ng an plas-of anctors in errorsecewise. Thepos-

imateeter-omo-sted ineroge-

turale areaades.][13]oderiesinearcopicnearoper-

Finite Deformation Plasticity for Composite Structures: Computational Models and Adaptive Strategies

Jacob Fish and Kamlun ShekDepartments of Civil, Mechanical and Aerospace Engineering

Rensselaer Polytechnic InstituteTroy, NY 12180

Abstract

We develop computational models and adaptive modeling strategies for obtainiapproximate solution to a boundary value problem describing the finite deformationticity of heterogeneous structures. A nearly optimal mathematical model consists averaging scheme based on approximating eigenstrains and elastic concentration faeach micro phase by a constant in the portion of the macro-domain where modelingare small, whereas elsewhere, a more detailed mathematical model based on a piconstant approximation of eigenstrains and elastic concentration factors is utilizedmethodology is developed within the framework of “statistically homogeneous” comite material and local periodicity assumptions.

1.0 Introduction

In this manuscript, we develop a theory and methodology for obtaining an approxsolution to a boundary value problem describing the finite deformation plasticity of hogeneous structures. The theory is developed within the framework of “statistically hgeneous” composite material and local periodicity assumptions. For readers interetheoretical and computational issues dealing with various aspects of nonperiodic hetneous media we refer to [7][9][28][37].

The challenge of solving structural problems with accurate resolution of microstrucfields undergoing inelastic deformation is enormous. This subject has been an activof research in the computational mechanics community for more than two decNumerous studies have dealt with the utilization of the finite element method [12[18][21][22][24][30][34], the boundary element method [11], the Voronoi cell meth[10], the spectral method [1], the transformation field analysis [5], and the Fourier sexpansion technique [26] for solving PDEs arising from the homogenization of nonlcomposites. The primary goals of these studies were twofold: (i) develop macrosconstitutive equations that would enable solution of an auxiliary problem with nonlihomogenized (smooth) coefficients, and (ii) establish bounds for overall nonlinear prties [2][29][32][33][34][35].

1

ion ofsmallpresen-epen-Gauss) equal

ains.

lysisnd 2.f free-main timeconds

me is

efor- justi-ials.nelastic defor-

of the [8] tovised formes. Inn intoation.erehin each 5 ise

mall,d in

Attempts at solving large scale nonlinear structural systems with accurate resolutmicrostructural fields are very rare [10][12][26] and successes were reported for problems and/or special cases. This is because for linear problems a unit cell or a retative volume problem has to be solved only once, whereas for nonlinear history ddent systems, it has to be solved at every increment and for each macroscopic (point. Furthermore, history data has to be updated at a number of integration pointsto the product of the number of Gauss points in the macro and micro (unit cell) dom

To illustrate the computational complexity involved we consider an elasto-plastic anaof the composite flap problem [8] with fibrous microstructure as shown in Figures 1 aThe structural problem is discretized with 788 tetrahedral elements (993 degrees odom), whereas fibrous microstructure is discretized with 98 elements in the fiber doand 253 elements in the matrix domain, totaling 330 degrees of freedom. The CPUon SPARC 10/51 workstation for this problem was over 7 hours, as opposed to 10 seif von Mises metal plasticity was used instead, which means that 99.9% of CPU tispent on stress updates.

With the exception of [6][12][19] most of the research activities focused on small dmation inelastic response of microconstituents and their interfaces. This is partiallyfied due to high stiffness and relatively low ductility of fibrous composite materHowever, when hardening is low and the stress measures are comparable to the itangent modulus, or in the case of thin structures undergoing large rotations, largemation formulation is required.

One of the objectives of the present manuscript is to extend the recent formulation mathematical homogenization theory with eigenstrains developed by the authors inaccount for finite deformation and thermal effects. In addition, adaptive strategy is deto ensure reliability and efficiency of computations. In Section 2 we derive a closedexpression relating arbitrary transformation fields to mechanical fields in the phasSections 3 and 4 we employ an additive decomposition of the rate of deformatioelastic rate of deformation, governed by hypoelasticty and inelastic rate of deformSection 3 focuses on the 2-point approximation scheme (for two phase materials), wheach point represents an average response within a phase. The local response witphase is then recovered by means of post-processing. In Section 4 we describe then-pointscheme model, where n denotes the number of elements in the microstructure. Sectiondevoted to modeling error estimation and adaptive strategy. We develop an adaptiv2/n-point model, where the 2-point scheme is used in regions where modeling errors are swhereas elsewhere the n-point scheme is employed. Numerical experiments conducteSection 6 investigate the 2-point, the n-point, and the adaptive 2/n-point schemes in thecontext of finite deformation plasticity.

2

4] for phasedy. Wes in arma-

iodic)menttor in

odic

the

g the

al

ative

lon

main-

semi-

right

equilib-ns atlem is

2.0 Mathematical Homogenization with Eigenstrains for SmallDeformations

In this section we generalize the classical mathematical homogenization theory [3][heterogeneous media to account for eigenstrains. We regard all inelastic strains,transformation and temperature effects as eigenstrains in an otherwise elastic bowill derive closed form expressions relating arbitrary eigenstrains to mechanical fieldmulti-phase composite medium. In this section attention is restricted to small defotions.

The microstructure of a composite material is assumed to be locally periodic (Y-perwith a period represented by a unit cell domain or a Representative Volume Ele(RVE), denoted by , as shown in Figure 3. Let be a macroscopic coordinate vec

macro domain and be a microscopic position vector in . For any Y-peri

function , we have in which vector is the basic period of

microstructure and is a 3 by 3 diagonal matrix with integer components. Adoptin

classical nomenclature, any Y-periodic function can be represented as

(1)

where superscript denotes a Y-periodic function . The indirect macroscopic spati

derivatives of can be calculated by the chain rule as

(2)

(3)

where the comma followed by a subscript variable or denotes a partial deriv

with respect to the subscript variable (i.e. and ). A semi-co

followed by a subscript variable denotes a partial derivative with respect to the re

ing x components (2), but a full derivative with respect to , and vice versa when a

colon is followed by subscript variable (3). Summation convention for repeated

hand side subscripts is employed, except for subscripts x and y.

We assume that micro-constituents possess homogeneous properties and satisfy rium, constitutive, kinematics and compatibility equations as well as jump conditiothe interface between the micro-phases. The corresponding boundary value probgoverned by the following equations:

(4)

(5)

Θ xΩ y x ς⁄≡ Θ

f f x y,( ) f x y ky+,( )= yk

f

fς x( ) f x y x( ),( )≡

ς f

fς

f,xi

ς x( ) f x; ix y,( )≡ f,xi

x y,( ) 1ς--- f,yi

x y,( )+=

f y; ix y,( ) f,yi

x y,( ) ς f,xix y,( )+ ςf x; i

x y,( )= =

xi yi

f,xi∂f ∂xi⁄≡ f,yi

∂f ∂yi⁄≡

xi

yi

yi

σi j x; j

ς bi+ 0 in Ω=

σi jς Lijkl εkl

ς µklς–( ) in Ω=

3

nd

y force

vector;

d anti-

ndary

h that

t the

t the

are

) and

(6)

(7)

(8)

(9)

where , and are components of stress, strain and rotation tensors; a

are components of elastic stiffness and eigenstrain tensors, respectively; is a bod

assumed to be independent of ; denotes the components of the displacement

the subscript pairs with regular and square parenthesizes denote the symmetric ansymmetric gradients defined as

(10)

denotes the macroscopic domain of interest with boundary ; and are bou

portions where displacements and tractions are prescribed, respectively, suc

and ; denotes the normal vector on . We assume tha

interface between the phases is perfectly bonded, i.e. and a

interface, , where is the normal vector to and is a jump operator.

In the following, displacements and eigenstrains

approximated in terms of double scale asymptotic expansions on :

(11)

(12)

Strain and rotation expansions on can be obtained by substituting (11) into (6(7) with consideration of the indirect differentiation rule (2)

(13)

(14)

where strain and rotation components for various orders of are given as

εi jς u i x; j( )

ς in Ω=

ωijς u i x; j[ ]

ς in Ω=

uiς ui on Γu=

σi jς nj ti on Γt=

σi jς εij

ς ωi jς Lijkl µij

ς

bi

y uiς

u i x; j( )ς 1

2--- ui x; j

ς uj x; i

ς+( )≡ , u i x; j[ ]ς 1

2--- ui x; j

ς uj x; i

ς–( )≡

Ω Γ Γu Γt

ui tiΓu Γt∩ ∅= Γ Γu Γt∪= ni Γ

σi jς nj[ ] 0= ui

ς[ ] 0=

Γint ni Γint •[ ]

uiς x( ) ui x y,( )= µi j

ς x( ) µij x y,( )=

Ω Θ×

ui x y,( ) ui0 x y,( ) ςui

1 x y,( ) …+ +≈

µij x y,( ) µij0 x y,( ) ςµij

1 x y,( ) …+ +≈

Ω Θ×

εi j x y,( ) 1ς---εi j

1– x y,( ) εi j0 x y,( ) ςεij

1 x y,( ) …+ + +≈

ωi j x y,( ) 1ς---ω ij

1– x y,( ) ωi j0 x y,( ) ςωi j

1 x y,( ) …+ + +≈

ς

4

(5)

of (2)

te-

due to

(15)

(16)

and

(17)

(18)

Stresses and strains for different orders of are related by the constitutive equation

(19)

The resulting asymptotic expansion of stress is given as

(20)

Inserting the stress expansion (20) into equilibrium equation (4) and making the useyields the following equilibrium equations for various orders:

(21)

(22)

(23)

(24)

Consider the equilibrium equation (21) first. Pre-multiplying it by and in

grating over a unit cell domain yields

(25)

and subsequently integrating by parts gives

(26)

where denotes the boundary of . The boundary integral term in (26) vanishes

Y-periodicity of boundary conditions on . Furthermore, since the elastic stiffness

is positive definite, we have

εij1– εyij u0( ), εi j

s εxij us( ) εyij us 1+( ), s+ 0 1 …, ,= = =

ωi j1– ωyij u0( ), ωi j

s ωxij us( ) ωyij us 1+( ), s+ 0 1 …, ,= = =

εxij us( ) u i ,xj( )s , εyij us( ) u i ,yj( )

s==

ωxij us( ) u i ,xj[ ]s , ωyij us( ) u i ,yj[ ]

s==

ς

σi j1– Lijkl εkl

1– , σi js Lijkl εkl

s µkls–( ), s= 0 1 …, ,= =

σi j x y,( ) 1ς---σij

1– x y,( ) σi j0 x y,( ) ςσi j

1 x y,( ) …+ + +≈

O ς 2–( ): σi j ,yj

1– 0=

O ς 1–( ): σi j ,xj

1– σi j ,yj

0+ 0=

O ς0( ): σij ,xj

0 σi j ,yj

1 bi+ + 0=

O ςs( ): σij ,xj

s σ i j ,yj

s 1++ 0, s 1 2 …, ,= =

O ς 2–( ) ui0

Θ

ui0σij ,yj

1– ΘdΘ∫ 0=

ui0σij

1– nj ΓΘdΓΘ

∫ u i ,yj( )0 Lijkl u k,yl( )

0 ΘdΘ∫– 0=

ΓΘ Θ

ΓΘ Lijkl

5

(19)

lting

rium

bina-

ider

inearod is

stem

s can

(27)

and

(28)

We proceed to the equilibrium equation (22) next. From equations (15) andfollows

(29)

To solve for (29) up to a constant we introduce the following separation of variables

(30)

where is a Y-periodic function, is a macroscopic portion of the solution resu

from eigenstrains, i.e. if then . It should be noted that both

and are symmetric with respect to indices and . Based on (30) equilib

equation takes the following form:

(31)

where

(32)

and is the Kronecker delta. Since equation (31) should be valid for arbitrary com

tion of macroscopic strain field and eigenstrain field , we first cons

, and then , which yields the following two

governing equations on :

(33)

(34)

Equation (33) together with Y-periodic boundary conditions comprise a standard lboundary value problem on . For complex microstructures the finite element meth

often employed for discretization of , which yields a set of linear algebraic sy

with six right hand side vectors [7]. In absence of eigenstrains, the asymptotic field

u i ,yj( )0 0= ⇒ ui

0 ui0 x( )=

σi j1– x y,( ) εi j

1– x y,( ) ωij1– x y,( ) 0= = =

O ς 1–( )

Lijkl εxkl u0( ) εykl u1( ) µkl0–+( ) ,yj

0 on Θ=

ui1 x y,( ) Hikl y( ) εxkl u0( ) dkl

µ x( )+ =

Hikl dklµ

µkl0 x y,( ) 0= dkl

µ x( ) 0= Hikl

dklµ k l O ς 1–( )

Lijkl I klmn Gklmn+( )εxmn u0( ) Gklmndmnµ x( )+ µkl

0–( ) ,yj

0 on Θ=

Iklmn12--- δmkδnl δnkδml+( )= , Gklmn y( ) H k,yl( )mn y( )=

δmk

εxmn u0( ) µkl0

µkl0 0≡ εxmn u0( ) 0≠ εxmn u0( ) 0≡ µkl

0 0≠

Θ

Lijkl I klmn H k,yl( )mn+( ) ,yj

0=

Lijkl H k,yl( )mndmnµ µkl

0–( ) ,yj

0=

ΘHikl y( )

6

ation

t the

lume

dic

qua-

ith

be written in terms of the macroscopic strain and the macroscopic rot

:

(35)

where

(36)

The terms and are known as polarization functions. It can be shown tha

integrals of the polarization functions on vanish due to periodicity conditions.

The elastic homogenized stiffness follows from equilibrium equation [7]:

(37)

where

(38)

is often referred to as an elastic strain concentration function and is the vo

of a unit cell.

After solving (33) for , we proceed to (34) for finding subjected to Y-perio

boundary conditions. Pre-multiplying (34) by and then integrating the resulting e

tion by parts with consideration of Y-periodic boundary conditions yields

(39)

Rewriting (39) in terms of strain concentration function and manipulating it w

(37) yields

(40)

where

(41)

εij εxij u0( )≡

ωi j ωxij u0( )≡

εij εij Gi jkl εkl O ς( ) , ωij+ + ω i j Gi jkl εkl O ς( )+ += =

Gijkl y( ) H i ,yj[ ]kl y( )=

Gijkl Gijkl

Θ

Lijkl O ς0( )

Lijkl1Θ------- LijmnAmnkl Θd

Θ∫≡ 1Θ------- AmnijLmnstAstkl Θd

Θ∫=

Aklmn Iklmn Gklmn+=

Aklmn Θ

Himn dklµ

Hist

GijstLijkl Gklmndmnµ x( ) µkl

0–( ) ΘdΘ∫ 0=

Aijkl

dijµ 1

Θ------- Lijkl Lijkl–( ) 1– GmnklLmnstµst

0 ΘdΘ∫=

Lijkl1Θ------- Lijkl Θd

Θ∫=

7

symp-

bles

n be

be

iven

The superscript denotes the reciprocal tensor. The approximation to the atotic strain (13) and rotation fields (14) reduces to:

(42)

(43)

Let be a set of continuous functions, then the separation of varia

for the eigenstrains is assumed to have the following decomposition:

(44)

The resulting asymptotic expansion of the strain and rotation fields (13), (14) caexpressed as follows:

(45)

(46)

where and are the eigenstrain influence functions, which can

expressed in terms of polarization functions and as follows

(47)

(48)

In particular, if is a set of piecewise constant functions defined as

(49)

and is the subdomain within a unit cell, the subdomain volume fraction g

by and satisfying , then (45) and (46) reduce to:

1– O ς0( )

εij εij Gijkl εkl dklµ+( ) O ς( )+ +=

ωi j ωi j Gijkl εkl dklµ+( ) O ς( )+ +=

ψ ψ η( ) y( ) 1n≡ C 1–

O ς0( )

µi j0 x y,( ) ψ η( ) y( ) µij

η( ) x( )η 1=

n

∑=

εi j x y,( ) εi j x( ) Gijkl y( )εkl x( ) Dijklη( ) y( ) µkl

η( ) x( ) O ς( )+η 1=

n

∑++=

ωi j x y,( ) ωij x( ) Gijkl y( )εkl x( ) Dijklη( ) y( ) µkl

η( ) x( ) O ς( )+η 1=

n

∑++=

Dijklη( ) y( ) Dijkl

η( ) y( )

Gijkl y( ) Gijkl y( )

Dijklη( ) y( ) 1

Θ-------Gijmn Lmnpq Lmnpq–( ) 1– GrspqLrsklψ η( ) Θd

Θ∫=

Dijklη( ) y( ) 1

Θ-------Gijmn Lmnpq Lmnpq–( ) 1– GrspqLrsklψ η( ) Θd

Θ∫=

ψ

ψ η( ) yρ( )1 if yρ Θ η( )∈

0 otherwise

=

Θ η( ) η c η( )

c η( ) Θ η( ) Θ⁄≡ c η( )η 1=n∑ 1=

8

for-

.

ensor

(50)

where

(51)

(52)

and

(53)

We will refer to the piecewise constant model defined by (50) as the n-point schememodel. Equation (50)a has been originally derived by Dvorak [5] on the basis of trans

mation field analysis. Finally, we integrate the equilibrium equation (23) over

The term vanishes due to periodicity and we obtain:

(54)

Substituting the constitutive relation (19) and the asymptotic expansion of strain t(42) into the above equation yields the macroscopic equilibrium equation

(55)

Finally, if we define the macroscopic stress as

(56)

then the equilibrium equations (54) and (55) can be further simplified as follows:

(57)

where is the overall eigenstrain given by

εi jρ( ) 1

Θ ρ( )------------- εi j Θd

Θ ρ( )∫ εi j Gijklρ( ) εkl Dijkl

ρη( )µklη( )

η 1=

n

∑ O ς( ) + + += =

ωi jρ( ) 1

Θ ρ( )------------- ωi j Θd

Θ ρ( )∫ ω i j Gijklρ( ) εkl Dijkl

ρη( )µklη( )

η 1=

n

∑ O ς( ) + + += =

Dijklρη( ) c η( )Gijmn

ρ( ) Lmnpq Lmnpq–( ) 1– Grspqη( ) Lrskl

η( )=

Dijklρη( ) c η( )Gijmn

ρ( ) Lmnpq Lmnpq–( ) 1– Grspqη( ) Lrskl

η( )=

Gijklη( ) Gijkl

η( ),( ) 1Θ η( )-------------- Gijkl Gijkl,( ) Θd

Θ η( )∫=

O ς0( ) Θ

σij ,yj

1 ΘdΘ∫

1Θ------- σi j

0 ΘdΘ∫

,xj

bi+ 0= on Ω

1Θ------- Lijkl Aklmnεmn Gklmndmn

µ µkl0–+( ) Θd

Θ∫

,xj

bi+ 0=

σi j

σi j1Θ------- σij

0 ΘdΘ∫≡

σi j ,xjbi+ 0= , Lijkl εkl µkl–( ) ,xj

bi+ 0=

µij

9

rall

ver-tion.

ases,

that

spec-

that

rs are

-

is the

sim-e.,

a-

the

(58)

Replacing by and manipulating (58) with (37) and (40), the ove

eigenstrain field can be expressed as

(59)

Equation (59) represents the well-known Levin’s formula [23] relating the local and oall eigenstrains, and is often referred to as the elastic stress concentration func

Remark 1: As a special case we consider a composite medium consisting of two ph

matrix and reinforcement, with respective volume fractions and such

. Superscripts and represent matrix and reinforcement phases, re

tively. and denote the matrix and reinforcement domains such

. We assume that eigenstrains and elastic strain concentration factoconstant within each phase. This yields the simplest variant of (50) where n=2. The corre-sponding approximation scheme is termed as the 2-point model. The overall elastic properties are given by [5]

(60)

and the overall stress reduces to:

(61)

3.0 2-Point Scheme for Finite Deformation Plasticity

For finite deformation analysis the left superscript denotes the configuration:

current configuration at time , whereas is the configuration at time . For plicity, we will often omit the left superscript for the current configuration, i.

. To extend the small deformation formulation to account for finite deformtion effects the following assumptions are made:

A1: Phase stress objectivity

We will assume that the principle of objectivity is satisfied for each phase. ThenCauchy stress rate for phase is given as:

µij1Θ-------– Lijkl

1– Lklmn Gmnpqdpqµ µmn

0–( ) ΘdΘ∫=

Gmnpq Amnpq Imnpq–

µij1Θ------- Bklij µkl

0 Θ, BijkldΘ∫ Lijmn y( )Amnpq y( )Lpqkl

1–= =

Bijkl

c m( ) c f( )

c m( ) c f( )+ 1= m f

Θ m( ) Θ f( )

Θ Θ m( ) Θ f( )∪=

Lijkl c r( )Lijmnr( ) Imnkl Gmnkl

r( )+( )r m=

f

∑=

σi j c m( )σi jm( ) c f( )σij

f( )+=

t t∆+

t t∆+t

t

t t∆+≡

r

10

mation

locity

aterial

rota-

rous

onfig-

the

ation

o [16]

ativek we

c rate

(62)

where the superposed dot represents the material time derivative. The rate of defor

and spin tensor components, denoted as and , respectively, are defined as

(63)

where is the phase velocity gradient. The asymptotic expansion of the phase ve

is given as

(64)

is the objective rate of the Cauchy stress in phase , which represents the m

response due to deformation, whereas represents the rate of

tion.

Remark 2: The optimal choice of rotation depends on the microstructure. For fib

composites it is natural to assume that , represents the fiber rotation from the c

uration aligned along the unit vector to the current configuration aligned along

vector . Thus

(65)

Following Lee [20] it can be shown that is related to the spin and rate of deformtensors by:

(66)

The choice of rotations in textile and particle composites is less obvious. We refer tfor the discussion on various choices.

A2: Additive decomposition of hypoelastic and inelastic rate of deformation

The theoretical and practical reasons favoring additive decomposition over multiplicdecomposition for fibrous composites were discussed in [27]. In the present wor

adopt the additive decomposition of rate of deformation into elastic and inelasti

of deformation , which gives

σ· i jr( ) σi j

r( )° σ·

ijr( )

+= where σ·

i jr( ) Λik

r( )σkjr( ) σik

r( )Λkjr( )–=

ε·ijς r( ) ω· i j

ς r( )

ε·ijς r( ) x( ) v i x; j( )

ς r( )≡ and ω· i jς r( ) x( ) v i x; j[ ]

ς r( )≡

vi x; j

ς r( )

viς r( ) x( ) vi

r( ) x y,( ) vi0 r( ) x y,( ) ςvi

1 r( ) x y,( ) …+ +≈≡

σi jr( )° r

Λ i jr( ) ℜ· ik

r( ) ℜkjr( ) 1–=

ℜijr( )

ℜijr( )

mt

i

mi

mi ℜi jr( ) m

tj= and m· i ℜ· ip

r( ) ℜpjr( ) 1–

mj Λi jr( )

mj≡=

Λi jr( )

Λi jr( ) ω· i j

r( ) ε·ikr( )mkmj ε·jk

r( )mkmi–+=

ε·i jr( )

e

µ· ijr( )

11

ective

y be

t incre-

on of

e sec-

us-

econd

ting

(67)

Furthermore, we will assume the hypoelastic constitutive equation relating the objCauchy stress rate with rate of elastic deformation:

(68)

A3: Midpoint integration scheme for micro- and macro-coordinates

In a typical time step , the configuration of the macro- and micro-structure ma

expressed as a sum of the configuration at the previous step and the displacemenment:

(69)

(70)

The macroscopic displacement increment is found from the incremental soluti

the macro-problem, whereas displacement increment in the RVE is given by:

(71)

The first term in (71) represents the contribution of macroscopic solution, whereas th

ond term accounts for oscillatory Y-periodic field. Figure 4 schematically ill

trates the decomposition of the deformation field in the RVE.

Strain and rotation increments are integrated using the midpoint rule to obtain a sorder accuracy:

(72)

where the midpoint coordinates are defined as

(73)

Similarly, the periodic portion of the solution increment is obtained by integra

(30) using the midpoint rule:

(74)

ε·ijr( ) ε·ij

r( )e µ· i j

r( )+=

σi jr( )° Lijkl

r( ) ε·klr( ) µ·kl

r( )–( )=

t ∆t+

t

xt ∆ t+

i xt

i ∆ui0+=

yt ∆t+

i yt

i ∆ui+=

∆ui0

∆ui x y,( ) εi j x( ) ω ij x( )∆+∆ yj ∆ui1 x y,( )+=

∆ui1 x y,( )

εi j∆ 12---

∂∆ui0

∂ xt ∆t 2⁄+

j

----------------------∂∆uj

0

∂ xt ∆t 2⁄+

i

----------------------+

= , ω ij∆ 12---

∂∆ui0

∂ xt ∆t 2⁄+

j

----------------------∂∆uj

0

∂ xt ∆t 2⁄+

i

----------------------–

=

xt ∆t 2⁄+

i12--- x

ti x

t ∆ t+i+( )≡ , y

t ∆t 2⁄+i

12--- y

ti y

t ∆t+i+( )≡

∆ui1

∆ui1 Himn yt ∆t 2⁄+( ) εmn x( )∆ ∆dmn

µ x( )+( )=

12

ion to as thed -

smalld byincre-dopted

d back

ations

aterial

,

thusulta-

tions fiberse thate first-istentan bew us to

where the increment of inelastic strain is defined in Section 4.

A4: Additive decomposition of material and rotational response

There are several formulations aimed at extending the small deformation formulataccount for large deformation effects. One of the most popular approaches is knownco-rotational method where all the fields of interest are transformed into the rotate

system [16]. In the -system, the form of constitutive equations is analogous to deformation theory. A simpler approach, proposed by Hallquist [14] and improveHughes and Winget [17] to preserve incremental objectivity, is based on the additive mental decomposition of material and rotational response. The latter procedure is ain the present manuscript.

For two phase materials, the integration scheme [17] decomposes stresses anstresses as follows:

(75)

(76)

where is the back stress. The midpoint rule is utilized to compute the phase rot

[17]

(77)

Remark 3: For homogeneous materials the integration scheme [17] uncouples the m

and rotational responses. In the present formulation phase rotations in each phase,

depend on phase eigenstrains, which are unknown prior to stress integration, andmaterial and rotational responses are fully coupled and have to be updated simneously.

A5: Constant phase volume fractions

For the 2-point scheme derived in Section 3 we will assume that phase volume fracremain constant throughout the analysis. This is apparently true in the case of elasticundergoing small strains and incompressible matrix material. In addition, we assumthe elastic properties of the phases are independent of temperature. Based on thorder approximation methods, such as the Mori-Tanaka method [25] and Self Consmethod [15], the strain concentration factors and eigenstrain influence functions cassumed to be constants throughout the entire analysis. These assumptions will allo

ℜℜ

σijr( )t ∆t+ σij

r( )t σ∆ ijr( ),+= σi j

r( )t ℜkir( ) σkl

r( )t ℜl jr( )=

αi jr( )t ∆t+ αi j

r( )t α∆ i jr( ),+= αi j

r( )t ℜkir( ) αkl

r( )t ℜl jr( )=

αi jr( )

ℜijr( ) δij δik

12---∆ωik

r( )– 1–

∆ωkjr( )+=

ℜi jr( )

13

he

ue to

n inth the

that

pansion

hown

cord-

face in

ress isojec-

ce:

carry out the entire analysis without updating the configuration of the unit cells. For tn-point scheme model, described in Section 4, these restrictions will be removed.

3.1 Implicit Integration of Constitutive Equation

For the elastically deforming reinforcement the only source of eigenstrain rate is d

temperature effects, i.e., where is the thermal rate of deformatioreinforcement domain. The eigenstrain rate in the matrix phase is comprised of bo

thermal, , and the plastic, , rate of deformation effects, such

. The phase thermal rate of deformation can be expressed as

(78)

where denotes the temperature and are components of the phase thermal ex

tensor.

Combining the rate form of (50), (68), (69), (75), Assumptions 3 and 4 it can be sthat the following relations for the phase stresses hold:

(79)

where is the overall phase eigenstrain increment and

(80)

Consider the yield function of the following form:

(81)

where is the yield stress of the matrix phase in a uniaxial test, which evolves ac

ing to the hardening laws assumed; corresponds to the center of the yield sur

the deviatoric stress space, or simply the back stress. Evolution of the back stassumed to follow the kinematic hardening rule. For von Mises plasticity, is a pr

tion operator which transforms an arbitrary second order tensor to the deviatoric spa

(82)

µ· ijf( ) ε·θ i j

f( )= ε·θ i jf( )

ε·θ i jm( ) ε·p ij

m( )

µ· ijm( ) ε·θ i j

m( ) ε·p ijm( )+=

ε·θ i jr( ) ξi j

r( )θ·=

θ ξi jr( )

σijr( )t ∆t+ σ ij

r( )tRijkl

r( )+ ∆εkl Qijklrs( )∆µkl

s( ) r,s m=

f

∑– m f,= =

∆µkls( )

Rijklr( ) Lijpq

r( ) Ipqkl Gpqklr( )+( ) =

Qijklrs( ) Lijpq

r( ) δrsIpqkl Dpqklrs( )–( )=

r s, m f,=

Φ m( ) σi jm( ) αi j

m( )– Y m( ),( ) 12--- σi j

m( ) αi jm( )–( )Pijkl σkl

m( ) αklm( )–( ) 1

3--- Y m( ) 2–=

Y m( )

αi jm( )

Pijkl

Pijkl I ijkl13---δi j δkl–=

14

ows

otro-eter

ening

(81).

n be

Zie-

efined

The

ack-

tion

For simplicity we assume that the plastic rate of deformation in the matrix phase follthe associative flow rule:

(83)

We adopt a modified version of the hardening evolution law [16] in the context of ispic, homogeneous, elasto-plastic matrix phase. A scalar material dependent param

is used as a measure of the proportion of isotropic and kinematic hard

and is a plastic parameter to be determined by the consistency condition

Accordingly, the evolution of the yield stress and the back stress ca

expressed as follows:

(84)

(85)

where corresponds to a pure isotropic hardening; is the widely used

gler-Prager kinematic hardening rule [36] for metals; is a hardening parameter das the ratio between effective stress rate and the effective plastic strain rate.

Integration of (83), (84) and (85) is carried out using the backward Euler scheme:

(86)

(87)

(88)

where , and is the rotated back stress defined in (76). phase rotation increment follows from (50), (78) and (83):

(89)

In the following we omit the left superscript for the current step . Using the b

ward Euler scheme for the rate form of in (79) and (86) yields the following rela

for the Cauchy stress in the matrix domain:

ε·i jm( )

pΦ∂

σi jm( )∂

--------------λ· m( ) ℵi jm( )λ· m( )= = , ℵij

m( ) Pijkl σklm( ) αkl

m( )–( )=

β0 β 1≤ ≤( )

λ m( )

Y m( ) αi jm( )

Y· m( ) 2βh3

----------Y m( )λ· m( )=

αi jm( )° 2 1 β–( )h

3-----------------------Pijkl σkl

m( ) αklm( )–( )λ· m( )=

β 1= β 0=

h

εijm( )t ∆t+

p εi jm( )t

p ℵijm( ) λ m( )∆t ∆t+

+=

Y m( )t ∆t+Y m( )t 2βh

3---------- Y m( )t ∆t+ ∆λ m( )+=

αi jm( )t ∆t+ αi j

m( )t 2 1 β–( )h3

-----------------------+ Pijkl σklm( )t ∆t+ αkl

m( )t ∆t+–( ) λ m( )∆=

λ m( )∆ λ m( )t ∆t+ λ m( )t–≡ αij

m( )t

∆ω ijr( ) ∆ωij Gijkl

r( ) ∆εkl Dijklrm( )Pklmn σmn

m( ) αmnm( )–( ) λ m( )∆ Dijkl

rs( )ξkls( )∆θ

s m=

f

∑+ + +=

t ∆t+

σi jm( )

15

s thatnt load

ndition

A

ired

then

(90)

where is a trial Cauchy stress in the matrix phase defined as

(91)

The process is termed elastic if:

(92)

Otherwise the process is plastic, which is the focus of our subsequent derivation.

Subtracting (88) from (90) we arrive at the following result:

(93)

where

(94)

The value of is obtained by satisfying the consistency condition which assurethe stress state in the plastic process lies on the yield surface at the end of the currestep. To this end, equations (87) and (93) are substituted into the consistency co

(81), , which produces a nonlinear equation for .

standard Newton’s method is applied to solve for :

(95)

where is the iteration count. It can be shown that the derivative requin (95) has the following form:

(96)

The expression for is derived in Appendix A. The converged value of is

used to compute the phase stresses. The overall stress is computed from (61).

σ ijm( ) σi j

m( )tr Qijkl

mm( )ℵklm( )∆λ m( )–=

σi jm( )

tr

σijm( )

tr σi jm( )t

Rijklm( ) εkl∆ Qijkl

ms( )ξkls( )∆θ

s m=

f

∑–+≡

σi jm( )

tr αijm( )–( )Pijkl σkl

m( )tr αkl

m( )–( ) 23---– Y m( ) 2

∆λ m( ) 0=

0<

σijm( ) αij

m( )– I ijkl ∆λ m( )℘i jkl+( ) 1– σklm( )

tr αklm( )t

–( )=

℘i jkl Qijstmm( )Pstkl

23--- 1 β–( )hPijkl+=

∆λ m( )

Φ m( ) σi jm( ) αi j

m( )– Y m( ),( ) 0= ∆λ m( )

∆λ m( )

∆λk 1+m( ) ∆λk

m( ) Φ m( )∂∂∆λ m( )-----------------

1–

Φ m( )–∆λk

m( )

=

k Φ m( )∂ ∂∆λ m( )⁄

Φ m( )∂∂∆λ m( )----------------- ℵij

m( )Cijklm( ) σkl

m( ) αklm( )–( ) 4βh Y m( ) 2

9 6βh∆λ m( )–---------------------------------–=

Cijklm( ) ∆λ m( )

16

e for-ential

for the

nd

3.2 Consistent Linearization

While integration of the constitutive equations affects the accuracy of the solution, thmation of a tangent stiffness matrix consistent with the integration procedure is essto maintain the quadratic rate of convergence if one is to adopt the Newton method solution of nonlinear system of equations on the macro level [31].

The starting point is the incremental form of the constitutive equations (79):

(97)

Taking material time derivative of (88), (89) and (97) yields:

(98)

(99)

(100)

Subtracting (98) from (100) for r = m yields:

(101)

where

(102)

Combining (99), (101), (102), (212), (213) with the consistent linearizations of a (given in Appendix B) yields:

(103)

σ i jr( ) σi j

r( )tR+ ijkl

r( ) ∆εkl Qijklrm( )ℵkl

m( )∆λ m( )– Qijklrs( )ξkl

s( )∆θs m=

f

∑–=

α· i jm( ) α

·i jm( ) 2 1 β–( )h

3----------------------- ℵi j

m( )λ· m( ) Pijpq σ· pqm( ) α· pq

m( )–( )∆λ m( )+ +=

∆ω· ijr( ) ∆ω· i j Gijkl

r( ) ∆ε·kl Dijklrm( ) ℵkl

m( )λ· m( ) Pklpq σ· pqm( ) α· pq

m( )–( )∆λ m( )+ + +=

Dijklrs( )ξkl

s( )θ·

s m=

f

∑+

σ· i jr( ) σ

·i jr( )t

Rijklr( ) ∆ε·kl Qijkl

rm( ) ℵklm( )λ· m( ) Pklpq σ· pq

m( ) α· pqm( )–( )∆λ m( )+ –+=

Qijklrs( )ξkl

s( )θ·

s m=

f

∑–

σ· i jm( ) α· ij

m( )– σ·

i jm( )t

α·

ijm( )t

– Rijklm( )∆ε·kl Qijkl

ms( )ξkls( )θ·

s m=

f

∑–+=

℘i jkl σklm( ) αkl

m( )–( )λ· m( ) σ· klm( ) α· kl

m( )–( )∆λ m( )+ –

σ·

i jm( )t

α·

i jm( )t

–∂ σij

m( ) αi jm( )–( )

∂∆ωklm( )

------------------------------------∆ω· klm( )=

∆ε·kl

∆ω· kl

σ· i jm( ) α· i j

m( )– I ijkl ∆λ m( )Wijklm( )+( ) 1– Sv klstvs xt,

0 Sθ klθ· Sλ klλ

· m( )+ +( )=

17

cy

form

ty gra-

where

(104)

(105)

(106)

and , and are defined in (217), (212) and (213), respectively. It

remains to eliminate from (103), by utilizing the linearized form of the consistencondition (81) and equation (87) which gives

(107)

Substituting (103) into (107) results in

(108)

where

(109)

and thus (103) can be simplified as

(110)

where

(111)

(112)

Finally, by substituting (108), (110), (212) and (228) into (100), we get a closed

expression relating the phase Cauchy stress rate with the macroscopic veloci

dient and the temperature rate

(113)

where

Sv klst Uklmnm( )

σ Uklmnm( )

α–( ) M mn[ ]st GmnuvM uv( )st+( ) RklmnM mn( )st+=

Sθ kl Uklpqm( )

σ Uklpqm( )

α–( )Dpqstms( ) Qklst

ms( )– ξsts( )

s m=

f

∑=

Sλ kl Wklstm( )– σst

m( ) αstm( )–( )=

Wijklm( ) Uklmn

m( )σ Uklmn

m( )α

λ· m( )

ℵi jm( ) σ· i j

m( ) α· i jm( )–( ) 4βh Y m( ) 2λ· m( )

9 6βh∆λ m( )–----------------------------------------– 0=

λ· m( ) Γklm( ) Sv klstvs xt,

0 Sθ klθ·+( )=

Γklm( ) 9 6βh∆λ m( )–( )ℵij

m( ) I ijkl ∆λ m( )Wijklm( )+( ) 1–

4βh Y m( ) 2 9 6βh∆λ m( )–( )ℵmnm( ) Imnst ∆λ m( )Wmnst

m( )+( ) 1– Sλ st–----------------------------------------------------------------------------------------------------------------------------------------------------------=

σ· i jm( ) α· i j

m( )– S˜v ijklvk xl,

0 S˜θ ij θ

·+=

S˜v ijkl I ijmn ∆λ m( )W+ i jmn( ) 1– Sv mnkl Sλ mnΓpq

m( ) Sv pqkl+( )=

S˜θ ij I ijmn ∆λ m( )W+ i jmn( ) 1– Sθ mn Sλ mnΓpq

m( ) Sθ pq+( )=

σ· i jr( )

vk xl,0 θ·

σ· i jr( ) Dijkl

r( ) vk xl,0 dij

r( )θ·+=

18

(61)

of the vec-

n

is

(114)

and

(115)

The overall consistent instantaneous stiffness is obtained from the rate form of

and Assumption A5:

(116)

where

(117)

The overall consistent tangent operator is derived from the consistent linearization weak form of the macroscopic equilibrium equation (57). Consider the internal forcetor expressed in terms of the quantities defined in the deformed configuration

(118)

where is a set of shape functions in the macroscale.

Prior to linearization, the internal force vector is defined in the reference configuratioas

(119)

where is the jacobian between the macro-configurations at times and ;

the macroscopic deformation gradient defined as

(120)

Linearization of (119) yields

Dijklr( ) Rijmn

r( ) M mn( )kl Uijpqr( )

σ M pq[ ]kl Gpqmnr( ) M mn( )kl+( )+=

Uijpqr( )

σ Dpqmn Qijmnrm( )–( ) ℵmn

m( )Γuvm( ) Sv uvkl ∆λ m( )Pmnpq S

˜v pqkl+( )+

dijr( ) Uijpq

r( )σ Dpqkl

rs( ) Qijklrs( )–( )ξkl

s( )

s m=

f

∑=

Uijpqr( )

σ Dpqmn Qijmnrm( )–( )+ ℵmn

m( )Γpqm( ) Sθ pq ∆λ m( )Pmnpq S

˜θ pq+( )

Dijkl

σ· i j Dijkl vk xl,0 dij θ

·+=

Dijkl c m( )Dijklm( ) c f( )Dijkl

f( ) dij c m( )dijm( ) c f( )dij

f( )+=,+=

fAint NiA xj, σ ij Ωd

Ω∫=

NiA

Ωt

fAint N

iA xt

m,Fmj

1– σij Jx d ΩtΩt∫=

Jx t t ∆t+ Fjm

Fjm xj x

tm,

= xt t∆+

j xt

m,≡ and Fmj

1– xt

m xj,= xt

m xt t∆+

j,≡

19

,

dom

onsis-

tions

and

15)

(121)

Substituting (116) into (121) and exploiting the kinematical relations

and the finite element discretization yields:

(122)

(123)

where and are defined in (117); denotes the velocity degrees-of-free

associated with the finite element mesh. The first integral in (122) represents the ctent tangent stiffness matrix for the macro-problem.

Remark 4: For the purpose of linearization it is convenient to approximate phase rota

within a unit cell by a constant field such that . The resulting rotated stressback stress rates are given as

(124)

Consequently, (104)-(106) can be simplified as

(125)

(126)

and in (103), (109), (111) and (112). and in (114) and (1

reduce to

(127)

and

(128)

tdd fA

int NiA x

tm,

F·mj1– σi j Jx Fmj

1– σ· ij Jx Fmj1– σi j Jx

·+ + d Ωt

Ωt∫=

Jx·

Jxvk xk,0=

F·mj1– Fml

1– vl xj,0–= vk xl,

0 NkB xl, q·B=

tdd fA

int NiA xj, Dijkl NkB xl, Ω q·BdΩ∫ NiA xj, dij Ω θ·d

Ω∫+=

Dijkl Dijkl δklσi j δkjσil–+=

Dijkl dij q·B

ω· i jr( ) ω· i j≈

σ·

i jr( ) ∆ω· ikσkj

r( ) σikr( )∆ω· kj–= , α

·ijm( ) ∆ω· ikαkj

m( ) αikm( )∆ω· kj–=

Sv ijkl RijmnM mn( )kl δin σmjm( ) αmj

m( )–( ) δjn σimm( ) αim

m( )–( )+ – M mn[ ]kl=

Sθ i j Qijklms( )ξkl

s( )

s m=

f

∑–= , Sλ i j ℘ijkl σklm( ) αkl

m( )–( )–=

Wijklm( ) ℘ijkl= Dijkl

r( ) dijr( )

Dijklr( ) Rijmn

r( ) M mn( )kl δinσmjr( ) δjnσim

r( )+( )– M mn[ ]kl=

Qijmnrm( )– ℵmn

m( )Γuvm( ) Sv uvkl ∆λ m( )Pmnpq S

˜v pqkl+( )

dijr( ) Qijmn

rm( )– ℵmnm( )Γpq

m( ) Sθ pq ∆λ m( )Pmnpq S˜θ pq+( ) Qijkl

rs( )ξkls( )

s m=

f

∑–=

20

constantctionsticitywed

(63),

mann

nt foritutive

elds

um

:

4.0 n-Point Scheme for Finite Deformation Plasticity

In this section we consider a unit cell model discretized with n elements. The n-pointscheme model assumes that eigenstrains are piecewise constant, i.e., they are within each element, but may vary from element to element. Our starting point (Se4.1) is a rate form of the governing equations representing the finite deformation plaof periodic heterogeneous media. Implicit integration of constitutive equations folloby consistent linearization are given in Sections 4.2 and 4.3.

4.1 Governing Equations

The governing equations consist of: equilibrium (4), kinematics in the rate form boundary conditions (8), (9), and the constitutive equation in the rate form

(129)

where

(130)

denotes the instantaneous stiffness properties. In the following, we adopt Jau

rate, i.e., .

Double scale asymptotic expansion of the velocity field (64) provides the starting poithe asymptotic analysis. Substituting the asymptotic expansions (20), (64) into constequation (130) based on the Jaumann rate yields:

(131)

where is the velocity gradient given as

(132)

Further assuming that Cauchy stress vanishes, , yi

provided that is not singular. We proceed to the equilibri

equation (22):

(133)

To solve for (133) up to a constant we introduce the following separation of variables

σ· i jς σij

ς° Λikς σkj

ς σikς Λkj

ς–+=

σi jς° Lijkl ε·kl

ς ξklθ·–( )=

Lijkl

Λkjς ω· kj

ς=

σ· i js σi j

s° 12--- σi l

r δjk σjlr δik σik

r δj l σjkr δil––+( ) l kl

s r– 1+

r 1–=

s

∑+= , s 1 0 …, ,–=

l kls

lkl1– vk xl,

0= and l kls vk xl,

s 1+ vk yl,s+= s, 0 1 …, ,=

O ς 1–( ) σ· i j1– Lijkl vk yj,

0 0= =

vi0 vi

0 x( )= Lijkl O ς 1–( )

σi j y j,0 x y,( ) 0=

21

d

Fort time

n

s (2)

ons

o

(134)

Note that plastic effects are now hidden in the Y-periodic function , whereas

accounts for temperature effects only.

Premultiplying (133) by the Y-periodic function , integrating over the deforme

unit cell domain and then carrying out integration by parts yields

(135)

Linearization of (135) is carried out by taking the material time derivative, . this purpose we express the integrand of (135) in the reference configuration, say a

, where denotes the deformatio

gradient in the unit cell and is the corresponding jacobian. By utilizing equation

and (3) it can be shown that .

Consequently, linearization of (135) yields:

(136)

Substituting (130), (131), (132) and (134) into (136) and exploiting kinematical relati

and gives:

(137)

where

(138)

(139)

Since (137) is satisfied for arbitrary macro-fields and we can obtain tw

integral equations in :

(140)

vi1 x y,( ) Hikl y( ) vk xl,

0 x( ) d·klθ x( )+ =

Hikl y( ) d·klθ

Hikl y( )

Θ

φ x y,( ) Hikl yj, σi j0 Θd

Θ∫ 0==

φ· 0=

t Hikl yj, σi j0 Θd H

ikl yt

m,Fmj

1– σi j0 Jy d Θt= Fjm y

j yt

m;=

Jy

yj y

tm;

xj x

tm;

=

Hikl yt m,

F·mj1– σij

0Jy Fmj1– σ· i j

0Jy Fmj1– σi j

0Jy·

+ +( )d ΘtΘt∫ 0=

Jy·

Jylkk0= F·mj

1– Fml1– l l j

0–=

Hikl yj, Lijmn Tijmn+( ) lmn0 Lijmnξmnθ

·– ΘdΘ∫ 0=

lmn0 δmsδnt Hmst yn, y( )+( )vs xt,

0 x( ) Hmst yn, y( )d·stθ x( )+=

Tijmn δmnσi j0 1

2--- δimσjn

0 δjm– σin0 δin– σjm

0 δjn– σim0( )+=

vs xt,0 x( ) d·st

θ x( )

Θ

Hikl yj, Lijmn Tijmn+( ) δmsδnt Hmst yn,+( ) ΘdΘ∫ 0=

22

ation

ocitytions.

s

(141)

Equation (140) is solved using the finite element method for . Note that equ

(140) is solved for nine right hand side vectors corresponding to nine uniform velgradient fields as opposed to six constant strain modes in the case of small deforma

After solving (140) for , can be obtained from (141) as

(142)

where

(143)

(144)

Once and are computed, the approximation of and , denoted a

and , are given as

(145)

(146)

where

(147)

(148)

(149)

(150)

4.2 Implicit Integration of Constitutive Equations

We start from the constitutive relation for a typical element in :

Hikl yj, Lijmn Tijmn+( )Hmst yn, d·stθ Lijmnξmnθ

·– ΘdΘ∫ 0=

Hikl

Hikl d·i jθ

d·ijθ 1

Θ------- Lijkl Lijkl–( ) 1– Hrkl ys, Lrsuvξuvθ

· ΘdΘ∫=

Lijkl1Θ------- Lijkl Tijkl+( ) Θd

Θ∫=

Lijkl1Θ------- Lijmn Tijmn+( )Hmkl yn, Θd

Θ∫=

Hikl d·i jθ O ς0( ) ε· i j

ς ω· ijς ε· i j

ω· ij

ε·ij Aijkl vk xl,0 aij θ

·+=

ω· ij Aijkl vk xl,0 aij θ

·+=

Aijkl y( ) 12--- δikδj l δjkδi l+( ) H i yj,( )kl y( )+=

Aijkl y( ) 12--- δikδj l δjk– δi l( ) H i yj,[ ]kl y( )+=

aij H i yj,( )kl Lklpq Lklpq–( ) 1– Hprs yq, Lrsuvξuvθ· Θd

Θ∫=

aij H i yj,[ ]kl Lklpq Lklpq–( ) 1– Hprs yq, Lrsuvξuvθ· Θd

Θ∫=

ρ Θ

23

6) and

uted

epend

t-

f

rs.

(151)

For elements in the matrix phase (151) can be written as

(152)

Applying the backward Euler integration scheme to (152) gives

(153)

and exploiting the equation for the back stress in element (88) yields

(154)

and

(155)

(156)

in which and are the rotated stress and back stress defined in (75) and (7

is given as

(157)

Note that the instantaneous concentration factors , , and comp

from (147) to (150) depend on the instantaneous material properties, which in turn d

on vector of plastic parameters in , . Substitu

ing (154) and (87) into the yield function (81) for each element in yields a set o

nonlinear equations with unknown plastic paramete

The system of nonlinear equations is solved by the Newton method:

(158)

A typical term in the Jacobian matrix is given as

σi jρ( )° Lijkl

ρ( ) ε·klρ( ) ξkl

ρ( )θ·–( ) if ρ Θ f( )∈

Lijklρ( ) ε·kl

ρ( ) ξklρ( )θ·– ε·kl

ρ( )p–( ) if ρ Θ m( )∈

=

σi jρ( )° Lijkl

ρ( ) Aklmnρ( ) vm xn,

0 aklρ( ) ξkl

ρ( )–( )θ· ℵklρ( )λ· ρ( )–+ =

σijρ( ) σij

ρ( )tLijkl

ρ( ) Aklmnρ( ) ∆εmn ∆ωmn+( ) akl

ρ( ) ξklρ( )–( )∆θ ℵkl

ρ( )∆λ ρ( )–+ +=

ρ

σi jρ( ) αij

ρ( )– I ijmn ∆λ ρ( )℘˜ ijmn

ρ( )+( ) 1– ℑ˜ mn

ρ( )=

℘˜ i jmn

ρ( ) Lijuvρ( ) Puvmn

23--- 1 β–( )hPijmn+=

ℑ˜ mn

ρ( ) σmnρ( )t αmn

ρ( )t– Lmnpq

ρ( ) Apqstρ( ) ∆εst ∆ωst+( ) apq

ρ( ) ξpqρ( )–( )∆θ+ –=

σmnρ( )t αmn

ρ( )t

∆ω ijρ( )

∆ωi jρ( ) Aijkl

ρ( ) ∆εkl ∆ωkl+( ) aijρ( )∆θ+=

Aijklρ( ) Aijkl

ρ( ) aijρ( ) aij

ρ( )

∆λ˜

Θ m( ) ∆λ˜

∆λ 1( ) ∆λ 2( ), … ∆λ ny( ), ,[ ]T≡

Θ m( ) ny

Φ˜

Φ 1( ) Φ 2( ) … Φ ny( ), , ,[ ]T≡ ny

∆λk 1+ρ( ) ∆λk

ρ( ) Φ ρ( )∂∂∆λ η( )-----------------

1–

Φ η( )–∆λk

ρ( )

=

24

ith

wing

stan-

oceeds

, yield

(88),

.

izatione of

(159)

where

(160)

(161)

In (161) depends on the derivatives of and w

respect to . Evaluation of these derivatives is not trivial and hence the folloapproximation is employed:

(162)

resulting in the block diagonal approximation of the jacobian matrix

(163)

where

(164)

At each modified Newton iteration step the residual vector is evaluated and the in

taneous concentration factors are recomputed from (140). The iterative process pr

until the residual norm vanishes up to a certain tolerance. The updated stress

stress and back stress for elements in are calculated from (153), (87) and

respectively. For elements in , stresses can be obtained using (153) with Finally, the macroscopic stress follows from (56).

4.3 Consistent Linearization

The instantaneous consistent stiffness properties are derived from consistent linearof incremental equations. For elements in , taking the material time derivativ(153) and (88), and making use of (228) yields:

∂Φ ρ( )

∂∆λ η( )----------------- ℵi j

ρ( ) I ijmn ∆λ ρ( )℘˜ i jmn

ρ( )+ 1– χmnρη( ) 4δρηβh Y ρ( ) 2

9 6βh∆λ ρ( )–--------------------------------------–=

χmnρη( )

∂ℑ˜ mn

ρ( )

∂∆λ η( )----------------- δρη℘

˜ mnpq– σpq

ρ( ) αpqρ( )–( )=

∂ℑ˜ mn

ρ( )

∂∆λ η( )-----------------

∂ σmnρ( )t αmn

ρ( )t–( )

∂∆λ η( )------------------------------------- Lmnpq

ρ( ) ∂Apqstρ( )

∂∆λ η( )----------------- ∆εst ∆ωst+( )

∂apqρ( )

∂∆λ η( )-----------------∆θ+

–=

∂ σmnρ( )t αmn

ρ( )t–( ) ∂∆λ η( )⁄ Apqst

ρ( ) apqρ( )

∆λ η( )

Apqstρ( ) Apqst

ρ( )t≈ , apqρ( ) apq

ρ( )t≈

∂Φ ρ( )

∂∆λ η( )----------------- δρη ℵij

ρ( )χi jρ( ) 4βh Y ρ( ) 2

9 6βh∆λ ρ( )–---------------------------------+

–≈

χijρ( ) I ijmn ∆λ ρ( )℘

˜ ijmnρ( )+( ) 1– ℘

˜ mnpqρ( ) σpq

ρ( ) αpqρ( )–( )=

Φ˜

Φ˜ 2

Θ m( )

Θ f( ) ∆λ ρ( ) 0≡

Θ m( )

25

s

also

(165)

and

(166)

Substituting (228) into (165), then subtracting (166) from the resulting equation yield

(167)

where in analogy to (162), we approximate and .

From equations (212), (213), (228) it follows that

(168)

(169)

Substituting (168) and (169) into (167) and collecting terms of gives

(170)

where

(171)

(172)

(173)

The value can be computed from the linearization of consistency conditions (seeSection 3.2) which yields

(174)

where

σ· i jρ( ) σ

·i jρ( )t

Lijmnρ( ) A·mnkl

ρ( ) ∆εkl ∆ωkl+( ) Amnpqρ( ) Mpqklvk xl,

0+ +=

Lijmnρ( ) a·mn

ρ( )∆θ amnρ( ) ξmn

ρ( )–( )θ· ℵmnρ( )λ· ρ( ) Pmnpq– σ· pq

ρ( ) α· pqρ( )–( )∆λ ρ( )–+ +

α· i jρ( ) α

·ijρ( )t 2 1 β–( )h

3----------------------- ℵi j

ρ( )λ· ρ( ) Pijpq σ· pqρ( ) α· pq

ρ( )–( )∆λ ρ( )+ +=

σ· i jρ( ) α· i j

ρ( )– σ·

i jρ( )t

α·

i jρ( )

Lijmnρ( )+

tAmnpq

ρ( ) Mpqklvk xl,0 amn

ρ( ) ξmnρ( )–( )θ·+ –=

℘˜ ijmn

ρ( )– σmnρ( ) αmn

ρ( )–( )λ· ρ( ) σ· mnρ( ) α· mn

ρ( )–( )∆λ ρ( )+

A·mnklρ( ) 0= a·mn

ρ( ) 0=

σ·

ijρ( )t

Uijmnρ( )

σ Amnstρ( ) Mstklvk xl,

0 amnρ( )θ·+( )=

α·

ijρ( )t

Uijmnρ( )

α Amnstρ( ) Mstklvk xl,

0 amnρ( )θ·+( )=

σ· i jρ( ) α· ij

ρ( )–

σ· i jρ( ) α· ij

ρ( )– I ijkl ∆λ ρ( )℘˜ i jkl

ρ( )+( ) 1– Ξv klstvs xt,0 Ξθ klθ

· Ξλ klλ· m( )+ +( )=

Ξv klst Uklmnρ( )

σ Uklmnρ( )

α–( )Amnuvρ( ) Lklmn

ρ( ) Amnuvρ( )+ Muvst=

Ξθ kl Uklmnρ( )

σ Uklmnρ( )

α–( )amnρ( ) Lklmn

ρ( ) amnρ( ) ξmn

ρ( )–( )+=

Ξλ kl ℘˜ klmn

ρ( ) σmnm( ) αmn

m( )–( )–=

λ· ρ( )

λ· ρ( ) ϒklρ( ) Ξv klstvs xt,

0 Ξθ klθ·+( )=

26

ept

tions

(175)

and then substituting (174) into (170) yields

(176)

where and have identical structure to and in (111) and (112) exc

that the symbols are replaced by , and by .

Substituting (174), (176) into (165) yields

(177)

where

(178)

and

(179)

Similarly, the stress rate for elements in is given by

(180)

where

(181)

(182)

The overall instantaneous stiffness is obtained from the rate form of (61), equa

(178), (179), (181) and (182):

(183)

where

ϒklρ( )

9 6βh∆λ ρ( )–( )ℵi jρ( ) I ijkl ∆λ ρ( )℘

˜ i jklρ( )+( ) 1–

4βh Y ρ( ) 2 9 6βh∆λ ρ( )–( )ℵmnρ( ) Imnst ∆λ ρ( )℘

˜ mnstρ( )+( ) 1– Ξλ st–

----------------------------------------------------------------------------------------------------------------------------------------------------------=

σ· i jρ( ) α· ij

ρ( )– Ξ˜v ijklvk xl,

0 Ξ˜θ i j θ

·+=

Ξ˜v ijkl Ξ

˜θ ij S˜v ijkl S

˜θ i j

S Ξ Γ m( ) ϒ ρ( )

σ· i jρ( ) Dijkl

ρ( ) vk xl,0 dij

ρ( )θ·+= for ρ Θ m( )∈

Dijklρ( ) Uijmn

ρ( )σ Amnst

ρ( ) Mstkl Lijmnρ( ) Amnpq

ρ( ) Mpqkl+=

Lijmnρ( )– ℵmn

ρ( )ϒpqρ( ) Ξv pqkl ∆λ ρ( )Pmnpq Ξ

˜v pqkl+( )

dijρ( ) Uijmn

ρ( )σ amn

ρ( ) Lijmnρ( ) amn

ρ( ) ξmnρ( )–( ) ℵmn

ρ( )ϒpqρ( ) Ξθ pq– ∆λ ρ( )Pmnpq Ξ

˜θ pq– +=

Θ f( )

σ· i jη( ) Dijkl

η( )vk xl,0 dij

η( )θ·+= for η Θ f( )∈

Dijklη( ) Uijmn

η( )σ Amnst

η( ) Mstkl Lijmnη( ) Amnpq

η( ) Mpqkl+=

dijη( ) Uijmn

η( )σ amn

η( ) Lijmnη( ) amn

η( ) ξmnη( )–( )+=

Dijkl

σ· i j Dijkl vk xl,0 dij θ

·+=

27

ell at

ss

on in

(184)

denotes the ratio between the volume of element and the volume of the unit ctime .

Finally, linearization of internal force vector yields:

(185)

(186)

where the first integral in (185) represents the consistent macroscopic tangent stiffnematrix for the n-point scheme model.

Remark 5: Approximating the piecewise constant phase rotations by a constant functithe entire unit cell such that , yields a simplified form of (171) and (172):

(187)

(188)

For elements in (178) and (179) reduce to:

(189)

and

(190)

whereas for elements in (181) and (182) are given by

(191)

and

(192)

Dijkl c η( )Dijklη( )

η 1=

n

∑ dij c η( )dijη( )

η 1=

n

∑=,=

c η( ) ηt t∆+

tdd fA

int NiA xj, Dijkl NkB xl, Ω q·BdΩ∫ NiA xj, dij θ

· ΩdΩ∫+=

Dijkl Dijkl δklσi j δkjσil–+=

ω· ijr( ) ω· ij≈

Ξv klst Lklmnρ( ) Amnuv

ρ( ) Muvst δkn σmlρ( ) αml

ρ( )–( ) δ ln σkmρ( ) αkm

ρ( )–( )+ M mn[ ]st–=

Ξθ kl Lklmnρ( ) amn

ρ( ) ξmnρ( )–( )=

Θ m( )

Dijklρ( ) Lijmn

ρ( ) Amnpqρ( ) Mpqkl δkn σml

ρ( ) αmlρ( )–( ) δln σkm

ρ( ) αkmρ( )–( )+ M mn[ ]st–=

Lijmnρ( )– ℵmn

ρ( )ϒpqρ( ) Ξv pqkl ∆λ ρ( )Pmnpq Ξ

˜v pqkl+( )

dijρ( ) Lijmn

ρ( ) amnρ( ) ξmn

ρ( )–( ) ℵmnρ( )ϒpq

ρ( ) Ξθ pq– ∆λ ρ( )Pmnpq Ξ˜θ pq– =

Θ f( )

Dijklη( ) Lijmn

η( ) Amnpqη( ) Mpqkl δinσmj

η( ) δjnσimη( )+( )M mn[ ]kl–=

dijη( ) Lijmn

η( ) amnη( ) ξmn

η( )–( )=

28

mpos-

the ase areo Fig-

to

cydel. Indel-ill

eriod-

, (iii)odel

e solu-

tion,

it is

the

erpretle and

5.0 Adaptive Model Construction

In Sections 3 and 4 we presented two schemes for modeling inelastic behavior of coite structures: the 2-point scheme and the n-point scheme. In the n-point scheme weemployed a piecewise constant approximation of the eigenstrain field, whereas in 2-point scheme the eigenstrain field and the elastic concentration factors in each phapproximated by a constant. For the Nozzle Flap problem considered in [8] (see alsure 1) the 2-point scheme is over three orders of magnitude faster than the n-point scheme.For linear problems the 2-point scheme with post-processing [5][8][9][13] is identical the n-point scheme, whereas for nonlinear problems there is no such guarantee.

If we assume that the n-point and the 2-point schemes are optimal in terms of accuraand speed, respectively, then it is natural to attempt to merge the two in a single mosuch a hybrid model, the 2-point scheme should be only used in regions where the moing errors are small, whereas elsewhere the n-point scheme should be employed. We wrefer to such a hybrid modeling strategy as the adaptive 2/n-point scheme.

The modeling error associated with the 2-point scheme can be defined as follows

(193)

where and

(194)

is an appropriate solution measure; the superscript ex refers to the exact solution withinthe framework of the mathematical homogenization theory, i.e., assuming solution picity. For estimation of errors resulting due to lack of periodicity we refer to [9][28].

The key questions are: (i) how to estimate , (ii) what is a suitable measure for how to make the process of error estimation efficient, and (iv) how to utilize the merror estimation for adaptive construction of the 2/n-point model.

It is appropriate to recall that as the number elements in the unit cell is increased th

tion obtained from the n-point scheme, denoted as , approaches the exact solu

i.e., . Even though the rate of convergence may not be monotonic,

reasonable to assume that for sufficiently large the modeling error associated with2-point scheme can be approximated as

(195)

We now turn to the second issue: the choice of . In this context it is essential to intthe 2-point scheme approach as consisting of two steps: analysis on the macrosca

e2-pt

e2-ptv

exv2-pt

–=

Ω Θ×=

f 2 1Θ---- f2 Θd Ωd

Ω∫

Θ∫=

v

vex

v

vn-pt

vn-pt

n ∞→lim v

ex→

n

e2-ptE2-pt≈ v

n-ptv2-pt

–=

v

29

arriede In theacro-xternals.

ic

other

.

(therepre-

astic

, aredeling

ator

ed ondnly

odel-ationn inantlyrocessng a

s, thef dis-e case

olvesains),

post-processing on the microscale. In the first step, a nonlinear macro-analysis is cout using the finite element method which utilizes the 2-point scheme. Consequently, thmacroscopic deformation history is stored in a database at macro-Gauss points.post-processing step, the deformation field in a unit cell corresponding to critical mpoints is extracted from the database, and then subjected onto the unit cell as an eloading. Finally, the n-point scheme is employed to solve for selected unit cell problem

Based on the above interpretation of the 2-point scheme, it follows that if the macroscopdeformation field obtained with the 2-point scheme is identical to one obtained with then-

point scheme, then the model error estimator, , should indicate zero error. In

words, should be a measure of the macroscopic deformation field, whereas

Possible deformation measures are: the macroscopic deformation gradient tensor, component form is defined in (120)), and/or incremental deformation measures sented by a pair . The former accounts for accumulation of errors

(196)

whereas the latter controls the incremental errors

(197)

In Section 6 we will show that in a confined deformation pattern, where small pl

zones are encompassed by elastically deforming solid, the modeling errors, very small, whereas in large plastic zones dominated by matrix deformation, the mo

errors, , might be significant. For simplicity, we adopt the incremental estim(197).

We now turn to the computational efficiency issue. Estimation of modeling error basequations (196) and (197) necessitates solution of the n-point scheme model. As indicateearlier the computational cost of the n-point scheme model is enormous, and hence, o

an estimate of , denoted , will be evaluated. The philosophy behind our ming error estimator is somewhat similar to that employed for estimation of discretizerrors, namely, if the mathematical model (or discretization) is locally altered, theabsence of the pollution errors the solution outside the local region is not significaffected, and thus the bulk of the error can be computed on the local level. This pavoids the need for solving an auxiliary global problem and replaces it by solvisequence of problems on small local domains.

When the aforementioned procedure is applied for estimation of discretization errorcomputational cost of solving the local problem is relatively low, reducing the cost ocretization error estimation to one of a manageable size. Unfortunately, this is not th

for estimation of modeling error . Even though the aforementioned process invmultiple solutions of small local problems (for example, on the macro-element dom

E2-pt

v Ω=

F

∆ε ∆ω,

EF2-pt Fn-pt F2-pt

– Ω=

E∆2-pt ε∆ n-pt ε∆ 2-pt

– Ω2

ω∆ n-pt ω∆ 2-pt– Ω

2+=

E2-pt

E2-pt

E2-ptE˜

2-pt

E˜

2-pt

30

rge

criti-eeringacro-

ementments,

ain

at

e

have

ure 5.

at

t time

le-

ated

the cost of applying the n-point scheme on each macro-element is formidable in a lascale computational environment. Therefore, the costly n-point scheme should be utilizedonly for those macro-elements which have been identified as “having potential to becal” by some simple cost-effective engineering-based criteria. One possible engincriterion is the magnitude of the deformation, measured by a norm of one of the mscopic strain measures. When the incremental deformation norm in a macro-eldomain exceeds a fraction of the average deformation among the macro-ele

i.e.,

(198)

then the corresponding macro element is tagged for a-posteriori model error estimation.

We now focus on the adaptive 2/n-point model construction. Consider the 2/n-point modelat time t, consisting of the 2-point scheme model in the portion of the macro-dom

and the n-point scheme model in the remainder such th

. The goal is to adaptively construct the 2/n-point model at time

, consisting of subdomains and . Let b

a subdomain in consisting of macro-element subdomains which

been tagged as critical by the aforementioned engineering criterion, as shown in Fig

Let and be the macro- strain and rotation increments on

. The first step in the adaptive process is to post-process the unit cell solution a

for all macro-elements on by utilizing the n-point scheme model outlined in

Section 4.

Let be the residual for all the elements on defined as

(199)

where is the corresponding internal force vector. In the second step, for all e

ments in the incremental nonlinear problem defined as

(200)

is solved twice: first, by using the 2-point scheme model, and second, by utilizing then-point scheme model with initial conditions obtained via post-processing. The estim

error on is computed by utilizing equation (197)

ζ N

ε∆ 2-pt ω∆ 2-pt+ Ωt e

ζ≥ ε∆ 2-pt ω∆ 2-pt+

ΩtN⁄

Ωt 2-pt Ωt⊂ Ωt n-pt

Ωt 2-pt Ωt n-pt∪ Ωt=

t t∆+ Ωt t∆+ 2-pt Ωt t∆+ n-pt Ωt 2-ptcr Ωt 2-pt

cr ee

Ncr

∪=

Ωt 2-ptNcr Ωt 2-pt

cr e

ε∆τ 2-ptcr e ω∆τ 2-pt

cr e Ωτ 2-ptcr e

τ t≤

t Ωt 2-ptcr e

r2-ptcr e Ωt 2-pt

cr e

r2-ptcr e f

t t∆+ 2-ptcr e f

t 2-ptcr e–=

ft 2-ptcr e

Ωt 2-ptcr e

r2-ptcr e 0=

Ωt 2-ptcr e

31

ted in

0%

to val- test

es ,

large dis-

omes (the

(201)

where the strain and rotation increments are evaluated by solving equation (200).

The total modeling error is then estimated as

(202)

To steer the process of adaptivity we define the modeling error indicator on

as

(203)

where

(204)

represents the average incremental deformation in a single element loca

. We replace the 2-point scheme model by the n-point scheme model for all the

elements on for which . A typical value for is between 1% to 1

depends on the accuracy requirement.

6.0 Numerical Experiments and Discussion

Our numerical experimentation agenda consists of three examples. The first is usedidate our finite deformation plasticity formulation. The second and the third examplesthe proposed adaptive 2/n-point scheme in a deformation pattern with large plastic zondominated by matrix deformation as well as in a typical confined deformation patternwhere a small plastic zone is encompassed by an elastically deforming solid.

6.1 Uniform Macro-Strain Loading

The objective of the first example is to carry out a qualitative assessment of thedeformation formulation. The primary “suspect” is equation (71) which decomposesplacement field in the microstructure into two parts: the macroscopic part which cfrom the integration of the nonperiodic macroscopic strain and rotation increments

E∆e2-pt ε∆ n-pt

cr e ε∆ 2-ptcr e– Ωt 2-pt

cr e

2ω∆ n-pt

cr e ω∆ 2-ptcr e– Ωt 2-pt

cr e

2+=

E∆2-pt E∆e

2-pt( )2

e Ωt 2 pt–cr e∈

Ncr

∑≈

η∆e Ωt 2-ptcr e

η∆e

E∆e2-pt

∆pav------------=

p∆av

∆ε2-pt ∆ω2-pt+ Ωt 2 pt–cr

2 E∆2-pt( )2+

Ncr----------------------------------------------------------------------------------=

p∆av

Ωt 2 pt–cr

Ωt 2-ptcr e η∆e tol≥ tol

32

that dis-copic

riodic

in fielded-w:

a,

loadings. Fig-efor-

aloneginal

e a testnatedThis is

con-edrals in then. Ination

main,. The

Pa,

e endit in all

ve the as

first term in (71)) and the periodic microscopic part (the second term in (71)). Notesolution update in the unit cell domain directly from the asymptotic expansion of theplacement field (11) is not feasible, because in the limit as , only the macros

part has contribution. On the other hand, if is considered only, then the nonpefinite deformation patterns are not accounted for.

As a test problem we select a macro problem subjected to the state of uniform stra(or linear displacement field). A unit cell consists of a stiff elastic cylindrical fiber embded in a compliant plastically deforming matrix. The phase properties are given belo

Fiber: Young’s modulus = 68.9 GPa, Poisson’s ratio = 0.21Matrix: Young’s modulus = 6.89 GPa, Poisson’s ratio = 0.33, yield stress = 24 MP

isotropic hardening modulus = 0.689 GPa, = 1.

We consider a uniform transverse tension, transverse shear and longitudinal shear conditions. The overall principal Green strain does not exceed 25% in all three caseures 6 to 8 show the contribution of macroscopic and microscopic fields to the total dmation field in the unit cell. It can be seen that each of the two contributing parts significantly distort the circular fiber cross section, but their sum recovers the orifiber shape, as expected in a matrix dominated loading condition.

6.2 The 3D Beam Problem

To validate the computational models and adaptive strategies proposed we compriscase, where a significant portion of the structure is subjected to the matrix domideformation in a load or stress control mode (as opposed to displacement control). a worse possible scenario in terms of accuracy for the 2-point scheme. The problemfiguration is shown in Figure 9. The macro problem is discretized with 5635 tetrahfinite elements. The geometry and the mesh for the microstructure are the same aprevious example. The fiber direction coincides with the beam’s longitudinal directiothe region of length from the fixed end the beam is subjected to the shear deform

(which is the matrix dominated mode) whereas in the remainder of the problem do length, the beam is in pure bending, which is a fiber dominated mode of loading

phase properties are summarized below:

Fiber: Young’s modulus = 37.92 GPa, Poisson’s ratio = 0.21Matrix: Young’s modulus = 6.89 GPa, Poisson’s ratio = 0.33, yield stress = 24 M

isotropic hardening modulus = 0.689 GPa, = 1.

The loading is applied in 15 load steps. The maximal vertical displacement at the freis over one third of the length of the beam and the stresses exceed the elastic limmacro-elements.

The problem is solved using the 2-point scheme with micro-history recovery, the adapti2/n-point scheme, and the n-point scheme for a comparison purpose. Figure 10 showsevolution of the normalized estimated local error in the vicinity of the fixed end

ς 0→u1

β

l1

l2

β

33

nor-gionof the) as 11.

he nor-rmal-

e unit

only ag theicro-

fabric fiber. Thematrix

ssure)llowedkes theeri-

ase of

atstruc-hingects error6.5%

ondsor the

obtained with the 2-point scheme (equation (203)). It can be seen that the maximalmalized local error in the region dominated by matrix deformation is 40%. In a redominated by the fiber deformation the error does not exceed 3%. The distribution local principal stress error in the critical unit cell (denoted by point A in Figure 10obtained with the 2-point scheme and micro-history postprocessing is shown in FigureIt can be seen that the normalized error in the unit cell is of the same magnitude as tmalized local error in the macrostructure. Figure 12 illustrates the evolution of the noized local error in the macrostructure obtained using the adaptive 2/n-point scheme model.The maximal normalized local error is less than 1% and the normalized error in thcell follows the same trend as shown in Figure 13.

We conclude that the adaptive 2/n-point scheme model outperforms the 2-point schememodel in terms of accuracy (0.8% maximal error as compared to 40%), and the n-pointscheme model in terms of CPU time as it is 14 times faster than the n-point scheme.

6.3 The Nozzle Flap Problem

For the final numerical example, we consider a typical aerospace component wheresmall region experiences inelastic deformation. The finite element mesh describinmacrostructure of the Nozzle Flap is shown in Figure 1. We consider two types of mstructures: (i) the fibrous unit cell (as in the previous example) and the plain weave microstructure shown in Figure 14. The fibrous unit cell contains 98 elements in thedomain and 253 elements in the matrix domain. The fiber volume fraction is 0.27plain weave microstructure has 370 elements in the fiber bundle and 1196 in the domain. The bundle volume fraction is 0.25. The phase properties are:

Fiber, fiber bundle: Young’s modulus = 379.2 GPa, Poisson’s ratio = 0.21Matrix: Young’s modulus = 68.9 GPa, Poisson’s ratio = 0.33,

yield stress = 24 MPa, isotropic hardening = 14 GPa, = 1.

The Nozzle Flap is subjected to an aerodynamic force (simulated by a uniform preon the back of the flap. We assume that the pin-eyes are rigid and a rotation is not aso that all the degrees of freedom on the pin-eye surfaces are fixed. The loading tasolution well into the inelastic region in the vicinity of the pins: 15% of elements expence inelastic deformation in the case of fibrous microstructure, and 29% in the cplain weave.

The problem is analyzed using the adaptive 2/n-point scheme model. Figure 15 shows ththe 2-point scheme model yields the maximum normalized local error in the macroture below 1% (for the plain weave microstructure). Hence, if the tolerance for switcfrom the 2-point scheme to the n-point scheme is higher than 1%, adaptive strategy selthe 2-point scheme model in the entire macro problem domain. The normalized localin the unit cell located at Point C of Figure 15 is 2.5% for fibrous microstructure and for the plain weave, as shown in Figures 16 and 17.

For the problem with the fibrous unit cell, the CPU time on a SPARC 10/51 is 30 secfor the macroscopic analysis and 120 seconds for postprocessing a single point. F

β

34

as post-7ave.

1200

ci-4-92-

os-

-

i-

l

Esti-

for tice,”

dic

ate-

ed

plain weave microstructure, the macroscopic analysis consumes 30 seconds, whereprocessing takes 510 seconds per point. On the other hand, the n-point scheme consumes hours of CPU time for fibrous composite and over 55 hours of CPU time for plain weMemory requirement ratios are approximately 1:250 for the fibrous unit cell and 1:for the plain weave in favor of the 2/n-point scheme (or 2-point scheme).

Acknowledgment

The authors gratefully acknowledge the support for this work by Air Force Office of Sentific Research under grant F49620-97-1-0090 and ARPA/ONR under grant N0001J-1779.

References

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2 M. L. Accorsi and S. Nemat-Nasser, “Bounds on the Overall Elastic and Instantaneous Elastoplastic Moduli of Periodic Composites,” Mechanics of Materials, 5, 1986.

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4 N. S. Bakhvalov and G. P. Panasenko, Homogenisation: Averaging Processes in Perodic Media, Kluwer Academic Publishers, 1989.

5 G.J. Dvorak, “Transformation Field Analysis of Inelastic Composite Materials,” Pro-ceedings Royal Society of London, A437, 1992.

6 N. Fares and G. J. Dvorak, “Large Elastic-Plastic Deformations of Fibrous MetaMatrix Composites,” Journal of the Mechanics and Physics of Solids, 39, 1991.

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8 J. Fish, K. Shek, M. Pandheeradi and M. S. Shephard, “Computational PlasticityComposite Structures Based on Mathematical Homogenization: Theory and PracComputer Methods in Applied Mechanics and Engineering, accepted for publication.

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10 S. Ghosh and S. Moorthy, “Elastic-Plastic Analysis of Arbitrary Heterogeneous Mrials with the Voronoi Cell Finite Element Method,” Computer Methods in Applied Mechanics and Engineering, 121, 1995.

11 M. Gosz, B. Moran and J. D. Achenbach, “Matrix Cracking in Transversely LoadFiber Composites with Compliant Interphases,” AMD-Vol. 150/AD-Vol. 32, Damage Mechanics in Composites, ASME, 1992.

12 J. M. Guedes, Nonlinear Computational Models for Composite Materials Using

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ased

en-edi-

of

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ent

and

Homogenization, PhD thesis, University of Michigan, 1990.

13 J. M. Guedes and N. Kikuchi, “Preprocessing and Postprocessing for Materials Bon the Homogenization Method with Adaptive Finite Element Methods,” Computer Methods in Applied Mechanics and Engineering, 83, 1990.

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15 R. Hill, “A Self Consistent Mechanics of Composite Materials,” Journal of the Mechanics and Physics of Solids, 13, 1965.

16 T. J. R. Hughes, “Numerical Implementation of Constitutive Models: Rate-Indepdent Deviatoric Plasticity,” in S. Nemat-Nasser, R. J. Asaro and G. A. Hegemier, tors, Theoretical Foundation for Large Scale Computations for Nonlinear MaterialBehavior, Martinus Nijhoff Publishers, 1983.

17 T. J. R. Hughes and J. Winget, “Finite Rotation Effects in Numerical Integration Rate Constitutive Equations Arising in Large Deformation Analysis,” International Journal of Numerical Methods in Engineering, 15, 1980.

18 S. Jansson, “Homogenized Nonlinear Constitutive Properties and Local Stress Ccentrations for Composites with Periodic Internal Structure,” International Journal of Solids and Structures, 29, 1992.

19 A. L. Kalamkarov, Composite and Reinforced Elements of Construction, John Wiley and Sons, 1992.

20 E. H. Lee, R. L. Mallett and T. B. Wertheimer, “Stress Analysis for Kinematic Haening in Finite Deformation Plasticity,” Journal of Applied Mechanics, 105, 1983.

21 F. Lene and D. Leguillon, “Homogenized Constitutive Law for a Partially CohesivComposite Material,” International Journal of Solids and Structures, 18, 1982.

22 F. Lene, “Damage Constitutive Relations for Composite Materials,” Engineering Fracture Mechanics, 25, 1986.

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24 C. J. Lissenden and C. T. Herakovich, “Numerical Modeling of Damage Developmand Viscoplasticity in Metal Matrix Composites,” Computer Methods in Applied Mechanics and Engineering, 126, 1995.

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27 S. Nemat-Nasser, “On Finite Plastic Flow of Crystalline and Geomaterials,” Journal of Applied Mechanics, 105, 1983.

36

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28 J. T. Oden and T. I. Zohdi, Analysis and Adaptive Modeling of Highly HeterogeneoElastic Structures, TICAM Report 56, University of Texas at Austin, 1996.

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30 E. Sanchez-Palencia and A. Zaoui, Homogenization Techniques for Composite Med, Springer-Verlag, 1987.

31 J. C. Simo and R. L. Taylor, “Consistent Tangent Operators for Rate-IndependeElastoplasticity,” Computer Methods in Applied Mechanics and Engineering, 48, 1985.

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33 P. M. Suquet, “Elements of Homogenization for Inelastic Solid Mechanics,” in ESanchez-Palencia and A. Zaoui, editors, Homogenization Techniques for CompositeMedia, Springer-Verlag, 1987.

34 J. L. Teply and G. J. Dvorak, “Bounds on Overall Instantaneous Properties of ElPlastic Composites,” Journal of the Mechanics and Physics of Solids, 36, 1988.

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7.0 Appendixes

A.0 Derivation of in (96)

Consider equation (93):

(205)

Taking the derivative of (205) with respect to yields:

(206)

where the last term can be written as

∂ σijm( ) αi j

m( )–( ) ∂∆λ m( )⁄

σijm( ) αij

m( )– I ijkl ∆λ m( )℘i jkl+( ) 1– σklm( )

tr αklm( )t

–( )=

∆λ m( )

∂∂∆λ m( )----------------- σi j

m( ) αijm( )–( ) I ijkl ∆λ m( )℘i jkl+( ) 1–=

℘klmn σmnm( ) αmn

m( )–( )– ∂∂∆λ m( )----------------- σkl

m( )tr αkl

m( )t–( )+

37

(207)

and

(208)

The rotation of phase is defined in (77) as

(209)

The derivative of is calculated using the chain rule:

(210)

in which

(211)

Consequently, equation (208) can be expressed as

(212)

Similarly, we have

(213)

Taking derivative of (89) with respect to yields:

(214)

∂ σklm( )

tr αklm( )t

–( )

∂∆λ m( )----------------------------------------

∂ σklm( )t αkl

m( )t–( )

∂∆λ m( )------------------------------------

∂ σklm( )t αkl

m( )t–( )

∂∆ωpqm( )

------------------------------------∂∆ωpq

m( )

∂∆λ m( )------------------= =

∂ σklr( )t

∂∆ωpqr( )

-----------------∂ℜsk

r( )

∂∆ωpqr( )

-----------------ℜtlr( ) ℜsk

r( ) ∂ℜtlr( )

∂∆ωpqr( )

-----------------+

σstr( )t

=

δmsδnkℜtlr( ) δmtδnlℜsk

r( )+( ) σstr( )t ∂ℜmn

r( )

∂∆ωpqr( )

-----------------=

ℜmnr( ) r

ℜmnr( ) δmn δmr

12---∆ωmr

r( )– 1–

∆ωrnr( )+=

ℜmnr( )

∂ℜmnr( )

∂∆λ m( )-----------------

∂ℜmnr( )

∂∆ωpqr( )

-----------------∂∆ωpq

r( )

∂∆λ m( )-----------------=

∂ℜmnr( )

∂∆ωpqr( )

----------------- 2δmp ∆ωmpr( )–( ) 1– δqn ℜqn

r( )+( )=

Uklpqr( )

σ∂ σkl

r( )t

∂∆ωpqr( )

----------------- δnkℜtlr( ) σmt

r( )t δnlℜskr( ) σsm

r( )t+( ) 2δmp ∆ωmp

r( )–( ) 1– δqn ℜqnr( )+( )=≡

Uklpqm( )

α∂ αkl

m( )t

∂∆ωpqm( )

------------------ δnkℜtlm( ) αmt

m( )t δnlℜskm( ) αsm

m( )t+( ) 2δmp ∆ωmp

m( )–( ) 1– δqn ℜqnm( )+( )=≡

∆λ m( )

∂∆ωpqm( )

∂∆λ m( )------------------ Dpqmn

mm( ) Pmnkl σklm( ) αkl

m( )–( ) ∆λ m( )∆λm∂∂ σkl

m( ) αklm( )–( )+

=

38

lt into

rate

n as

Substituting equations (212), (213) and (214) into (207), and then inserting the resu(206), gives

(215)

where

(216)

(217)

B.0 Consistent Linearization of and

We derive the equations for and consistent with the midpoint integration of

of deformation and rotation. The left superscript is omitted.

Taking the material time derivative of (72) yields:

(218)

The material time derivative of the first term in the parenthesis of (218) can be writte

(219)

where

(220)

Consequently, (219) can be expressed as

(221)

where

(222)

∂∂∆λ m( )----------------- σij

m( ) αi jm( )–( ) Cijkl

m( ) σklm( ) αkl

m( )–( )=

Cijklm( ) I ijmn ∆λ m( )Wijmn

m( )+( ) 1– Wmnklm( )–=

Wmnklm( ) ℘mnkl Umnst

m( )σ Umnst

m( )α–( )Dstpq

mm( )Ppqkl–=

∆ε i j ∆ω ij

∆ε·i j ∆ω· ij

t ∆t+

tdd ∆εij

12--- d

dt-----

∂∆ui0

∂ xt ∆t 2⁄+

j

----------------------∂∆uj

0

∂ xt ∆t 2⁄+

i

----------------------+

=

tdd ∂∆ui

0

∂ xt ∆t 2⁄+

j

---------------------- ∂vi

0

∂ xt

k

---------∂ x

tk

∂ xt ∆t 2⁄+

j

----------------------∂∆ui

0

∂ xt

k

-------------td

d ∂ xt

k

∂ xt ∆t 2⁄+

j

----------------------

+=

tdd ∂ x

tk

∂ xt ∆t 2⁄+

j

---------------------- ∂ x

tk

∂ xt ∆ t 2⁄+

m

------------------------td

d ∂ xt ∆t 2⁄+

m

∂ xt

n

------------------------ ∂ x

tn

∂ xt ∆t 2⁄+

j

----------------------–=

tdd ∂∆ui

0

∂ xt ∆t 2⁄+

j

---------------------- ∂vi

0

∂ xt ∆t 2⁄+

j

----------------------∂∆ui

0

∂ xt ∆t 2⁄+

m

------------------------td

d ∂ xt ∆ t 2⁄+

m

∂ xt

n

------------------------ ∂ x

tn

∂ xt ∆ t 2⁄+

j

----------------------–=

tdd ∂ x

t ∆t 2⁄+m

∂ xt

n

------------------------ ∂

∂ xt

n

---------td

d xt ∆t 2⁄+

m ∂

∂ xt

n

---------td

d xm xt

m+

2-------------------

1

2---

∂vm0

∂ xt

n

---------= = =

39

the

Substituting (222) into (221) gives

(223)

Equation (223) can be further simplified as

(224)

where the following equality has been utilized.

(225)

Defining as

(226)

We have

(227)

Substituting (227) into (218) and preforming the same procedure for , we obtain

final expressions for and as

(228)

Note that in the case of backward Euler integration .

tdd ∂∆ui

0

∂ xt ∆t 2⁄+

j

----------------------

δim12---

∂∆ui0

∂ xt ∆t 2⁄+

m

------------------------– ∂vm

0

∂ xt ∆t 2⁄+

j

----------------------=

tdd ∂∆ui

0

∂ xt ∆t 2⁄+

j

---------------------- ∂ x

ti

∂ xt ∆t 2⁄+

m

------------------------∂vm

0

∂ xt ∆t 2⁄+

j

----------------------=

δim12---

∂∆ui0

∂ xt ∆t 2⁄+

m

------------------------–x

t ∆ t 2⁄+m∂

∂ xt ∆t 2⁄+

i12---∆ui

0– ∂ x

ti

∂ xt ∆t 2⁄+

m

------------------------= =

Mijkl

Mijkl

∂ xt

i

∂ xt ∆t 2⁄+

k

-----------------------∂xl

∂ xt ∆t 2⁄+

j

----------------------≡

tdd ∂∆ui

0

∂ xt ∆t 2⁄+

j

----------------------

Mijkl vk xl,0=

∆ω· ij

∆ε·i j ∆ω· ij

∆ε·i j M ij( )kl vk xl,0= , ∆ω· ij M ij[ ]kl vk xl,

0=

Mijkl δikδj l=

40

Figure 1: Finite element mesh for the nozzle flap problem

Figure 2: Finite element mesh for the fibrous unit cell

41

Figure 3: Macroscopic and microscopic structures

Figure 4: Decomposition of deformation in the microstructure

Figure 5: Adaptive model construction

Ω

t

y1

y2

x1

x2

ΘZoomΘ Θ

ΘΓt

Γu

Unit cells

y x ς⁄≡

Composite structure

b

Θ...

...

(a) Undeformed configuration

(b) Macroscopic strain contribution

(c) Final deformed configuration

Ωt n-pt

Ωt 2-pt

Ωt 2-ptcr Ωt 2-pt⊆

Ωt ∆t+ n-pt

Ωt ∆t+ 2-pt

Ωt ∆t+ 2-ptcr Ωt ∆t+ 2-pt⊆

Time t= Time t ∆t+=

42

Figure 6: Deformation of unit cell under transverse tension

Figure 7: Deformation of unit cell under transverse shear

+ =

u1 uε∆ ∆ω+( )y td∫

u1

+ =

uε∆ ∆ω+( )y td∫

u1

+ =

uε∆ ∆ω+( )y td∫

43

Figure 8: Deformation of unit cell under longitudinal shear

Figure 9: Finite element mesh for the 3D beam problem

P

Fixed end

M

l1

l2

u 0=

l2l1---- 15=

44

Figure 10: Distribution of the normalized local error with the 2-point model

Figure 11: Effective stress and normalized error at point A with the 2-point model

Step 5

Step 10

Step 15

A

Fixed end l1

l1

l1

Effective stress Normalized error of effective stress

45

Figure 12: Distribution of the normalized local error with the 2/n-point model

Figure 13: Effective stress and normalized error at point B with the 2/n-point model

Step 5

Step 10

Step 15

B

Fixed end l1

l1

l1

Effective stress Normalized error of effective stress

46

Figure 14: Geometric model and FE mesh of the plain weave unit cell

(a) Geometric model (b) Finite element mesh

47

Figure 15: Nozzle flap problem / plain weave RVE: distribution of the normalized local error in the macrostructure as obtained with the 2-point model

Step 5

Step 1

Step 3

Step 7C

48

Figure 16: Effective stress and normalized error for fibrous unit cell as obtained with the 2-point model

Figure 17: Effective stress and normalized error for plain weave unit cell as obtainedwith the 2-point model

Effective stress Normalized error of effective stress

Effective stress Normalized error of effective stress

49