variation of the eﬀective elastic moduli with number of hard particles/voids, and it is concluded that if the number of particles/voids in the matrix are more than 245 then there will be no signiﬁcant change in the eﬀective property. Hence, in the present work, 245 randomly distributed particles/voids of non-uniform size are modelled in the RVE to evaluate the homogenized elastic properties. The dimensions of square RVE is evaluated based on number of discontinuities and volume fraction.

given in Table 1. The mean and standard deviation of the evaluated values of the eﬀective shear and Young's modulus of the composite as a function of the number of hard particles and voids are presented in 281

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numerically calculated strain energy of the heterogeneous medium under the action of uniform shear strain. Similarly, to compute Young's modulus (E ), a uniaxial uniform tensile strain (say x -direction) is applied then the strain energy of homogenous medium is given as,

Uxx =

1 2

∫V σxxεxxdV

where,

σxx =

(20)

E (1 − ν 2)

εxx

for

plane

stress

and

Poisson's

ratio

ν = (E /2G ) − 1. Equating the strain energy of the homogenous medium with the strain energy of the heterogeneous composite, the Young's modulus of homogenized composite can be obtained as, E = 4G −

Various RVEs with diﬀerent amount of reinforcement volume fractions (5–50%) are taken for analysis. The RVE is analyzed in the multiscale framework without scale separation. There are two meshes in multiscale framework, micro mesh and macro mesh as shown in Fig. 6. In one macro element, ﬁve discontinuities (particles/voids) are modelled using XFEM at the micro scale. The equivalent stiﬀness matrix of such macro elements is derived as explained in Section 2. An RVE consists of 49 macro elements and each macro element contains ﬁve randomly distributed discontinuities (particles and/or voids) which ensures random distribution of 245 discontinuities in the RVE.

3.3. Periodic boundary conditions Suquet [43] proposed the periodic boundary conditions to solve the RVE for extracting the macroscopic mechanical response of the composite materials. A general formulation for the periodic boundary conditions is given as,

ui = εijnj + δi,

Particulate and porous composites are essentially heterogeneous in nature. Hence, it is diﬃcult to get the analytical solutions for such composite materials. Thus, an RVE is used to evaluate the homogenized properties of particulate and porous composites. The principle of energy equivalence [42] is used to evaluate eﬀective elastic properties of the composites. In this approach, the strain energy of the actual heterogeneous composite is compared with the strain energy of the equivalent homogeneous material under same load and boundary conditions. The MsXFEM is used to compute strain energy of the actual heterogeneous material, and the strain energy of the equivalent homogeneous material is obtained analytically in terms of unknown material properties. Thus, the unknown material properties of the equivalent homogeneous material can be obtained by equating the two strain energies. The equivalent homogenous material is assumed to be isotropic in nature. The total strain energy of a body is given by

∫V σij εijdV

T

1 2

∫V σxyγxydV

T

vΓ = vΓ

B

(23)

where, Δx is the applied unidirectional unit displacement along x direction. Similarly, the displacements at left edge Γ L and right edge Γ R is expressed as, R

L

R

L

uΓ = uΓ vΓ = vΓ

(24)

The Eqs. (23) and (24) represent the periodic boundary conditions applied on the RVE to determine the equivalent shear modulus as shown in Fig. 7(a). Similarly, to obtain equivalent Young's modulus, the periodic boundary conditions (as shown in Fig. 7(b)) are deﬁned as,

(17)

T

B

R

L

uΓ = uΓ ,

T

B

R

L

vΓ = vΓ

u Γ = u Γ + Δx, v Γ = v Γ

(25)

3.4. Analytical homogenization techniques

(18)

Using stress-strain relationship σxy = Gγxy and equating strain energy of homogenous medium with strain energy of heterogeneous composite, the shear modulus of homogenized composite can be obtained as,

G=

B

u Γ = u Γ + Δx

where, σij and εij are the local stress and strain components respectively. To ﬁnd shear modulus (G ), a uniform shear strain is applied on a homogenous medium, then above equation reduces to

Uxy =

(22)

where, εij is the average strain, nj is a position vector from a reference point to a point at the boundary and δi can be considered as the periodic part of the displacement along the boundary surfaces and it depends on the applied load. A more simpliﬁed form of this equation given by Xia et al. [44] is used to solve a 2-D RVE. To evaluate the equivalent shear modulus, the displacements at top edge Γ T and bottom edge Γ B are constrained as,

3.2. Elastic property evaluation

1 2

(21)

where, Uxx is the numerically calculated strain energy of the heterogeneous medium due to the action of uniaxial uniform tensile strain. Finally, the eﬀective Poisson's ratio of the homogenized composite is obtained using the relation, ν = (E /2G ) − 1.

Fig. 6. (a) MsXFEM model of RVE containing 245 particles and (b) Finite element model of particular macro element containing 5 particles.

Uij =

2εxx2 G2l 2 Uxx

The equivalent properties obtained using MsXFEM are compared with the analytical and semi-analytical models. Voigt [2] and Reuss [3] models are used for two-phase composite in which matrix and particles are assumed to be isotropic.

2Uxy γxy2 l 2

3.4.1. Voigt model It is based on the assumption that both matrix and particles are subjected to same uniform strain. Hence, the average strain in the matrix

(19)

where, l is the length of the side of the square RVE, and Uxy is the 282

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R.U. Patil et al.

y

y

x

x (a)

(b) Fig. 7. RVE subjected periodic boundary condition.

with each other. The CPU time is the total time required by MsXFEM (i.e. MsFEM and XFEM) to solve a complete problem in MATLAB 2016a solver on a HP Elitedesk machine with Intel (R) i7-6700 CPU @ 3.40 GHz processor and 32 GB RAM.

and particles is equal to the average strain of the composite. As per this model, the equivalent property (CC ) of two-phase composite is given as,

CC = VI CI + (1 − VI )CM

(26)

where, VI is the volume fraction of particles, CI is the property of particles and CM is the property of the matrix. 4.1. Case-I: RVE with hard particles 3.4.2. Reuss model In contrast to the Voigt model, the Reuss model assumes that the matrix and particles experiences the same uniform stress. As per this model, the equivalent property (CC ) of two-phase composite is given as,

(27)

It is known that Voigt model gives upper bound and Reuss model gives lower bound of the equivalent property. The relation for eﬀective property of porous composite (foam) is given by Ramakrishnan and Arunachalam [7]. The eﬀective Young's modulus of porous material is given as,

CC = C (1 − θ )2 /(1 + b0θ )

(28)

where b0 = (11 − 19ν )/{4(1 + ν )} to evaluate eﬀective shear modulus and b0 = 2 to 3ν to evaluate eﬀective Young's modulus of porous material, θ is the porosity volume fraction, C and ν are the modulus and the Poisson's ratio of the corresponding dense matrix. In this paper, the value of b0 = 3ν is taken to evaluate eﬀective Young's modulus for comparison of the results.

138.65

Effective shear modulus (GPa)

51.39

4. Results and discussion In this section, few representative example problems are solved using MsXFEM to evaluate the eﬀective elastic moduli of heterogeneous material containing hard particles and voids. The results obtained using MsXFEM are compared with those obtained from analytical and semianalytical models. To examine the validity of the MsXFEM, few numerical examples are also solved by standard XFEM under plane stress condition. In standard XFEM, the entire domain is discretized with a ﬁne mesh to resolve the heterogeneities. Note that the size of elements in the standard XFEM (applied directly over entire domain) is kept same as that of the micro-scale elements in the MsXFEM. In the entire manuscript, the standard XFEM applied directly over the entire domain at ﬁne scale will be referred as XFEM. Moreover, the computation time i.e. CPU time required by MsXFEM and XFEM is compared

51.38

Effective shear modulus Effective Young's modulus

138.55

51.37

51.36

138.60

26×26

28×28

30×30

32×32

34×34

36×36

38×38

40×40

Effective Young's modulus (GPa)

−1 ⎛V 1 − VI ⎞ CC = ⎜ I + ⎟ CM ⎠ ⎝ CI

In this case, the equivalent shear and Young's moduli of the particulate composite containing SiC hard particles and Zr-2.5Nb matrix is obtained for diﬀerent reinforcement volume fractions i.e. 5% to 50%. The RVE is studied using MsXFEM framework consisting of 49 macro-elements (8×8 nodes) and each macro element contains ﬁve randomly distributed circular particles as shown in Fig. 6. The diameter (d ) of non-overlapping circular particles varies from 40 to 100 µm. Several numerical trials were conducted on RVE in the presence of randomly distributed particles for an average particle size (diameter) of 70 µm. The ratio of RVE size (l ) to the average particle size (d ) remains nearly constant for various distributions. The RVE size is evaluated by using volume fraction and number of particles. The elastic properties of matrix and reinforcing particles are given in Table 1. A mesh size test is performed on a RVE having 245 particles to determine the optimum micro-mesh size. For this purpose, the equivalent shear modulus and Young's modulus are calculated for various

138.50

Micro-mesh density Fig. 8. Variation of eﬀective shear modulus (G) and eﬀective Young's modulus (E) with micro-mesh for 30% volume fraction of particles.

283

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400

Table 4 A comparison of computation time between MsXFEM and XFEM in case of composite with hard particles.

0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05

MsXFEM

XFEM (nodes×nodes)

Macro mesh (nodes×nodes)

Micro Mesh (nodes×nodes)

8×8 8×8 8×8 8×8 8×8 8×8 8×8 8×8 8×8 8×8

56×56 50×50 44×44 38×38 32×32 34×34 36×36 42×42 48×48 56×56

386×386 344×344 302×302 260×260 218×218 232×232 246×246 288×288 330×330 386×386

Computation time (min) MsXFEM

XFEM

63.1 45.3 26.3 18.0 15.2 15.4 16.3 22.1 26.5 39.9

375.1 237.8 198.3 11.5 106.3 108.4 118.8 121.6 142.7 168.9

MsXFEM XFEM

300 Computation time (min)

Volume Fraction

200

100

0

0.05

0.10

0.15

0.20

0.25 0.30 0.35 Volume fraction

0.40

0.45

0.50

Fig. 11. Variation of computation time with the volume fraction of hard particles.

120

90

37.3 Effective shear modulus Effective Young's modulus

80 70 60 50

37.1 13.90 37.0 13.85 36.9

40 30

37.2

13.95

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

36.8

13.80

Volume fraction

Effective Young's modulus (GPa)

Effective shear modulus G (GPa)

100

14.00

Voigt [2] Reuss [3] MsXFEM XFEM

Effective shear modulus (GPa)

110

28×28

30×30

32×32

34×34

36×36

38×38

40×40

42×42

Micro-mesh density

Fig. 9. Variation of eﬀective shear modulus (G) with the volume fraction of hard particles.

Fig. 12. Variation of eﬀective shear modulus (G) and eﬀective Young's modulus (E) with micro-mesh for 30% volume fraction of voids.

280 260

Effective Young's modulus (GPa)

240 220

Table 5 A comparison of computation time between MsXFEM and XFEM in case of composite with voids.

Voigt [2] Reuss [3] MsXFEM XFEM

Volume Fraction

MsXFEM

XFEM (nodes×nodes)

Computation time (min)

200 180 160

0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05

140 120 100 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 Volume fraction

Macro mesh (nodes×nodes)

Micro Mesh (nodes×nodes)

8×8 8×8 8×8 8×8 8×8 8×8 8×8 8×8 8×8 8×8

58×58 52×52 46×46 40×40 34×34 36×36 38×38 44×44 50×50 58×58

400×400 358×358 316×316 274×274 232×232 246×246 260×260 302×302 344×344 400×400

MsXFEM

XFEM

63.9 46.1 27.1 18.7 15.6 15.9 16.9 22.8 27.3 40.7

378.6 214.3 201.7 114.5 109.8 111.9 121.4 124.5 145.2 171.4

Fig. 10. Variation of eﬀective Young's modulus (E) with the volume fraction of hard particles.

similar optimum mesh density test is performed for remaining volume fractions of particles and the number of micro-mesh nodes for a converged solution are listed in Table 4. The variation of equivalent shear and Young's moduli with volume fraction is shown in Figs. 9 and 10 respectively. The numerical results lie in the upper and lower bounds calculated using Voigt and Reuss

micro-mesh densities for 30% volume fraction of the reinforcing particles as shown in Fig. 8. The values of eﬀective moduli converged for a micro-mesh of 32×32 nodes. The number of particles and their average size are kept constant in the analysis. The RVE size increases with the decrease in the volume fraction of the particles. Hence, a 284

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24.00

Ramakrishnan and Arunachalam [7] MsXFEM XFEM

30

Effective shear modulus (GPa)

Effective shear modulus (GPa)

35

25 20 15

66.10

Effective shear modulus Effective Young's modulus

66.05

66.00

23.95

65.95

Effective Young's modulus (GPa)

40

10 5

65.90

23.90 28×28

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

30×30

32×32

34×34

36×36

38×38

40×40

42×42

Micro-mesh density

Volume fraction

Fig. 16. Variation of eﬀective shear modulus (G) and eﬀective Young's modulus (E) with micro-mesh for 30% volume fraction of particles and voids.

Fig. 13. Variation of eﬀective shear modulus (G) with the volume fraction of voids.

100

Effective Young's modulus E (GPa)

Table 6 A comparison of computation time between MsXFEM and XFEM in case of composite with hard particles and voids.

Ramakrishnan and Arunachalam [7] MsXFEM XFEM

90 80

Volume Fraction

70

MsXFEM

60

XFEM (nodes×nodes)

Macro mesh (nodes×nodes)

Micro Mesh (nodes×nodes)

8×8 8×8 8×8 8×8 8×8 8×8 8×8 8×8 8×8 8×8

58×58 52×52 46×46 40×40 34×34 36×36 38×38 44×44 50×50 58×58

Computation time (min) MsXFEM

XFEM

69.2 50.3 30.2 20.9 17.1 17.4 18.8 25.1 30.1 43.9

399.9 257.9 214.8 122.2 118.2 115.9 129.6 132.8 156.7 183.5

50 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05

40 30 20 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 Volume fraction

400×400 358×358 316×316 274×274 232×232 246×246 260×260 302×302 344×344 400×400

Fig. 14. Variation of eﬀective Young's modulus (E) with the volume fraction of voids.

400

35 MsXFEM XFEM

30

Effective shear modulus (GPa)

Computation time (min)

300

200

100

25

20

15

MsXFEM XFEM

10 0

0.05

0.10

0.15

0.20

0.25 0.30 0.35 Volume fraction

0.40

0.45

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

0.50

Volume fraction Fig. 17. Variation of eﬀective shear modulus (G) with the volume fraction of particles and voids.

Fig. 15. Variation of computation time with the volume fraction of voids.

models. The results of MsXFEM are in good agreement with XFEM results. A comparison of CPU time required for MsXFEM and XFEM is provided in Table 4 and Fig. 11. For the comparison of memory requirement of both the approaches, consider a composite with volume fraction of 0.30. For this problem, the total number of macro elements are 49 (8×8) and micro-elements in each macro elements are

961(32×32). In case of XFEM (where XFEM is applied directly on micro-scale model), total elements required are 49×961 and memory required is a function of fn(a × 49 × 961), where a is the degrees of freedom per node (e.g. a = 2 for 2-D). In case of MsXFEM, the memory required is a function of fn(a × 49) + f (a × 961), which is quite less as

285

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100

particles is taken same as discussed in case-I. The optimum micro-mesh density for 30% volume fraction of hard particles and voids together (12% particles and 18% voids) is shown in Fig. 16. For other volume fractions, the same is tabulated in Table 6. The equivalent shear and Young's moduli for diﬀerent volume fractions are presented in Figs. 17 and 18 respectively. From the results presented in ﬁgures, it is observed that the results obtained by MsXFEM agree well with XFEM. The computation time required by MsXFEM and XFEM are presented in Table 6. The computation time required by MsXFEM and XFEM is also represented graphically in Fig. 19. Thus, it can be seen that the CPU time required by MsXFEM is quite small as compared to XFEM.

Effective Young's modulus (GPa)

90 80 70 60 50 MsXFEM XFEM

40

5. Conclusions

30

In the present work, a new MsXFEM scheme is proposed to evaluate the eﬀective macroscopic properties of heterogeneous composites. A MATLAB code has been developed to analyze the heterogeneous RVE. The multiscale base functions are constructed numerically to capture the small scale features induced due to the heterogeneities. These small scale features at micro-scale have been modelled by XFEM. Based on these simulations, the following conclusions are drawn,

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 Volume Fraction Fig. 18. Variation of eﬀective Young's modulus (E) with the volume fraction of particles and voids.

400

• The elastic moduli of the heterogeneous composite increase with the

MsXFEM XFEM

Computation time (min)

300

• •

200

100

0

0.05

0.10

0.15

0.20

0.25 0.30 0.35 Volume fraction

0.40

0.45

• •

0.50

Fig. 19. Variation of computation time with the volume fraction of particles and voids.

compared to XFEM. Thus, it is observed that the MsXFEM is computationally more eﬃcient in comparison to XFEM.

increase in the volume fraction of the hard particles. As expected, the evaluated values of elastic moduli are close to Reuss model, which gives a better estimate of the elastic property as compared to Voigt model. For porous composite, the results obtained by MsXFEM are in good agreement with Ramakrishnan and Arunachalam [7]. In case of composites having both voids and particles (volume fraction of voids is 1.5 times the volume fraction of particles), the eﬀective elastic moduli of the composite decrease with the increase in the volume fraction of voids and particles together. However, the rate of decrease is lower as compared to the case with only voids. MsXFEM results are in good agreement with the standard XFEM applied directly. The proposed MsXFEM is computationally very eﬃcient for the materials having large number of heterogeneities, as it requires less memory and CPU time as compared to standard XFEM.

Acknowledgement This work does not receive any funding from any funding agency.

4.2. Case-II: RVE with voids References In this case, an RVE containing 245 circular voids is analyzed using MsXFEM. The average size and distribution of voids is taken same as in case-I. The mesh convergence of eﬀective moduli in the presence of 30% voids (by volume) lying in the matrix is shown in Fig. 12. The converged micro-mesh sizes for other void volume fractions are listed in Table 5. The values of equivalent shear and Young's moduli are compared with the same obtained by Ramakrishnan and Arunachalam [7] for diﬀerent volume fractions are presented. The results are given in Figs. 13 and 14. It is observed that the homogenized properties obtained by MsXFEM are very close to the properties obtained from Ramakrishnan and Arunachalam [7] and XFEM applied directly. The computational time required to solve the RVE using MsXFEM and XFEM is provided in Table 5 and Fig. 15.

[1] Ma J, Temizer I, Wriggers P. Random homogenization analysis in linear elasticity based on analytical bounds and estimates. Int J Solids Struct 2011;48(2):280–91. [2] Voigt W. Ueber die beziehung zwischen den beiden elasticittsconstanten isotroper korper. Ann Phys 1889;274(12):573–87. [3] Reuss A. Berechnung der ﬂiegrenze von mischkristallen auf grund der plastizittsbedingung fr einkristalle. J Appl Math Mech 1929;9(1):49–58. [4] Mori T, Tanaka K. Average stress in matrix and average elastic energy of materials with misﬁtting inclusions. Acta Met Mater 1973;21(5):571–4. [5] Hashin Z, Shtrikman S. A variational approach to the theory of the elastic behavior of multiphase materials. J Mech Phys Solids 1963;11(2):127–40. [6] Milton GW. Bounds on the elastic and transport properties of two-component composites. J Mech Phys Solids 1982;30:177–91. [7] Ramakrishnan N, Arunachalam VS. Eﬀective elastic moduli of porous solids. J Mater Sci 1990;25:3930–7. [8] Kari S, Berger H, Rodriguez–Ramos R. Computational evaluation of eﬀective material properties of composites reinforced by randomly distributed spherical particles. Compos Struct 2007;77:223–31. [9] Singh IV, Shedbale AS, Mishra BK. Material property evaluation of particle reinforced composites using ﬁnite element approach. J Compos Mater 2016;50(20):2757–71. [10] Brassart L, Doghri I, Delannay L. Homogenization of elasto-plastic composites coupled with a nonlinear ﬁnite element analysis of the equivalent inclusion problem. Int J Solids Struct 2010;47:716–29. [11] Kaddouri W, Moumen A, Kanit T, Madani S, Imad A. On the eﬀect of inclusion shape on eﬀective thermal conductivity of heterogeneous materials. Mech Mater

4.3. Case-III: RVE with hard particles and voids In this case, the RVE containing both hard particles and voids is analyzed for combined volume fractions ranging from 5% to 50%. A heterogeneous domain is taken such that volume fraction of voids contribute 60% and particles contribute 40%. The diameter of voids varies from 60 to 150 µm with an average size of 105 µm. The size of 286

International Journal of Mechanical Sciences 122 (2017) 277–287

R.U. Patil et al. 2016;92:28–41. [12] Hiriyur B, Waisman H, Deodatis G. Uncertainty quantiﬁcation in homogenization of heterogeneous microstructures modeled by XFEM. Int J Numer Methods Eng 2011;88:257–78. [13] Savvas D, Stefanou G, Papadrakakis M, Deodatis G. Homogenization of random heterogeneous media with inclusions of arbitrary shape modeled by XFEM. Comput Mech 2014;54:1221–35. [14] Moes N, Dolbow J, Belytschko T. A ﬁnite element method for crack growth without remeshing. Int J Numer Methods Eng 1999;46:131–50. [15] Singh IV, Mishra BK, Bhattacharya S, Patil RU. The numerical simulation of fatigue crack growth using extended ﬁnite element method. Int J Fatigue 2012;36:109–19. [16] Sharma K, Singh IV, Mishra BK, Shedbale AS. The eﬀect of inhomogeneities on an edge crack: a numerical study using XFEM. Int J Comput Methods Eng Sci Mech 2013;14:505–23. [17] Kumar S, Shedbale AS, Singh IV, Mishra BK. Elasto-plastic fatigue crack growth simulations in the presence of ﬂaws using XFEM. Front Struct Civ Eng 2015;9:420–40. [18] Sukumar N, Chopp DL, Moes N, Belytschko T. Modeling holes and inclusions by level sets in the extended ﬁnite element method. Comput Methods Appl Mech Eng 2001;190:6183–200. [19] Gal E, Suday E, Waisman H. Homogenization of materials having inclusions surrounded by layers modeled by the extended ﬁnite element method. Int J Multiscale Com 2013;11(3):239–52. [20] Hou TY, Wu XH. A multiscale ﬁnite element method for elliptic problems in composite materials and porous media. J Comput Phys 1997;134:169–89. [21] Kouznetsova V, Brekelmans WAM, Baaijens FPT. An approach to micro–macro modelling of heterogeneous materials. Comput Mech 2001;27:37–48. [22] Kouznetsova VG, Geers MGD, Brekelmans WAM. Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme. Int J Numer Methods Eng 2002;54(8):1235–60. [23] Kaczmarczyk L, Pearce CJ, Bicanic N. Scale transition and enforcement of RVE boundary conditions in second-order computational homogenization. Int J Numer Methods Eng 2008;74(3):506–22. [24] Geers MGD, Kouznetsova VG, Brekelmans WAM. Multi-scale computational homogenization: trends and challenges. J Comput Appl Math 2010;234(7):2175–82. [25] Mergheim J. A variational multiscale method to model crack propagation at ﬁnite strains. Int J Numer Methods Engng 2009;80:269–89. [26] Loehnert S, Belytschko T. A multiscale projection method for macro/microcrack simulations. Int J Numer Methods Eng 2007;71:1466–82. [27] Belytschko T, Song JH. Coarse-graining of multiscale crack propagation. Int J Numer Methods Eng 2010;81:537–63. [28] Verhoosel CV, Remmers JJC, Gutierrez MA, deBorst R. Computational homogeni-

[29]

[30]

[31] [32]

[33]

[34]

[35] [36]

[37] [38] [39] [40] [41] [42] [43]

[44]

287

sation for adhesive and cohesive failure in quasibrittle solids. Int J Numer Methods Eng 2010;83:1155–79. Nguyen VP, Stroeven M, Sluys LJ. An enhanced continuous discontinuous multiscale method for modelling mode-I failure in random heterogeneous quasi-brittle materials. Eng Fract Mech 2012;79:78–102. Nguyen VP, Lloberas-Valls O, Stroeven M, Sluys LJ. Computational homogenization for multiscale crack modelling. Implementational and computational aspects. Int J Numer Methods Eng 2012;89:192–226. Karamnejad A, Nguyen VP, Sluys LJ. A multi-scale rate dependent crack model for quasi-brittle heterogeneous materials. Eng Fract Mech 2013;104:96–113. Hettich T, Hund A, Ramm E. Modeling of failure in composites by X-FEM and level sets within a multiscale framework. Comput Methods Appl Mech Eng 2008;197:414–24. Kastner M, Haasemann G, Ulbricht V. Multiscale XFEM‐modelling and simulation of the inelastic material behaviour of textile‐reinforced polymers. Int J Numer Methods Eng 2011;86:477–98. Hughes TJR, Feijoo GR, Mazzei L, Quincy JB. The variational multiscale method-a paradigm for computational mechanics. Comput Methods Appl Mech Eng 1998;166:3–24. Zhang HW, Wu KJ, Lu J, Fu ZD. Extended multiscale ﬁnite element method for mechanical analysis of heterogeneous materials. Acta Mech Sin 2010;26:899–920. Zhang HW, Wu JK, Fu ZD. Extended multiscale ﬁnite element method for elastoplastic analysis of 2D periodic lattice truss materials. Comput Mech 2010;45:623–35. Zhang HW, Wu JK, Lv J. A new multiscale computational method for elasto-plastic analysis of heterogeneous materials. Comput Mech 2012;49:149–69. Li Z, Lin T, Wu X. New Cartesian grid methods for interface problems using the ﬁnite element formulation. Numer Math 2003;96:61–98. Cook RD, Malkus DS, Plesha ME, Witt RJ. Concepts and applications of ﬁnite element analysis. 4th ed. US: John Wiley & Sons; 1989. Galvanetto U, Ferri Aliabadi MH. Multiscale modeling in solid mechanics computational approaches. London UK: Imperial College Press; 2010. Hill R. Elastic properties of reinforced solids: some theoretical principles. J Mech Phys Solids 1963;11:357–72. Agarwal A, Singh IV, Mishra BK. Numerical prediction of elasto-plastic behaviour of interpenetrating phase composites by EFGM. Comp Part B 2013;51:327–36. Suquet P. Elements of homogenization theory for inelastic solid mechanics. In: Sanchez-Palencia E, Zaoui A (eds.), Homogenization techniques for composite media; 1987. 194–275. Xia ZH, Zhou CW, Yong QL. On selection of repeated unit cell model and application of uniﬁed periodic boundary conditions in micro-mechanical analysis of composites. Int J Solids Struct 2006;43(2):266–78.

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