ijnme 87 0149 peric eadsn et al on micro-to-macro transitions

8/13/2019 IJNME 87 0149 Peric Eadsn Et Al on Micro-To-macro Transitions

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150 D. PERICET AL.

capability. Therefore a means of continuous interchange of information between scales is needed

if better predictive modelling of material behaviour is to be attempted.

As the basic principles for the micromacro modelling of heterogeneous materials were intro-

duced (see [13]), this technique has proved to be a very effective way to deal with arbitrary

physically non-linear and time-dependent material behaviour at microlevel. During the last decade

various approaches and techniques for the micromacro modelling and simulation of heteroge-

neous materials have been proposed. Among these we highlight the contributions by Oden andco-workers [4, 5], Ghoshet al. [6], Suquet and co-workers [7, 8], Fish and co-workers [911],

Smitet al. [12], Feyel and Chaboche [13], Miehe and co-workers [1416], Pellegrinoet al. [17],

Kouznetsova et al. [18, 19], Ladevzeet al. [20, 21], Terada and co-workers [2224], Markovic

and Ibrahimbegovi c [25], Zohdi and Wriggers [26], Belytschkoet al. [27, 28] and Hund and

Ramm [29].

The present article discusses issues related to a computational strategy for homogenization

of microstructures with non-linear material behaviour undergoing small strains. As the aim is to

provide the basic ingredients of the computational strategy allowing for concurrent simulation at

different scales of the model, a simple model is considered comprising two scales arising, for

instance, in the modelling of heterogeneous composite materials.

The aims of this article are two fold: first, a unified variational basis is outlined, which, apart

from the continuum-based variational formulation at both micro- and macroscales of the problem,

also includes the variational formulation governing micro-to-macro transitions. The unified varia-

tional formulation provides clear axiomatic basis and hierarchy related to the choice of boundary

conditions at the microscale. In this way, issues and ambiguities are avoided that often arise when

the microscale boundary conditions are imposed in anad hoc manner.

This unified variational basis leads naturally to a generic finite element-based framework for

homogenization-based multi-scale analysis of heterogenous solids. Hence, as the second impor-

tant aim of this article, the finite element implementation arising from the multi-scale varia-

tional framework is developed. The attention is restricted to deformation-driven microstructures,

which have been proven to provide a convenient computational format [14, 16, 30]. Four types of

boundary conditions are imposed over the Representative Volume Element (RVE): (i) Taylor model,

(ii) linear displacements, (iii) periodic displacement fluctuations with antiperiodic tractions and

(iv) uniform boundary tractions. These boundary conditions satisfy the fundamental Hill-Mandel

averaging principle, which equates microscopic and macroscopic stress power [3134]. Theresulting computational strategy is characterized by the NewtonRaphson solution of the discrete

boundary value problem, and incorporates the appropriate exact tangent operators.

Numerical examples of both microscale and two-scale finite element simulations are presented

to illustrate the scope and the benefits of the described computational strategy.

2. MULTI-SCALE CONSTITUTIVE THEORY: VARIATIONAL FRAMEWORK

This section summarizes the (kinematical) variational basis of the family of homogenization-

based multi-scale constitutive theories addressed in this paper. The variational setting summarized

here is described in further detail by de Souza Neto and Feijo [35]. It provides a structured

axiomatic framework whereby a clear distinction is made between the basic assumptions and theirconsequences to the resulting theory. As often seen in the literature, these are usually blurred and

difficult to grasp when the theory is presented on the basis ofad hoc arguments.

The starting point of these theories is the assumption that any material point x of the (macro-

scopic) continuum is associated to a local RVE whose domain , with boundary , (refer to

Figure 1), has a characteristic length, l, much smaller than the characteristic length, l , of the

macro-continuum. The domain of the RVE is assumed to consist in general of a solid part, s,

and a void part, v:

=s

v. (1)

Copyright 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2011; 87:149170

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ON MICRO-TO-MACRO TRANSITIONS FOR MULTI-SCALE ANALYSIS 151

macroscopiccontinuum

(macro-scale)

x

representativevolume element

(micro-scale)

y

l

l


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152 D. PERICET AL.

variational framework used here to cast this family of multi-scale models, this constraint is promptly

incorporated by simply requiring the (as yet not defined) set K of kinematically admissible RVE

displacement fields to be a subset of the minimally constrained set of kinematically admissible

microscopic displacements, denoted K :

K K v, sufficiently regular

vsndA = V e (6)withsufficiently regularmeaning that the relevant functions have the sufficient degree of regularity

so that all operations in which they are involved make sense.

Alternatively, by splitting u, without loss of generality, into a sum

u(y, t) =e(t)y+ u(y, t) (7)

of a displacement due to a homogeneous strain field, e(t)y, and adisplacement fluctuation field,

u, the above constraint is made equivalent to requiring that the functional set K of kinematicallyadmissible displacement fluctuations of the RVE be a subset of the minimally constrained space

of kinematically admissible displacement fluctuations, K

:

K K

v, sufficiently regular

vsndA =0 . (8)At this point, a further fundamental assumption is introduced: The (as yet not defined) set K

is required to be asubspaceof K

. For application within a virtual work-based variational setting,

it is worth noting that in this case the associated space of virtual kinematically admissible of RVE

displacements, defined as

V {g=v1 v2|v1,v2 K}, (9)

trivially coincides with the space of kinematically admissible displacement fluctuations:

V = K. (10)

Further, the same arguments applied to the rate form

u = ey+u (11)

of the additive split (7) establish that anykinematically admissible fluctuation velocity, u, satisfies

u V = K. (12)

Following the split (7) the microscopic strain (3) can be expressed as the sum

e(y, t) =e (t)+su(y, t) (13)

of a homogeneous strain field (coinciding with the macroscopic, average strain) and a field suthat represents a fluctuation about the average strain.

2.3. Equilibrium of the RVEThe next basic axiom of the theory establishes that the RVE must be in equilibrium at any instant

of its deformation history. Assuming that the RVE is subjected in general to a body force field

b=b(y, t) and an external traction field te =te(y, t) exerted upon the RVE across its externalboundary , thePrinciple of Virtual Workestablishes that the RVE is in equilibrium if and only

if the variational equation

s

r(y, t) : sgdV

s

b(y, t)gdV

te(y, t) gdA = 0

gV (14)


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holds at eacht, where V is an appropriate space of virtual displacements of the RVE, subjected

to the general constraints set out in (8,10). In establishing the RVE equilibrium, we have assumed

for simplicity that possible non-zero tractions exerted upon the solid part of the RVE across its

solid/void interfaces (which could arise, for instance, in the presence of a pressurized fluid in the

voids) are absent. These can be incorporated into the theory, however, in a straightforward manner.

2.4. Macroscopic stress

Similar to the macroscopic strain definition (3), the macroscopic stress tensor, r, is taken as the

volume average of the microscopic stress field, r, over the RVE:

r(t) 1

V

s

r(y, t) dV. (15)

2.5. The Hill-Mandel principle of macro-homogeneity

Another important axiom underlying models of the present type is the Hill-Mandel Principle of

Macro-homogeneity [31, 36] which requires the macroscopic stress power to equal the volume

average of the microscopic stress power for any kinematically admissible motion of the RVE. This

is expressed by the equation

r : e= 1

V

s

r : e dV (16)

that must hold for any kinematically admissible microscopic strain rate field, e. By combining(16) with (14) and taking (1012) into account together with the fact that V is a vector space, we

can establish after straightforward manipulations that (16) is equivalent to the following variational

equation:

te gdA = 0,

s

b gdV= 0 gV (17)

in terms of the RVE boundary traction and body force fields denoted, respectively, te and b. That

is, the virtual work of the RVE body force and surface traction fields vanishthey are the reaction

forces associated to the imposed kinematical constraints embedded in the choice ofV.

2.6. The RVE equilibrium problem

As a consequence of the above, the variational equilibrium statement (14) for the RVE is reduced to

s

r : sgdV= 0 gV. (18)

Further, we assume that at any timet the stress at each point y of the RVE is delivered by a

generic constitutive functional Sy of the strain history et(y) at that point up to time t:

r(y, t) =Sy(et(y)). (19)

This constitutive assumption, together with the equilibrium equation (18) leads to the definition

of the RVE equilibrium problem which consists in finding, for a given macroscopic strain e (a

function of time), a displacement fluctuation functionu V such that

s

Sy{[e+su(y)]

t} : sgdV= 0 gV. (20)


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2.7. Characterization of the multi-scale constitutive model

The general multi-scale constitutive model in the present context is defined as follows. For a given

macroscopic strain history, we must first solve the RVE equilibrium problem defined by (20). With

the solution u at hand, the macroscopic stress tensor is determined according to the averagingrelation (15), i.e. we have

r(t) =S(et) 1V

s

Sy{[e+su]t} dV, (21)

where S denotes the resulting (homogenized) macroscopic constitutive response functional.

2.7.1. The choice of kinematical constraints. The characterization of a multi-scale model of the

present type is completed with the choice of a suitable space of kinematically admissible displace-

ment fluctuations, V K

, so as to make the RVE equilibrium problem (18) well-posed. In

general, different choices lead to different macroscopic constitutive response functionals. The

following choices are frequently employed:

(i) The Taylor model, or rule of mixtures [37] :

V

= TaylorV

{0}. (22)

The strain in this case is uniform over the RVE:

e(y) =e y. (23)

The (generally non-zero) RVE body force and boundary surface traction fields in this case

are reactions to the imposed uniform strain field.

(ii) Linear boundary displacements (or zero boundary fluctuations) model:

V =linV {v, sufficiently regular |v(y) =0 y}. (24)

The displacements of the boundary of the RVE for this class of models are fully prescribed

as

u(y) =ey y. (25)

The boundary surface traction field of the RVE is this case is a reaction to the prescribed

boundary displacements of the RVE. The reactive body force field in this case is

b(y) =0 y. (26)

(iii) Periodic boundary fluctuations. This assumption is typically associated with the description

of media with periodic microstructure. The macrostructure in this case is generated by

the periodic repetition of the RVE [8]. For simplicity, we will focus the description on

two-dimensional problems and we shall follow the notation adopted by Mieheet al. [14].

Consider, for example, the square or hexagonal RVEs, as illustrated in Figure 2. In this

case, each pairi of sides consists of equally sized subsets

+

i and

i

of, with respective unit normals

n+i and ni ,

such that

ni = n

+i . (27)

A one-to-one correspondence exists between the points of+i and i . That is, each point

y+ +i has a corresponding pairy i .


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Figure 2. RVE geometries for periodic media. Square and hexagonal cells.

The key kinematical constraint for this class of models is that the displacement fluctuation

must be periodic on the boundary of the RVE. That is, for each pair {y+,y}of boundarymaterial points we have

u(y+, t) = u(y

, t). (28)

Accordingly, the space V is defined as

V =perV {u,suff. reg.u(y

+, t) = u(y, t) pairs{y+,y}}. (29)

The variational relations (17) in this case imply anti-periodic boundary surface tractions:

t(y+, t) = t(y, t) pairs {y+,y}, (30)

and zero body force field b=0.(iv) The minimally constrained(or uniform boundary traction) model:

V =uniV K

. (31)

It can be shown [35] that the distribution of stress vector on the RVE boundary that satisfies

(17)1 is given by

r(y, t)n(y) =r(t)n(y) y. (32)

The reactive body force field, in turn, is zero. As for the Taylor model and linear boundary

displacements assumption, there are no restrictions on the geometry of the RVE in the

present case.

Remark 1

The use of different definitions ofV for a given RVE produces, in general, different estimates

of the corresponding macroscopic constitutive response. The general rule is as follows: First note

that upon inspection of definitions (22), (24), (29) and (31) it follows that

TaylorV

linV

perV

uniV K

. (33)

That is, the Taylor model gives the stiffest (most kinematically constrained) solution to the

microscopic equilibrium problem, followed in order of decreasing stiffness, by the linear boundary

displacement, the periodic displacement fluctuation and the uniform boundary traction model.

The uniform traction model produces the most compliant (least kinematically constrained)

solution.

3. NUMERICAL APPROXIMATION

This section provides a brief description of the computational implementation of multi-scale

constitutive theories of the above type within a non-linear finite element framework.


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3.1. The incremental equilibrium problem

At the outset, we shall assume the constitutive behaviour at the RVE level to be described by

conventional internal variable-based dissipative constitutive theories, whereby the stress tensor is

obtained by integrating a set of ordinary differential equations in time (or pseudo-time) for the given

strain tensor history. Elasto-plasticity and visco-plasticity are classical, widely used examples of

such specializations of (19). In these cases, numerical approximations to the initial value problem

defined by the constitutive equations of the model are usually obtained by Euler-type difference

schemes. For a typical time (or pseudo-time) interval [tn, tn+1], with known set an of internal

variables attn , the stress rn+1 attn+1 is a (generally implicit) function of the (prescribed) strain

en+1 att

n+1 (refer, for instance, to [3841] for a detailed account of procedures of this kind in

the context of plasticity and visco-plasticity). This can be symbolically represented as

rn+1 = ry(e

n+1 ;a

n), (34)

where ry denotes the integration algorithm-related implicit incremental constitutive function at thepoint of interest, y.

The above leads to the definition of an incremental version of the homogenized constitutive

function defined in (21), obtained by replacing Sy with its time-discrete counterpart ry :

rn+1 = r(en+1; an)

1

V

s

ry(en+1 +sun+1 ;a

n) dV, (35)

wherean denotes the field of internal variable sets over s at timetn and un+1 is the displacement

fluctuation field of the RVE at tn+1the solution to the time-discrete version of equilibrium

problem (20):

s

ry(en+1 +sun+1 ;a

n) : sgdV= 0 gV. (36)

3.2. The incremental constitutive function: homogenized constitutive tangent

3.2.1. The incremental constitutive function. Consider first the simplest multi-scale constitutivemodelthe Taylor model. In this case, we have

V = {0} un+1 =en+1y un+1

=0, (37)

and the homogenized constitutive function can be written as

Taylorr(en+1,t; an)

1

V

s

r(en+1,t;an) dV, (38)

where an denotes the field of variables an over theentire domain s. That is, at each point y

s,

an = an(y). (39)

In the general case, the homogenized incremental constitutive function for the stress is definedimplicitly through the incremental microscopic equilibrium equation (36). The stress rn+1 is

obtained by first solving (36) and then, with u|n+1 at hand, computing

rn+1 = 1

V

s

r(en+1 +su|n+1,t;an) dV. (40)

That is, the incremental macroscopic stress constitutive function is defined as

homr(en+1,t; an)

1

V

s

r(en+1 +su|n+1,t;an) dV, (41)


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where u|n+1 is itself a function solely of en+1 (for given t and an) defined as the solutionof (36). Note that, under the Taylor assumption (37), definition (41) recovers the Taylor model

incremental constitutive function (38).

3.2.2. Homogenized constitutive tangent. Let

e

=e

n+

1+e (42)

be the perturbed macroscopic strain, where is a scalar and edenotes a generic incremental strain

tensor. The incremental tangent, denoted homD is afourth-order tensorthat expresses the tangential

relationship between the macroscopic stress and macroscopic strain tensor at tn+1, consistently

with the homogenized incremental constitutive function (41). That is, for any macroscopic strain

direction e, we have

homr(e,t; an) =

homr(en+1,t; an)+

homD :e+o(), (43)

where homD :e is the directional derivative of the incremental constitutive function hom r in the

direction e:

homD :e

d

d=0

homr(en+1 +e,t; an). (44)

The operator homD is simply

homD

e

n+1

homr(e,t; an). (45)

Taylor model. For the incremental version of the Taylor model, we obtain, by differentiating (38),

the following homogenized tangent operator:

homD= TaylorD

1

V

s

D dV, (46)

where

D =

e

|n+1

r(e,t;an) (47)

is the tangent operator consistent with the microscopic incremental constitutive law. That is, for the

Taylor model, the incremental homogenized tangent tensor is the volume average of the microscopic

incremental constitutive tangent tensor.

The general case. To obtain a canonical expression for the general case, let us first consider the

perturbed microscopic displacement fluctuation

u(y) un+1

(y)+u(y). (48)

The tangential relation between e and u is obtained from the linearization of the incremental

equilibrium problem defined by (36). Given e, find the field u V that solves the linearvariational equation

s

sg :D : su dV=

s

sg :D dV

:e gV. (49)

It can be shown that the following compact canonical formula for the general homogenized

incremental constitutive tangent operator, can be obtained:

homD= TaylorD+D. (50)

Here, only the contribution D depends on the choice of space V.


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158 D. PERICET AL.

3.3. Finite element discretization and solution

We now focus on the finite element solution of the time-discrete equilibrium problem (36)a

crucial step in the definition of the approximate homogenized constitutive functional. Following

a standard notation, the finite element approximation to problem (36) for a given discretizationh

consists in determining the unknown vector un+1 Vh of global nodal displacement fluctuations

such that

Gh(un+1 )

s,h

BTy(

n+1 +Bun+1 ) dV

= 0 Vh, (51)

where s,h denotes the discretized RVE domain, B the global straindisplacement matrix (or

discrete symmetric gradient operator), n+1 is the fixed (given) array of macroscopic engineering

strains attn+1, y (with upright ) is the functional that delivers the finite element array of stress

components, denotes global vectors of nodal virtual displacements of the RVE and Vh is the

finite-dimensional space of virtual nodal displacement vectors associated with the finite element

discretizationh of the domain s.

The solution to the (generally non-linear) problem (51) can be efficiently undertaken by the

NewtonRaphson iterative scheme, whose typical iteration (k) consists in solving the linearized

form, F

(k1) +K(k1) u(k)

= 0 Vh , (52)

for the unknown iterative nodal displacement fluctuations vector, u(k) V

h where

F(k1)

s,h

BTy(

n+1 +Bu(k1) ) dV, (53)

and

K(k1) s,h B

TD

(k1)BdV (54)

is the tangent stiffness matrix of the RVE with

D(k1)

dy

d

=n+1+Bu

(k1)

(55)

denoting the consistent constitutive tangent matrix field over the RVE domain. In the above the

bracketed superscript denotes the Newton iteration number and the time station superscript n +1

has been dropped whenever convenient for ease of notation. With the solutionu(k) at hand, the new

guess u(k) for the displacement fluctuation attn+1 is obtained according to the NewtonRaphson

update formula:

u(k) =u

(k1) +u

(k) . (56)

4. FINITE ELEMENT IMPLEMENTATION

Trivially, for the Taylor model, the choice of space of admissible fluctuations already defines

the solution to the equilibrium problem (52) so that no numerical procedures are required in

this case, with the macro-stress being computed simply as the volume-fraction weighted average

of the microstresses resulting from the application of the macro-strain history to the different

phases of the RVE. Under the assumption of linear boundary displacements, the solution of

problem (52) follows the conventional route of general linear solid mechanics problemshere


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y y+ -y-

y+

i-i+

i+

i-

Figure 3. Discretized RVEs for periodic media.

with the degrees of freedom (fluctuations) of the boundary fully prescribed as zero. Hence, the

finite element implementation of this class of multi-scale models requires no further consideration.

For theperiodic boundary condition andminimally constrained models, however, the kinematic

boundary conditions of the RVE are non-conventional. The main difference lies in the finite element-

generated finite dimensional spaces of admissible fluctuations and virtual displacements whose

constraints, here, are not simply described in terms of either fully constrained or completely free

nodal degrees of freedom. Thus, some ambiguities exist in the literature regarding the imposition

of the non-conventional boundary conditions, which are often prescribed in an ad hoc manner.It is demonstrated below that finite element implementation of the periodic boundary condition

and minimally constrained models follows naturally from the unified variational framework for

multi-scale analysis described in Section 2.

4.1. Periodic boundary fluctuations model

For the periodic boundary displacement fluctuations model, the RVE geometry must comply

with the constraints set out in item (iii) listed in Section 2.7.1. In this case, it is convenient to

assume, further, that each boundary nodei+, with coordinatesy+i , has a pairi, with coordinates

yi , as schematically illustrated in Figure 3. Under this assumption, the space V

h of discretized

kinematically admissible nodal displacement fluctuation vectors (with periodicity on the boundary)

can then be defined as

Vh=

v=

vi

v+

v

|v+ =v

, (57)

where vi , v+ and v denote the vectors containing, respectively, the degrees of freedom of the

RVE interior and the portions + and of the RVE boundary. Here we adopt the direct approach

suggested by Michel et al. [8] whereby the periodicity constraint is enforced exactly in the

discretized space of fluctuations and virtual displacements. This is at variance with the treatment

adopted by Miehe and Koch [16] who used a Lagrange multiplier method to enforce the discrete

space constraint.By splitting F, K, u and in the same fashion as v in the above and taking into account

definition (57) as well as the fact that both and u(k)

belong to space Vh

, the linearizedequation (52) takes the form

Fi

F+

F

(k1)

+

kii ki+ ki

k+i k++ k+

ki k+ k

(k1)

ui

u+

u+

(k)

i

+

+

= 0 i ,+. (58)

This assumption is not necessary, but simplifies considerably the finite element implementation of the periodicfluctuations model.


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160 D. PERICET AL.

Straightforward manipulations, considering the repetition ofu+ and + in the relevant vectors of

nodal degrees of freedom, reduce the linearized discrete equilibrium equation (58) to the following

form:

Fi

F+ +F

(k1)

+kii ki+ +ki

k+i +ki k++ +k+ +k+ +k

(k1)

uiu+

(k)

i

+= 0

i ,+, (59)

which, finally, in view of the arbitrariness of i and +, yields the linear system of algebraic

equations for the unknown vectors u(k)i andu

(k)+ ,

kii ki+ +ki

k+i +ki k++ +k+ +k+ +k

(k1)ui

u+

(k)=

Fi

F+ +F

(k1). (60)

4.2. Minimally constrained model

A procedure completely analogous to the one above is followed to obtain the final NewtonRaphson

set of algebraic finite element equations under the assumption of minimally constrained kinematics

(or uniform RVE boundary traction). We then start by defining the discrete counterpart of the

minimally constrained space of fluctuations and virtual displacements (8, 31):

Vh

v=

vi

vb

h

Nbvb sndA =0

, (61)

wherevbis the vector containing the boundary degrees of freedom and Nbis the global interpolation

matrix associated solely with the boundary nodes of the discretized RVE.

It can be easily established that the integral constraint on vb can be equivalently written inmatrix form as

Cvb =0, (62)

where Cis theconstraint matrix. For an RVE mesh withk interior nodes andm boundary nodes,

in the two-dimensional case vb is a vector of dimension 2m and C is the 32m matrix given by

C

=

h

Nk+1 n1 dA 0

h

Nk+mn1 dA 0

0 h

Nk+1 n2 dA 0 h

Nk+mn2pacedA

h

Nk+1 n2 dA

h

Nk+1 n1 dA

h

Nmn2 dA

h

Nmn1 dA

, (63)

where n1 and n2 denote the components of the outward unit normal field along the global

orthonormal basis{e1,e2}and Nj ,j = 1, ,m, are the global shape functions associated with theboundary nodes. In this case, Equation (62) poses three linear constraints upon the total number

of 2m boundary degrees of freedom of the discrete RVE. For three-dimensional RVEs, vb is of

dimension 3m and matrix Chas dimension 6 3m.


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In practice, rather than using global shape functions, matrix C is obtained by assembling

elemental matrices which in two dimensions, for an element e with p nodes on the intersection

(e) between the boundary of the element and the boundary of the RVE, read

C(e) =

(e)N

(e)1 n1 dA 0

(e)N(e)p n1 dA 0

0

(e)N(e)1 n2 dA 0

(e)N(e)p n2 dA

(e)

N(e)1 n2 dA

(e)N

(e)1 n1 dA

(e)N(e)p n2 dA

(e)N(e)p n1 dA

, (64)

where we have assumed that the nodes of element e lying on (e) are locally numbered 1 to p

andN(e)j ,j = 1, . . . ,p, are the associated local shape functions. For example, a conventional eight-

noded bilinear quadrilateral element (of the type employed in Section 5), having a single straight

side of lengthl(e) withn=e1 and three equally spaced nodes intersecting the RVE boundary, has

C(e) = l(e)

16

0 23

0 16

0

0 0 0 0 0 0

0 16

0 23

0 16

. (65)

In order to handle constraint (62) upon the discrete space of fluctuations and virtual displacements

it is convenient to split vb as

vb =

vf

vd

vp

, (66)

where the subscripts f, d and p stand, respectively, forfree,dependentandprescribeddegrees

of freedom on the boundary of the discrete RVE. Accordingly, the global constraint matrix is

partitioned as

C= [Cf Cd Cp], (67)

so that the constraint equation (62) reads as

[Cf Cd Cp]

vf

vd

vp

=0. (68)

Prescribed degrees of freedom are needed here to remove rigid body displacements of the RVE and

make the corresponding discrete equilibrium problem (51) well-posed. Trivially, we then prescribe

vp =0, (69)

where, in two and three dimensions, vpcontains, respectively, three and six suitably chosen degrees

of freedom. The constraint equation is now reduced to

Cf Cd

vfvd

=0. (70)

In two dimensions, the above represents three scalar equations involving 2m 3 variables, whereasin the three-dimensional case, we have six scalar equations and 3m 6 variables. Hence, thenumber ofdependentvariablesthe dimension ofvdand of the square sub-matrix Cdis 3 and

6 for the two- and three-dimensional cases, respectively. The total number offree variablesthe


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162 D. PERICET AL.

dimension ofvfand number of columns ofCfis 2m 6 and 3m 12, respectively, in two andthree dimensions. Finally, following a trivial manipulation of (70), vdcan be expressed in terms

ofvf as

vd=Rvf, (71)

where

R C1d Cf. (72)

Note that the dependent degrees of freedom (corresponding to vd) must be chosen such that Cd is

invertible.

With the above considerations at hand, we can re-define the discrete space (61) of fluctuations

and virtual displacements of the RVE as

Vh

v=

vi

vf

vd

|vd=Rvf

, (73)

which, for convenience, contains now only the non-prescribed degrees of freedom.

The particularization of the linearized finite element equation (52) for the present case is obtained,analogously to (58), by splitting the corresponding vectors and tangential stiffness matrix according

to the above partitioning and taking (73) into account. This gives

Fi

Ff

Fd

(k1)

+

kii kif kid

kfi kff kfd

kdi kdf kdd

(k1)

ui

uf

Ruf

(k)

i

f

Rf

= 0 i ,f, (74)

which, after straightforward matrix manipulations taking into account the arbitrariness ofi and

f, is reduced to the final form

kii kif+kidR

kfi +RTkdi kff+kfdR+R

Tkdf+R

TkddR

(k1)ui

uf

(k)=

Fi

Ff+RTFd

(k1). (75)

5. NUMERICAL EXAMPLES

In this section numerical examples are presented to illustrate the scope and benefits of the described

computational strategy. The first set of numerical examples focuses on microstructure simulations

and discusses some important issues regarding numerical analysis at the microlevel, such as the

effect of boundary conditions, topology and distribution of heterogeneities. The second numerical

example considers a full two-scale simulation of a boundary value problem and incorporates all

the computational ingredients described in this paper. This example also includes a comparison

with a detailed single-scale analysis. The standard computational algorithms for elasto-plasticityand viscoplasticity are employed that underly the multi-scale approach, and we refer the reader to

[40, 4246] for detailed descriptions of the computational schemes behind the present simulations.

It should be emphasized that in order to ensure the robustness and efficiency of the multi-scale

simulations the recently proposed sub-stepping procedure is utilized [47].

5.1. Effect of cavity distribution on homogenized properties

5.1.1. Problem specifications. A square cell is considered representing an RVE at the microlevel.

The cell is composed of an elasto-plastic material with heterogeneities in the form of cavities.

Two models are considered: (i) a regular cell with a single circular void embedded in a soft matrix


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Figure 4. Regular cavity model.

depicted in Figure 4 and (ii) randomly generated distribution of voids surrounded by the soft matrix

given in Figure 6. For both models the void volume fraction of the cell is 15%.

Two types of finite elements are employed: linear three-noded triangle element and eight-noded

quadrilateral element. The matrix in all models is assumed to be composed of a von Mises elasto-

plastic material with linear strain hardening. The material properties assigned are: Youngs modulus

E= 70GPa, Poissons ratio= 0.2, the initial yield stress Y0 = 0.243GPa and the strain hardening

modulusH= 0.2GPa.

5.1.2. Analysis approach. All simulations in this section have been performed by employing the

computational homogenization under the plane stress assumption in small strain regime. The

average stress is obtained by imposing the macro-strain over the RVE and subsequently solving

the microscopic initial boundary value problem for the boundary condition assumed. The generic

imposed macro-strain tensor is expressed by

[11, 22,212] = [0.001,0.001,0.0034].

The loading programme considered is proportional in the sense that the final strain at the end of

each step is defined by multiplying the above strain array by the relevant load factor. The analysis

is performed under three different boundary conditions considered in this paper: (i) linear boundary

displacements, (ii) periodic boundary displacement fluctuations and (iii) uniform boundary traction.

5.1.3. Study of the regular cavity model. A mesh of 350 8-noded isoparametric quadrilateral

elements (see Figure 4) is employed in this simulation. The mesh contains a total number of 1158

nodes.

Figures 5(a) and (b) show, respectively, the deformed mesh and the equivalent plastic strain

distribution for the linear displacement boundary condition. This plastic zone is clearly positioned

along the diagonal side of the unit cell in the direction of the imposed shear. The corresponding

results for the periodic boundary condition and uniform boundary traction assumption are given in

Figures 5 (c, d) and Figures 5 (e, f), respectively. From Figure 5, it can be seen that the plastic zone

has a distinctively different pattern under different boundary conditions. To ease visualization, all

deformed meshes of Figure 5 have been plotted with exaggerated displacements.

The overall stressstrain response is presented in terms of the effective homogenized stress andthe Euclidean norm of the average strain, given, respectively as

eff=

3J2, =

211 +

222 +2

212.

Figure 8 shows the resulting average stressstrain curves for this model. The obtained results

indicate that under linear displacement boundary condition the overall response of the regular cavity

model shows significantly stiffer homogenized behaviour than that predicted under the periodic

boundary condition. On the other hand, the results obtained for the uniform boundary traction

assumption show the softest behaviour.


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164 D. PERICET AL.

(a) (b)

(c) (d)

(e) (f )

Figure 5. Regular cavity model under linear displacement boundary condition (a)(b), periodic condition(c)(d) and uniform traction boundary condition (e)(f). (a), (c) and (e) represent the deformed mesh,

whereas (b), (d) and (f) are corresponding effective plastic strain contour plots.

Figure 6. RVE with randomly generated voids.

5.1.4. The RVE with randomly generated voids. The analysis carried out here is identical to the

one described above except for the RVE which now contains a randomly generated distribution of

void placements and sizes (see Figure 6). A mesh of standard three-node linear triangular elements

is employed in this simulation.

Figure 7 shows the equivalent plastic strain distribution for all the prescribed boundary condi-

tions. The occurrence of localized bands with significant plastic straining can be observed on all

contour plots. Significantly, unlike the case of the single cavity model all boundary conditions give


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Figure 7. Effective plastic strain contours for the unit cell with randomly generatedvoids for different boundary conditions.

a similar distribution of the plastic strain indicating the convergence of the results at the microlevel

with the increase of the statistical sample of heterogeneities.Figure 8 shows the average stressstrain curves for this model. It can be observed that the

microcell with randomly generated void distribution results in the stressstrain behaviour that

shows a small difference between the three different boundary conditions imposed at the microlevel.

This clearly indicates the convergence of the average properties with the increase of the statistical

sample representing the heterogeneities at the microlevel.

5.2. Two-scale analysis of stretching of an elasto-plastic perforated plate

In this section the analysis of the stretching of a perforated plate is performed. The perforated plate

problem is often used as a benchmarking example in computational plasticity. Here we consider


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166 D. PERICET AL.

Regular cavity. Linear

Regular cavity. Periodic

Regular cavity. Uniform traction

Random distribution. Linear

Random distribution. Uniform traction

Random distribution. Periodic

Figure 8. Effective stressstrain norm curves. Regular (single) cavity model and random cavity distributionmodel under different RVE boundary conditions.

R=

Figure 9. Plane-stress strip with a circular hole. Geometry and boundary conditions.

the plate composed of an elasto-plastic material and containing regularly distributed voids. The

plate has width 10 mm, length 18 mm and uniform thickness of 1 mm (see Figure 9). For obvious

symmetry reasons only one-quarter of the specimen is considered (see Figure 9). The simulation is

performed by imposing uniform displacement along the upper boundary. The elasto-plastic material

properties are identical to those used in the RVE simulations described above. Two simulations are

performed: (i) a coupled two-scale analysis using a relatively coarse macroscopic finite element

mesh with the constitutive response at each Gauss quadrature point defined by the computational

homogenization of an RVE consisting of a single cavity model (as discussed above) embedded

in an elasto-plastic matrix and (ii) a single-scale detailed analysis of the plate where the regular

distribution of cavities is discretized directly at the macroscale.

We remark that this example is presented merely to illustrate the potential applicability of

coupled two-scale analysis. It should be pointed out that we are not looking into the convergence

properties of such solutions, which would require a much more detailed analysis.


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Figure 10. Single-scale analysis of an elasto-plastic perforated plate: (a) finite element mesh and(b) distribution of equivalent plastic strain.

Figure 11. FE meshes at macro- and microlevels for multi-scale analysis.

5.2.1. Single-scale analysis. Single-scale analysis is used for comparative purposes and is

performed on a detailed finite element mesh of the problem given in Figure 10(a). The mesh is

composed of 11 216 4-node quadrilateral elements and 12 147 nodes. Figure 10(b) illustrates the

distribution of an equivalent plastic strain at the latter stages of the simulation.

5.2.2. Two-scale analysis. For the multi-scale analysis the perforated plate is defined as a homo-

geneous structure at the macrolevel, whereas at the microlevel an RVE is defined with side length

equal to 1 and a centred single void giving a void volume fraction of 50%. Linear three-nodedtriangle elements are employed at both macro- and microlevels (see Figure 11). The mesh at the

macrolevel is composed of 25 elements and 21 nodes, whereas at the microlevel the FE mesh is

composed of 603 elements and 352 nodes.

The multi-scale analysis has been performed under the three types of RVE boundary constraints

considered in this paper. In addition, an analysis using the Taylor assumption (the uniform RVE

strain or rule of mixtures) is carried out for comparison. Figure 12 plots the reaction force against

the prescribed top edge displacement for the four RVE assumptions considered. The different

The actual size of the RVE is immaterial here since size effects are not accounted for within the present approach.


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168 D. PERICET AL.

Regular cavity. Linear

Single scale

Taylor assumption

Multi-scale. Linear

Multi-scale. Uniform traction

Multi-scale. Periodic

Single scale

Taylor assumption

Multi-scale. Linear

Multi-scale. Uniform traction

Multi-scale. Periodic

Edge

reactio

n

[kN]

Prescribed edge displacement [mm]

Figure 12. Perforated plate. Reactiondisplacement curve.

assumptions produce markedly different forcedisplacement diagrams. As expected, the results

obtained for the Taylor model show substantially stiffer behaviour than all the others. The resulting

overall behaviour for the periodic boundary condition shows very good correspondence with the

results obtained by the detailed single-scale analysis. This is expected since the actual heterogeneity

is periodic in the macrostructure. Finally, uniform boundary traction assumption generates the

softest response.

6. CONCLUSIONS

A compact and efficient computational framework has been developed for the homogenization-

based multi-scale finite element analysis of solids. The computational framework relies on an

elegant variational formulation, which, in a natural way, introduces a hierarchy of boundary condi-tions at the microscale, and allows for direct treatment of micro-to-macro transitions.

Details of the novel implementation procedure for three different types of RVE kinematical

constraints have been described, including the corresponding tangent constitutive operators. The

implementation procedure is based on an algebraic factorization and, importantly, does not involve

any artificial parameters. As a result an efficient and robust overall scheme for multi-scale analysis

of heterogeneous materials has been developed.

The presented numerical tests confirm the successful implementation of the computational

procedure and efficient solution of the discrete multi-scale problem. The examples clearly illustrate

the bounding properties related to the choice of the microscale boundary conditions, which emanate

from the described variational structure. The significance of the statistical sample representing the

heterogenities at the microlevel is also illustrated.

The ongoing research is concerned with the analysis of more general non-linear materialbehaviour at the microscale at both small and finite strains (see [48] for representative application

to bio-materials) and design and optimization of microstructures [4951].

REFERENCES

1. Sanchez-Palencia E. Comportement local et macroscopique dun type de milieux physique htrognes.

International Journal of Engineering Science 1974; 12:231251.2. Suquet PM. Local and global aspects in the mathematical theory of plasticity. InPlasticity Today: Modelling

Methods and Applications, Sawczuk A (ed.). Elsevier Applied Science Publishers: Amsterdam, 1985.3. Suquet PM. Elements of homogenization for inelastic solid mechanics. In Homogenization Techniques for

Composite Media, Sanchez-Palencia E, Zaoui A (eds). Springer: Berlin, 1987.


DOI: 10.1002/nme


21/22


4. Oden JT, Vemaganti K, Moes N. Hierarchical modelling of heterogenous solids. Computer Methods in Applied

Mechanics and Engineering 1999; 172:225.

5. Zohdi TI, Oden JT, Rodin GJ. Hierarchical modeling of heterogeneous bodies.Computer Methods in Applied

Mechanics and Engineering 1996; 138:273298.

6. Ghosh S, Lee K, Murthy S. Two scale analysis of heterogeneous elastic-plastic materials with asymptotic

homogenization and Voronoi cell finite element method.Computer Methods in Applied Mechanics and Engineering

1996; 132:63116.

7. Moulinec H, Suquet P. A numerical method for computing the overall response of nonlinear composites withcomplex microstructure.Computer Methods in Applied Mechanics and Engineering 1998; 157:6994.

8. Michel JC, Moulinec H, Suquet P. Effective properties of composite materials with periodic microstructure: a

computational approach.Computer Methods in Applied Mechanics and Engineering 1999; 172:109143.

9. Fish J, Shek K, Pandheeradi M, Shephard MS. Computational plasticity for composite structures based on

mathematical homogenization: theory and practice. Computer Methods in Applied Mechanics and Engineering

1997; 148:5373.

10. Fish J, Zheng Y. Multiscale enrichment based on partition of unity. International Journal for Numerical Methods

in Engineering 2005; 62:13411359.

11. Yuan Z, Fish J. Toward realization of computational homogenization in practice. International Journal for

Numerical Methods in Engineering 2008; 73:361380.

12. Smit RJM, Brekelmans WAM, Meijer HEH. Prediction of the mechanical behaviour of nonlinear heterogeneous

systems by multi-level finite element modeling.Computer Methods in Applied Mechanics and Engineering 1998;

155:181192.

13. Feyel F, Chaboche J-L. FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre

SiC/Ti composite materials.Computer Methods in Applied Mechanics and Engineering 2000; 183

:309330.14. Miehe C, Schotte J, Schrder J. Computational micro-macro transitions and overall tangent moduli in the analysis

of polycrystals at large strains. Computational Materials Science 1999; 16:372382.

15. Miehe C, Schrder J, Schotte J. Computational homogenization analysis in finite plasticity. Simulation of texture

development in polycrystalline materials. Computer Methods in Applied Mechanics and Engineering 1999;

171:387418.

16. Miehe C, Koch A. Computational micro-macro transitions of discretized microstructures undergoing small strains.

Archives of Applied Mechanics 2002; 72:300317.

17. Pellegrino C, Galvanetto U, Schrefler BA. Numerical homogenization of periodic composite materials with non-

linear material components.International Journal for Numerical Methods in Engineering 1999; 46:16091637.

18. Kouznetsova VG, Brekelmans WAM, Baaijens FPT. An approach to micro-macro modelling of heterogeneous

materials.Computational Mechanics 2001; 27:3748.

19. Kouznetsova VG. Computational homogenization for multiscale analysis of multi-phase materials.Ph.D. Thesis,

TU University Eindhoven, Eindhoven, 2002.

20. Ladevze P, Loiseau O, Dureisseix D. A micro-macro and parallel computational strategy for highly heterogeneous

structures.International Journal for Numerical Methods in Engineering 2001; 52:121138.

21. Ladevze P. Multiscale modelling and computational strategies for composites.International Journal for Numerical

Methods in Engineering 2004; 60:233253.

22. Terada K, Kikuchi N. A class of general algorithms for multi-scale analyses of heterogeneous media. Computer

Methods in Applied Mechanics and Engineering 2001; 190:54275464.

23. Matsui K, Terada K, Yuge K. Two-scale finite element analysis of heterogeneous solids with periodic

microstructures.Computers and Structures 2004; 82:593606.

24. Watanabe I, Terada K, de Souza Neto EA, Peric D. Characterization of macroscopic tensile strength of

polycrystalline metals with two-scale finite element analysis.Journal of the Mechanics and Physics of Solids

2008; 56:11051125.

25. Markovic D, Ibrahimbegovic A. On micro-macro interface conditions for micro-scale based FEM for inelastic

behaviour of heterogeneous materials.Computer Methods in Applied Mechanics and Engineering2004; 193:5503

5523.

26. Zohdi TI, Wriggers P.Introduction to Computational Micromechanics. Springer: Berlin, 2005.

27. Belytschko T, Loehnert S, Song JH. Multiscale aggregating discontinuities: a method for circumventing loss of

material stability.International Journal for Numerical Methods in Engineering 2008; 73:869894.

28. Belytschko T, Song JH. Coarse-graining of multiscale crack propagation. International Journal for Numerical

Methods in Engineering 2010; 81:537563.

29. Hund A, Ramm E. Locality constraints within multiscale model for non-linear material behaviour. International

Journal for Numerical Methods in Engineering 2007; 70:16131632.

30. Swan CC. Techniques for stress- and strain-controlled homogenization of inelastic periodic composites. Computer

Methods in Applied Mechanics and Engineering 1994; 117:249267.

31. Mandel J. Plasticit Classique et Viscoplasticit.CISM Courses and Lectures No. 97. Springer: Udine, Italy,

1971.

32. Hill R. On constitutive macro-variables for heterogeneous solids at finite strain.Proceedings of the Royal Society

of London, Series A 1972; 326:131147.

33. Willis JR. Variational and related methods for the overall properties of composites. Advances in Applied Mechanics

1981; 21:178.


DOI: 10.1002/nme


22/22

170 D. PERICET AL.

34. Nemat-Nasser S, Hori M. Micromechanics: Overall Properties of Heterogeneous Materials. North-Holland:

Amsterdam, 1993.

35. de Souza Neto EA, Feijo RA. Variational foundations of multi-scale constitutive models of solid: small and

large strain kinematical formulation.National Laboratory for Scientific Computing(LNCC/MCT), Brazil,Internal

Research &Development Report No. 16/2006, 2006.

36. Hill R. A self-consistent mechanics of composite materials. Journal of the Mechanics and Physics of Solids

1965; 13(4):213222.

37. Taylor GI. Plastic strains in metals. Journal Institute of Metals 1938; 62

:307324.38. Simo JC, Hughes TJR. Computational inelasticity. Springer: New York, 1998.

39. Belytschko T, Liu WK, Moran B. Nonlinear Finite Elements for Continua and Structures. Wiley: New York,

2000.

40. de Souza Neto EA, Peric D, Owen DRJ.Computational Methods for Plasticity:Theory and Applications. Wiley:

Chichester, 2008.

41. Wriggers P.Nonlinear Finite Element Methods. Springer: Berlin, 2008.

42. de Souza Neto EA, Peric D. A computational framework for a class of fully coupled models for elastoplastic

damage at finite strains with reference to the linearization aspects. Computer Methods in Applied Mechanics and

Engineering 1996; 130:179193.

43. de Souza Neto EA, Peric D, Owen DRJ. Continuum modelling and numerical simulation of material damage at

finite strains.Archives of Computational Methods in Engineering 1998; 5:311384.

44. Dutko M, Peric D, Owen DRJ. Universal anisotropic yield criterion based on superquadric functional

representation1: algorithmic issues and accuracy analysis. Computer Methods in Applied Mechanics and

Engineering 1993; 109:7393.

45. Peric D. On a class of constitutive equations in viscoplasticity: formulation and computational issues. InternationalJournal for Numerical Methods in Engineering 1993; 36:13651393.

46. Peric D, de Souza Neto EA. A new computational model for Tresca plasticity at finite strains with an optimal

parametrization in the principal space. Computer Methods in Applied Mechanics and Engineering 1999; 171:

463489.

47. Somer DD, de Souza Neto EA, Dettmer WG, Peric D. A sub-stepping scheme for multi-scale analysis of solids.

Computer Methods in Applied Mechanics and Engineering 2009; 198:10061016.

48. Speirs DCD, de Souza Neto EA, Peric D. An approach to the mechanical constitutive modelling of arterial tissue

based on homogenization and optimization. Journal of Biomechanics 2008; 41:26732680.

49. Giusti SM, Novotny AA, de Souza Neto EA, Feijo RA. Sensitivity of the macroscopic elasticity tensor to

topological microstructural changes. Journal of the Mechanics and Physics of Solids 2009; 57:555570.

50. Giusti SM, Novotny AA, de Souza Neto EA, Feijo RA. Sensitivity of the macroscopic thermal conductivity

tensor to topological microstructural changes. Computer Methods in Applied Mechanics and Engineering 2009;

198:727739.

51. Giusti SM, Novotny AA, de Souza Neto EA. Sensitivity of the macroscopic response of elastic microstructures

to the insertion of inclusions. Proceedings of the Royal Society A 2010; 466:17031723.

Copyright 2010 John Wiley & Sons Ltd Int J Numer Meth Engng 2011; 87:149170

ijnme 87 0149 peric eadsn et al on micro-to-macro transitions

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