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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
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(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
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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 ( ):
Equilibrium equation: on
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
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(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
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: (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)):
Equilibrium equation: on
Constitutive equation: on
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)
Boundary conditions: on
on
For the global initial-boundary value problem (using (46)):
Equilibrium equation: on
Constitutive equation: on
Initial condition: on (48)
Boundary conditions: on
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
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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⁄=
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(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⁄
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(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)
homogenized solutionreference solution
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(ς0) smooth solutionO(ς0) order solution reference solution
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)
O(ς0) smooth solutionO(ς0) order solution reference solution
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(ς0) smooth solutionO(ς0) order solution reference solution
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
homogenized solutionreference solution
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
homogenized solutionreference solution
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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