Computational Framework for Multiphysics Problems with Multiple Spatial and Temporal Scales

Qing Yu and Jacob FishDepartments of Civil Engineering, Mechanical and Aeronautical Engineering

Rensselaer Polytechnic Institute, Troy, NY 12180

Abstract: A systematic approach for analyzing multiple physical processes interacting at multiplespatial and temporal scales is developed. The proposed computational framework is applied to the cou-pled thermo-viscoelastic composites with microscopically periodic mechanical and thermal properties.A rapidly varying spatial and temporal scales are introduced to capture the effects of spatial and tem-poral fluctuations induced by spatial heterogeneities at diverse time scales. The initial-boundary valueproblem on the macroscale is derived by using the double scale asymptotic analysis in space and time.It is shown that an extra history-dependent long-term memory term introduced by the homogenizationprocess in space and time can be obtained by solving a first order initial value problem. This is in con-trast to the long-term memory term obtained by the classical spatial homogenization, which requiressolutions of the initial-boundary value problem in the unit cell domain. The validity limits of the pro-posed spatial-temporal homogenized solution are established. Numerical example shows a good agree-ment between the proposed model and the reference solution obtained by using a finite element meshwith element size comparable to that of material heterogeneity.

1.0 Introduction

The primary objective of the manuscript is to develop a systematic approach for analyzing multiplephysical processes interacting at multiple spatial and temporal scales. The interacting physical pro-cesses may include mechanical, thermal, diffusion, chemical and electromagnetic fields. Most oftenthese phenomena are treated as being uncoupled; hence, few separate analyses of the same system aretypically performed for the complete prediction of the response. It is, however, understood that suchtreatments should be regarded as first-order approximations to the real complex interactions.

The coupling of mechanical, thermal, diffusion, chemical and electromagnetic fields (stress/strain,temperature, concentration, current) occurs through diverse phenomena, some of which are depicted inthe interaction matrix shown in Table 1. The interaction matrix is “non-symmetric,” with cell (i, j) rep-resenting the phenomenon induced by the process corresponding to field i and which influences field j.For instance, cell (1, 2) represents heating due to plastic deformation, while cell (2, 1) corresponds tothermal expansion and thermal stresses. A fully coupled analysis would consider all processes shownin the matrix, while fully uncoupled approach would only consider the diagonal entries. A one-way (or

1

partially) coupled approach would consider a lower (or upper) triangular entries in the interactionmatrix.

TABLE 1. Physical processes interaction matrix

It is important to note that coupling of various physical processes in the mathematical model can becarried out either by considering additional terms in the field equations (equation coupling) or byallowing the constitutive law to depend on the interacting field (constitutive law coupling). An exam-ple of the first category is the effect of the temperature gradient on the stress field, which can be cap-tured by adding a thermal stress term to equilibrium equations. A less familiar example within thesame class of problems is the effect of diffusion (clustering) and chemical reactions (reaction products)on the stress field, which can be accounted for by the eigenstrain formulation [4]. An example from thesecond category is the influence of temperature upon diffusion which can be captured by simply con-sidering the diffusion coefficients to be temperature dependent. Symbols E and C in Table 1 stand forthe technique to be used for coupling the respective phenomenon, with E standing for “equation cou-pling” and C designating “constitutive law coupling”.

As an illustrative example, we consider an initial-boundary value problem for the thermo-viscoelasticcomposite. It consists of two spatial scales (micro-constituents and the macro-domain), two temporalscales (the time scale associated with an applied loading and the intrinsic time scale of the rate-depen-dent material) and two fully coupled physical processes (thermal and mechanical). Both the constitu-tive law coupling due to thermally sensitive material properties and the equation coupling induced bythermal stresses are taken into account for mechanical fields. For thermal fields, on the other hand, weassume that only the equation coupling occurs due to the mechanical dissipation and dilation effects.

Deformation(displacements)

Deformation induced heat production

E

Strain controlleddiffusion

C

Strain controlledchemical activity

C

Deformation induced electric

and magnetic fluxE

Thermal expansion

E

Heat transfer(temperature)

Temperature controlled diffusion

C

Temperature controlled chemical

activityC

Temperature con-trolled electric and magnetic behavior

CClustering,

precipitation (eigenstrains)

E

Diffusant concentra-tion controlled heat

transferC

Diffusion (diffusant concen-

tration)

Transport of reactants

E

Diffusion induced electrical and

magnetic potentialsE

Reaction productsaccommodation(eigenstrains)

E

Heat productionduring chemical

reactionsE

Transport of reaction products

E

Chemical reactions

Reaction products induced electric and

magnetic fluxE

Deformation due to Lorentz force

E

Heat input from Joule heating

E

Transport of charged particles

E

electro-magnetic field controlled

chemical activityC

electro-magnetism

2

To model the local oscillations of mechanical and thermal fields induced by spatial heterogeneities atdiverse time scales, an asymptotic homogenization theory for multiple physical processes with multi-ple spatial and temporal scales is developed. When the loading is highly oscillatory in comparison withthe material intrinsical time scale it is natural to incorporate a rapidly varying time scale in the asymp-totic analysis. This fast time variable is defined to characterize the fast varying features of mechanicaland thermal response fields in time domain.

Homogenization with multiple temporal scales could be traced back to Bensousan et al. [1] where theconvergence analyses of the hyperbolic equations with oscillatory coefficients were established.Francfort [4] generalized the conventional spatial homogenization method to the case of thermo-elasticcomposites. For the hyperbolic conservation law with rapid spatial fluctuations, Kevorkin and Bosley[7] showed that the continuous initial data which is independent on the fast temporal scale may intro-duce a dependence on both fast spatial and temporal scales in the homogenized solutions. A handful ofrecently publications on this topic has been briefly reviewed in [3], where the role of multiple temporalscales in wave propagation in heterogeneous solids was investigated.

For the homogenization of viscoelastic heterogeneous media, it is well known that in addition to theoriginal memories due to the viscosity of micro-constituents, an extra long-term fading memory isinduced by the homogenization process. This phenomenon has been illustrated in [5], [8], [9] and [11]for either Kelvin-Voigt or Maxwell viscoelastic model with only instantaneous memories in the micro-constituents. This extra long-term memory in the homogenized constitutive equation arises due to theinteractions between fast spatial variation and time dependence of the coefficients of partial differen-tial equations. We remark that all these studies were based on the spatial homogenization only and thehistory-dependent integral kernel associated with the long-term memory is determined by a local ini-tial-boundary value problem in the unite cell domain [6][13]. In this manuscript we show that the extralong-term memory term resulting from the homogenization process in space and time can be obtainedby solving a first order initial value problem. This is in contrast to the long-term memory term obtainedby the classical spatial homogenization, which requires solution of the initial-boundary value problemin the unit cell domain. This gives rise to an elegant homogenized solution which can be easily imple-mented into the numerical setting.

In Section 2 we start our presentation with a general setting of the initial-boundary value problem forthe fully coupled Kelvin-Voigt thermo-viscoelastic composite. In addition to the usual space-timecoordinates rapidly varying spatial and temporal scales are introduced to capture the effects of spatialand temporal fluctuations. The macroscopic initial-boundary value problem is obtained by the doublescale asymptotic analysis in space and time. It is shown that an extra long-term memory obtained fromsolving a first order initial value problem in macroscopic field is introduced into the homogenizedsolution. Section 3 discusses various relations between the temporal and spatial scales as well as thevalidity of the proposed model. Numerical experiment comparing the proposed model with the refer-ence solution obtained by using a finite element mesh with element size comparable to that of materialheterogeneity is given in Section 4.

3

2.0 Space-time multiple scale analysis for the coupled thermo-viscoelastic composite

In the present work, the Kelvin-Voigt viscoelastic model is considered for micro-constituents. In con-trast to the multiscale analysis conducted by Boutin and Wong [2], where a single frequency quasi-har-monic displacement field has been assumed so that the constitutive equation can be transformed to theelastic-like form, we consider a general setting of the coupled initial-boundary value problem. The dis-tributed heat source arises due to the mechanical dissipation and the thermal dilation in micro-constitu-ents. The thermally sensitive mechanical properties (stiffness and viscosity) as well as the thermaldilation term in the constitutive equation lead to the full coupling between mechanical response andthermal diffusion.

2.1 Definition of multiple spatial and temporal scales

The microstructure of a composite material is assumed to be locally periodic ( -periodic) with a scaleparameter defined by the Representative Volume Element (RVE or unit cell). The macroscopicdomain is represented by while denotes the unit cell domain. We assume that RVE exists and itscharacteristic size is small enough in comparison with the intrinsic reference length on themacro-scale so that

(1)

Thus, the coordinate vector for macro- and micro- scales can be related by

(2)

In addition to the distinct spatial scales, we can identify at least two temporal scales in a typicalthermo-viscoelastic problem: the time scale associated with the applied loading and the intrinsic timescale of the rate-dependent material. In the present work, we introduce a fast varying temporal coordi-nate to represent the fast oscillations of mechanical and thermal fields in time domain induced by thehighly oscillatory loading. For linear systems and weakly nonlinear systems, the characteristic lengthof is of the same order as the period of a loading profile [2]. We assume that the intrinsic timescale , which is determined by material properties, describes a relatively long-term behavior, andthus the following relations hold:

(3)

where and denote the fast varying and natural time coordinates, respectively; is the characteris-tic length of the natural time scale; is the small scale parameter defined in the time domain. We startby considering the special case of

(4)

Yεl

Ωε Θl lr

εl l lr⁄= and εl <<1

x( ) y( )

y x εl⁄=

τ

τ τ 0tr

ετ τ0 tr , ετ <<1⁄= and τ t ετ⁄=

τ t trετ

εl ετ ε≈ ≈

4

It is important to note that the definitions of the multiple spatial and temporal length scales are physi-cally distinct for the mechanical and thermal fields. Further discussion on the relation between the tem-poral and spatial scales as well as on the validity of the homogenized solution is left to Section 3.

With the definition of the fast varying variables and as well as the local -periodicity assumption,all the mechanical and thermal response quantities denoted by can be defined as

(5)

where is the basic period vector of the microstructure and is a 3 by 3 diagonal matrix with arbi-trary integer components. The corresponding -periodic function can be defined by using the con-ventional nomenclature:

(6)

The differentiations with respect to space and time variables can be expressed using the chain rule:

and (7)

where the comma followed by a subscript variable denotes a partial derivative and superscribed dotdenotes the time derivative. Summation convention for repeated subscripts is adopted except for thesubscripts and .

2.2 Initial-boundary value problem statement for the coupled thermo-viscoelastic composites

Attention is restricted to small deformations and small temperature increases. The microscopic constit-uents are assumed to be homogeneous and their thermo-viscoelastic behavior can be described by thefollowing initial-boundary value problem in the macroscopic domain [6][7]:

(1) Equation of motion

(8)

where is the density; and the displacement and stress components, respectively; thebody force component assumed to be independent of the fast varying coordinates.

(2) Constitutive equation

(9)

y τ Yφ

φ y τ,( ) φ y Ky+ τ,( )=

y KεY

φε x t,( ) φ x y t τ, , ,( )=

φ,iε φ,xi

ε 1– φ,yi+= φ· ε φ,t ε 1– φ,τ+=

x y

Ωε

ρεu··iε σij j,

ε bi+=

ρε uiε σij

ε bi

σijε Lijkl

ε eklε Vijkl

ε e·klε βij

ε θε–+=

5

where and denote the strain and the strain rate components, respectively; the temperaturechange from the initial temperature; and the elastic stiffness and the viscosity tensor com-ponents, respectively; where denotes the coefficient of thermal expansion and

stands for the components of the thermal stress. We assume that the fourth-rank tensor compo-nents and as well as the second-rank tensor components and satisfy conditions ofsymmetry and positivity. We further assume that and are thermally sensitive which lead tothe constitutive law coupling between the mechanical and thermal fields, while is assumed to beinsensitive to the temperature change, i.e.

(3) Kinematic equation

(10)

(4) Energy equation

(11)

where and is the specific heat per unit mass; denotes heat flux; the initial tem-perature and the heat supply. Both and are assumed to be independent of the fast varyingcoordinates. The total temperature is given by . The mechanical dissipation term

and the dilation induced heat supply fall into the category of equation coupling forthe thermal fields.

(5) Linear thermal diffusion

(12)

where denotes the thermal conductivity tensor components assumed to be symmetric and positivedefinite. Since is insensitive to temperature changes, it is also assumed to be time-independent, i.e.,

.

(6) Initial and boundary conditions

The non-oscillatory initial conditions at and are imposed on both time scales, i.e., theinitial state is assumed to be spatially and temporally smooth [5]. The thermo-mechanical boundaryconditions are also assumed to be non-oscillatory and the interfaces between different microscopicconstituents are perfectly bonded.

eklε e·kl

ε θε

Lijklε Vijkl

ε

βijε Lijkl

ε αklε= αkl

ε

βijε θε

Lijklε Vijkl

ε αklε βkl

ε

Lijklε Vijkl

ε

αklε

α ijε α ij x y,( )≡

eijε 1

2--- ui j,

ε uj i,ε+( )=

λεθ·ε

qi i,ε Q Vijkl

ε e· ijε e·kl

ε T0βijε e· ij

ε–+ +=

λε ρεcε= cε qiε T0

Q T0 QTε Tε T0 θε+=

Vijklε e· ij

ε e·klε T0βij

ε e· ijε

qiε kij

ε θ, jε=

kijε

kijε

kijε kij x y,( )≡

t 0= τ 0=

6

2.3 Double scale asymptotic analysis in space and time

To solve for the initial-boundary value problem described in Section 2.2, we start by introducing thefollowing double scale asymptotic expansions:

; (13)

where and are -periodic functions and denotes the order of the associated component inthe expansion. According to (13) and the chain rule in (7), the asymptotic expansions of strain (10) andthe strain rate can be expressed as

; (14)

where, with the definition of symmetric displacement gradients

and , (15)

the strain and strain rate components for various orders of in (14) are given as:

, , (16)

and

(17)

Consequently, the expansion of the stress field is obtained by substituting the expansions in (15) intothe constitutive equation (9), which gives

(18)

where

(19)

uiε εmui

m x y t τ, , ,( )m 0 1 …, ,=∑= θε εmθm x y t τ, , ,( )

m 0 1 …, ,=∑=

uim θm Y m

eijε εmeij

m x y t τ, , ,( )m 1 0 …, ,–=∑= e· ij

ε εme· ijm x y t τ, , ,( )

m 2 1– …, ,–=∑=

eijxn 1

2--- ui xj,

n uj xi,n+( )= eijy

n 12--- ui yj,

n uj yi,n+( )= n 0 1 2 …, , ,=

ε

eij1– x y t τ, , ,( ) eijy

0= eijn x y t τ, , ,( ) eijx

n eijyn 1++= n 0 1 2 …, , ,=

e· ij2– x y t τ, , ,( ) eijy τ,

0 = e· ij1– x y t τ, , ,( ) eijy t,

0 eijx0 eijy

1+( ),τ+=,

e· ijn x y t τ, , ,( ) eijx

n eijyn 1++( ),t eijx

n 1+ eijyn 2++( ),τ += n 0 1 2 …, , ,=,

σijε εmσij

m x y t τ, , ,( )m 2 1– …, ,–=

∑=

σij2– x y t τ, , ,( ) Vijkl

ε e·kl2– = σij

1– x y t τ, , ,( ) Lijklε ekl

1– Vijklε e·kl

1–+=,

σijn x y t τ, , ,( ) Lijkl

ε ekln βij

ε θn– Vijklε e·kl

n += n 0 1 2 …, , ,=,

7

Similarly, the expansion of the heat flux is obtained by using (12) and (13) such that

(20)

where

and , (21)

Having defined the asymptotic expansions for the mechanical and thermal fields, the equation ofmotion (8) and the energy equation (11) can be stated in terms of two sets of equations with increasingorder of starting from for the energy equation and for the equation of motion. Solv-ing these equations successively yields the initial-boundary value problem and the homoge-nized constitutive equations.

2.3.1 and equations

We first consider the energy equation

(22)

Due to symmetry and positivity of the viscosity tensor as illustrated by equation (22), along withequations (17) and (19), leads to

; (23)

Since the initial conditions are non-oscillatory it implies and the first equation in(23) gives , i.e., is independent of

(24)

With (23), it can be easily shown that the order equation of motion and energy equation areautomatically satisfied.

2.3.2 equation

The order equations take the following form:

qiε εmqi

m x y t τ, , ,( )m 1 0 …, ,–=∑=

qi1– x y t τ, , ,( ) kij

ε θ,yj

0= qin x y t τ, , ,( ) kij

ε θ,xj

n θ,yj

n 1++( )= n 0 1 2 …, , ,=

ε O ε 4–( ) O ε 3–( )O ε0( )

O ε 4–( ) O ε 3–( )

O ε 4–( )

Vijklε e· ij

2– e·kl2– 0=

Vijklε

e· ij2– eijy τ,

0 0= = σij2– 0=

eijy0 x t τ 0=, ,( ) 0=

eijy0 0= ui

0 y

ui0 ui

0 x t τ, ,( )≡

O ε 3–( )

O ε 2–( )

O ε 2–( )

8

(25)

Integrating the first equation in (25) over the unit cell domain and making use of the -periodicity of and the -independence of as shown in (24), as well as the non-oscillatory initial conditions,

yields

(26)

Apparently, is independent of fast spatial and temporal variables and thus it represents the macro-scopic displacement field while its symmetrical gradient represents the macroscopic strain field.With this in mind along with (17) and (19), the order equation of motion in (25) is reduced to

(27)

which can be further reduced to due to the fact that is symmetric and positive defi-nite and is -periodic. Due to the non-oscillatory initial conditions it follows that

(28)

indicating that is independent of . Based on (26) and (27), the following identities for the strainand stress fields can be identified:

; ; (29)

Thus the first non-vanishing terms in the expansion of strain, strain rate and stress fields are all order.

As for the order energy equation in (25), the simplified form can be obtained by exploiting(21) and (29) which yields

(30)

Once again, the symmetry and positivity of as well as the -periodicity of provides the solutionof (30) in the form of

and (31)

Therefore the first non-vanishing term in the expansion of heat flux also starts from order.

ρεui ττ,0 σij yj,

1–=

qi yi,1– Vijkl

ε e· ij1– e·kl

1–+ 0=

Yσij

1– Y ui0

ui ττ,0 0 = ui

0 ui0 x t,( )≡⇒

ui0

eijx0

O ε 2–( )

σij yj,1– Vijkl

ε ekly τ,1( ),yj

0= =

ekly τ,1 0= Vijkl

ε

ui τ,1 Y

eijy1 0 = ui

1 ui1 x t τ, ,( )≡⇒

ui1 y

eij1– 0= e· ij

1– 0= σij1– 0=

O ε0( )

O ε 2–( )

kijε θ,yj

0( ), yi0=

kijε Y θ0

θ,yj

0 0 = θ0 θ0 x t τ, ,( )≡⇒ qi1– 0=

qiε O ε0( )

9

2.3.3 equation

With the solutions obtained from the lower order equations, (29) and (31), the order equationstake the following form:

(32)

We first consider the order energy equation. Averaging it over the unit cell domain and utiliz-ing the -periodicity of and -independence of , we have

(33)

Thus, is independent of the fast varying variables and can be viewed as the macroscopic tempera-ture change

Along with (21) and (33), order energy equation turns into

(34)

where, as we assumed in (12), is independent of temperature change, i.e., . Due to thelinearity of (34) ( is independent of ), the solution of can be expressed by the followingdecomposition

(35)

where is an arbitrary -independent function and is determined by

in (36)

where is Kronecker delta. Equation (36) along with the corresponding periodic boundary condi-tions represents a typical linear unit cell problem. Finite element method can be used to solve for thisequation (see, for example, [9]). Upon the solutions of (36) and (35), order heat flux defined in(21) can be written as

(37)

O ε 1–( )

O ε 1–( )

ρεui ττ,1 σij yj,

0=

λεθ,τ0 qi yi,

0=

O ε 1–( )Y qi

0 Y θ0

θ,τ0 0 = θ0 θ0 x t,( )≡⇒

θ0

O ε 1–( )

kijε θ,xj

0 θ,yj

1+( )

, yi

0=

kijε kij

ε kij x y,( )≡θ0 y θ1

θ1 x y t τ, , ,( ) µi y( )θ,xj

0 P x t τ, ,( )+=

P x t τ, ,( ) Y µi y( )

kijε δjm µm yj, y( )+( )

, yi

0= Θ

δjm

O ε0( )

qi0 kij

ε δjm µm yj, y( )+( )θ,xm

0=

10

We now consider order equation of motion in (32). Using the same considerations as for equa-tion (25) and recalling that is independent of , the order equation of motion provides

(38)

It can be seen that is independent of the fast varying variables. Consequently, the orderequation of motion, combined with definitions in (17) and (19) as well as the relations in (26) and (33),turns into

(39)

According to (26) and (33), we conclude that , and represent the macroscopic strain,strain rate and temperature fields which are independent of and . To solve (39) for in termsof these macroscopic response quantities, we recall that and have been assumed to be ther-mally sensitive in the present work. Following [2] and noting that our attention is restricted to smalltemperature changes in comparison with the initial temperature , the thermally sensitive materialproperties are defined as

(40)

where and are elastic stiffness and viscosity tensor components evaluated at the initial tem-perature; and represent the thermal sensitivity tensor components associated with the elas-tic stiffness and viscosity evaluated at the initial temperature, respectively.

As stated in Section 2.1 the thermal expansion coefficient is assumed to be independent of temper-ature change, i.e., . Since all the quantities in (39), except for ,are known to be independent of . It follows that should be also independent of . Followingthe approach developed in [4] and considering the temperature change as a state variable deter-mined by the energy equation (evolution law) we introduce the following decomposition

(41)

where and are the third-rank tensors which are symmetric with respect to ; and represent the temperature induced macroscopic strain and

strain rate (eigenstrain and eigenstrain rate), respectively; is an arbitrary function. Thesymmetric gradient of (41) is given as

O ε 1–( )ui

1 y O ε 1–( )

ui ττ,1 0 = ui

1 ui1 x t,( )≡⇒

ui1 O ε 1–( )

σij yj,0 Lijkl

ε eklx0 Vijkl

ε eklx t,0 ekly τ,

2+( ) βijε θ0–+

,yj

0= =

eklx0 eklx t,

0 θ0

y τ ekly τ,2

Lijklε Vijkl

ε

T0

Lijklε Lijkl

ini x y,( ) θ0 x t,( )Aijklini x y,( )+≈

Vijklε Vijkl

ini x y,( ) θ0 x t,( )Bijklini x y,( )+≈

Lijklini Vijkl

ini

Aijklini Bijkl

ini

αklε

αklε αkl

ε x y,( )≡ βijε Lijkl

ε αklε= ekly τ,

2

τ ekly τ,2 τ

θ0

uk τ,2 x y t τ, , ,( ) ςkmn y( ) emnx

0 wmn0+( ) ϑ kmn y( ) emnx t,

0 wmn t,0+( ) Uk x t τ, ,( )+ +=

ςkmn y( ) ϑ kmn y( ) m n,wmn

0 wmn0 x t,( )≡ wmn t,

0 wmn t,0 x t,( )≡

Uk x t τ, ,( )

11

(42)

where and denote the symmetric gradient of and with respect

to , i.e.

(43)

To solve for and as well as for the temperature change induced macroscopicresponse and , we start by substituting (42) into (39), which yields

(44)

Since and are both assumed to be independent of time and thus temperature change,these two local concentration functions can be evaluated by the following two linear unit cell problemsobtained from (44) when so that and :

in (45)

in (46)

Again, similarly to (36), the above two linear unit cell problems along with the periodic boundary con-ditions can be solved for and , respectively, by using finite element method. Owingto the -periodicity of and , both and are also -periodic. It can be readilyproved that and are polarization functions whose integrations over the unit cell domainvanish due to the periodicity. The temperature change induced macroscopic response fields, and

, are obtained by multiplying (44) with the periodic function and then integrating it byparts over the unit cell domain, which leads to the initial value problem for :

(47)

with the initial condition: at . The time-dependent coefficients are given as

ekly τ,2 x y t, ,( ) ψklmn y( ) emnx

0 wmn0+( ) χklmn y( ) emnx t,

0 wmn t,0+( )+=

ψklmn y( ) χklmn y( ) ςkmn y( ) ϑ kmn y( )

y

ψklmn ςkmn,ylςlmn,yk

+ 2⁄=

χklmn ϑ kmn,ylϑ lmn,yk

+ 2⁄=

ςkmn y( ) ϑ kmn y( )wmn

0 dmn t,0

Lijmnε Vijkl

ε ψklmn+( )emnx0 Vijkl

ε ψklmnwmn0 βij

ε θ0–+ +

Vijklε δkmδln χklmn+( )emnx t,

0 Vijklε χklmnwmn t,

0+

, yj

0=

ψklmn χklmn

t 0= θ0 0= wmn0 wmn t,

0 0= =

Lijmnini Vijkl

ini ψklmn+

, yj

0= Θ

Vijklini δkmδln χklmn+( )

, yj

0= Θ

ςkmn y( ) ϑ kmn y( )Y Lijkl

ini Vijklini ψklmn χklmn Y

ψklmn χklmnwmn

0

wmn t,0 ϑ kmn y( )

wmn0

apqmn x t,( )wmn t,0 bpqmn x t,( )wmn

0+ fpqmn x t,( )emnx0

– gpqmn x t,( )emnx t,0– hpq x t,( )θ0+=

wmn0 0= t 0=

12

(48)

and the spatial averaging operator is defined as

(49)

where is the volume of the unit cell. The solution of from (47) is a history-dependent func-tion which leads to the long-term fading memory in the macroscopic constitutive equation. In contrastto the long-term memory induced by the classical spatial homogenization process [6][9][13], whichinvolves solving an initial-boundary value problem in the unit cell domain (see (A16) in Appendix),the present long-term memory is obtained by solving the first order initial value problem at eachGauss-point in the macro-domain.

To this end, (42) can be expressed in a concise form as

(50)

where

(51)

Finally, we remark that the time-dependence of these four parameters is due to the temperature-depen-dence of and .

2.3.4 equation

The order equations of motion and energy, along with (26) and (29), can be written as

apqmn χijpqVijklε χklmn⟨ ⟩=

bpqmn χijpqVijklε ψklmn⟨ ⟩=

fpqmn χijpq Lijmnε Vijkl

ε ψklmn+( )⟨ ⟩=

gpqmn χijpqVijklε δkmδln χklmn+( )⟨ ⟩=

hpq χijpqβijε⟨ ⟩=

•⟨ ⟩

•⟨ ⟩ 1Θ------- Θd•

Θ∫=

Θ wmn0

ekly τ,2 x y t, ,( ) ξklmn x y t, ,( )emnx

0 ζklmn x y t, ,( )emnx t,0 ηkl x y t, ,( )θ0– γklmn x y t, ,( )wmn

0+ +=

ξklmn ψklmn χklpq apqij( ) 1– fijmn–=

ζklmn χklmn χklpq apqij( ) 1– gijmn–=

ηkl χ– klpq apqij( ) 1–hij=

γklmn ψklmn χklpq apqij( ) 1– bijmn–=

Lijklε Vijkl

ε

O ε0( )

O ε0( )

13

(52)

where the order stress and strain rate can be obtained from (17), (19), (38) and (50)which gives

(53)

It can be seen that both and are independent of .

For the order equation of motion in (52), the volume average over the unit cell domain provides

(54)

where the macroscopic stress is determined by the homogenized constitutive equation

(55)

and the homogenized coefficients are given by

(56)

The homogenized energy equation is obtained by averaging the second equation in (52) overthe unit cell domain and making use of (33) and (37)

(57)

where is given in (53); is the homogenized thermal conductivity given by

(58)

ρεu··i0 σij xj,

0 σij yj,1 bi+ +=

λεθ·0 qi xi,

0 qi yi,1 Q Vijkl

ε e· ij0 e·kl

0 T0βijε e· ij

0–+ + +=

O ε0( ) σij0 e· ij

0

σij0 Lijkl

ε ekl0 Vijkl

ε e·kl0 βij

ε θ0–+=

e· ij0 ξ ijmnemnx

0 δimδjn ζ ijmn+( )emnx t,0 η ijθ

0– γijmnwmn0+ +=

σij0 e· ij

0 τ

O ε0( )

ρu··i0 σij xj,

0bi+=

σij0

Lijkleklx0 Vijkleklx t,

0 βijθ0– Λijklwkl

0+ +=

ρ ρε⟨ ⟩=

Lijkl Lijklε Vijmn

ε ξmnkl+⟨ ⟩=

Vijkl Vijmnε δmkδnl ζmnkl+( )⟨ ⟩=

βij βijε Vijkl

ε ηkl+⟨ ⟩=

Λijkl Vijmnε γmnkl⟨ ⟩=

O ε0( )

λθ· 0

kijθ,xj

0( ),xiQ Vijkl

ε e· ij0 e·kl

0⟨ ⟩ T0 βijε e· ij

0⟨ ⟩–+ +=

e· ij0 kij

kij kimε δjm µj ym,+( )⟨ ⟩=

14

and the average volumetric specific heat is defined as

(59)

Finally, we remark that the initial and boundary conditions for the components in asymptotic expan-sions are defined to satisfy the imposed conditions of the source problem defined in Section 2.2. Thusthe initial and boundary conditions for the components coincide with those imposed on thesource problem, while the components of higher order of follow trivial initial and boundary condi-tions.

3.0 Validity of the Homogenized Solutions

In the previous section we have shown that the extra long-term memory in the homogenized constitu-tive equation (55) can be obtained by solving a first order initial value problem in the macroscopicdomain. Therefore, the proposed homogenized constitutive model (55) offers significant computa-tional advantages by eliminating the need for evaluating the local initial-boundary problem associatedwith the long-term memory in homogenization processes [6][13].

Despite the remarkable simplicity of the present model, it is important to investigate under what cir-cumstances the fast temporal scale exists and the homogenized solution is valid. Recall that thehomogenized solution obtained in Section 2.3 is based on the assumption . We have inves-tigated other relations between the temporal and spatial scales and results of these findings are summa-rized in Table 2. The derivation details corresponding to various combinations of multiple spatial andtemporal scales are presented in Appendix. It is shown that once the fast temporal scale in mechanicalfields exists, the present solution is unconditionally valid; the multiple temporal scales in thermalfields have no effect on the homogenized solutions. On the other hand, when the fast temporalscale does not exist in mechanical fields, our approach is not valid and the homogenized solutions takethe classical form obtained by spatial homogenization only [6][9].

TABLE 2. Validity of the homogenized solutions (m, n = 1,2,3...; )

λ

λ ρεcε⟨ ⟩=

O ε0( )ε

εl ετ ε≈ ≈

O ε0( )

, V

alid V

a lid Inv a lid

, V

a lid V

a lid Inv a lid

, V

a lid V

a lid Inv a lid

l

m=

=

l

= m

=

l

=

0 t r

1

l

n =

l

= n=

l

n=

0

1

mechanicalfields

thermal fields

m

=

rt

ε<<1

15

The existence of fast varying temporal scale is determined by the characteristic temporal length ofresponse fields and material itself. The intrinsic temporal scale can be estimated by requiring the twomajor terms in the equation of motion (8), i.e. elastic and viscous contributions, to be of the sameorder, which yields

(60)

where means the norm of ; is the characteristic length of macroscopic reference timescale. Physically, the ratio defined in (60) characterizes the rate of creep behavior. Note that the ther-mal dilation effect is typically very small and inertial force is assumed to be not in dominance.

For the energy equation, we denote the characteristic length of the intrinsic temporal scale in thermalfields as . To quantify , we require the first two terms in the energy equation (11), i.e. the specificheat increase and thermal conductivity term, are of the same order, which yields

(61)

where represents the heat front advance after time elapse . Then, we can infer by such reason-ing that, when , the heat front has swept through the unit cell so that all the heat fluctuationsgenerated before the start of has been smoothed out in the unit cell, and thus the homogenized ther-mal fields could be reached provided that no more heat is generated during . In this sense, can beapproximated as

(62)

4.0 Numerical Examples

As shown in Figure 1, we consider a single ply of unidirectionally reinforced fibrous composite sub-jected to the uniform pressure. The fibers are assumed to be aligned in direction and the ply is sup-ported by a rigid foundation. We further assume that the thickness ( in direction) of the ply isvery small compared with the size in other two dimensions ( and directions). Thus the thermo-mechanical response of the composite ply can be described by a stack of unit cells along directionsubjected to the periodic boundary conditions in and directions. The finite element model of theunit cell is shown in Figure 1.

tr O min Vijklε Lijkl

ε⁄( )

=

• • tr

tr tr

ladv min tr kijε λε⁄

1 2⁄

=

ladv trladv l≥

trtr tr

tr O l2 min kijε λε⁄

⁄

≈

X10mm Z

X YZ

X Y

16

Figure 1. Composite plate and the associated unit cell

We assume that each phase in the unit cell is isotropic and homogeneous. The volume fractions of thetwo phases are denoted by for phase on and for phase on , such that and

. Material properties of each phase are denoted as follows: Young’s moduli at initialtemperature , Poisson’s ratio at initial temperature , viscosity at initial temperature , ther-mal expansion coefficient , thermal conductivity , density , and specific heat per unit mass .The material properties are summarized as follows:

Fiber: , , , ,

, , ,

Matrix: , , , ,

, , ,

We assume that the thermal sensitivities for elastic stiffness and viscosity are: and . The adiabatic boundary conditions are imposed on the ply, and the bodyforce and the heat supply are set to zero. The initial temperature is assumed to beuniform throughout the ply. As shown in Figure 1, the displacement boundary condition in the form of

(63)

is applied in direction on the free boundary, where and are the loading parameters.

unit cell

10mm

zoom

XZ

Yzoom

XZ

Y

00.1

00.10

0.05

0.1 We

Q

composite ply reinforced in x direction

macro problem

u3(t)

0.1*0.1mm

z (m

m)

x (mm)

y (mm)

d1 Θ1 d2 Θ2 d1 d2+ 1=Θ1 Θ2∪ Θ=

Eini νini Vini

α k ρ c

E1ini 37.92GPa= ν1

ini 0.21= V1ini 6GPa= α1 1 6–×10 K 1–=

ρ1 2200kg/m3= c1 1kJ/kg-K= k1 0.2W/m-K= d1 0.267=

E2ini 6.89GPa= ν2

ini 0.33= V2ini 3GPa= α2 5 5–×10 K 1–=

ρ2 1100kg/m3= c2 2kJ/kg-K= k2 1W/m-K= d2 0.733=

Aijklε 0.01– Lijkl

ini=Bijklε 0.05V– ijkl

ini=bi Q T0 300K=

u x 0= t,( ) 0.5u 1 2πωtcos–( )=

Z u ω

17

In this example, the inertia force is set to be very small in comparison with the elastic and viscouseffects so that the quasi-static problem is considered. The reference solution obtained by deploying avery fine mesh in the three-dimensional strip is compared against the homogenized solution. The sim-ulation results of one-cycle 10-Hz loading are plotted in Figure 2 and Figure 3, where the loading his-tory and the corresponding comparisons between the reference solutions and the homogenizedsolutions of end force, temperature change and stress component are illustrated. It can be seen that thedilation effect offsets the temperature increase in the unloading phase. In Figure 4, the distribution ofthe residual stress in the unit cell right after the one-cycle loading, which is reconstructed from thehomogenized solution (53), compares well with the reference solution.

Figure 2. Loading history and the end force obtained with the homogenized and reference solutions

σ330

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−2

−1.5

−1

−0.5

0x 10

−3

t (s)

u 3(x=

0,t)

(m

m)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−40

−20

0

20

40

t (s)

end

forc

e (M

Pa)

homogenized solutionreference solution

18

Figure 3. Temperature and stress obtained with the homogenized and reference solutions

Figure 4. Distribution of the local residual stress right after the loading

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−40

−20

0

20

40

t (s)

σ 33 (

MP

a)

homogenized reference in matrixreference in fiber

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

1

2

3

4x 10

−3

t (s)

Θ0 (

K)

homogenized reference in matrixreference in fiber

σ330

homogenized solutionreference solution

u3 t )( )

XZ

Y

S33 VALUE+2.66E+00

+2.88E+00

+3.10E+00

+3.33E+00

+3.55E+00

+3.77E+00

+3.99E+00

+4.22E+00

+4.44E+00

+4.66E+00

+4.88E+00

multiscale solutionr efer ence solution

2.4

3

3.6

4.2

4.8

0

0.05

0.1

00.05

0.10

0.05

0.1

x (mm)y (mm)

z (m

m)

XZ

Y

u3(t)

σ330

19

5.0 Summary and future research directions

A computational framework for analyzing multiple physical processes interacting at multiple spatialand temporal scales is developed and applied to the coupled thermo-viscoelastic composites. Rapidlyvarying spatial and temporal scales are introduced to capture the oscillations induced by local hetero-geneities at diverse time scales. The homogenized initial-boundary value problem along with thehomogenized constitutive equations are derived using the double scale asymptotic analysis in spaceand time. It is shown that the additional long-term memory induced by homogenization process can beobtained by solving a first order initial value problem as opposed to solving initial-boundary valueproblem in the unit cell domain in the case of the classical spatial homogenization.

We have identified two diverse time scales resulting from the input excitations and rate dependentmaterial behavior. Further investigation reveals higher order terms in the asymptotic expansions growunbounded in time and will affect the accuracy of the first order homogenized solutions when theobservation time window is long enough. A regularization scheme to suppress this secular time depen-dence has been recently proposed for wave propagation problems in elastic heterogeneous solid [3].The applicability of this regularization scheme for the present model will be investigated in the futurework.

6.0 Acknowledgement

This work was supported by the Sandia National Laboratories under Contract DE-AL04-94AL8500and the Office of Naval Research through grant number N00014-97-1-0687.

References

1 Bensoussan, A., Lions, J.L. and Papanicolaou, G., Asymptotic Analysis for Periodic Structures,North-Holland, Amsterdam, 1978

2 Boutin, C. and Wong, H., Study of thermosensitive heterogeneous media via space-time homoge-nisation, Eur. J. Mech. A/Solids 17 (1998) 939-968

3 Chen, W. and Fish, J., A dispersive model for Wave propagation in periodic composites based onhomogenization with multiple spatial and temporal scales, J. Appl. Mech., in print (2000)

4 Fish, J., Yu, Q. and Shek, K., Computational damage mechanics for composite materials based onmathematical homogenization, Int. J. Num. Methods Engrg., 45 (1999) 1657~1679.

5 Francfort, G.A., Homogenization and linear thermoelasticity, SIAM J. Math. Anal. 14 (1983) 696-708

6 Francfort, G.A. and Suquet, P.M., Homogenization and mechanical dissipation in thermoviscoelas-ticity, Arch. Rational Mech. Anal. 96 (1986) 265-293.

7 Iesan, D. and Scalia, A., Thermoelastic Deformation, Kluwer Academic Publishers, Amsterdam,1996

8 Kevorkian, J. and Bosley, D.L., Multiple-scale homogenization for weakly nonlinear conservation

20

laws with rapid spatial fluctuations, Studies Appl. Math., 101 (1998) 127-183.9 Sanchez-Palencia, E., Non-Homogeneous Media and Vibration Theory, Springer-Verlag, Berlin,

1980.10 Suquet, P.M., Elements of homogenization for inelastic solid mechanics, In: Homogenization

Techniques for Composite Media, eds. Sanchez-Palencia, E. and Zaoui, A., Springer-Verlag, Ber-lin. 1987, pp193-278.

11 Tartar, L., Memory effects and homogenization, Arch. Rational Mech. Anal. 111 (1990) 121-133.12 Terada, K., Ito, T. and Kikuchi, N., Characterization of the mechanical behaviors of solid-fluid

mixture by the homogenization method, Comput. Methods Appl. Mech. Engrg. 153 (1998) 223-257.

13 Yi, Y.M., Park, S.H. and Youn, S.K., Asymptotic homogenization of viscoelastic composites withperiodic microstructures, Int. J. Solids Struct. 35 (1998) 2030-2055.

Appendix

In this section, we summarize the results of homogenized solutions corresponding to differentcombinations of spatial and temporal length scales as shown in Table 2. We start by introducing theaveraging operator in the time domain with fast temporal oscillating variable . According to [5], itrequires

exist and be finite (A1)

In the Laplace transformed domain, (A1) is equivalent to

exist and be finite (A2)

where is the variable in the transformed domain corresponding to ; is Laplace transformationof function .

A.1 Thermal Fields:

A.1.1 Mechanical Fields:

The order equation of motion and energy equation can be expressed as

(A3)

O ε0( )

τ

1τ---

τ ∞→lim f τ( ) τd

0

τ

∫

s f s( )s 0→lim

s τ f s( )f τ( )

εl εn= ετ, ε=

εl εm= ετ, ε=

O ε0( )

ρεu··i0 σij xj,

0 σij yj,m bi+ +=

λε θ·0

θ,τ1+( ) qi xi,

0 qi yi,1 Vijkl

ε e· ij0 e·kl

0 T0βijε e· ij

0–+ +=

21

where and are both independent of the fast varying variables and represent the macroscopic dis-placement and temperature change; the order stress and strain rate take the followingform

(A4)

where is obtained by solving a linear unit cell problem with periodic boundary conditions

(A5)

Following (39) and (50), the solution of (A5) is given as

(A6)

where the four time-dependent parameters and are defined in (51). To thisend, we conclude that the present local constitutive equation (A4) is identical to (53). Making use ofthe homogenization process for equation of motion in Section 2.3.4 leads to the same homoge-nized equation of motion:

(A7)

and the associated homogenized constitutive equation

(A8)

where the homogenized coefficients in (A8) are defined in (56).

For the energy equation, the heat flux is determined by

(A9)

where is obtained by solving a linear unit cell problem

(A10)

Following (34)-(37), we have

ui0 θ0

O ε0( ) σij0 e·kl

0

σij0 Lijkl

ε eklx0 Vijkl

ε e·kl0 βij

ε θ0–+=

e·kl0 eklx t,

0 ekly τ,m 1++=

ekly τ,m 1+

Lijklε eklx

0 Vijklε eklx t,

0 ekly τ,m 1++( ) βij

ε θ0–+

,yj

0=

ekly τ,m 1+ x y t, ,( ) ξklmn x y t, ,( )emnx

0 ζklmn x y t, ,( )emnx t,0 ηkl x y t, ,( )θ0– γklmn x y t, ,( )wmn

0+ +=

ξklmn ζklmn ηkl, , γklmn

O ε0( )

ρu··i0 σij xj,

0bi+=

σij0

Lijkleklx0 Vijkleklx t,

0 βijθ0– Λijklwkl

0+ +=

O ε0( ) qi0

qi0 kij

ε θ,xj

0 θ,yj

n+( )=

θ,yj

n

kijε θ,xj

0 θ,yj

n+( )

, yi

0=

22

and (A11)

where is defined in (36). The homogenized energy equation is obtained by applying the spatialand temporal averaging processes, (49) and (A2), to the second equation in (A3) and making use of(A11) so that

(A12)

where all the quantities have the same definitions as those in (57).

A.1.2 Mechanical Fields:

Comparing this case to Section A.1.1, the only difference is in the order equation of motion

(A13)

where and are both independent of the fast varying variables and represent the macroscopic dis-placement and temperature change; is defined in (A4). Since vanishes due to the temporalaveraging (A2), and both in (A3) and in (A13) have no contribution to the homogenizedequation of motion due to local periodicity, the order homogenized equations in the presentcase are identical to those in Section A.1.1.

A.1.3 Mechanical Fields:

The order equation of motion and energy equation can be expressed as

(A14)

where and are both independent of the fast varying variables and represent the macroscopic dis-placement and temperature change; the order stress tensor and strain rate tensor takethe following form

(A15)

where is determined by a local initial-boundary problem

θ,yj

n x y t, ,( ) µi yj, y( )θ,xi

0= qi0 kij

ε δjk µk yj, y( )+( )θ,xk

0=

µi y( )

λθ·0

kijθ,xj

0( ),xiQ Vijkl

ε e· ij0 e·kl

0⟨ ⟩ T0 βijε e· ij

0⟨ ⟩–+ +=

εl ε= ετ, εm=

O ε0( )

ρε u··i0 ui ττ,

2m+( ) σij xj,0 σij yj,

1 bi+ +=

ui0 θ0

σij0 ui ττ,

2m

σij yj,m σij yj,

1

O ε0( )

εl εm= τ0 tr⁄ 1≥ ,

O ε0( )

ρεu··i0 σij xj,

0 σij yj,m bi+ +=

λε θ· 0 θ,τ1+( ) qi xi,

0 qi yi,1 Vijkl

ε e· ij0 e·kl

0 T0βijε e· ij

0–+ +=

ui0 θ0

O ε0( ) σij0 e·kl

0

σij0 Lijkl

ε Vijklε ∂

t∂-----+

eklx0 ekly

m+( ) βijε θ0–=

e·kl0 eklx t,

0 ekly τ,m+=

eklym

23

(A16)

(A16) is in the similar form as those obtained by spatial homogenization [6][9][11]. For the energy equation, the heat flux is determined by (A11). The homogenized energy equation has thesame form as (A12), but the definition of follows (A15). In this case, it is shown that the homoge-nized solutions obtained in Section 2.3.4 are not valid.

A.2 Thermal Fields:

A.2.1 Mechanical Fields:

In this case, the order equation of motion and its homogenized solutions are the same as thosein Section A.1.1. For the energy equation,

(A17)

where strain rate is given in (A4); the heat flux is determined by

(A18)

and can be obtained by solving a linear unit cell problem

(A19)

Following (34)-(37), we have

and (A20)

where is defined in (36). The homogenized energy equation is obtained by applying the spatialand temporal averaging processes to (A17) and making use of (A20) so that

(A21)

where all the quantities have the same definitions as those in (57).

A.2.2 Mechanical Fields:

Lijklε Vijkl

ε ∂t∂

-----+ eklx

0 eklym+( ) βij

ε θ0–

,yj

0=

O ε0( )qi

0

e· ij0

εl ε= ετ, εn=

εl εm= ετ, ε=

O ε0( )O ε0( )

λε θ· 0 θ,τn+( ) qi xi,

0 qi yi,1 Vijkl

ε e· ij0 e·kl

0 T0βijε e· ij

0–+ +=

e· ij0 qi

0

qi0 kij

ε θ,xj

0 θ,yj

1+( )=

θ,yj

1

kijε θ,xj

0 θ,yj

1+( )

, yi

0=

θ,yj

1 x y t, ,( ) µi yj, y( )θ,xi

0= qi0 kij

ε δjk µk yj, y( )+( )θ,xk

0=

µi y( )

λθ·0

kijθ,xj

0( ),xiQ Vijkl

ε e· ij0 e·kl

0⟨ ⟩ T0 βijε e· ij

0⟨ ⟩–+ +=

εl ε= ετ, εm=

24

In this case, the order equation of motion and its homogenized solutions are the same as thosein Section A.1.2, while the order energy equation and its homogenized solutions are the sameas those in Section A.2.1.

A.2.3 Mechanical Fields:

The homogenized equation of motion and energy equation in this case are the same as those obtainedin Section A.1.3. Our solutions in Section 2.3.4 are not valid in this case.

A.3 Thermal Fields:

A.3.1 Mechanical Fields:

In this case the order equation of motion and its homogenized solutions are the same as thoseobtained in Section A.1.1. The order energy equation can be obtained by removing in (A5)and the homogenized energy equation is the same as that in Section A.1.1 since vanishes due totemporal averaging.

A.3.2 Mechanical Fields:

In this case the order equation of motion and its homogenized solutions are the same as thoseobtained in Section A.1.2. The order energy equation can be obtained by removing in (A5)and the homogenized energy equation is the same as that in Section A.1.1.

A.3.3 Mechanical Fields:

Similar to Section A.2.3, our homogenized solutions in Section 2.3.4 are not valid in this case.

O ε0( )O ε0( )

εl εm= τ0 tr⁄ 1≥ ,

εl εn= τ0 tr⁄ 1≥ ,

εl εm= ετ, ε=

O ε0( )O ε0( ) θ,τ

1

θ,τ1

εl ε= ετ, εm=

O ε0( )O ε0( ) θ,τ

1

εl εm= τ0 tr⁄ 1≥ ,

25