Multiple scale eigendeformation-based reduced order homogenization

Comput. Methods Appl. Mech. Engrg. 198 (2009) 2016–2038

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Comput. Methods Appl. Mech. Engrg.

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Multiple scale eigendeformation-based reduced order homogenization

Zheng Yuan, Jacob Fish *

Multiscale Science and Engineering Center, Rensselaer Polytechnic Institute, CII 7129 110 8th St., Troy, NY 12180, United States

a r t i c l e i n f o

Article history:Received 24 February 2008Received in revised form 29 August 2008Accepted 31 December 2008Available online 23 January 2009

Keywords:Homogenization model-reductioneigendeformation

0045-7825/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.cma.2008.12.038

* Corresponding author. Tel.: +1 518 276 6191.E-mail address: [email protected] (J. Fish).

a b s t r a c t

Multiple scale eigendeformation-based reduced order homogenization method, which provides consider-able computational cost saving in comparison to the direct homogenization method, has been developed,verified against the direct homogenization method, and validated against experimental data. The salientfeature of the method is in the formulation of the unit cell problem in terms of residual-free stresses, andtherefore eliminating the need for costly equilibrium calculations required by the direct homogenizationmethod. This is accomplished by introducing discretized eigendeformation fields at different scales,which are collectively referred to as eigenstrains and eigenseparations, and then precomputing asequence of elasticity solutions of the unit cell problem subjected to the eigendeformation fields priorto nonlinear analysis.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

Since the pioneering works of and Hill [1], Babuska [2], Benssousan [3] and Sanchez-Palencia [4], homogenization methods made sig-nificant strides, recently finding their way into industrial arena. The new comers are the so-called direct computational homogenizationand direct mathematical homogenization approaches. The direct computational homogenization approach, based on the Hill–Mandel rela-tion [5] in combination with numerical methods (see [6–24] for closely related methods), followed the homogenization approach set fourthby Hill [1]. The direct mathematical homogenization approaches, based on the multiple scale asymptotic expansion methods in combina-tion with numerical methods [25–30] follow the formulation pathway set in [2–4].

While computational cost of the direct computational and mathematical homogenization approaches is a small fraction compared tothe direct numerical simulation where a characteristic mesh size is of the heterogeneity order they remain computationally prohibitivefor large scale nonlinear problems. This is because a nonlinear unit cell problem for a two-scale problem has to be solved number of timesequal to the product of number of quadrature points at a coarse scale times the number of load increments and iterations at the coarsescale. With each additional scale the number of times a finest scale unit cell problem has to be solved is increased by a factor equal tothe product of number of quadrature points, load increments and iterations at a previous scale.

It is therefore necessary to reduce the computational complexity of the fine scale problem without significantly compromising on thesolution accuracy of interest, which is typically at the system level. The method presented in this manuscript is a generalization of the two-scale eigendeformation-based approach developed in [31] in the following three respects:

1. The two-scale mathematical framework in [31] is extended to multiple-scales with equations explicitly derived for three scales denotedhereafter as macro, meso and micro.

2. The formulation in [31] was hard-wired to continuum damage mechanics at the fine scale. The present manuscript extends it to arbi-trary inelastic deformation of micro phases and interfaces.

3. Numerical Greens-functions in [31] were developed on the element-by-element basis. The present manuscript is based on the partition-by-partition representation of numerical Greens functions, which results in considerable computational cost savings.

The manuscript is organized as follows. In Section 2, the three-scale mathematical homogenization formulation is developed. The gov-erning equations are reformulated in terms of residual-free stresses based on analytical influence (Green’s) functions in Section 3. Section 4presents a reduced order model based on the partition-by-partition representation of numerical influence functions. Solution of a system ofnonlinear equations corresponding to the micro and meso scales is described in Section 5. Derivation of the two-scale formulation as a

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Z. Yuan, J. Fish / Comput. Methods Appl. Mech. Engrg. 198 (2009) 2016–2038 2017

special case of the multiple scale formulation is considered in Section 6. Convergence studies of the reduced order model to the directhomogenization are conducted in Section 7. Validation studies on tube crush tests conclude the manuscript.

2. Three-scale mathematical homogenization

Mathematical homogenization is an upscaling technique by which one starts with governing equations at the finest scale of interest andthen scale-by-scale derives coarser scale equations. The strong form of the boundary value problem on the finest (micro) scale of interest isgiven by

rfij;jðxÞ þ bf

i ðxÞ ¼ 0 x 2 X; ð1Þ

rfijðxÞ ¼ Lf

ijklðxÞ efklðxÞ �

XI

IlfklðxÞ

" #x 2 X; ð2Þ

efijðxÞ ¼ uf

ði;jÞðxÞ �12ðuf

i;j þ ufj;iÞ x 2 X; ð3Þ

ufi ðxÞ ¼ �uiðxÞ x 2 Cu; ð4Þ

rfijðxÞnjðxÞ ¼ �tiðxÞ x 2 Ct; ð5Þ

dfi ðxÞ � suf

i ðxÞt ¼ ufi jSf�� uf

i jSfþ

x 2 Sf; ð6Þ

rfijnjjSf

þþ rf

ijnjjSf�¼ tf

i jSfþþ tf

i jSf�¼ 0; ð7Þ

where the superscript f denotes dependence of the response field on the microstructural heterogeneities. Constitutive relations described inEq. (2) are assumed to admit an additive decomposition of total strains ef

ij into elastic and inelastic components (or more generally stated aseigenstrains Ilf

klÞ; the left superscript I indicates different types of eigenstrains, such as inelastic deformation (damage, plasticity, etc.), ther-mal changes, moisture effects and phase transformation. Sf in Eqs. (6) and (7) denotes the interface between the micro-constituents with Sf

þand Sf

� denoting the two sides of the interface. dfi in Eq. (6) is the displacement jump along the interface, and s � t is the jump operator. The

traction continuity along the interface is given by Eq. (7).We assume existence of material heterogeneity at two different scales as schematically illustrated in Fig. 1. Making the usual assump-

tion of scale separation, an approximate solution of Eqs. (12), (3)–(7) is sought of within the framework of the mathematical homogeni-zation theory where material heterogeneity is expressed in terms of locally periodic representative volume elements (RVEs or unitcells) at the meso- and the micro-scales. The volume of the unit cells at the two scales are denoted by Hy and Hz, respectively. The cor-responding interfaces in the meso and micro unit cells are denoted by Sy and Sz, respectively. Two additional position vectors, y � x=fin Hy and z � y=f in Hz, are introduced to track the position of a point at any scale, where 0 < f� 1. Any locally periodic function canbe represented in terms of the two position vectors as

/fðxÞ � /ðx; y; zÞ: ð8Þ

The indirect spatial derivative is calculated by the chain rule as

/f;iðxÞ ¼ /;xi

ðx; y; zÞ þ 1f

/;yiðx; y; zÞ þ 1

f2 /;ziðx; y; zÞ; ð9Þ

where a comma followed by a subscript variable xi, yi or zi denotes the partial derivative with respect to the macro, meso or micro positionvector.

A three-scale asymptotic expansion is employed to approximate the displacement field

uiðx; y; zÞ ¼ u0i ðxÞ þ fu1

i ðx; yÞ þ f2u2i ðx; y; zÞ þ � � � ð10Þ

Using the chain rule of the differentiation (9) and the kinematical relation (3), the asymptotic expansion of the strain field can be expressedas

eijðx; y; zÞ ¼ e0ijðx; y; zÞ þ fe1

ijðx; y; zÞ þ f2e2ijðx; y; zÞ þ � � � ð11Þ

Fig. 1. Three-scale representation of a heterogeneous material [27].

2018 Z. Yuan, J. Fish / Comput. Methods Appl. Mech. Engrg. 198 (2009) 2016–2038

with the leading order strain given as

e0ijðx; y; zÞ ¼ u0

ði;xjÞðxÞ þ u1ði;yjÞðx; yÞ þ u2

ði;zjÞðx; y; zÞ: ð12Þ

The macro strain is defined as an average strain over Hy �Hz. With consideration of periodicity at the two scales it is given as

emacij ðxÞ �

1jHyj

1jHzj

ZHy

ZHz

e0ijðx; y; zÞdHzdHy ¼ u0

ði;xjÞðxÞ: ð13Þ

The meso strain is defined as an average strain over Hz. Assuming periodicity at the micro scale it is given as

emesoij ðx; yÞ � 1

jHzj

ZHz

e0ijðx; y; zÞdH ¼ u0

ði;xjÞðxÞ þ u1ði;yjÞðx; yÞ ¼ emac

ij ðxÞ þ u1ði;yjÞðx; yÞ: ð14Þ

Thus the micro scale strain can be expressed as

emicij ðx; y; zÞ � e0

ijðx; y; zÞ ¼ u0ði;xjÞðxÞ þ u1

ði;yjÞðx; yÞ þ u2ði;zjÞðx; y; zÞ ¼ emeso

ij ðx; yÞ þ u2ði;zjÞðx; y; zÞ: ð15Þ

Throughout the manuscript the superscripts mic, meso and mac are used to denote quantities at different scales.The asymptotic expansion of the stress field is given as

rijðx; y; zÞ ¼ r0ijðx; y; zÞ þ fr1

ijðx; y; zÞ þ f2r2ijðx; y; zÞ þ � � � ð16Þ

where the leading order stress is given by

r0ijðx; y; zÞ ¼ Lijklðy; zÞ e0

klðx; y; zÞ �X

I

Ilklðx; y; zÞ" #

: ð17Þ

The micro-, meso- and macro-stresses are defined as

rmicij ðx; y; zÞ � r0

ijðx; y; zÞ; ð18Þ

rmesoij ðx; yÞ � 1

jHzj

ZHz

rmicij ðx; y; zÞdH; ð19Þ

rmacij ðxÞ �

1jHyj

ZHy

rmesoij ðx; yÞdH: ð20Þ

Inserting the asymptotic expansion of the stress field (16) into the equilibrium (1) and identifying terms with equal powers of f gives variousorder equilibrium equations

Oðf�2Þ : r0ij;zjðx; y; zÞ ¼ 0; ð21Þ

Oðf�1Þ : r0ij;yjðx; y; zÞ þ r1

ij;zjðx; y; zÞ ¼ 0; ð22Þ

Oðf0Þ : r0ij;xjðx; y; zÞ þ r1

ij;yjðx; y; zÞ þ r2

ij;zjðx; y; zÞ þ bi ¼ 0: ð23Þ

Using the definition (18) and Eq. (21), the micro equilibrium equation can be restated as

Oðf�2Þ : rmicij;zjðx; y; zÞ ¼ 0 : ð24Þ

Integrating Eq. (22) over Hz, exploiting periodicity at the microscale and using the definition (19) yields the meso equilibrium

Oðf�1Þ : rmesoij;yjðx; yÞ ¼ 0 : ð25Þ

Integrating Eq. (23) over Hz and Hy, exploiting periodicity at the micro and meso scales and using the definition (20) yields the equilibriumequation at the macro scale

Oðf0Þ : rmacij;xjðxÞ þ �bi ¼ 0 ; ð26Þ

where

�bi ¼1jHyj

1jHzj

ZHy

ZHz

bi dHz dHy: ð27Þ

For more details on the three-scale mathematical homogenization the reader is referred to [27].

3. Residual-free governing equations

In this section, we formulate equilibrium equations in terms of equilibrated (or residual-free) micro and meso stresses. By this approachthe computational bottle neck of solving large discrete system equilibrium equations corresponding to the unit cell problems is removed.

Using the definitions (18) and (15) together with the constitutive relation (16), the equilibrium equation at the micro scale (24) can beexpressed as

Lmicijkl ðy; zÞ emeso

kl ðx; yÞ þ u2ðk;zlÞðx; y; zÞ �

XI

Ilklðx; y; zÞ" #( )

;zj

¼ 0: ð28Þ


Following [31], the micro displacement field is decomposed using the elastic influence function Hmicikl , the eigenstrain influence function hmic l

ikl

and the eigenseparation influence function hmic din as

u2i ðx; y; zÞ ¼ u2e

i ðx; y; zÞ þX

I

Iu2li ðx; y; zÞ þ u2d

i ðx; y; zÞ

¼ Hmicikl ðy; zÞemeso

kl ðx; yÞ þZ

Hz

hmic likl ðy; z; zÞ

XI

Ilklðx; y; zÞdHþZ

Sz

hmic din ðy; z; zÞdnðx; y; zÞdS; ð29Þ

where the influence (or Green’s) functions Hmicikl , hmic l

ikl and hmic din are computed by solving a sequence of elastic boundary value problems

independent of and prior to the nonlinear macro analysis. The subscript n in the eigenseparation influence function denotes the componentin the local Cartesian coordinate system of the interface. The expression similar to the second term in (29) can be found in [32]. These func-tions are chosen to satisfy the equilibrium Eq. (28) for arbitrary emeso

kl ðx; yÞ, Ilklðx; y; zÞ and dnðx; y; zÞ.The governing equation for the elastic influence function Hmic

ikl is obtained by substituting (29) into (28) for vanishing eigendeformationIlklðx; y; zÞ, dnðx; y; zÞ and arbitrary meso strain emeso

kl ðx; yÞ

fLmicijkl ðy; zÞ½Iklmnðy; zÞ þ Gmic

klmnðy; zÞ�g;zj¼ 0; y 2 Hy; z 2 Hz; ð30Þ

where

Gmicijkl ðy; zÞ ¼ Hmic

ði;zjÞklðy; zÞ ð31Þ

and Iijkl is the fourth order identity tensor.The governing equation for the eigenstrain influence function hmic l

ikl is obtained by substituting (29) into (28), assuming vanishingdnðx; y; zÞ and emeso

kl ðx; yÞ, and eigenstrain Ilklðx; y; zÞ in the form of the Dirac delta function, which gives

fLmicijkl ðy; zÞ½g

mic lklmn ðy; z; zÞ � Iklmnðy; zÞdðy; z� zÞ�g;zj

¼ 0; y 2 Hy; z; z 2 Hz; ð32Þ

where

gmic lijkl ðy; z; zÞ ¼ hmic l

ði;zjÞklðy; z; zÞ ð33Þ

and d is the Dirac delta function.Similarly, the governing equation for the eigenseparation influence function hmic d

in is obtained by substituting (29) into (28), assumingvanishing eigenstrain and meso strain, and eigenseparation in the form of the Dirac delta function, which gives

fLmicijkl ðy; zÞgmic d

kln ðy; z; zÞg;zj¼ 0; y 2 Hy; z 2 Hz; z 2 Sz;

E:B:C: dnðy; ~zÞ ¼ dðy; ~z� zÞ; ~z 2 Sz;ð34Þ

where

gmic dijn ðy; z; zÞ ¼ hmic d

ði;zjÞn ðy; z; zÞ ð35Þ

and dnðy; ~zÞ is the displacement jump along the interface at the micro scale defined in the local interface Cartesian coordinate system.Note that the relation between elastic and eigenstrain polarization functions can be found by integrating (32) over z 2 Hz, which gives
Z
Hz

gmic lijkl ðy; z; zÞdH ¼ �Gmic

ijkl ðy; zÞ; ð36Þ

where we have exploited the relationR

HzIijklðy; zÞdðy; z� zÞdH ¼ Iijklðy; zÞ.

The residual-free micro stress ðrmicij;zj¼ 0Þ follows from Eqs. (28) and (29)

rmicij ¼ Lmic

ijkl ðy; zÞ

Amicklmnðy; zÞemeso

mn ðx; yÞþR

Szgmic d

kln ðy; z; zÞdnðx; y; zÞdS

þR

Hzgmic l

klmn ðy; z; zÞP

I

Ilmnðx; y; zÞdH�P

I

Ilklðx; y; zÞ

26664

37775; ð37Þ

where

Amicijkl ðy; zÞ ¼ Iijklðy; zÞ þ Gmic

ijkl ðy; zÞ: ð38Þ

Using the definitions (19), (14) and the residual-free micro stress (37), the meso equilibrium Eq. (25) can be expressed in terms of the mesoeigenstrains as

fLmesoijkl ðyÞ½emac

kl ðxÞ þ u1ðk;ylÞðx; yÞ � lmeso

kl ðx; yÞ�g;yj¼ 0; ð39Þ

where

Lmesoijkl ðyÞ ¼

1jHzj

ZHz

Lmicijmnðy; zÞA

micmnklðy; zÞdH: ð40Þ


The meso eigenstrain is related to the micro eigendeformation by

lmesoij ðx; yÞ ¼ �½Lmeso

ijst ðyÞ��1 1jHzj

ZHz

Lmicstmnðy; zÞ

RSz

gmic dmnn ðy; z; zÞdnðx; y; zÞdS

þR

Hzgmic l

mnkl ðy; z; zÞP

I

Ilklðx; y; zÞdH

�P

I

Ilmnðx; y; zÞ

266664

377775

8>>>><>>>>:

9>>>>=>>>>;

dH: ð41Þ

Note that while the meso eigenstrain is completely determined by micro eigendeformation, the meso eigenseparation is independent of microeigendeformation. The meso displacement u1

i ðx; yÞ assumes a similar decomposition to that of the micro deformation u2i ðx; yÞ given in (29)

u1i ðx; yÞ ¼ u1e

i ðx; yÞ þ u1li ðx; yÞ þ u1d

i ðx; yÞ ¼ Hmesoikl ðyÞemac

kl ðxÞ þZ

Hy

hmeso likl ðy; yÞlmeso

kl ðx; yÞdHþZ

Sy

hmeso din ðy; yÞdmeso

n ðx; yÞdS: ð42Þ

The meso influence functions are defined to satisfy equilibrium at mesoscale (39) for arbitrary emackl ðxÞ, lmeso

kl ðx; yÞ and dmeson ðx; yÞ.

For the elastic influence function Hmesoikl ðyÞ, we have

fLmesoijkl ðyÞ½IklmnðyÞ þ Gmeso

klmn ðyÞ�g;yj¼ 0; y 2 Hy; ð43Þ

where

Gmesoijkl ðyÞ ¼ Hmeso

ði;yjÞklðyÞ: ð44Þ

The eigenstrain influence function hmeso likl ðy; yÞ are given as

fLmesoijkl ðyÞ½g

meso lklmn ðy; yÞ � IklmnðyÞdðy � yÞ�g;yj

¼ 0; y; y 2 Hy; ð45Þ

where

gmeso lijkl ðy; yÞ ¼ hmeso l

ði;yjÞkl ðy; yÞ: ð46Þ

Similarly to Eq. (36), we have the following relation between the elastic and eigenstrain polarization functions
ZHz
gmeso lijkl ðy; z; zÞdH ¼ �Gmeso

ijkl ðy; zÞ: ð47Þ

For the eigenseparation influence function hmeso din ðy; yÞ, we have

fLmesoijkl ðyÞgmeso d

kln ðy; yÞg;yj¼ 0; y 2 Hy; y 2 Sy;

E:B:C: dnð~yÞ ¼ dð~y � yÞ; ~y 2 Sy;ð48Þ

where

gmeso dijn ðy; yÞ ¼ hmeso d

ði;yjÞn ðy; yÞ: ð49Þ

Using Eqs. (39) and (42) the resulting residual-free meso stress ðrmesoij;yj¼ 0Þ is given as

rmesoij ¼ Lmeso

ijmn ðyÞAmeso

mnkl ðyÞemackl ðxÞ

þR

Sygmeso d

mnn ðy; yÞdmeson ðx; yÞdS

þR

Hygmeso l

mnkl ðy; yÞlmesokl ðx; yÞdH� lmeso

mn ðx; yÞ

2664

3775; ð50Þ

where

Amesoijkl ðyÞ ¼ IijklðyÞ þ Gmeso

ijkl ðyÞ: ð51Þ

4. Formulation of the reduced order model

In this section, we focus on reducing computational complexity of the micro and meso unit cell problems derived in Section 3. This isaccomplished by discretizing the eigendeformation and formulating the residual-free discrete (or reduced order) governing equations.

At the micro scale, the micro eigenstrains are discretized in terms of piecewise constant shape function NðazÞðzÞ as

Ilijðx; y; zÞ ¼Xnz

az¼1

NðazÞðzÞIlmicðazÞij ðx; yÞ; ð52Þ

where

NðazÞðzÞ ¼1; z 2 HðazÞ

z ;

0; z R HðazÞz ;

(ð53Þ

IlmicðazÞij ðx; yÞ ¼ 1

jHðazÞz j

ZHðazÞ

z

Ilijðx; y; zÞdH ð54Þ

in which the total volume of the micro unit cell is partitioned into nz nonoverlapping subdomains denoted by HðazÞz . Partitions at various

scales are denoted by a superscript enclosed in parenthesis: ðazÞ; ðbzÞ and ðayÞ; ðbyÞ denote the phase (volume) partitions at the micro andmesoscale, respectively; ðnzÞ; ðgzÞ and ðnyÞ; ðgyÞ denote the interface partitions at the micro and mesoscale, respectively.


The eigenseparation dnðx; y; ~zÞ is discretized in terms of C0 continuous interface partition shape function NðnzÞð~zÞ as

dnðx; y; ~zÞ ¼Xmz

nz¼1

NðnzÞð~zÞdmicðnzÞn ðx; yÞ; ð55Þ

where NðnzÞð~zÞ is a linear combination of piecewise linear finite element shape functions defined over partition nz

NðnzÞð~zÞ ¼

Pa2Sðnz Þ

z

Na ~zð Þ; ~z 2 SðnzÞz ;

0; ~z R SðnzÞz ;

8><>: ð56Þ

dmicðnzÞn ðx; yÞ ¼ 1

jSðnzÞz j

ZSðnzÞ

z

dnðx; y; ~zÞdS ð57Þ

in which the total interface at the microscale is divided into mz partitions denoted by SðnzÞz , and Nað~zÞ is a linear shape function associated with

a finite element mesh node a along the micro interface. Note that in contrast to the volume partitions, the surface partitions are overlappingas illustrated in Fig. 2 for the case of fibrous unit cell. Fig. 2a depicts the unit cell finite element mesh. Fig. 2b shows the two interface par-titions. The overlapping areas are shown in brown. Fig. 2c and d illustrate the two interface partition shape functions.

Substituting Eqs. (52) and (55) into Eq. (37) yields the reduced order residual-free micro stress

ð58Þ

where

Q micðnzÞijn ðy; zÞ ¼

ZSz

gmic dijn ðy; z; zÞNðnzÞðzÞdS; ð59Þ

SmicðazÞijkl ðy; zÞ ¼ PmicðazÞ

ijkl ðy; zÞ � ImicðazÞijkl ðy; zÞ ð60Þ

and

PmicðazÞijkl ðy; zÞ ¼

ZHz

gmic lijkl ðy; z; zÞN

ðazÞðzÞdH;¼Z

Hðaz Þz

gmic lijkl ðy; z; zÞdH; ð61Þ

ImicðazÞijkl ðy; zÞ ¼

Iijkl; z 2 HðazÞz ;

0; z R HðazÞz :

(ð62Þ

The reduced order influence functions, PmicðazÞklmn ðy; zÞ and QmicðnzÞ

kln ðy; zÞ, can be computed in two ways: (i) directly from Eqs. (59) and (61) incombination with Eq. (32), or by requiring the reduced order micro stress (58) to be residual-free.

By integrating (32) over partition z 2 HðazÞz and exploiting (61) yields

fLmicijkl ðy; zÞ½P

micðazÞklmn ðy; zÞ � ImicðazÞ

klmn ðy; zÞ�g;zj¼ 0; y 2 Hy; z 2 Hz ð63Þ

from which the reduced order eigenstrain influence function can be solved for. Precisely the same expression can be obtained from (58)assuming vanishing meso strain and partitioned eigenseparation dmicðnzÞ

n ðx; yÞ.The governing equation for the reduced order eigenseparation influence function is given by

fLmicijkl ðy; zÞQ

micðnzÞkln ðy; zÞg;zj

¼ 0; y 2 Hy; z 2 Hz;

E:B:C: dnðy; ~zÞ ¼ NðnzÞð~zÞ; ~z 2 Sz:ð64Þ

At the meso scale, the relation between the meso eigenstrain and the reduced order micro eigendeformation is obtained by substituting Eqs.(52) and (55) into Eq. (41), which gives

lmesoij ðx; yÞ ¼

Xnz

az¼1

SmesoðazÞijkl ðyÞ

XI

IlmicðazÞkl ðx; yÞ þ

Xmz

nz¼1

TmesoðnzÞijn ðyÞdmicðnzÞ

n ðx; yÞ; ð65Þ

where

SmesoðazÞijkl ðyÞ ¼ �½Lmeso

ijst ðyÞ��1 1jHzj

ZHz

fLmicstmnðy; zÞS

micðazÞmnkl ðy; zÞgdH; ð66Þ

TmesoðnzÞijn ðyÞ ¼ �½Lmeso

ijmn ðyÞ��1 1jHzj

ZHz

Lmicmnklðy; zÞQ

micðnzÞkln ðy; zÞdH: ð67Þ

Fig. 2. Interface partitions: (a) unit cell mesh; (b) interface partitions; (c) shape function of partition 1; (d) shape function of partition 2.


The variation of the micro eigendeformation fields IlmicðazÞij ðx; yÞ and dmicðnzÞ

n ðx; yÞ over mesoscale is assumed to be governed by C�1 continuousmesoscale shape functions NðayÞ as

IlmicðazÞij ðx; yÞ ¼

Xny

ay¼1

NðayÞðyÞIlmicðay :azÞij ðxÞ; ð68Þ

dmicðnzÞn ðx; yÞ ¼

Xny

ay¼1

NðayÞðyÞdmicðay :nzÞn ðxÞ; ð69Þ

where

NðayÞðyÞ ¼1; y 2 HðayÞ

y ;

0; y R HðayÞy ;

(ð70Þ

Ilmicðay :azÞij ðxÞ ¼ 1

jHðayÞy j

ZHðay Þy

IlmicðazÞij ðx; yÞdH; ð71Þ

dmicðay :nzÞn ðxÞ ¼ 1

jHðayÞy j

ZHðay Þy

dmicðnzÞn ðx; yÞdH ð72Þ

in which the total volume of the meso unit cell is partitioned into ny nonoverlapping subdomains denoted by HðayÞy . Note that the micro par-

titions az and nz occupy the domain within the meso partition ay and that there could be different micro partitions embedded within variousmeso partitions. Hereafter, such parent–children relation is denoted by colon ‘‘:”.

Similarly to the micro eigenseparations, the meso eigenseparations are discretized as

dmeson ðx; ~yÞ ¼

Xmy

ny¼1

NðnyÞð~yÞdmesoðnyÞn ðxÞ; ð73Þ

where

NðnyÞð~yÞ ¼

Pa2S

ðny Þy

Nað~yÞ; ~y 2 SðnyÞy ;

0; ~y R SðnyÞy ;

8><>: ð74Þ

dmesoðnyÞn ðxÞ ¼ 1

jSðnyÞy j

ZSðny Þy

dmeson ðx; ~yÞdS ð75Þ

in which the meso interface is divided into my partitions denoted by SðnyÞy .

A woven unit cell at the mesoscale and a fibrous unit cell at the microscale depicted in Fig. 3 is used to demonstrate various partitions.At the mesoscale, there are two phase partitions (matrix-1, weave-2) and one interface partition. At the microscale, there are two phasepartitions (matrix-1, fiber-2) and one interface partition. All microscale partitions are within the weave partition at the mesoscale.

Fig. 3. Unit cell partitions at the mesoscale (left) and microscale (right).


Using Eq. (65), and substituting Eqs. (73), (68) and (69) into Eq. (50) yields the reduced order residual-free meso stress

rmesoij ¼ Lmeso

ijmn ðyÞ

Amesomnkl ðyÞemac

kl ðxÞ þPmy

ny¼1Q mesoðnyÞ

mnn ðyÞdmesoðnyÞn ðxÞ

þPny

ay¼1

Pnz

az¼1Smesoðay :azÞ

mnkl ðyÞP

I

Ilmicðay :azÞkl ðxÞ

þPny

ay¼1

Pmz

nz¼1Tmesoðay :nzÞ

mnn ðyÞdmicðay :nzÞn ðxÞ

26666666664

37777777775; ð76Þ

where

Q mesoðnyÞijn ðyÞ ¼

ZSðny Þy

gmeso dijn ðy; yÞNðnyÞðyÞdS; ð77Þ

Smesoðay :azÞijkl ðyÞ ¼ Pmesoðay :azÞ

ijkl ðyÞ � ImesoðayÞijmn ðyÞSmesoðazÞ

mnkl ðyÞ; ð78Þ

Tmesoðay :nzÞijn ðyÞ ¼ Q mesoðay :nzÞ

ijn ðyÞ � ImesoðayÞijkl ðyÞTmesoðnzÞ

kln ðyÞ ð79Þ

and

Pmesoðay :azÞijkl ðyÞ ¼

ZHðay Þy

gmeso lijmn ðy; yÞSmesoðazÞ

mnkl ðyÞdH; ð80Þ

Q mesoðay :nzÞijn ðyÞ ¼

ZHðay Þy

gmeso lijkl ðy; yÞTmesoðnzÞ

kln ðyÞdH; ð81Þ

ImesoðayÞijkl ðyÞ ¼

Iijkl; y 2 HðayÞy ;

0; y R HðayÞy :

(ð82Þ

In the above, ‘‘Q” is used to indicate polarization functions associated with the two eigenseparation fields. The superscripts ðay : nzÞ and ðnyÞare used to distinguish between the association with the micro eigenseparations and meso displacements, respectively.

The reduced order eigenstrain influence functions are computed by solving the following elasticity problem

fLmesoijkl ðyÞ½P

mesoðay :azÞklmn ðyÞ � ImesoðayÞ

klst ðyÞSmesoðazÞstmn ðyÞ�g;yj

¼ 0; y 2 Hy; ð83Þ

fLmesoijkl ðyÞ½Q

mesoðay :nzÞkln ðyÞ � ImesoðayÞ

klmn ðyÞTmesoðnzÞmnn ðyÞ�g;yj

¼ 0; y 2 Hy: ð84Þ

It can be seen that solution of the meso eigenstrain influence function requires a priori solutions of the micro scale eigendeformation influ-ence functions (recall the definitions (66) and (67)).

The reduced order eigenseparation influence functions are computed from

fLmesoijkl ðyÞQ

mesoðnyÞkln ðyÞg;yj

¼ 0; y 2 Hy;

E:B:C: dnð~yÞ ¼ NðnyÞð~yÞ; ~y 2 Sy:ð85Þ

Finally, from the reduced order meso stress (76) and the definition (20), the macro stress can be expressed in terms of the micro- and meso-eigendeformation as

rmacij ¼ Lmac

ijkl emackl ðxÞ þ

Pmy

ny¼1Q macðnyÞ

ijn dmesoðnyÞn ðxÞ þ

Pny

ay¼1

Pnz

az¼1Smacðay :azÞ

ijkl

PI

Ilmicðay :azÞkl ðxÞ þ

Pny

ay¼1

Pmz

nz¼1Tmacðay :nzÞ

ijn dmicðay :nzÞn ðxÞ ; ð86Þ

Table 1Summary of the influence function problems for the three-scale analysis.

Governing equation Solution tensor Calculated tensor(s)

Micro elastic (30) Gmicijkl Amic

ijkl (38), Lmesoijkl (40)

Micro eigenstrain (63) PmicðazÞijkl SmicðazÞ

ijkl (60), SmesoðazÞijkl (66)

Micro eigenseparation (64) QmicðnzÞijn TmesoðnzÞ

ijn (67)

Meso elastic (43) Gmesoijkl Ameso

ijkl (51), Lmacijkl (87)

Meso eigenstrain (83); (84) Pmesoðay :azÞijkl ; Qmesoðay :nzÞ

ijn Smesoðay :azÞijkl (78), Smacðay :azÞ

ijkl (89); Tmesoðay :nzÞijn (79), Tmacðay :nzÞ

ijn (90)

Meso eigenseparation (85) QmesoðnyÞijn QmacðnyÞ

ijn (88)


where

Lmacijkl ¼

1jHyj

ZHy

Lmesoijmn ðyÞA

mesomnkl ðyÞdH; ð87Þ

QmacðnyÞijn ¼ 1

jHyj

ZHy

Lmesoijkl ðyÞQ

mesoðnyÞkln ðyÞdH; ð88Þ

Smacðay :azÞijkl ¼ 1

jHyj

ZHy

Lmesoijmn ðyÞS

mesoðay :azÞmnkl ðyÞdH; ð89Þ

Tmacðay :nzÞijn ¼ 1

jHyj

ZHy

Lmesoijkl ðyÞT

mesoðay :nzÞkln ðyÞdH: ð90Þ

To this end, Table 1 summarizes all the coefficient tensors needed to be precomputed for the reduced order model. The salient feature of themethod is that the influence functions can be obtained by approximately solving the corresponding elastic boundary value problem using thefinite element method independent of and prior to the nonlinear macro analysis. Once the influence functions have been computed, allthe coefficient tensors of the reduced order model can be calculated. Since these computations are independent of the nonlinear macroanalysis, they are precomputed in the preprocessing stage.

5. Solving the reduced order unit cell problems

Since all the coefficient tensors of the reduced order model can be precomputed prior to macro analysis, only partitioned eigendefor-mations d

mesoðnyÞn ðxÞ, Ilmicðay :azÞ

kl ðxÞ and dmicðay :nzÞn ðxÞ need to be updated at each increment of macro analysis. The constitutive relations at the

finest scale of interest are assumed to be known based on the material models of phases and interfaces

Ilmicij ðx; y; zÞ ¼ f micðemic

ij ðx; y; zÞ;rmicij ðx; y; zÞÞ; ð91Þ

tmicn ðx; y; zÞ ¼ gmicðdmic

n ðx; y; zÞÞ; ð92Þtmeso

n ðx; y; zÞ ¼ gmesoðdmeson ðx; y; zÞÞ; ð93Þ

where

Ilmicij ðx; y; zÞ � Ilijðx; y; zÞ; ð94Þ

dmicij ðx; y; zÞ � dijðx; y; zÞ ð95Þ

and tmicn and tmeso

n are the micro and meso tractions in the corresponding local interface coordinate system.Consider the micro strain field first. Referring to the reduced order micro stress (58)the micro strain is given as

emicij ðx; y; zÞ ¼ Amic

ijkl ðy; zÞemesokl ðx; yÞ þ

Xnz

az¼1

PmicðazÞijkl ðy; zÞ

XI


Xmz

nz¼1

Q micðnzÞijn ðy; zÞdmicðnzÞ

n ðx; yÞ: ð96Þ

Applying the averaging operator 1jHðbzÞ

z j

RHðbzÞ

z�dH on both sides of (96) yields

emicðbzÞij ðx; yÞ ¼ AmicðbzÞ

ijkl ðyÞemesokl ðx; yÞ þ

Xnz

az¼1

Pmicðbz�azÞijkl ðyÞ

XI


Xmz

nz¼1

Q micðbz�nzÞijn ðyÞdmicðnzÞ

n ðx; yÞ; ð97Þ

where

Pmicðbz�azÞijkl ðyÞ ¼

1jHðbz Þ

z j

RHðbz Þ

zPmicðazÞ

ijkl ðy; zÞdH if bz;az 2 the same Hz;

0 else;

(ð98Þ

Q micðbz�nzÞijn ðyÞ ¼

1jHðbz Þ

z j

RHðbz Þ

zQ micðnzÞ

ijn ðy; zÞdH if bz; nz 2 the same Hz;

0 else:

(ð99Þ

Note that the integrals in (98) and (99) are defined only when bz and az (or nz) belong to the same micro unit cell. We use the symbol ‘‘�” toindicate this sibling relation.


Referring to the reduced order meso stress (76), the meso strain is given as

emesoij ðx; yÞ ¼ Ameso

ijkl ðyÞemackl ðxÞ þ

Xmy

ny¼1

Q mesoðnyÞijn ðyÞdmesoðnyÞ

n ðxÞ þXny

ay¼1

Xnz

az¼1

Pmesoðay :azÞijkl ðyÞ

XI

Ilmicðay :azÞkl ðxÞ

þXny

ay¼1

Xmz

nz¼1

Q mesoðay :nzÞijn ðyÞdmicðay :nzÞ

n ðxÞ: ð100Þ

Substituting Eqs. (100), (68) and (69) into Eq. (97) yields

emicðbzÞij ðx; yÞ ¼ AmicðbzÞ

ijkl ðyÞAmesoklmn ðyÞemac

mn ðxÞ þXmy

ny¼1

AmicðbzÞijkl ðyÞQmesoðnyÞ

kln ðyÞdmesoðnyÞn ðxÞ

þXny

ay¼1

Xnz

az¼1

AmicðbzÞijmn ðyÞPmesoðay :azÞ

mnkl ðyÞ

þPmicðbz�azÞijkl ðyÞNðayÞðyÞ

24

35X

IIlmicðay :azÞ

kl ðxÞ

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Xmicðbz Þ

ij

þXny

ay¼1

Xmz

nz¼1

AmicðbzÞijkl ðyÞQ mesoðay :nzÞ

kln ðyÞ

þQ micðbz�nzÞijn ðyÞNðayÞðyÞ

24

35dmicðay :nzÞ

n ðxÞ: ð101Þ

The partitioned micro eigenstrain Ilmicðay :azÞkl on emicðbzÞ

ij can be better understood by writing out the third term of RHS of Eq. (101) as

XmicðbzÞij ðx; yÞ ¼

Pmicðbz�azÞijkl ðyÞþAmicðbzÞ

ijmn ðyÞPmesoðay :azÞmnkl ðyÞ

" #PI

Ilmicðay :azÞkl ðxÞ for bz � az;


mnkl ðyÞP

I

Ilmicðay :azÞkl ðxÞ otherwise;

8>>><>>>: ð102Þ

where bz � az denote that partitions az and bz belong to the same unit cell. It can be seen that the partitioned micro eigenstrain from thesame micro unit cell ðbz � azÞ influences the micro strain emicðbzÞ

ij ðx; yÞ in the following two ways: (i) through a direct influence within themicro unit cell ðPmicðbz�azÞ

ijkl Þ and (ii) by indirect influence through meso scale ðAmicðbzÞijmn Pmesoðay :azÞ

mnkl Þ. All the partitioned micro eigenstrains fromother micro unit cell(s) influence through the meso scale ðAmicðbzÞ

ijmn Pmesoðay :azÞmnkl Þ only. A similar observation follows for the last term in the

RHS of Eq. (101).Applying 1

jHðby Þy j

RHðby Þy�dH (where bz partition in embedded in byÞ on both sides of (101) yields the expression for the partitioned micro

strain in terms of the macro strain and partitioned eigendeformations

emicðby :bzÞij ðxÞ¼A

mixðby :bzÞijkl emac

kl ðxÞþPmy

ny¼1Q

mixðby :bzÞðnyÞijn d

mesoðnyÞn ðxÞþ

Pny

ay¼1

Pnz

az¼1P

mixðby :bzÞðay :azÞijkl

PI

Ilmicðay :azÞkl ðxÞþ

Pny

ay¼1

Pmz

nz¼1Q

mixðby :bzÞðay :nzÞijn dmicðay :nzÞ

n ðxÞ ;

ð103Þ

where

Amixðby :bzÞijkl ¼ 1

jHðbyÞy j

ZHðby Þy

AmicðbzÞijmn ðyÞAmeso

mnkl ðyÞdH; ð104Þ

Qmixðby :bzÞðnyÞijn ¼ 1

jHðbyÞy j

ZHðby Þy

AmicðbzÞijkl ðyÞQ mesoðnyÞ

kln ðyÞdH; ð105Þ

Pmixðby :bzÞðay :azÞijkl ¼ 1

jHðbyÞy j

ZHðby Þy


mnkl ðyÞþPmicðbz�azÞ

ijkl ðyÞNðayÞðyÞ

" #dH; ð106Þ

Qmixðby :bzÞðay :nzÞijn ¼ 1

jHðbyÞy j

ZHðby Þy

AmicðbzÞijkl ðyÞQ mesoðay :nzÞ

kln ðyÞþQmicðbz�nzÞ

ijn ðyÞNðayÞðyÞ

" #dH: ð107Þ

Next, we focus on the micro traction field. Substituting Eqs. (100), (68) and (69) into the reduced order micro stress (58) yields

rmicij ðx; y; zÞ ¼ Lmic

ijkl ðy; zÞAmicklmnðy; zÞA

mesomnst ðyÞemac

st ðxÞ þ Lmicijkl ðy; zÞA

micklmnðy; zÞ

Xmy

ny¼1

Q mesoðnyÞmnn ðyÞdmesoðnyÞ

n ðxÞ

þXny

ay¼1

Xnz

az¼1

Lmicijkl ðy; zÞ Amic

klmnðy; zÞPmesoðay :azÞmnst ðyÞ þ Smixðay :azÞ

klst ðy; zÞh iX

I

Ilmicðay :azÞst ðxÞ

þXny

ay¼1

Xmz

nz¼1

Lmicijkl ðy; zÞ Amic

klmnðy; zÞQmesoðay :nzÞmnn ðyÞ þ Q mixðay :nzÞ

kln ðy; zÞh i

dmicðay :nzÞn ðxÞ; ð108Þ

where

Smixðay :azÞijkl ðy; zÞ ¼ SmicðazÞ

ijkl ðy; zÞNðayÞðyÞ; ð109Þ

Q mixðay :nzÞijn ðy; zÞ ¼ QmicðnzÞ

ijn ðy; zÞNðayÞðyÞ: ð110Þ

The corresponding traction along the micro interface is given by

tmicm ðx; y; zÞ ¼ amic

mi tmici ðx; y; zÞ ¼ amic

mi rmicij ðx; y; zÞnmic

j ðy; zÞ; z 2 Sz; ð111Þ

where amicmi is the transformation matrix from the global coordinates system to the local interface.


Substituting Eq. (108) into (111) and applyingR Rðby :gzÞ

� averaging operator on both sides of (111) yields the partitioned micro traction interms of the macro strain and the partitioned eigendeformations

tmicðby :gzÞm ðxÞ¼B

mixðby :gzÞmkl emac

kl ðxÞþPmy

ny¼1W

mixðby :gzÞðnyÞmn d

mesoðnyÞn ðxÞþ

Pny

ay¼1

Pnz

az¼1V

mixðby :gzÞðay :azÞmkl

PI

Ilmicðay :azÞkl ðxÞþ

Pny

ay¼1

Pmz

nz¼1W

mixðby :gzÞðay :nzÞmn dmicðay :nzÞ

n ðxÞ ;

ð112Þ
where Z Z
ðby :gzÞ� � 1

jHðbyÞy j

ZHðby Þy

1

jSðgzÞz j

ZSðgzÞ

z

�dSdH; ð113Þ

Bmixðby :gzÞmkl ¼

Z Zðby :gzÞ

amicmi Lmic

ijmnðy; zÞAmicmnstðy; zÞA

mesostkl ðyÞnmic

j ðy; zÞ; ð114Þ

Wmixðby :gzÞðnyÞmn ¼

Z Zðby :gzÞ

amicmi Lmic

ijkl ðy; zÞAmicklmnðy; zÞQ

mesoðnyÞmnn ðyÞnmic

j ðy; zÞ; ð115Þ

Vmixðby :gzÞðay :azÞmkl ¼

Z Zðby :gzÞ

amicmi Lmic

ijmnðy; zÞAmic

mnstðy; zÞPmesoðay :azÞstkl ðyÞ

þSmixðay :azÞmnkl ðy; zÞ

" #nmic

j ðy; zÞ; ð116Þ

Wmixðby :gzÞðay :nzÞmn ¼

Z Zðby :gzÞ

amicmi Lmic

ijkl ðy; zÞAmic

klmnðy; zÞQmesoðay :nzÞmnn ðyÞ

þQ mixðay :nzÞkln ðy; zÞ

" #nmic

j ðy; zÞ: ð117Þ

To this end, we consider the meso traction field. With the reduced order meso stress given by (76), the corresponding traction field along themeso interface is given as

tmesom ðx; yÞ ¼ ameso

mi tmesoi ðx; yÞ ¼ ameso

mi rmesoij ðx; yÞnmeso

j ðyÞ y 2 Sy: ð118Þ

Applying 1

jSðgy Þy j

RSðgy Þy�dS on both sides of (118) yields the partitioned meso traction in terms of the macro strain and the partitioned

eigendeformations

tmesoðgyÞm ðxÞ ¼ B

mesoðgyÞmkl emac

kl ðxÞ þPmy

ny¼1W

mesoðgyÞðnyÞmn d

mesoðnyÞn ðxÞ þ

Pny

ay¼1

Pnz

az¼1V

mesoðgyÞðay :azÞmkl

PI

Ilmicðay :azÞkl ðxÞ þ

Pny

ay¼1

Pmz

nz¼1W

mesoðgyÞðay :nzÞmn dmicðay :nzÞ

n ðxÞ ;

ð119Þ

where

BmesoðgyÞmkl ¼ 1

jSðgyÞy j

ZSðgy Þy

amesomi Lmeso

ijmn ðyÞAmesomnkl ðyÞnmeso

j ðyÞdS; ð120Þ

WmesoðgyÞðnyÞmn ¼ 1

jSðgyÞy j

ZSðgy Þy

amesomi Lmeso

ijmn ðyÞQmesoðnyÞmnn ðyÞnmeso

j ðyÞdS; ð121Þ

VmesoðgyÞðay :azÞmkl ¼ 1

jSðgyÞy j

ZSðgy Þy

amesomi Lmeso

ijmn ðyÞSmesoðay :azÞmnkl ðyÞnmeso

j ðyÞdS; ð122Þ

WmesoðgyÞðay :nzÞmn ¼ 1

jSðgyÞy j

ZSðgy Þy

amesomi Lmeso

ijmn ðyÞTmesoðay :nzÞmnn ðyÞnmeso

j ðyÞdS: ð123Þ

The partitioned variant of the constitutive relations (91)–(93) can be expressed as

Ilmicðay :azÞij ðxÞ ¼ f micðemicðay :azÞ

ij ðxÞ;rmicðay :azÞij ðxÞÞ; ð124Þ

tmicðay :nzÞn ðxÞ ¼ gmicðdmicðay :nzÞ

n ðxÞÞ; ð125Þ

tmesoðnyÞn ðxÞ ¼ gmesoðdmesoðnyÞ

n ðxÞÞ; ð126Þ

where emicðay :azÞij is the partitioned micro strain; tmicðay :nzÞ

n the partitioned micro traction; and tmesoðnyÞn the partitioned meso traction. Eqs. (103),

(112) and (119) along with the partitioned constitutive relations (124)–(126) form a system of nonlinear equations for the unknowns(emicðay :azÞ

ij ðxÞ; dmicðay :nzÞn ðxÞ and d

mesoðnyÞn ðxÞÞ. The number of equations (and unknowns) is given as

6ðny : nzÞ þ 3ðny : mzÞ þ 3my;

where ny : nz ¼Xny

ay¼1

nzjay; ny : mz ¼

Xny

ay¼1

mzjay:

ð127Þ

The partitioned eigendeformations are obtained by solving the system of equations within each macro increment. Once partitioned eigende-formations have been determined, Eq. (86) is used to update the macro stress.

6. Two-scale reduced order homogenization

In the previous sections we derived a three-scale reduced order homogenization formulation, which can be generalized to arbitrarynumber of scales. From the practical point of view, the most interesting case is that of two-scales, which is considered in this section.


We start with the three-scale boundary value problem defined by Eqs. (1)–(7). For two-scales, the composite material is represented bya locally periodic unit cell on the mesoscale denoted by Hy with the interface denoted by Sy. A single additional position vector y � x=f isintroduced. For any locally periodic function, the spatial derivative reduces to

/fðxÞ � /ðx; yÞ; ð128Þ

/f;iðxÞ ¼ /;xi

ðx; yÞ þ 1f

/;yiðx; yÞ: ð129Þ

The two-scale asymptotic expansion of the displacement field is given as

uiðx; yÞ ¼ u0i ðxÞ þ fu1

i ðx; yÞ þ � � � ð130Þ

and the asymptotic expansion of the strain field is

eijðx; yÞ ¼ e0ijðx; yÞ þ fe1

ijðx; yÞ þ � � � ð131Þ

with the leading order strain

e0ijðx; yÞ ¼ u0

ði;xjÞðxÞ þ u1ði;yjÞðx; yÞ: ð132Þ

The macro strain is defined as an average strain over Hy

emacij ðxÞ �

1jHyj

ZHy

e0ijðx; yÞdHy ¼ u0

ði;xjÞðxÞ: ð133Þ

Thus the meso strain can be expressed as

emesoij ðx; yÞ � e0

ijðx; yÞ ¼ u0ði;xjÞðxÞ þ u1

ði;yjÞðx; yÞ ¼ emacij ðxÞ þ u1

ði;yjÞðx; yÞ: ð134Þ

The asymptotic expansion of the stress field is given as

rijðx; yÞ ¼ r0ijðx; yÞ þ fr1

ijðx; yÞ þ � � � ð135Þ

where

r0ijðx; yÞ ¼ LijklðyÞ e0

klðx; yÞ �X

I

Ilklðx; yÞ" #

: ð136Þ

The meso- and macro-stresses are defined as

rmesoij ðx; yÞ � r0

ijðx; yÞ; ð137Þ

rmacij ðxÞ �

1jHyj

ZHy

rmesoij ðx; yÞdH: ð138Þ

The resulting meso and macro equilibrium equations are identical to those obtained in the three-scale case

Oðf�1Þ : rmesoij;yjðx; yÞ ¼ 0 ; ð139Þ

Oðf0Þ : rmacij;xjðxÞ þ �bi ¼ 0 ; ð140Þ

where

�bi ¼1jHyj

ZHy

bidH: ð141Þ

Due to the homogeneity of the meso-phases the following can be deduced:

1. All the terms related to the micro eigenseparation dmicn vanish since they no longer exist; we denote dn � dmeso

n to simplify the notation.2. The micro elastic polarization function Gmic

ijkl , the micro eigenstrain polarization function gmic lijkl and the corresponding partitioned Pmic

ijkl areall zero tensors.

3. The superscripts ‘‘mic”, ‘‘: az” and ‘‘: bz” can be dropped since there are no partitions at the micro scale.

The resulting reduced order residual-free meso stress can be directly deduced from Eq. (76)

rmesoij ¼ Lmeso

ijmn ðyÞAmeso

mnkl ðyÞemackl ðxÞ þ

Pmy

ny¼1Q mesoðnyÞ

mnn ðyÞdðnyÞn ðxÞ

þPny

ay¼1SmesoðayÞ

mnkl ðyÞP

I

IlðayÞkl ðxÞ

266664

377775 ; ð142Þ

where

SmesoðayÞijkl ðyÞ ¼ PmesoðayÞ

ijkl ðyÞ � ImesoðayÞijmn ðyÞ ð143Þ

Table 2Summary of influence function problems for the two-scale analysis.

Governing equation Solution tensor Calculated tensor(s)

Meso elastic (43) Gmesoijkl Ameso

ijkl (51), Lmacijkl (87)

Meso eigenstrain (145) PmesoðayÞijkl SmesoðayÞ

ijkl (143), SmacðayÞijkl (147)

Meso eigenseparation (85) QmesoðnyÞijn QmacðnyÞ

ijn (88)


and

Table 3Coeffici

Macro

Lmac ¼ E

Smacð1Þ ¼

�E

Qmacð1Þ ¼

PmesoðayÞijkl ðyÞ ¼

ZHðay Þy

gmeso lijmn ðy; yÞdH: ð144Þ

The elasticity problem at the mesoscale from which the reduced order eigenstrain influence are determined directly follows from Eq. (83)

fLmesoijkl ðyÞ½P

mesoðayÞklmn ðyÞ � ImesoðayÞ

klst ðyÞ�g;yj¼ 0 y 2 Hy: ð145Þ

The reduced order meso stress follows from Eq. (86)

rmacij ¼ Lmac

ijkl emackl ðxÞ þ

Pmy

ny¼1QmacðnyÞ

ijn dðnyÞn ðxÞ þ

Pny

ay¼1SmacðayÞ

ijkl

PI

IlðayÞkl ðxÞ ; ð146Þ

where

SmacðayÞijkl ¼ 1

jHyj

ZHy

Lmesoijmn ðyÞS

mesoðayÞmnkl ðyÞdH: ð147Þ

Table 2 summarizes the coefficient tensors, which follow from Table 1.

Fig. 4. A one-dimensional model problem.

ent tensors for the 1D model problem.

Phase Interface

2E1 E2

1þE2Amesoð1Þ ¼ 2E2

E1þE2; Ameso

ð2Þ ¼ 2E1E1þE2

Bmesoð1Þ ¼

2E1 E2E1þE2

E1 E2

1þE2; Smac

ð2Þ ¼�E1 E2E1þE2

Pmesoð1Þð1Þ ¼

E1E1þE2

; Pmesoð1Þð2Þ ¼

�E2E1þE2


�E1E1þE2


E2E1þE2

Vmesoð1Þð1Þ ¼

�E1 E2E1þE2

; Vmesoð1Þð2Þ ¼

�E1 E2E1þE2

�2E1E2E1þE2

Qmesoð1Þð1Þ ¼

�2E2E1þE2

; Qmesoð2Þð1Þ ¼

�2E1E1þE2

Wmesoð1Þð1Þ ¼

�2E1 E2E1þE2

Fig. 5. The unit cell for 1D model problem.

Fig. 6. The fibrous unit cell for convergence study.

Fig. 7. The stress–strain curves for longitudinal loading.


The system of equations for calculating the meso eigendeformations is obtained from Eqs. (103) and (119). The partitioned strain field isgiven as

eðbyÞij ðxÞ ¼ A

mesoðbyÞijkl emac

kl ðxÞ þPmy

ny¼1Q

mesoðbyÞðnyÞijn d

ðnyÞn ðxÞ þ

Pny

ay¼1P

mesoðbyÞðayÞijkl

PI


where

AmesoðbyÞijkl ¼ 1

jHðbyÞy j

ZHðby Þy

Amesoijkl ðyÞdH; ð149Þ

QmesoðbyÞðnyÞijn ¼ 1

jHðbyÞy j

ZHðby Þy

Q mesoðnyÞijn ðyÞdH; ð150Þ

PmesoðbyÞðayÞijkl ¼ 1

jHðbyÞy j

ZHðby Þy

PmesoðayÞijkl ðyÞdH: ð151Þ

The partitioned traction field becomes

tmesoðgyÞm ðxÞ ¼ B

mesoðgyÞmkl emac

kl ðxÞ þPmy

ny¼1W

mesoðgyÞðnyÞmn d

ðnyÞn ðxÞ þ

Pny

ay¼1V

mesoðgyÞðayÞmkl

PI


Fig. 8. The stress–strain curves for transverse loading.

Fig. 9. Braid architectures considered for validation.

Table 4The virgin properties of micro phases (manufacture’s values).

E (GPa) m

Matrix 3.170 0.35Fiber 234.0 (231.0 for bias tow) � (assume 0.1 for calibration)

Table 5Initial properties of the tow.

E22 (GPa) E11 (GPa) m23 m12 G23a (GPa) G12 (GPa)

43.89 210.9 0.40 0.12 15.71 18.45

a Where G23 ¼ E22=ð2ð1þ m23ÞÞ.

Table 6Elastic calibration of tow properties.

Lower bound Upper bound Calibrated value

E22 (GPa) 10.0 100.0 31.62E11 (GPa) 100.0 1000 348.8



where

Table 7Identifie

EA (GPaET (GPa

Table 8Multi-s

Step #

123

BmesoðgyÞmkl ¼ 1

jSðgyÞy j

ZSðgy Þy

amesomi Lmeso

ijmn ðyÞAmesomnkl ðyÞnmeso

j ðyÞdS; ð153Þ

WmesoðgyÞðnyÞmn ¼ 1

jSðgyÞy j

ZSðgy Þy

amesomi Lmeso

ijmn ðyÞQmesoðnyÞmnn ðyÞnmeso

j ðyÞdS; ð154Þ

VmesoðgyÞðayÞmkl ¼ 1

jSðgyÞy j

ZSðgy Þy

amesomi Lmeso

ijmn ðyÞSmesoðayÞmnkl ðyÞnmeso

j ðyÞdS: ð155Þ

d macro properties.

Experiment Simulation

) 60.3 ± 2.95 60.42) 8.7 ± 0.24 8.76

Fig. 11. The overall stress–strain curves in tow tension and compression.

Fig. 10. Partitions for the braided unit cell (45�).

tep calibration of inelastic parameters.

Active parameter Used tests of composite Overall ultimate strength of composite (MPa)

Experiment Simulation

Smatrix Transverse tension 65.2 62.7Stow Longitudinal tension 654.3 651.7Cmatrix and Sb

tow Longitudinal compression 375.8 380.1


The constitutive relations for the partitioned eigendeformations are deduced from Eqs. (124) and (126)

IlðayÞij ðxÞ ¼ f ðeðayÞ

ij ðxÞ;rðayÞij ðxÞÞ; ð156Þ

tðnyÞn ðxÞ ¼ gðdðnyÞ

n ðxÞÞ: ð157Þ

Eqs. (148) and (152) along with the partitioned constitutive relations (156) and (157) form a system of nonlinear equations for the unknowns(eðayÞ

ij ðxÞ and dðnyÞn ðxÞÞ. The number of equations (and unknowns) is 6ny þ 3my, which follows from Eq. (127).

Partitioned eigendeformations are obtained by solving a system of equations within each macro increment. Once partitioned eigende-formations have been determined, Eq. (146) is used to update the macro stress.

7. Verification

In this section, a one-dimensional model is used for verification of the reduced order model influence functions. Convergence of the re-duced order model to the direct homogenization is studied next.

We consider a one-dimensional heterogeneous bar made of a periodic arrangement of two different materials (E1 on the left and E2 onthe right) as shown in Fig. 4.

Fig. 12. Circular tube model.

Fig. 13. Comparison of the multiscale simulation and experimental results for circular tube made of 45� braid architecture.




For the 1D model problem, we can obtain an analytical solution of coefficient tensors, which are given in Table 3.We select several pairs of Young’s Modulus values for the two phases (Fig. 5) to verify the values of coefficient tensors. The simulation

results coincide with the analytical solutions given in Table 3 for all pairs of Young’s Modulus selected.Next, we study the convergence of the reduced order model to the direct homogenization model. The reduced order model should con-

verge to direct homogenization model when partition number approaches the number of elements in the unit cell. For this study, we con-sider a fibrous unit cell with 226 elements as shown Fig. 6. Only the matrix phase (160 elements) is assumed to be damageable. Weincrease the partition number for matrix phase from 1, 4, 8, and up to 160 and compare the overall stress–strain curves with the directhomogenization results. The results of two loading cases, longitudinal loading and transverse loading, are shown in Figs. 7 and 8, respec-tively. It can bee seen that for both cases, the reduced order model converges to the direct homogenization model. However, the compu-tational cost of one matrix partition is three orders of magnitude smaller than of the direct cost homogenization. For more details see [31].

8. Validation

In this section the reduced order formulation is validated on the tube crush simulation of braided composite.We consider a triaxially braided composite unit cell model with three different angles (30�, 45� and 60�) as shown in Fig. 9.In each unit cell model we identify bias tows with plus/minus (clockwise/counterclockwise) angles with respect to the axial tow as two

different inclusion phases. Hence there are four phases (matrix, axial tow, bias tow 1 and bias tow 2) and three interfaces associated witheach inclusion phase.

Fig. 16. Size effect studies for three different mesh sizes.

Fig. 17. Square tube model.


Since each tow in the unit cell consists of thousands of filaments of carbon fiber (axial tow: 80k1 Fortafil 511; bias tow: 12k Grafil 34-700)and epoxy matrix (Ashland Hetron 922), we have a three-scale representation of the composite: fibrous micro unit cell model, braided mesounit cell model (constructed in the previous step) and macro tube model.

To identify the elastic properties of the braided unit cell, we assume that the matrix properties are known from manufacture’s values,and calibrate the tow properties based on a series of coupon tests. The material symmetry of tow is assumed to be transversely isotropic(x1 ¼ xA, 5 independent values) and the initial values of tow for the calibration process (Table 5) are determined using Mori–Tanaka method[33] for the micro fibrous unit cell (volume fraction of fiber 87.7%). The virgin properties of micro phases are given in Table 4.

During calibration process, the two Young’s Moduli of a tow were selected as active parameters. The lower and upper bounds and result-ing calibrated values are summarized in Table 6. The identified macro properties of the braided composite are compared with experimentaltest data in Table 7.

We use one partition per phase strategy and assume no interface. This results in four phase partitions (one matrix partition, one axialtow partition and two bias tow partitions) as shown in Fig. 10. There are 24 (4 partitions time 6 modes) eigenstrain components that needto be updated within each macro increment (reduced order unit cell problem).

1 An 80k tow nominally contains 80,000 individual fibers.

Fig. 18. Comparison of the multiscale simulation and experimental results for square tube made of 45� braid architecture.



The constitutive behavior of phases at the mesoscale is modeled using continuum damage mechanics (see Appendix for details). Modelparameters, the ultimate strength of matrix, Smatrix, the ultimate strength of tow, Stow, the compression factor of matrix, Cmatrix, and the buck-ling strength of tow Sb

tow (see discussion below) are calibrated using inverse methods [34]. The values of the identified parameters and theircomparison to the experimental values are summarized in Table 8.

The buckling strength Sb is introduced to account for tow buckling in the following way

if rtow 6 Sbtow < 0;

wtow ¼ wmatrix;ð158Þ

i.e., when Sbtow is reached, the damage state of tow is associated with the damage state of matrix phase. As an example, in Fig. 11 we give the

overall axial stress–strain curves of a fibrous unit cell subjected to axial tension and compression with displacement control. It can be seenthat under compression loading and with a relative small value of Sb, the damage state of fiber is associate with the damage state of matrixand the entire unit cell loses its loading capacity.

Note, that elastic and inelastic parameters are calibrated based on 45� coupon tests. The 30� and 60� unit cells models use the same setof parameters identified in 45� coupon tests.

Once the model parameters have calibrated, we proceed with crush simulations of circular and square tubes on three different materialarchitectures and compare the simulation results to the experimental data (all test data are from [35]). For all tube crush simulations weconsider a quasi-static loading (load rate = 0.5 in./min), two layers of braided composite.



First we conduct crush simulations of circular tube as shown in Fig. 12. The inner diameter of the tube is 2.35 in. An initiator plug with 5/16 in. fillet radius is considered.

Fig. 13 compares the multiscale simulations to the experimental results for the 45 braid architecture shown in Fig. 9. Excellent agree-ment with the experiment can be observed. Fig. 14 compares the multiscale simulations to the experimental results for the 30 braid archi-tecture shown in Fig. 9. This time the agreement with experiment is not as good. Possibly, this can be attributed to the fact that we have notaccounted for interface failure. Fig. 15 compares the multiscale simulations to the experimental results for the 60 braid architecture shownin Fig. 9. Again, an excellent agreement with the experiment can be observed.

For the circular tube, we also study the size effect – that is the influence of mesh size on simulation results. Fig. 16 summarizes thefindings of these studies. Interestingly, the model shows very little if any influence to the mesh size. For coarser meshes, however, the oscil-lations are higher, which is due to the fact that when the macro element is eroded the load is reduced. The larger element is eroded, thelarger jump is observed.

Next we conduct crush simulations of square tube as shown in Fig. 17. The inner dimensions of the tube are 2 in. � 2 in. and inner cornerradius of 0.33 in. An initiator plug with 1/4 in. fillet radius is considered.

Figs. 18–20 compare the multiscale simulation to the experimental results for the 45, 30 and 60 braid architectures, respectively. Theagreement with experiments is reasonable at best, suggesting the need for considering interface failure. Note that implementation of inter-face failure will affect the calibrated properties of micro-constituents. This is because the failure properties of micro-constituents were cal-ibrated to fit the coupon data.

Acknowledgements

The financial supports of National Science Foundation under grants CMS-0310596, 0303902, 0408359, Rolls-Royce Contract 0518502,Automotive Composite Consortium Contract 606-03-063L, and AFRL/MNAC MNK-BAA-04-0001 contract are gratefully acknowledged.

Appendix

In the present manuscript, continuum damage mechanics with isotropic damage law is used to model constitutive behavior of micro-constituents. By this approach, the constitutive relation is given by

rij ¼ ð1�wphÞLijklekl; ð159Þ

where w 2 ½0;1� is a damage state variable. Thus the partitioned eigenstrain is defined as

lðayÞij ¼ wðayÞ

ph eðayÞij : ð160Þ

The damage state variable wðayÞph is taken to be a piecewise-continuous function of damage equivalent strain jðayÞ

ph . The evolution of phase dam-age may be expressed as

wðayÞph ¼

0; jðayÞph 6

1jðayÞph ;

UðjðayÞph Þ; 1jðayÞ

ph < jðayÞph 6

2jðayÞph ;

1; jðayÞph > 2jðayÞ

ph ;

8>>><>>>: ð161Þ

where 1jðayÞph and 2jðayÞ

ph are model parameters corresponding to the initial and fully damaged state, respectively.


The damage equivalent strain jðayÞph is assumed to be a function of the principal strain

jðayÞph ðtÞ ¼max

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX3

I¼1

heðayÞI ðsÞi

2

vuut ; s < t

8<:

9=;; ð162Þ

where

hxi ¼x; x P 0;Cx; x < 0

�ð163Þ

and C is the compression factor (one of inelastic material parameters).1jðayÞ

ph can be expressed in terms of material strength S (another inelastic material parameter) and stiffness E

1jðayÞph ¼

SE: ð164Þ

Based on the selection of the damage evolution function, 2jðayÞph can be expressed in terms of physical-based material parameters (E, S and G –

the total strain energy). For various evolution functions UðjðayÞph Þ see [34].

Similarly, eigenseparation are modeled using damage mechanics based cohesive law

tðnyÞn ¼ ð1�wðnyÞ

int ÞKðnyÞd

ðnyÞn ; ð165Þ

where KðnyÞ is the interface stiffness. In the normal direction, wðnyÞint is considered only when d

ðnyÞn > 0.

The interface damage state variable wðnyÞint is assumed to be a function of damage equivalent displacement jðnyÞ

int defined as

jðnyÞint ðtÞ ¼max

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihdðnyÞ

n ðsÞi2þ þ ðdðnyÞt1 ðsÞÞ

2 þ ðdðnyÞt2 ðsÞÞ

2q

; s < t� �

; ð166Þ

where

hxiþ ¼x; x P 0;0; x < 0:

�ð167Þ

The evolution of interface damage can be expressed as

wðnyÞint ¼

0; jðnyÞint 6

1jðnyÞint ;

UðjðnyÞint Þ; 1jðnyÞ

int < jðnyÞint 6

2jðnyÞint ;

1; jðnyÞint > 2jðnyÞ

int :

8>><>>: ð168Þ

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Multiple scale eigendeformation-based reduced order homogenization

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