temporal homogenization of viscoelastic and viscoplastic solids subjected … · 2014. 4. 30. ·...

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Temporal Homogenization of Viscoelastic and Viscoplastic Solids Subjected to Locally Periodic Loading

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

Rensselaer Polytechnic Institute, Troy, NY 12180

Abstract: As a direct extension of the asymptotic spatial homogenization method we develop a tem-poral homogenization scheme for a class of homogeneous solids with an intrinsic time scale signifi-cantly longer than a period of prescribed loading. Two rate-dependent material models, the Maxwellviscoelastic model and the power-law viscoplastic model, are studied as an illustrative examples. Dou-ble scale asymptotic analysis in time domain is utilized to obtain a sequence of initial-boundary valueproblems with various orders of temporal scaling parameter. It is shown that various order initial-boundary value problems can be further decomposed into: (i) the global initial-boundary value prob-lem with smooth loading for the entire loading history, and (ii) the local initial-boundary value prob-lem with the remaining (oscillatory) portion of loading for a single load period. Large time incrementscan be used for integrating the global problem due to smooth loading, whereas the integration of thelocal initial-boundary value problem requires a significantly smaller time step, but only locally in asingle load period. The present temporal homogenization approach has been found to be in good agree-ment with a closed-form analytical solution for one-dimensional case and with a numerical solution inmultidimensional case obtained by using a sufficiently small time step required to resolve the loadoscillations.

1.0 Introduction

Mathematical homogenization method has been widely used for solving initial-boundary value prob-lems with oscillatory coefficients. The validity of the asymptotic homogenization depends on the exist-ence of distinct multiple length scales in the physical processes so that a small positive scalingparameter quantifying the ratio between the scales can be identified. In general, multiple length scalesmay exist in both space and time domains, although most of the recent research efforts have beenfocussing on the spatial homogenization (see, for instance, [12][14]). In contrast to the spatial scaleseparation, which is typically induced by spatial heterogeneities, the multiple temporal scales can beattributed to at least three sources (or their combinations):

• the interaction of multiple physical processes

Different physical processes, such as mechanical, thermal, diffusion, and chemical reaction, mayevolve along different time frames. Interaction between multiple physical processes requires consid-eration of relevant time frames within a single reference time coordinate. An example problem fall-

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ing into this category is a coupled thermo-mechanical process, which has been studied by Boutinand Wong [2] using spatial-temporal homogenization approach. Most recently, a general setting forthe spatial-temporal asymptotic homogenization theory has been established by Yu and Fish [15].

• existence of spatial heterogeneities

Spatial heterogeneities may cause dispersion of high frequency waves traveling in heterogeneousmedia. The time frame corresponding to the successive reflection and refraction of waves betweenthe interfaces in microstructure could be significantly different from the time frame of the macro-scopic wave motion. The earliest work on the multiple temporal scales induced by spatial heteroge-neities is often attributed to Benssousan et al. [1] who studied parabolic equations with oscillatorycoefficients. Francfort [8] utilized multiple temporal scales to analyze thermo-elastic composites.Kevorkin and Bosley [9] introduced an additional fast time scale to study the hyperbolic conserva-tion laws with rapid spatial fluctuations. In the recent work Fish, Chen and Nagai [4][5] introducedmultiple slow temporal scales to alleviate the problem of secularity caused by high order terms inthe asymptotic analysis of wave propagation in heterogeneous solids and established a nonlocal con-tinuum approach to capture dispersion effects [6][7].

• existence multiple time scale within a single physical process on a single spatial scale

In many engineering problems multiple temporal scales arise in a single physical process taking placein a homogeneous medium. For example, slow degradation of materials properties due to creep, relax-ation and fatigue, subjected to rapidly oscillatory loading exhibit multiple temporal scales. This cate-gory of problems possess an intrinsic slow time scale, which may significantly differ from thefrequency of external input.

In the present manuscript, we focus on the third category of problems. Attention is restricted to theasymptotic homogenization of rate-dependent solids. The prediction of long-term behavior of rate-dependent solids subjected to oscillatory loading requires significant computational resources, in par-ticular, for nonlinear solids subjected to non-harmonic loading. This is because the resolution of highfrequency loading requires time integration increments, which are much smaller than the observationtime window. The primary objective of this manuscript is to develop a temporal homogenizationscheme by which the initial-boundary value problem with locally periodic loading in time domain canbe approximated by: (i) the global initial-boundary value problem with smooth loading for the entireloading history, and (ii) the local initial-boundary value problem with the remaining (oscillatory) por-tion of loading for a single load period in selected region(s) of the time domain.

For the global initial-boundary value problem a large time increment can be used, whereas the integra-tion of the local initial-boundary value problem requires a significantly smaller time step, but onlylocally in the time domain, where a full response is sought. It is apparent that the present temporalhomogenization approach closely resembles the classical spatial homogenization scheme. The global

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initial value problem is equivalent to the macroscopic boundary value problem with homogenizedcoefficients, whereas the postprocessing of local fields within the Representative Volume Element(RVE) is equivalent to the local initial-boundary value problem in the temporal homogenizationscheme. The main conceptual difference between the two will be shown to exist for nonlinear prob-lems. For nonlinear heterogeneous solids both the macroscopic and the RVE problems are nonlinear,whereas the temporal homogenization of viscoplastic solids gives rise to nonlinear global initial-boundary value problem and a linear local initial-boundary value problem. It will be shown that non-linearities do reappear in the higher order initial-boundary value problems, which can be used toimprove the quality of the global-local approximation, but are rarely used in practice.

Two rate-dependent material models, the Maxwell viscoelastic model and the power-law viscoplasticmodel [10][11], are considered as illustrative examples. We start with the definition of multiple tempo-ral scales in Section 2.1. In Section 2.2, the temporal homogenization scheme for the linear Maxwellviscoelastic model is presented. It is shown that a long-term response can be obtained by solving thetime-averaged zero-order homogenized initial-boundary value problem along with the smooth portionof external loading. The deviation from the smooth solution is obtained by solving a local linear initial-boundary value problems within one period of load cycle. In Section 2.3, we extend the temporalhomogenization scheme to the power-law viscoplastic solid. In Section 3 the temporal homogenizationapproach is verified against the closed-form reference solution for one-dimensional viscoelastic andviscoplastic solids. In multidimensions, two numerical examples comparing the temporal homoge-nized approach with the reference solutions obtained with a time step sufficiently small to resolve thelocal load fluctuations are described in Section 4.

2.0 Temporal homogenization of the rate-dependent solids subjected to locally periodic loading

2.1 Definition of multiple temporal scales

In the present work, we assume that the intrinsic time scale , which is determined by material proper-ties and serves as the characteristic length of the natural time coordinate , describes a relatively long-term behavior compared with a single period of loading. To characterize the fast varying features ofresponse fields induced by the locally periodic loading as shown in Figure 1, we assume that thereexists a small positive scaling parameter so that a fast time coordinate can be identified anddefined as

(1)

We further assume that the response fields are locally periodic in the time domain with respect to , orat least in the statistical sense. The period of external loading denoted by serves as characteristiclength of the fast time coordinate. Thus, the scaling parameter can be defined as

trt

ς τ

τ t ς⁄=

ττ0

ς

4

(2)

With the definition of the fast varying variable as well as the -periodicity assumption, all theresponse fields denoted by can be defined by using the conventional nomenclature:

(3)

where denotes the position vector in space. The time differentiations in this case can be expressedusing the chain rule:

(4)

where the comma followed by a subscript variable denotes a partial derivative and superscribed dotdenotes the time derivative.

Figure 1. Natural and fast time coordinates

2.2 Temporal homogenization of the Maxwell viscoelastic solids under cyclic loading

The initial-boundary value problems for the Maxwell viscoelastic model is summarized below:

Equilibrium equation: on (5)

Constitutive equation: on (6)

Kinematic equation: on (7)

Initial conditions: on (8)

Boundary conditions: on (9)

on (10)

ς τ0 tr , ς

5

where is a body force; and are the components of stress and strain tensors, respectively; represents the components of elastic compliance tensor and denotes the components of the

inverse of viscosity tensor; both and are assumed to be symmetric and positive definite; represents the components of displacement vector; is the observation time in the natural time coor-dinates; is the load period in the fast (scaled) time coordinates as shown in Figure 1; denotes thespatial domain while and are the corresponding boundary portions where displacements andtractions are prescribed, respectively; denotes the normal vector component on the boundary; is the initial displacement. Summation convention for repeated subscripts is adopted.

To solve the initial-boundary value problem (5)-(10), we start by approximating the displacement fieldin terms of the double temporal scales asymptotic expansion

(11)

where ( ) are -periodic functions and denotes the order of the terms in the expan-sion. Note that the first term in the asymptotic expansion (11) is a function of both, and , to reflectthe fact that the smooth and oscillatory parts of the displacement field could be of the same order ofmagnitude. According to (7) and the chain rule in (4), the corresponding expansions of strains and thestrain rates can be expressed as

; (12)

and

; and (13)

Consequently, the expansion of stresses is obtained by substituting expansions in (12) and (13) into theconstitutive equation (6), which gives

(14)

where the stress components in the asymptotic expansion are determined from various order constitu-tive equations:

: on (15)

bi σijς eij

ς

Cijkl SijklCijkl Sijkl uiς

Tτ0 Ω

Γu Γf uifi ni ũi

uiς ςmui

m x t τ, ,( )m 0 1 …, ,=∑=

uim m 0 1 …, ,= τ m

t τ

eijς ςmeij

m x t τ, ,( )m 0 1 …, ,=∑= eij

m uj i,m ui j,

m+( ) 2⁄=

e· ijς ςm 1– e· ij

m 1– x t τ, ,( )m 0 1 …, ,=∑= e· ij

1– eij τ,0= e· ij

m eij t,m eij τ,

m 1++=

σijς ςmσij

m x t τ, ,( )m 0 1 …, ,=∑=

O ς 1–( ) eij τ,0 Cijklσkl τ,

0= Ω 0 T,( ) 0 τ0,( )××

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: on (16)

Having defined the expansions of response fields, the asymptotic expansion of the equilibrium equa-tion can be obtained by substituting (14) into (5) which gives

: on (17)

: on (18)

where . From (8)-(10), along with the asymptotic expansion (11) and (14) for displace-ments and stresses, the initial and boundary conditions (ICs and BCs) for the order initial-boundary problems (15)(17) are given by

ICs: on

BCs: on (19)

on

For the high order problems defined in (16)(18), both initial and boundary conditions are trivial.

To solve (15)-(18) along with the appropriate initial and boundary conditions for various order ofresponse fields, we introduce the temporal averaging operator , defined as

(20)

as well as the following decompositions:

(21)

where and represent the oscillatory portion of the stress, strain and displacement fields,respectively. From (12), we have

; (22)

O ςm( ) eij t,m eij τ,

m 1++ Cijkl σkl t,m σkl τ,

m 1++( ) Sijklσklm+= Ω 0 T,( ) 0 τ0,( )××

O ς0( ) σij0

,j bi x t τ, ,( )+ 0= Ω 0 T,( ) 0 τ0,( )××

O ςm 1+( ) σij j,m 1+ 0= Ω 0 T,( ) 0 τ0,( )××

m 0 1 …, ,=O ς0( )

O ς0( ) ui0 x t τ 0= =,( ) ũi x( )= Ω

O ς0( ) ui0 ui x t τ, ,( )= Γu 0 T,( ) 0 τ0,( )××

σij0 nj fi x t τ, ,( )= Γf 0 T,( ) 0 τ0,( )××

< >•

< >• 1τ0----- • τd

0

τ0

∫=

Φijm x t τ, ,( ) σij

m σijm〈 〉–=

Ψijm x t τ, ,( ) eij

m eijm〈 〉–=

χim x t τ, ,( ) ui

m uim〈 〉–=

Φijm Ψij

m, χim

Ψijm χj i,

m χi j,m+( ) 2⁄= m 0 1 …, ,=

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For the smooth portion of the order homogenized solution, the constitutive relation and the fieldequation can be obtained by applying temporal averaging operator (20) to (16) and (17) in the case of

, which yields

on (23)

on (24)

where -periodicity of and has been exploited. The corresponding initial and boundary condi-tions for the global initial-boundary value problem are given by averaging (18) over a singleload cycle, which yields

ICs: on

BCs: (25)

Solutions of , and for the order initial-boundary value prob-lem represent the non-oscillatory long-term behavior of the response fields, which is independent ofthe fast time variable .

For the oscillatory portion of the order homogenized solution, equations (15) and (21) lead tothe following constitutive relation:

on (26)

The corresponding equilibrium equation is obtained by substructing (24) from (17) and exploiting thedefinition in (21), which gives

on (27)

The initial and boundary conditions corresponding to equations (26) and (27) are given as:

ICs: on

BCs: on (28)

O 1( )

m 1=

eij0〈 〉 ,t Cijkl σkl

0〈 〉 ,t Sijkl σkl0〈 〉+= Ω 0 T,( )×

σij0〈 〉 ,j bi x t τ, ,( )〈 〉+ 0= Ω 0 T,( )×

τ ekl1 σkl

1

O ς0( )

ui0〈 〉 x t 0=,( ) ũi x( )= Ω

ui0〈 〉 ui x t τ, ,( )〈 〉= on Γu 0 T,( )×

σij0〈 〉nj fi x t τ, ,( )〈 〉= on Γf 0 T,( )×

ui0〈 〉 x t,( ) eij0〈 〉 x t,( ) σij0〈 〉 x t,( ) O ς0( )

τ

O ς0( )

Ψij τ,0 CijklΦkl τ,

0= Ω 0 τ0,( )×

Φij0

,j bi bi〈 〉–+ 0= Ω 0 τ0,( )×

χi0〈 〉 0= Ω

χi0 ui ui〈 〉–= Γu 0 τ0,( )×

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on

It is worth noting that the initial-boundary value problem described by (22) and (26)-(28) is defined on, i.e, it has to be solved for one load cycle only. This is because the response fields are

assumed to be periodic functions of and the constitutive equation (26) need to be integrated withrespect to only.

In summary, the order initial-boundary problem (15)-(17) defined on hasbeen decomposed into two initial-boundary problems: one for the smooth long term behavior definedon which is independent of the fast time variable , and the second one on , fora single load period evolving around the smooth solution.

A similar two-step scheme is used for solving high order initial-boundary value problems. The highorder initial-boundary value problems can be obtained from (16) and (18)-(21), which yields:

For the global initial-boundary value problem ( ):

Equilibrium equation: on

Constitutive equation: on (29)

Trivial initial and boundary conditions.

For the local initial-boundary value problem ( ):


Constitutive equation: on

Initial conditions: on (30)

Boundary conditions: on

on

The solution of (29) is trivial, i.e., the only contribution from the high order equations comes from thelocal initial-boundary value problem. Hence, , and where .

Φij0 nj fi fi〈 〉–= Γf 0 τ0,( )×

Ω 0 τ0,( )×τ

τ

O 1( ) Ω 0 T,( ) 0 τ0,( )××

Ω 0 T,( )× τ Ω 0 τ0,( )×

O ςm 1+( ) m 0 1 …, ,=

σijm 1+〈 〉 ,j 0= Ω 0 T,( )×

eijm 1+〈 〉 ,t Cijkl σkl

m 1+〈 〉 ,t Sijkl σklm 1+〈 〉+= Ω 0 T,( )×

O ςm 1+( ) m 0 1 …, ,=

Φij ,jm 1+ 0= Ω 0 τ0,( )×

Ψij τ,m 1+ CijklΦkl τ,

m 1+ CijklΦklm

,t SijklΦklm Ψij

m,t–+( )+= Ω 0 τ0,( )×

χim 1+〈 〉 0= Ω

χim 1+ 0= Γu 0 τ0,( )×

Φijm 1+ nj 0= Γf 0 τ0,( )×

Φijm 1+ σij

m 1+= Ψijm 1+ eij

m 1+= χim 1+ ui

m 1+=m 0 1 …, ,=

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2.3 Temporal homogenization of the viscoplastic solid subjected to locally periodic loading

In this section, we develop a temporal homogenization scheme for the power-law viscoplastic solid[10]. The initial-boundary value problem in this case takes a similar form to that described in Section2.2 (see equation (5), (7)-(9)), except for the constitutive equation which is given as

(31)

where denotes the elastic strain components defined in (12) and is postulated as a viscoplasticstrain which follows the power-law form flow rule:

(32)

where and are material constants; is termed as the effective stress defined as

; (33)

where is a projector, which transfers to the deviatoric space; is

the Kronecker delta and ; is often referred to as the back stress while

in (32) is the drag stress. For simplicity, we assume that and follow linear hardening rules [13]:

(34)

where and are material constants assumed to be independent of the viscoplastic flow.

To obtain the asymptotic expansion of the stress fields, we introduce the following expansion for theviscoplastic strain along with the assumption (11) for the displacement :

σ· ijς

Lijkl e·klς µ· kl

ς–( )=

eklς µkl

ς

µ· klς

λςNklς=

λς a 32--- ξij

ς Yς⁄

1 c⁄

=

Nklς ξkl

ς ξklς⁄=

a c ξklς

ξklς Pklij σij

ς βijς–( )= ξkl

ς ξklς ξkl

ς=

Pklij Iklij δijδkl 3⁄–= σijς βij

ς–( ) δij

Iklij12--- δikδjl δilδjk–( )= βij

ς Yς

βijς Yς

β· ijς 2

3---Hµ· ij

ς=

Yς Ŷ ας α·ς

;– 23---Ĥλς= =

H Ĥ, Ŷ

µijς ui

ς

10

(35)

where all the components in the expansion are assumed to be locally periodic functions of the fast timevariable . From the definition in (35), together with the constitutive equation (31), the flow rule (32)and the hardening rule (34), it can be shown that the asymptotic expansion of and are givenas:

(36)

The expansion of the norm of the effective stress defined in (33) is given by

(37)

where

; (38)

Since equation (37) can be further expanded as

(39)

Similarly, the expansion of the viscoplastic flow parameter and the flow direction vector defined in (32) can be expressed as

(40)

(41)

and thus the two leading order expansions of the flow rule (32) take the following form:

µijς ςmµij

m x t τ, ,( )m 0 1 …, ,=∑=

τσijς βij

ς, ας

σijς ςmσij

m x t τ, ,( )m 0 1 …, ,=∑=

βijς ςmβij

m x t τ, ,( )m 0 1 …, ,=∑=

ας ςmαm x t τ, ,( )m 0 1 …, ,=∑=

ξklς

ξklς ξkl

0 1 ςR+( )1 2⁄=

ξkl0 Pklij σij

0 βij0–( )= R 2ξkl

0 ξkl1 ξkl

0⁄ O ς( )+=

ςR

11

: (42)

: (43)

where (42) indicates that the leading order viscoplastic strain is independent of , i.e. .Furthermore, as a result of (40) and (42), the back stress and drag stress defined in (34)are also independent of so that the expansion of (34) is given by

(44)

where and .

Having defined the expansions in (12), (36), and (40)-(44), we can obtain the asymptotic expansion ofthe constitutive equation (31):

: on (45)

: on (46)

where and the definition of the elastic strain components is given in (13).

We note that the asymptotic expansions of the equilibrium equation and initial-boundary conditions inthis case are the same as those for the Maxwell viscoelastic model derived in Section 2.2 (see equa-tions (17)-(19)). To solve the various order initial-boundary problems, we follow the decompositionsdefined in (21) so that various order initial-boundary value problems can be again solved in two steps,first for the whole loading history and the second for one period of load cycle. Following the proce-dure described in Section 2.2, the initial-boundary value problems for the responses fields of variousorder can be summarized as follows:

For the local initial-boundary value problem (using (45)):



O ς 1–( ) µkl ,τ0 0=

O ς0( ) µkl ,t0 µkl,τ

1+ λ0Nkl0=

τ µkl0 µkl

0 x t,( )≡O ς0( ) βij

0 Y0

τ O ς0( )

βij ,t0 βij ,τ

1+ 23---Hλ0Nij

0=

Y0 Ŷ α0 αij ,t0 αij ,τ

1+;– 23---Ĥλ0= =

βij0 βij

0 x t,( )≡ αij0 αij

0 x t,( )≡

O ς 1–( ) σij τ,0 Lijklekl τ,

0= Ω 0 T,( ) 0 τ0,( )××

O ςm( ) σij t,m σij τ,

m 1++ Lijkl eklm µkl

m–( ),t eklm 1+ µkl

m 1+–( ),τ+{ }= Ω 0 T,( ) 0 τ0,( )××

m 0 1 …, ,= eklm

O ς0( )

Φij0

,j biς bi

ς〈 〉–+ 0= Ω 0 τ0,( )×

Φij τ,0 LijklΨkl τ,

0= Ω 0 τ0,( )×

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Initial condition: on (47)


on

For the global initial-boundary value problem (using (46)):



Initial condition: on (48)


on

where the plastic strain rate is obtained by averaging (43) over one period of load cycle, which

yields

(49)

with and defined in (40) and (41). The corresponding back stress and drag stress are governed

by the temporal average of (44):

(50)

where the -periodicity has been applied. Note that the flow rule is a function of the total stress, which provides a one-way coupling between the global and local initial-boundary value

problems. In the one-way coupled scheme, is computed first by solving the local problem (47) ateach time increment of the global problem (48), except when the loading in (47) is independent of thenatural time variable in which case the local contribution can be precomputed ahead of global analy-sis. Subsequently, the global initial-boundary value is solved for the next time increment.

χi0〈 〉 0= Ω 0 τ0,( )×

χi0 ui ui〈 〉–= Γu 0 τ0,( )×

Φij1 nj fi fi〈 〉–= Γf 0 τ0,( )×

O ς0( )

σij0〈 〉 ,j bi

ς〈 〉+ 0= Ω 0 T,( )×

σij0〈 〉 ,t Lijkl ekl

0〈 〉 ,t µkl0〈 〉 ,t–{ }= Ω 0 T,( )×

ui0〈 〉 x t 0=,( ) ũi x( )= Ω 0 T,( )×

ui0〈 〉 ui x t τ, ,( )〈 〉= Γu 0 T,( )×

σij0〈 〉nj fi x t τ, ,( )〈 〉= Γf 0 T,( )×

µkl0〈 〉 ,t

µkl0〈 〉 ,t λ

0Nkl0〈 〉=

λ0 Nkl0

βij ,t0 2

3---H λ0Nij

0〈 〉=

Y0 Ŷ α0 αij ,t0;– 2

3---Ĥ λ0〈 〉= =

τσkl

0〈 〉 Φkl0+

Φkl0

t

13

Note that the constitutive equation for the oscillatory portion of homogenized solutions in (47)is linear while the constitutive equations for the high order oscillations remain nonlinear according to(46). Also, the smooth portion of the high order homogenized solution is generally non-trivial in con-trast to the solution for the Maxwell viscoelastic model.

3.0 One-dimensional verification examples

In this section, the analytical and numerical solutions for the one-dimensional homogenized problemsare compared with the reference solutions in order to verify the present temporal homogenizationscheme.

3.1 One-dimensional solution of the Maxwell viscoelastic model

Consider a one-dimensional bar clamped at one end ( ) and subjected to loading at the other end( ) as shown in Figure 1. A sinusoidal displacement with a period of superimposedon the constant field, , is chosen as a prescribed displacement. According to [15],the material intrinsic time scale for the one-dimensional Maxwell viscoelastic model can be defined as

(51)

where denotes the viscosity, is elastic stiffness, and is the creep time. We assume that theperiod of loading is much smaller than the material intrinsic time scale so that . Theprescribed displacement expressed in terms of the fast time coordinate is given as

(52)

where is the amplitude of the prescribed displacement and is the radial frequency of the load.

Figure 2. One-dimensional bar and the oscillatory loading

Following (5)-(9), the reference solution for the strain field in one-dimensional viscoelastic problemcan be obtained by solving

O ς0( )

x 0=x d= τ0 2π ω⁄=

uς U0 ωtsin 1+( )=

tr V L⁄=

V L trς τ0 tr⁄=

14

(53)

where denotes the axial stress. The solution of (53) is given as

(54)

where follows from (51). Since the scaling parameter is , equation (54) can beapproximated as:

(55)

The homogenized solutions for the one-dimensional viscoelastic problem can be obtained by reducingthe equations in Section 2.2 to the one-dimensional case. Noting that according to (1), theleading order initial-boundary problem can summarized as follows:

The global initial-boundary value problem:

(56)

The local initial-boundary value problem:

(57)

The analytical solution of the initial-boundary value problem (56)-(58) is given by:

σ·ς

L----- σ

ς

V-----+

U0ωd

----------- ωtcos σς t 0=( )LU0

d----------=;=

σς

σςLU0

d---------- 1

ωtr1 ω2tr

2+--------------------–

t

tr---–

expω2tr

2

1 ω2tr2+

-------------------- ωtsin 1ωtr-------- ωtcos+

+

=

tr ς ς 2π ωtr⁄

15

(58)

which coincides with the corresponding reference solution (55) provided that .

3.2 One-dimensional solution for the power-law viscoplastic model

The loading is assumed to be the same as in (52) defined in Section 3.1. Following Section 2.3, thesource initial-boundary value problem can be stated as:

(59)

The closed form solution of (59) exists only when and . For this case, the refer-ence solution of the stress field can be obtained by solving the linear initial value problem

(60)

It can be seen that equation (60) is similar to (53) and thus the solution can be expressed in the form ofequation (55) where the material intrinsic time scale is defined as and the scaling parame-ter is given as . In the second part of this section, we will consider a general case of(59) with nonzero hardening parameter and .

Following Section 2.3, the leading order one-dimensional homogenized solution can be obtained bysolving the following two initial-boundary value problems.

The local initial-boundary value problem:

σςLU0

d---------- t

tr---–

exp ωtsin+

O ς( )+=

ς

16

(61)

The solution of (61), which is a linear problem, can be easily obtained as

(62)

The global initial-boundary value problem:

(63)

Similarly to (59), the analytical solution of (61) can be found for and . In thiscase, order smooth stress field is given by

(64)

Thus the total stress field obtained by adding the contributions from equations (62) and (64)coincides with the reference solutions given in (54) and (55).

To this end we consider a more general viscoplastic material model where all the nonlinearities aretaken into account. The geometry of the one-dimensional bar is shown in Figure 2. The loading is

assumed to be in the form of prescribed displacement . The amplitude of the

loading is taken as and the radial frequency so that . We

select material properties as , , , , , and

. Numerical solution for source problem (59) is obtained by using a very fine time increment

Φ0,x 0 on 0 τ0,( )=

Φ,τ0 LΨ,τ

0 Ψ0 χ,x0=;=

χ0 τ 0=( ) is defined in such a way that Φ0〈 〉 0=

χ0 x 0=( ) 0 χ0 x d=( ) U0 ωtsin=; U02πtr

------τsin= =

Φ0LU0

d---------- 2π

tr------τsin

LU0d

---------- ωtsin= =

O 1( )

σ0〈 〉 ,x 0 on 0 T,( )=

σ0〈 〉 ,t L e0〈 〉 ,t µ0〈 〉 ,t–{ } e0〈 〉 u0〈 〉 ,x= µ0〈 〉 ,t a ξ0 Y0⁄( )1 c⁄

ξ0( )sgn〈 〉=;;=

ξ0 σ0〈 〉 Φ0 β0–+= ; Y0 Ŷ α0–=

β,t0 H µ,t

0〈 〉= ; α,t0 Ĥ µ,t

0 ξ0( )sgn〈 〉=

u0 t 0=( ) U0= u0; x 0=( ) 0 u0 x d=( ) U0=;=

H Ĥ 0= = c 1=O 1( )

σ0〈 〉LU0

d---------- t

tr---–

exp=

O 1( )

uς U0 0.1 ωtsin 1+( )=

U0 d⁄ 23–×10= ω 20π s 1–= τ0 0.1s=

L 50GPa= Ŷ 100MPa= H 4GPa= Ĥ 0= a 5 4–×10 s 1–=

c 0.7=

17

for the entire loading history. The comparison between the smooth homogenization solution

and the reference solution is given in Figure 3. It can be seen that captures well the non-oscilla-

tory long-term behavior. In Figure 4, we show the oscillatory stress field for two load cycles,

one at the early stage of the loading at and second, at the end of the loading .Good agreement with the reference solution can be observed.

Figure 3. global solution in comparison with reference solutions for the one-dimensional viscoplastic model

Figure 4. homogenized solution in comparison with reference solutions for the one-dimensional viscoplastic model

O 1( )

σ0〈 〉

O 1( )

0.9 1.0s,[ ] 9.9s 10s,[ ]

0 2 4 6 8 100

20

40

60

80

100

120

σς

time (s)

stre

ss (

MP

a)

O 1( )

9.9 9.92 9.94 9.96 9.98 1010

15

20

25

30

35

σς

σ0

time (s)

stre

ss (

MP

a)

homogenized solutionreference solution

0.9 0.92 0.94 0.96 0.98 160

65

70

75

80

85

σς

σ0

time (s)

stre

ss (

MP

a)


O 1( )

18

4.0 Numerical Examples in 3D

4.1 Four-point bending of viscoelastic beam

We first consider a four-point bending problem with a configuration shown in Figure 5. The beam ismade of isotropic viscoelastic material of Maxwell type. The material properties are selected as

, and , where is Young’s modulus, denotes Poisson’sratio, and is viscosity. The load applied to the cross heads is in the form of prescribed displacement

(65)

where is the amplitude and is the radial frequency. The load period is given by .According to (2), defines the ratio between the loading period and the intrinsic time scale forthe Maxwell viscoelastic model, where can be estimated by [15]:

(66)

where represents the norm of . Thus, the intrinsic time scale in this example is estimatedas and the load frequency is chosen as , i.e. 5 cycles per hour, so that

. The load amplitude is chosen as .

Numerical results for the maximum tensile strain component, , at the bottom surface in the midspan, as well as it’s temporal average, , are shown in Figure 6. As in the one-dimensionalcase, provides a good approximation of the non-oscillatory portion of the long-term solution.Similar observations can be made for the stress field shown in Figure 7. In Figure 8 and Figure 9, weshow the total strains and stresses recovered by postprocessing in the two time windows. It can be seenthat the leading order homogenized solution agrees well with the reference solution.

E 50GPa= ν 0.3= V 200GPa-hr= E νV

u2ς U0 0.1 ωtsin 1 t– 2tr⁄( )exp–+{ }=

U0 ω τ0 2π ω⁄=ς τ0 tr

tr

tr O Vijkl Lijkl⁄{ }=

• •tr 1.6 hr= ω 10π hr

1–=ς 2π ωtr⁄ 0.125= = U0 U0 0.4mm–=

e33ς

O 1( ) e330〈 〉

e330〈 〉

19

Figure 5. Configuration and FE mesh for the four-point bending problem

Figure 6. Reference solution versus global solution for the maximum strain

1

2

3 1

2

3

40

1.6

6.4

6.3525.4

0 3 6 9 12 15−1

0

1

2

3

4

5

6x 10

−3

e33ς

time (hr)

stra

in

O 1( )

20

Figure 7. Reference solution versus global solution for the maximum stress

Figure 8. Reference solutions versus homogenized solution for the maximum strain

0 3 6 9 12 15−40

0

40

80

120

160

σ33ς

time (hr)

stre

ss (

MP

a)

O 1( )

0.8 0.85 0.9 0.95 10.6

0.8

1

1.2

1.4

1.6

1.8x 10

−3

e33ς

e330

time (hr)

stra

in

O(ς0) smooth solutionO(ς0) order solution reference solution

14.8 14.85 14.9 14.95 154.2

4.4

4.6

4.8

5

5.2

5.4x 10

−3

e33ς

e330

time (hr)

stra

in


O 1( )

21

Figure 9. Reference solutions versus homogenized solution for the maximum stress

4.2 The nozzle flap problem for the power-law viscoplastic model

The finite element mesh of the half of the nozzle flap (due to symmetry) is shown in Figure 10. Theflap is subjected to an aerodynamic force which is simulated by a superposition of the uniform con-stant pressure and a sinusoidal loading with an amplitude equal to 10% of the value of the constantpressure and the load period of one cycle per minute, i.e., . The loading is applied onthe flat surface. We assume that the pin eyes are rigid and not allowed to rotate so that all degrees offreedom on the pin eye surfaces are fixed. The nozzle flap is made of type 316 stainless steel whichexhibits a viscoplastic behavior in room temperature. Material constants in the power-law viscoplasticmodel are obtained by fitting the creep test data provided in [3]. Material properties are summarizedbelow:

Type 316 stainless steel: , , , , ,, and

Figure 10. FE mesh for the nozzle flap

0.8 0.85 0.9 0.95 130

40

50

60

70

80

σ33ς

σ330

time (hr)

stre

ss (

MP

a)


14.8 14.85 14.9 14.95 15−10

0

10

20

30

40

50

σ33ς

σ330

time (hr)

stre

ss (

MP

a)


O 1( )

ω 120π hr 1–=

L 185GPa= ν 0.3= Y 95MPa= H 320GPa= Ĥ 0=a 2 8–×10 hr 1–= c 0.1=

1

2

3 1

2

3

fixed pin eyes

f2ς F0 0.1 ωtsin 1+( )=

uniform pressure on the flat surface

A

B2

13

22

Numerical results reveal that the maximum stress and strain components (in direction 2) occur on theinner surface of the pin eye A. Figure 11 and Figure 12 depict the history of maximum stresses andstrains as obtained with the reference solution (using very small time increment step) and the temporal homogenization (with postprocessing) solutions. In Figure 13, the response fields atthe end of loading, including displacement, stress and strain fields, are compared with the correspond-ing reference solutions. In all the cases considered the response fields agree well with the refer-ence solution.

Figure 11. Reference solution versus global homogenized solution at the pin eye A

Figure 12. Reference solution versus homogenized solution at the pin eye A

O 1( )O 1( )

O 1( )

9 9.2 9.4 9.6 9.8 100

0.2

0.4

0.6

0.8

1x 10

−3

time (hr)

stra

in

e22ς

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1x 10

−3

time (hr)

stra

in

e22ς

initial loading

…

e22ς O 1( ) e22

0〈 〉

0.084 0.086 0.088 0.09 0.092 0.094 0.096 0.098 0.17.4

7.6

7.8

8

8.2

8.4

8.6

8.8

9

9.2

9.4x 10

−4

e22ς

e220

time (hr)

stra

in


9.984 9.986 9.988 9.99 9.992 9.994 9.996 9.998 107.6

7.8

8

8.2

8.4

8.6

8.8

9

9.2

9.4

9.6x 10

−4

e22ς

e220

time (hr)

stra

in


e22ς O 1( ) e22

0

23

Figure 13. Reference solution versus homogenized solution for the nozzle flap problem

33

U2 VALUE-3.59E-01

-3.22E-01

-2.84E-01

-2.46E-01

-2.08E-01

-1.70E-01

-1.32E-01

-9.41E-02

-5.62E-02

-1.83E-02

+1.97E-02

u2ς

33

U2 VALUE-3.60E-01

-3.22E-01

-2.84E-01

-2.46E-01

-2.08E-01

-1.70E-01

-1.32E-01

-9.41E-02

-5.62E-02

-1.82E-02

+1.97E-02

u20

33

E22 VALUE-1.70E-04

-7.76E-05

+1.46E-05

+1.07E-04

+1.99E-04

+2.91E-04

+3.83E-04

+4.75E-04

+5.68E-04

+6.60E-04

+7.52E-04

e22ς

E22 VALUE-1.70E-04

-7.74E-05

+1.51E-05

+1.08E-04

+2.00E-04

+2.93E-04

+3.85E-04

+4.78E-04

+5.70E-04

+6.63E-04

+7.55E-04

e220

33

S22 VALUE-4.29E+01

-2.43E+01

-5.72E+00

+1.29E+01

+3.15E+01

+5.01E+01

+6.87E+01

+8.73E+01

+1.06E+02

+1.25E+02

+1.43E+02

σ22ς

S22 VALUE-4.29E+01

-2.43E+01

-5.73E+00

+1.29E+01

+3.14E+01

+5.00E+01

+6.86E+01

+8.72E+01

+1.06E+02

+1.24E+02

+1.43E+02

σ220

O 1( )

24

5.0 Concluding Remark

The asymptotic temporal homogenization formulation for viscoelastic and viscoplastic models hasbeen developed to resolve multiple temporal scales. The scaling parameter is defined as the ratiobetween the material intrinsic time and the frequency of load period. It is shown that a long-termresponse can be obtained by solving the temporally averaged leading order homogenized initial-boundary value problem along with the smooth portion of the external loading. The leading orderoscillatory behavior, which represents the deviation from the smooth solutions, is obtained by solvinga linear initial-boundary value problems for one period of load cycle. The global and local initial-boundary value problems for the linear Maxwell viscoelastic model are decoupled, whereas for the vis-coplastic model, local analysis has to be performed at each global time increment. In both cases, largetime increments can be used for the global problem with smooth loading, while the integration of thelocal initial-boundary value problem requires a significantly smaller time step, but only locally in asingle load period. The present temporal homogenization approach has been found to be in good agree-ment with the reference solution as long as the scaling parameter remains small.

In our future work the present temporal homogenization scheme will be extended to fatigue of homo-geneous solids. If successful, the methodology will be then generalized to fatigue analysis of heteroge-neous solids, which are characterized by multiple temporal and spatial scales.

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.

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3 Chaboche, J.L. and Rousselier, G., 1983. On the plastic and viscoplastic constitutive equation-PartII: Application of internal variable concept to the 316 stainless steel, Journal of Pressure VesselTechnology, 105, 159-164.

4 Chen, W., Fish, J., 2001. A dispersive model for Wave propagation in periodic composites basedon homogenization with multiple spatial and temporal scales. Journal of Applied Mechanics,68(2), 153-161.

5 Fish, J. and Chen, W., 2001. Uniformly Valid Multiple Spatial-Temporal Scale Modeling for WavePropagation in Heterogeneous Media. Mechanics of Composite Materials and Structures, 8, 81-99.

6 Fish, J. and Chen, W., and Nagai, G., 2001. Nonlocal dispersive model for wave propagation inheterogeneous media: One-Dimensional Case. International Journal for Numerical Methods in

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10 Odqvist, F.K.G., 1974. Mathematical Theory of Creep and Creep Rupture, Clarendon Press,Oxford.

11 Pierce, D, Shih, C.F., and Needleman, A., 1984. A tangent modulus method for rate dependent sol-ids, Computers and Structures, 18, 875-887.

12 Sanchez-Palencia, E., 1980. Non-Homogeneous Media and Vibration Theory, Springer-Verlag,Berlin.

13 Simo, J.C. and Hughes, T.J.R., 1998. Computational Inelasticity, Springer-Verlag New York, Inc.,New York.

14 Suquet, P.M., 1987. Elements of homogenization for inelastic solid mechanics. In: Homogeniza-tion Techniques for Composite Media, eds. Sanchez-Palencia, E. and Zaoui, A., Springer-Verlag,Berlin, pp193-278.

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