On the scaling from statistical to representative volume element in thermoelasticity of random materials

Abstract

Under consideration is the finite-size scaling of effective thermoelas- tic properties of random microstructures from a Statistical Volume Element (SVE) to a Representative Volume Element (RVE), without invoking any pe- riodic structure assumptions, but only assuming the microstructure's statistics to be spatially homogeneous and ergodic. The SVE is set up on a mesoscale, i.e. any scale finite relative to the microstructural length scale. The Hill con- dition generalized to thermoelasticity dictates uniform Neumann and Dirichlet boundary conditions, which, with the help of two variational principles, lead to scale dependent hierarchies of mesoscale bounds on effective (RVE level) prop- erties: thermal expansion and stress coefficients, effectivestiffness, and specific heats. Due to the presence of a non-quadratic term in the energy formulas, the mesoscale bounds for the thermal expansion are more complicated than those for the stiffness tensor and the heat capacity. To quantitatively assess the scaling trend towards the RVE, the hierarchies are computed for a planar matrix-inclusion composite, with inclusions (of circular disk shape) located at points of a planar, hard-core Poisson point field. Overall, while the RVE is attained exactly on scales infinitely large relative to the microscale, depend- ing on the microstructural parameters, the random fluctuations in the SVE response may become very weak on scales an order of magnitude larger than the microscale, thus already approximating the RVE.

Full text

NETWORKS AND HETEROGENEOUS MEDIA Website: http://AIMsciences.org
c
American Institute of Mathematical Sciences
Volume 1, Number 2, June 2006 pp. 259–274
ON THE SCALING FROM STATISTICAL TO REPRESENTATIVE
VOLUME ELEMENT IN THERMOELASTICITY
OF RANDOM MATERIALS
Xiangdong Du
McGill University, Department of Mechanical Engineering
Montreal, QC H3A 2K6, CANADA
Martin Ostoja-Starzewski
University of Illinois at Urbana-Champaign
Department of Mechanical and Industrial Engineering
Urbana, IL 61801, USA
Abstract. Under consideration is the finite-size scaling of effective thermoelas-
tic properties of random microstructures from a Statistical Volume Element
(SVE) to a Representative Volume Element (RVE), without invoking any pe-
riodic structure assumptions, but only assuming the microstructure’s statistics
to be spatially homogeneous and ergodic. The SVE is set up on a mesoscale,
i.e. any scale finite relative to the microstructural length scale. The Hill con-
dition generalized to thermoelasticity dictates uniform Neumann and Dirichlet
boundary conditions, which, with the help of two variational principles, lead to
scale dependent hierarchies of mesoscale bounds on effective (RVE level) prop-
erties: thermal expansion and stress coefficients, effective stiffness, and specific
heats. Due to the presence of a non-quadratic term in the energy formulas,
the mesoscale bounds for the thermal expansion are more complicated than
those for the stiffness tensor and the heat capacity. To quantitatively assess
the scaling trend towards the RVE, the hierarchies are computed for a planar
matrix-inclusion composite, with inclusions (of circular disk shape) located at
points of a planar, hard-core Poisson point field. Overall, while the RVE is
attained exactly on scales infinitely large relative to the microscale, depend-
ing on the microstructural parameters, the random fluctuations in the SVE
response may become very weak on scales an order of magnitude larger than
the microscale, thus already approximating the RVE.
1. Introduction. The problem of effective properties of material micro-structures
has received considerable, and ever growing, attention over the past thirty years.
By effective (or overall, macroscopic) is meant the constitutive response assuming
the existence of a Representative Volume Element (RVE). In the case of spatial
disorder having no microstructural periodicity, the RVE concept implies that there
must be a scale (much) larger than the microscale (e.g., single heterogeneity size)
to ensure a homogenization limit in the sense of Hill [1].
In general, the RVE is involved in the so-called
d < L ≪Lmacro (1.1)
2000 Mathematics Subject Classification. 35R60, 74F05, 74Q05, 74Q20, 74A60.
Key words and phrases. random media, thermoelasticity, homogenization, scale effects, repre-
sentative volume element.
259

260 XIANGDONG DU AND MARTIN OSTOJA-STARZEWSKI
on which all of the deterministic continuum mechanics and physics hinges. Here,
Lis the RVE size, dis the microscale (average size of heterogeneity), and Lmacr o
is the macroscale on which we set up, say, boundary value problems of elasticity
theory. While the latter occurs for L≪Lmacro (i.e. the range of continuum
mechanics of homogeneous media), it is not clear what exactly is meant by d < L.
Indeed, sometimes d≪Lis implied. Without intending to sound critical, typical
prescriptions of continuum solid mechanics vaguely state that domains roughly from
10 up to 100 times larger than the heterogeneity size should be taken, e.g. [2].
Overall, most studies of effective properties simply assume that the RVE is attained
and do not specify or assess its size, e.g. [3-5].
Generally, the effective media studies usually follow one of four methods: (i)
rigorous energy bounds (e.g., of Hashin-Shtrikman type), (ii) perturbation methods
(e.g. [6]), (iii) effective medium models, and (iv) computational methods. Be-
ginning with the late 1980s, we have seen the introduction of a fifth approach to
determination of structure-property relations: bounds that explicitly involve (a) the
size of a mesoscale domain relative to the microscale, and (b) the type of boundary
conditions applied to this domain, e.g. [7-9]. That fifth approach was also driven
by the need to derive continuum random field models of heterogeneous materials
from actual microstructures [10,11].
Quantification of the tendency to asymptotically approach the RVE in the sense
of Hill [1] as the size of the mesoscale domain increases - that domain also being
called a Statistical Volume Element (SVE) - is of great practical interest. Random
microstructures studied in the vein of the fifth approach mentioned above progressed
from linear elastic to other types: physically and geometrically nonlinear elastic,
viscoelastic, permeability phenomena, elastic-plastic and rigid-plastic; e.g. [12-17].
In general, the size of the RVE depends on several characteristics of microstructures,
principal of which are: random microgeometries, mismatch in mechanical properties
of individual phases, and the actual physics involved. The present research extends
the above methodology to mesoscale bounds in a coupled field problem: thermal
expansion coefficients, effective stiffness, and specific heat of linear thermoelastic
microstructures; a short version was first given in [18].
Our analysis is based on variational principles of thermoelasticity, combined with
the assumption of statistical homogeneity and ergodicity of the random medium,
without any spatial periodicity assumptions. On that basis we derive hierarchies of
scale-dependent tensors, bounding the RVE responses from above (and below) under
uniform Dirichlet (respectively, uniform Neumann) boundary conditions. Using
computational mechanics, we then quantitatively demonstrate the trend of these
hierarchies to converge toward the RVE responses on the example of a two-phase
random matrix-inclusion composite in two dimensions (2-D).
2. Theoretical Fundamentals.
2.1. Random microstructure. We take the random material to be a set B={B(ω);
ω∈Ω}of realizations B(ω), defined over the sample space Ω. Thus, for an ele-
mentary event ω∈Ω, we have a realization B(ω), theoretically of infinite extent.
From it we isolate an arbitrary mesoscale specimen Bδ(ω), of finite size L=δd
and volume Vδ, bounded by a surface Sδ≡∂Vδ, Fig. 1(a). Here we employ a
non-dimensional mesoscale parameter defined as
δ=L
d.(2.1)

SCALING IN THERMOELASTICITY OF RANDOM MATERIALS 261
Ideally, δ→ ∞ leads to homogeneous properties of heterogeneous materials. How-
ever, depending on the actual microstructural parameters, can be anything from
a moderate to a very large number such that RVE is achieved within a required
accuracy. It is the assessment of such a trend that is of interest to us here.
δ
δ′
Figure 1. Planar model of a random composite material on a
mesoscale δ, formed from a non-overlapping distribution of circular
disks (Poisson hard-core process of disk centers); one deterministic
configuration Bδ(ω) at 40% nominal volume fraction of inclusions
is shown. (b) Domain Bδ(ω) of size L(= δd), and its partitioning
into four subdomains Bs
δ′(ω) of size L′=L/2.
We require all the statistics of material properties to be spatially homogeneous
and mean-ergodic. While the first property implies the invariance of all n-point
distributions with respect to arbitrary shifts in spatial domain, the latter property
means that the spatial average for any fixed realization B(ω) equals the ensemble
average at any fixed point in the material domain
F(ω)≡lim
V→∞
1
VZV
F(ω, x)dV =ZΩ
F(ω, x)dP (ω)≡ hF(x)i,(2.2)
In (2.2) we introduce the overbar ·to denote a spatial average, and the brackets
h·i to denote an ensemble average; Pis the probability distribution over Ω. More
rigorous statements of the required ergodic properties of the material was given by
Sab [8,9]. We assume the composite to be made of perfectly bonded phases, i.e.
there are no displacement discontinuities.
2.2. Constitutive laws in thermoelastic problems. Locally (i.e. in each phase),
the linear thermoelastic constitutive response is written as either
εij =Sijkl (ω, x)σkl +αij (ω, x)θ, (2.3)
or
σij =Cijkl (ω, x)εkl + Γij (ω, x)θ, (2.4)
where the thermal stress coefficient Γij is linked to the thermal expansion coefficient
αij by
Γij (ω, x) = −Cijkl (ω , x)αkl(ω, x),(2.5)
and θ(= T−T0) is a temperature change from the reference temperature T0;
Sijkl (ω, x) and Cij kl (ω, x) are the compliance and stiffness tensors, respectively.

262 XIANGDONG DU AND MARTIN OSTOJA-STARZEWSKI
[Hereafter we use the index and symbol notations for tensors interchangeably, as
the need arises.]
At the microscale Cis an inverse of S, and a nice relation like (2.5) holds.
However, when we go to the mesoscale δ > 1, this reciprocity is no longer assured,
and this topic will be discussed in Section 3 below. When we reach a medium
homogenized at the level of of RVE (δ→ ∞), the reciprocities are recovered [19],
and we have
S∗
ijkl =C∗−1
ijkl Γ∗
ij =−C∗
ijkl α∗
kl c∗
p−c∗
v=T0C∗
ijkl α∗
ij α∗
kl.(2.6)
Hereafter, the superscript ∗indicates an effective property of a homogenized mate-
rial (i.e. at the RVE, or macroscopic, level). The key question is: What mesoscale
δis needed to actually approximate the RVE level with a good accuracy?
2.3. Energy formulations for heterogeneous materials. In mechanics of elas-
tic and inelastic heterogeneous materials, energetic approaches are widely used,
especially when setting up variational principles used to bound the effective proper-
ties. Energy concepts appear also in the so-called Hill (or Hill-Mandel) condition [1]
σ:ε=σ:ε, which guarantees the equivalence of the mechanical and energetic ap-
proaches when determining the constitutive response, both at the macroscale and
the mesoscale levels. Thus, if the stress and strain tensors are decomposed into
their spatial average and fluctuating parts (σij =σij +σ′
ij ,εij =εij +ε′
ij ), the Hill
condition is equivalent to a statement regarding the boundary values
σ:ε=σ:ε⇐⇒ Z∂Vδ
(t−σ·n)·(u−ε·x)dS = 0.(2.7)
This shows that, when the boundary conditions satisfy the right hand side above,
the average of the product of the strain tensor and the stress tensor is equal to the
product of their averages. For the sake of general reference, the derivation of (2.7)
is given in the Appendix A.
With reference to the fundamentals of thermoelasticity theory, the Helmholtz
free energy density Aper unit volume at small temperature changes is
A=1
2Cijkl εij εkl + Γij εij θ−1
2cv
θ2
T0
.(2.8)
This gives stress and entropy
σij =∂A
∂εij T
S=−∂A
∂T εij
.
The potential energy is defined as
UP=1
V ZVδ
AdV −ZVδ
biuidV −ZSt
δ
tiuidV !,(2.9)
where St
δis the part of boundary Sδwith traction tiprescribed on it, while biis
the body force.
On the other hand, by a Legendre transformation, the Gibbs free energy is
G=−1
2Sijkl σij σkl −αij σij θ−1
2cp
θ2
T0
,(2.10)
and the constitutive laws are given by
εij =−∂G
∂σij T0
S=−∂G
∂T σij
.

SCALING IN THERMOELASTICITY OF RANDOM MATERIALS 263
The complementary energy is defined as
UC=−1
V ZVδ
GdV +ZSu
δ
tiuidV !,(2.11)
where Su
δis the part of Sδwith displacement uiprescribed on it.
2.4. Variational principles. Again with reference to [19], we recall that the vari-
ational principles in a thermoelastic problem can be stated in terms of the potential
and/or complementary energies. First, the variational principle in terms of the
potential energy states that of all the admissible displacement fields, those which
satisfy the equilibrium equations make the potential energy minimum. The minimum
potential energy principle can be applied to a displacement controlled (or essential,
Dirichlet) boundary value problem
UP d ≤g
UP d (2.12)
where g
UP d is calculated for an admissible strain field [20]. Here we insert the super-
script ‘d’ on purpose so as to indicate the application of a displacement boundary
condition.
The variational principle in terms of the complementary energy states that of all
the admissible stress fields, those which satisfy the compatibility equation of strain
make the complementary energy minimum. This can be applied to a traction con-
trolled (or natural, Neumann) boundary value problem as follows
UCt ≤g
UCt (2.13)
where g
UCt denotes an admissible stress field, and ‘t’ indicates a traction boundary
condition.
Also, under the same kind of boundary condition, either traction or displacement,
we have UPt =−UCt or UP d =−UCd, respectively.
3. Mesoscale Bounds on Thermal Effects.
3.1. Scale dependent hierarchy on the specific heat. Under the hypothesis
of spatially homogeneous and ergodic statistics of the material microstructure, we
select an arbitrary realization Bδ(ω) from B. Henceforth, Bδ(ω) is taken as a
square-shaped mesoscale domain in 2-D, and a cubic-shaped one in 3-D. We next
consider partitioning of Bδ(ω) into 2msubdomains (m= 2 and 3 in 2-D and 3-
D, respectively), each subdomain Bs
δ′(ω) being characterized by scale δ′=δ/2;
s= 1,2,...,2m; see Fig. 1(b). Obviously,
Bδ(ω) = ∪2m
s=1Bs
δ′(ω) (3.1)
First, under consideration are two uniform displacement boundary conditions
consistent with (2.7)1: one of an unrestricted type
ui(x) = ε0
ij xj∀x∈∂Bδ,(3.2)
and another of a restricted type applied to the partition (3.1)
ui(x) = ε0
ij xj∀x∈∂Bs
δ′, s = 1, ..., 4.(3.3)
The latter amounts to loading the entire domain Bδboth on its external boundary
and on the internal cross partitioning Bδinto four subdomains Bs
δ′. Given the
perfect bonding throughout the material, ε0=ε.

264 XIANGDONG DU AND MARTIN OSTOJA-STARZEWSKI
We now observe that the solution under the restricted condition (3.3) on all four
subdomains Bs
δ′is an admissible solution with respect to the unrestricted condition
(3.2) on Bδ, but not vice versa. Thus, in view of the principle of minimum potential
energy, the following inequality is obtained
UP d
δ≤
4
X
s=1
UP d
δ′(ω) (3.4)
The boundary condition (3.2) satisfies the Hill condition, so that, in view of (3.24)
given below, (3.4) can be written as follows
Vδ1
2Cd
ijkl,δ ε0
ij ε0
kl + Γd
ij,δε0
ij θ−1
2cd
v,δ
θ2
T0
≤
4
X
s=1
Vs
δ′1
2Cds
ijkl,δ′ε0
ij ε0
kl + Γds
ij,δ′ε0
ij θ−1
2cd
v,δ′
θ2
T0.
(3.5)
Here, and in the following, Cd
ijkl,δ and Cds
ijkl,δ′are the apparent stiffness tensors for
mesoscale domains Bδand Bs
δ′, respectively. [Thus, for example, Cds
ijkl,δ′is subscript
notation for the mesoscale tensor Cds
δ′in symbolic notation, with “, δ′” indicating
the mesoscale, not a partial differentiation.] These tensors are parametrized by a
particular realization ωfor a given B(ω), but we do not indicate that dependence
for the sake of clarity. In the inequality (3.5) the elastic deformations and thermal
effects are coupled. However, the case of zero temperature change θreduces (3.5) to
the well-known pure elasticity case hCd
δi ≤ hCd
δ′i. This then leads to a hierarchy of
mesoscale stiffness tensors under displacement boundary conditions bounding the
RVE response from above
C∗≡Cd
∞≤... ≤ hCd
δi ≤ hCd
δ′i ≤ ... ≤ hCd
1i ≡ CV∀δ′=δ/2,(3.6)
where CVis the Voigt bound. In view of (2.2), the ensemble averaging of Cd
∞has
been dropped.
On the other hand, setting the applied strain to zero (ε0
ij = 0), leads to an
inequality between ensemble averages on mesoscales δand δ′:hcd
v,δ i ≥ hcd
v,δ′i. By
extension to an arbitrary sequence of mesoscales, we arrive at the scale dependent
hierarchy of volume specific heat at constant volume
c∗
v≡cd
v,∞≥... ≥ hcd
v,δ i ≥ hcd
v,δ′i ≥ ... ≥cd
v,1≡cV
v∀δ′=δ/2,(3.7)
where cV
vis the Voigt bound.
We now turn to the reciprocal expression for the lower bound. We shall now load
the mesoscale domain through either one of two types of uniform traction boundary
conditions: one of an unrestricted type
ti(x) = σ0
ij nj∀x∈∂Bδ,(3.8)
and another of a restricted type applied to the partition (3.1)
ti(x) = σ0
ij nj∀x∈∂Bs
δ′, s = 1, ..., 4.(3.9)
We note that σ0=σ.
Considering the variational principle for complementary energy under both bound-
ary conditions (3.8) and (3.9), an inequality similar to (3.4) can be obtained
UCt
δ≤
4
X
s=1
UCt
δ′(ω),(3.10)

SCALING IN THERMOELASTICITY OF RANDOM MATERIALS 265
which, with reference to the Hill condition and (3.26), leads to
Vδ1
2St
ijkl,δ σ0
ij σ0
kl +αt
ij,δσ0
ij θ+1
2ct
p,δ
θ2
T0
≤
4
X
s=1
Vs
δ′1
2St
ijkl,δ σ0
ij σ0
kl +αt
ij,δ′σ0
ij θ+1
2ct
p,δ′
θ2
T0.
(3.11)
Again, for no thermal effects (θ= 0), we arrive at the inequality hSt
δi ≤ hSt
δ′i,
under the traction boundary condition, which leads to a well-known scale dependent
hierarchy on the effective (RVE level) compliance tensor
S∗≡St
∞≤... ≤ hSt
δi ≤ hSt
δ′i ≤ ... ≤ hSt
1i ≡ SR∀δ′=δ/2.(3.12)
where SRis the Reuss bound, and the same remark as that following (3.6) applies.
On the other hand, when σ0
ij = 0, we obtain an inequality between ensemble
averages on mesoscales δand δ′:hct
p,δ i ≤ hct
p,δ′i. Again by extension to an arbi-
trary sequence of mesoscales, this leads to the scale dependent hierarchy of pressure
specific heat under the traction boundary condition as follows
c∗
p≡ct
p,∞≤... ≤ hct
p,δ i ≤ hct
p,δ′i ≤ ... ≤ct
p,1≡cV
p∀δ′=δ/2.(3.13)
This is the hierarchy of upper bounds on the pressure specific heat under the traction
boundary condition (3.8). According to the hierarchies (3.11) and (3.12), we can see
that the constitutive relation (2.6)3does not hold unless the domain size reaches the
RVE size. Thus, combining (2.6), (3.7) and (3.13) we obtain a hierarchy including
both upper and lower bounds:
hct
p,1i − hT0Ct
ijkl,1αt
ij,1αt
kl,1i ≥ ... ≥ hct
p,δ′i − hT0Ct
ijkl,δ′αt
ij,δ′αt
kl,δ′i
≥ hct
p,δ i − hT0Ct
ijkl,δ αt
ij,δαt
kl,δ i ≥ ... ≥ hc∗
pi − hT0C∗
ijkl α∗
ij α∗
kl i
≥c∗
v≥... ≥ hcd
v,δ i ≥ hcd
v,δ′i ≥ ... ≥cd
v,1∀δ′=δ/2.
(3.14)
where Ct
ijkl,1= (St
ijkl,1)−1. This hierarchy shows the upper and lower bound char-
acter on the effective specific heat and the scaling trend to RVE of both mesoscale
bounds. In Section 4 a numerical simulation quantitatively demonstrates the con-
vergence trends.
3.2. Scale effects on the thermal expansion coefficient. We now turn to scale
dependent hierarchical bounds on effective thermal expansion coefficients. This case
is different from that of the stiffness and compliance tensors as well as the specific
heat capacities under constant pressure/displacement because of the presence of
a non-quadratic term in both energy formulations (2.8) and (2.10). Clearly, this
term couples the elastic and thermal effects. Therefore, the hierarchical bounds
on thermal expansion coefficients do not simply mimic the upper/lower bounds on
stiffness and compliance tensors under displacement/traction boundary conditions.
However, we may use the latter to obtain the former.
We now consider a two-phase composite material with properties of phase 1
being C(1)
ijkl , α(1)
ij , ... and those of phase 2 being C(2)
ijkl , α(2)
ij , ... . It was shown earlier
by Christensen [19] that, under the traction boundary condition (3.8) and by setting
Pklmn(St
mnij−S(2)
mnij ) = Iklij , the thermal expansion coefficient can be expressed as
αt
ij = (α(1)
kl −α(2)
kl )Pklmn (St
mnij −S(2)
mnij ) + α(2)
ij .(3.15)

266 XIANGDONG DU AND MARTIN OSTOJA-STARZEWSKI
Similarly, under the displacement boundary condition (3.2), and by setting
Pklmn(S(1)
mnij−S(2)
mnij ) = Iklij , for the thermal stress coefficient we have
Γd
ij = (Γ(1)
kl −Γ(2)
kl )Pklmn(Cd
mnij −C(2)
mnij ) + Γ(2)
ij .(3.16)
From the scale dependence of compliance and stiffness tensors (St
mnij and Cd
mnij )
we can next derive the scale dependence of thermal expansion coefficients since
Pklmn is scale independent by definition. Thus, considering each phase to be
isotropic and macroscopically isotropic, from (3.15) we have a simplified relation
αt
ij = (α(1) −α(2))St
nnij −δij /κ(2)
1/κ(1) −1/κ(2) +α(2)δij ,(3.17)
where St
mnij is the mesoscale compliance tensor under traction boundary condition,
and κ(1) denotes the bulk modulus of phase 2. Next, taking an ensemble average
over (3.17), and noting the scale dependent hierarchy (3.12), leads to two scale
dependent hierarchies for the isotropic part αt
δof αt
ij,δ:
(i) α(1) ≥α(2) ≥0 and κ(1) > κ(2):
α∗≥... ≥ hαt
δi ≥ hαt
δ′i ≥ ... ≥αt
1≡αR∀δ′=δ/2.(3.18)
(ii) α(1) ≥α(2) ≥0 and κ(1) < κ(2):
α∗≤... ≤ hαt
δi ≤ hαt
δ′i ≤ ... ≤αt
1≡αV∀δ′=δ/2.(3.19)
where αRis the Reuss-type bound on α∗.
Using an expression for Γd
ij entirely analogous to (3.17), we find
Γd
ij = (Γ(1) −Γ(2))Cd
nnij −δij κ(2)
κ(1) −κ(2) −Γ(2)δij .(3.20)
With the help of (3.6) we derive two hierarchical relations for the isotropic part Γd
δ
of Γd
ij,δ:
(i) 0 ≥Γ(1) ≥Γ(2) ≥0 and κ(1) > κ(2):
Γ∗≤... ≤ hΓd
δi ≤ hΓd
δ′i ≤ ... ≤Γd
1≡ΓV∀δ′=δ/2.(3.21)
(ii) 0 ≥Γ(1) ≥Γ(2) ≥0 and κ(1) < κ(2):
Γ∗≥... ≥ hΓd
δi ≥ hΓd
δ′i ≥ ... ≥Γd
1≡ΓV∀δ′=δ/2.(3.22)
where ΓVis the Voigt-type bound on Γ∗.
A closer inspection reveals that taking the special case κ(1) =κ(2) does not
present a singularity in (3.18-19) and (3.21-22). Note that, since the Hill condition is
satisfied on any mesoscale with either (3.2) or (3.8), we can use apparent properties
in the constitutive relation (2.6)2so as to arrive at upper and lower mesoscale
bounds and scaling towards the RVE. The numerical simulation results of Section
4 demonstrate the aforementioned bounds.
3.3. Legendre transformations. For a homogeneous continuum – or also (as
mentioned at the end of Section 2.2) for a material homogenized at the RVE level
- the complementary energy under a prescribed traction boundary condition and
the potential energy under a prescribed displacement boundary condition are the
negative of each other
UP=−UC(3.23)

SCALING IN THERMOELASTICITY OF RANDOM MATERIALS 267
We also have this classical Legendre transformation linking the Helmholtz and Gibbs
energies
A(εij , θ) = G(σij , θ) + σij εij .(3.24)
Here we simply write A,G,σij and εij. This, of course, is one pair out of all
four possible Legendre transformations in a quartet linking the internal energy,
Helmholtz free energy, enthalpy, and Gibbs energy when the temperature (θ) and
entropy (s) are kept as passive variables, e.g. [20]. In a random medium (i.e., at
the SVE level below the RVE), the quartet must be interpreted carefully according
as either displacement or traction boundary conditions are applied. Following [21],
we examine these possibilities.
First, assuming loading via (3.2), we must consider the spatial fluctuation terms
and thus, we write the Helmholtz and Gibbs energies in the following forms
Aδ(ε0
ij , θ, ω) = 1
2Cijkl,δ (ω)ε0
ij ε0
kl + Γij,δ(ω)ε0
ij θ−1
2cv,δ (ω)θ2
T0
,(3.25)
and
Gδ(σ0
ij , θ, ω) = −1
2Sijkl,δ (ω)σ0
ij σ0
kl −αij,δ (ω)σij θ−1
2cp,δ(ω)θ2
T0
.(3.26)
Upon ensemble averaging, (3.25-26) become
hAδ(ε0
ij , θ)i=1
2hCijkl,δ iε0
ij ε0
kl +hΓij,δ iε0
ij θ−1
2hcv,δ iθ2
T0
,(3.27)
and
hGδ(σ0
ij , θ)i=−1
2hSijkl,δ iσ0
ij σ0
kl − hαij,δ iσ0
ij θ−1
2hcp,δ iθ2
T0
.(3.28)
It follows that, under displacement boundary conditions (ε0
ij controlled), the
volume average stress is random (i.e., σij (ω)), so that
Gδ(σij (ω), θ) = Aδ(ε0
ij , θ, ω)−σij (ω)ε0
ij ,(3.29)
and hence, the ensemble average Gibbs energy on mesoscale δshould be calculated
from hAδ(ε0
ij , θ)iaccording to
Gδ(hσij i, θ) = hAδ(ε0
ij , θ)i − hσij iε0
ij ,(3.30)
rather than as hGδ(σij (ω), θ)i.
Similarly, under traction boundary conditions (σ0
ij controlled), the volume aver-
age strain is random (i.e., ε0
ij (ω)), so that
Aδ(εij (ω), θ) = Gδ(σ0
ij , θ, ω) + σ0
ij εij (ω),(3.31)
and hence, the ensemble average Helmholtz energy on mesoscale δshould be calcu-
lated from according to
hAδ(εij , θ)i=hGδ(σ0
ij , θ)i+σ0
ij hεij i,(3.32)
rather than as hAδ(εij (ω), θ)i. When the mesoscale SVE reaches the RVE, the
dependence on the type of boundary conditions vanishes, and we recover the classical
relation (3.24) for a homogeneous material.

268 XIANGDONG DU AND MARTIN OSTOJA-STARZEWSKI
E[GPa]ν α [1/K]ci
p[Jg/ K]
Aluminum matrix (phase 1) 71 0.3 2.4 10−50.900
Steel inclusions (phase 2) 211 0.3 1.2 10−50.452
Mismatches 3 1 0.5 0.5
Table 1. Material properties of steel and aluminum
δ2 4 8 16 32
number of specimens Bδ(ω) 160 81 49 32 16
Table 2. Number of specimens in the Monte Carlo simulation;
increasing the numbers of the second row had no noticeable effects
on the averages.
4. Numerical Simulations of Scaling Trends.
4.1. Methodology. In order to demonstrate the hierarchies of bounds derived in
Section 3, a series of numerical simulation is carried out on different mesoscales and
under different boundary conditions. Our attention is focused on a 2-D, two-phase
matrix-inclusion composite material with circular shaped inclusions, everywhere
perfectly bonded. A finite element mesh, finer than the single inclusion, is employed.
The inclusion and matrix phases have different Young’s moduli E, Poisson ratios
ν, thermal expansion coefficients αand specific heats cp. In general, this leads to a
characterization of the composite in terms of four mismatches (contrasts, or ratios
of the inclusion property to the matrix property): Ei/Em,νi/νm,αi/αm, and
ci
p/cm
p. Each realization is generated by a hard-core Poisson point field (Fig. 1(a)),
with the inter-point distance being 1.2 times the disk diameter (so as to avoid the
special problem of very narrow necks between the inclusions), and this process is
repeated in a Monte Carlo sense to simulate an ensemble. The material properties
and mismatches are listed in Table 1. Perfect bonding between the inclusion and
matrix phases is assumed.
To compute all the hierarchies of Section 3, we proceed in the following steps:
(a) Choose a specific mesoscale δand a nominal disk volume fraction;
(b) Generate a mesoscale realization Bδ(ω) of the disordered medium;
(c) For a specific Bδ(ω), compute the stiffness tensor Cd
δ(ω), the thermal expan-
sion stress coefficient Γd
δ(ω) and the constant-volume specific heat cv ,δ (ω) numeri-
cally, Fig. 2. Cd
δ(ω) is computed under a non-zero displacement boundary condition
(3.2) using various settings of ε0
ij ;Γd
δ(ω) and cv,δ (ω) are computed under ε0
ij = 0 and
the temperature change θ= 5Kup from T0= 293.15K. Then, ensemble averages
are taken to compute mesoscale bounds on the given mesoscale δ.
(d) For the same Bδ(ω), compute the compliance tensor St
δ(ω), the thermal
expansion strain coefficient αt
δ(ω) and the constant-pressure specific heat cp,δ(ω)
numerically, Fig. 2. St
δ(ω) is computed under a non-zero traction boundary con-
dition (3.8), using various settings of σ0
ij ;αt
δ(ω) and cp,δ (ω) are computed under
σ0
ij = 0 and the temperature change θ= 5Kup from T0= 293.15K.
(e) Proceed from (a) to (d) in a Monte Carlo sense over the sample space Ω, ac-
cording to the Table 2, so as to generate an ensemble of results at a fixed mesoscale
δ:{Cd
δ(ω),Γd
δ(ω), cd
v,δ (ω); ω∈Ω}and {Sd
δ(ω), αd
δ(ω), ct
p,δ (ω); ω∈Ω}. Then, en-
semble averages are taken to compute mesoscale bounds on the given mesoscale
δ.

SCALING IN THERMOELASTICITY OF RANDOM MATERIALS 269
(f) Change the mesoscale and repeat steps (a) to (e). We then compute the
first moments in function of δ(i.e. scale dependent bounds) for all the mesoscale
properties. Of course, higher moments may also be computed in the same fashion.
Figure 2. Numerical results on ε= 2 and ε= 32 under the
displacement boundary condition (3.2) at ε0
ij ; disks do not touch.
Figure 3. Numerical results on δ= 2 and δ= 32 under the
traction boundary condition (3.8) at σ0
ij ; disks do not touch.
Figures 2 and 3 demonstrate sample results on a very small mesoscale and a
rather large mesoscale. In Fig. 3(a) for δ= 2, the boundary deformation is very
pronounced. On the other hand, for δ= 32 in Fig. 3(b), the boundary deforma-
tion is hardly seen. This illustrates the scaling trend from SVE towards the RVE.
Since we deal with infinitesimal strains in linear thermoelasticity, in order to visual-
ize the differences between responses under displacement versus traction boundary
conditions, displacements in these figures are plotted with an amplification factor
of 200.
4.2. Numerical results.
4.2.1. Scaling of specific heats cvand cp.The hierarchy (3.7) states that the en-
semble average of the specific heat cvunder constant volume increases when the
mesoscale δincreases. This is brought out in Fig. 4(a). On the other hand, Fig
4(b) confirms that the ensemble average of cpdecreases with increasing δ. All these
results are combined in Fig. 5, where we use the notation ct
v=ct
p−T0C∗
ijkl α∗
ij α∗
kl .

270 XIANGDONG DU AND MARTIN OSTOJA-STARZEWSKI
For rather small mismatches in material properties chosen here, the numerical dif-
ferences between hcd
v,δ iand hct
p,δiare (very) small, but the methodology developed
here can be applied to study higher contrast materials.
0 8 16 24 32 40
2.86792
2.86794
2.86796
2.86798
2.86800
δ (L/d)
cv
δ x10−6 (J/m3K)
Steel Inclusion(40%)
in Aluminum Matrix
Property Mismatches:
Ei
__
Em=3, νi
__
νm=1, αi
__
αm=0.5, cp
i
__
cp
m=2.
0 8 16 24 32 40
2.87864
2.87868
2.87872
2.87876
δ (L/d)
cp
δ x10−6 (J/m3K)
Steel Inclusion(40%)
in Aluminum Matrix
Property Mismatches:
Ei
__
Em=3, νi
__
νm=1, αi
__
αm=0.5, cp
i
__
cp
m=2.
(a) (b)
Figure 4. Hierarchical trends for: (a) cd
vunder displacement
boundary condition; (b) ct
punder traction boundary condition.
0 8 16 24 32 40
2.8679
2.868
2.8681
2.8682
δ (L/d)
cv
δ x10−6 (J/m3K)
Steel Inclusion(40%)
in Aluminum Matrix
Property Mismatches:
Ei
__
Em=3, νi
__
νm=1, αi
__
αm=0.5, cp
i
__
cp
m=2.
〈cv
d〉
〈cv
t〉
Figure 5. Scaling of specific heat capacities from SVE towards
RVE; the same results were shown separately in Fig. 4.
4.2.2. Scaling of thermal expansion coefficients. Thermal expansion coefficients are
also scale dependent due to their relation with the elastic properties stated in (3.15)
and (3.16). We henceforth carry out numerical simulation to exhibit their scale
dependent trends as per (3.18-17) and (3.21-22). Figure 6 shows the results. With
reference to Table 1, we see that α(1) > α(2) ≥0 and κ(1) < κ(2) so that the
scale dependence trend of hαt
ij ifollows the hierarchy (3.19): it decreases with δ
increasing. On the other hand, considering the displacement boundary condition,
the composite material model has the mismatches 0 ≥Γ(1) > α(1) > α(2) ≥0 and
κ(1) > κ(2). Therefore, the scale dependent trend of hΓd
ij imust follow the hierarchy
(3.21). Figure 6(b) shows that the trace of hΓd
ij iincreases with increasing δ.
In order to obtain the convergence of hΓd
ij iand hαt
ij itowards the RVE, two sepa-
rate hierarchical trends may not be sufficient to assess the size of RVE - both (upper

SCALING IN THERMOELASTICITY OF RANDOM MATERIALS 271
and lower) mesoscale bounds are jointly necessary. Thus, we can apply equation
(2.5) to set up bounds (upper and lower) in terms of displacement and traction
boundary conditions. Figure 6(c) shows numerical result on the upper and lower
bounds for the trace of the stiffness tensor under displacement and traction bound-
ary conditions. Consequently, modifying the formula (2.5) to Γt
kl =−Ct
klij αt
ij , and
using the Legendre transformations of Section 3.3, the thermal stress coefficient cor-
responding to the traction boundary condition is obtained at any given mesoscale.
Figure 6(d) shows that both thermal stress coefficients under displacement and trac-
tion boundary conditions converge toward each other as δ→ ∞. At δ= 32, the
difference between Γd
ij and Γt
ij is less than 0.2%. Thus, δ= 32 may be adopted
as the RVE scale, the actual choice of that difference being, of course, a matter of
taste.
As stated in the derivation of (3.18-19) and (3.21-22), different mismatches in
composites may present different hierarchical trends. We therefore complete this
paper with a study of two different kinds of composites having these mismatches:
composite #1: Ei
Em= 3, νi
νm= 1, αi
αm= 2;
composite #2: Ei
Em= 3, νi
νm= 1, αi
αm= 0.2.
In Fig. 7, these two ‘opposite’ cases of mismatches are compared. In particular,
Fig. 7(a) shows that the composite displays a hierarchical trend of hαt
ij iaccording
to (3.18), while Fig. 7(b) presents the trend according to (3.19). Furthermore, Fig.
7(c) shows that hΓd
ij iincreases with the increase of δ, as given in (3.22). With
different mismatches of materials, Fig 7(d) shows the trend stated in (3.21). Once
again, employing the Legendre transformations of Section 3.3, we compute the upper
and lower bounds presented in Figs. 7(e) and (f). In the case of the composite #1,
the traction boundary condition provides the upper bound, while the displacement
boundary condition provides the lower bound. In contradistinction to this, in the
case of the composite #2, the traction boundary condition gives the lower bound
while the displacement boundary condition gives the upper bound. In both cases,
we clearly see that both bounds converge toward each other with the mesoscale
increasing.
5. Conclusions. Results of this paper may be summarized as follows:
i) The Hill condition is employed so as to develop the equivalence between the
energetic and mechanical formulations of constitutive laws of thermoelastic random
heterogeneous materials at arbitrary mesoscale.
ii) Using the potential and complementary energy principles, we obtain scale
dependent hierarchies for the upper and lower bounds on the specific heat capacity
at the RVE level. Application of displacement (respectively, traction) boundary
condition results in a hierarchy of lower bounds on cd
v(upper bounds on ct
p). With
the increasing mesoscale, both bounds converge to one another in the sense that
c∗
p−c∗
v=T0C∗
ijkl α∗
ij α∗
kl is attained.
iii) Due to the presence of a non-quadratic term in the energy formulas, the
mesoscale bounds on the thermal expansion are more complicated than those on
the stiffness tensor and the heat capacity. In general, the upper and lower bounds
correspond to loading of mesoscale domains by displacement and traction boundary
conditions. Depending on the property mismatches, the upper and lower bounds
can be provided either by displacement boundary condition or traction boundary
conditions.

272 XIANGDONG DU AND MARTIN OSTOJA-STARZEWSKI
0 8 16 24 32 40
1.83
1.84
1.85
1.86
1.87
1.88
δ (L/d)
Tr(αij)/2 x10−5 /oC
Steel Inclusion(40%)
in Aluminum Matrix
Property Mismatches:
Ei
__
Em=3, νi
__
νm=1, αi
__
αm=0.5, cp
i
__
cp
m=2.
← 〈αt〉
0 8 16 24 32 40
−2.75
−2.74
−2.73
−2.72
−2.71
−2.7
δ (L/d)
Tr(Γij)/2 x106 Pa/oC
Steel Inclusion(40%)
in Aluminum Matrix
Property Mismatches:
Ei
__
Em=3, νi
__
νm=1, αi
__
αm=0.5, cp
i
__
cp
m=2.
← 〈Γd〉
(a) (b)
0 8 16 24 32 40
108
110
112
114
116
118
δ (L/d)
C11+C22)/2 GPa/oC
Steel Inclusion(40%)
in Aluminum Matrix
Property Mismatches:
Ei
__
Em=3, νi
__
νm=1, αi
__
αm=0.5, cp
i
__
cp
m=2.
〈Cd〉
(〈St〉)−1
0 8 16 24 32 40
−2.75
−2.73
−2.71
−2.69
−2.67
−2.65
δ (L/d)
Tr(Γij)/2 x106 Pa/oC
Steel Inclusion(40%)
in Aluminum Matrix
Property Mismatches:
Ei
__
Em=3, νi
__
νm=1, αi
__
αm=0.5, cp
i
__
cp
m=2.
〈Γd〉
〈Γt〉
(c) (d)
Figure 6. Numerical results for the aluminum-steel composite of
Fig. 1. (a) scale effect on thermal strain coefficient under zero
traction boundary condition; (b) scale effect on thermal stress co-
efficient under zero displacement boundary condition; (c) mesoscale
bounds (upper and lower) on stiffness; (d) mesoscale bounds (upper
and lower) on thermal stress coefficient.
iv) In the proposed scale-dependent homogenization of thermoelastic properties,
the RVE properties are attained approximately on a finite scale with whatever
desired accuracy. In other words, the condition d≪Lin the separation of scales
(1.1) is not always necessary, and it is possible that the RVE is attained simply at
d < L, i.e., only an order of magnitude smaller than L.
Appendix A. The Hill conditon. The Hill condition (2.7)1is a prerequisite
for setting up the equivalence between the mechanical and energetic approaches to
constitutive laws. In elasticity, along with the average strain/stress theorems, it
provides a foundation for mesoscale bounds on effective moduli of heterogeneous
materials. To derive the Hill condition one starts from the observation that, for
any volume V, the difference between the volume average of strain energy and the
strain energy calculated from the volume average strss and strain fields (where we
drop the factor 1/2 for simplicity) is
σij εij −σij εij =σ′
ij ε′
ij (A.1)
In (A.1) σ′
ij and ε′
ij are the fluctuations in stress and strain fields, respectively.
Next, noting that εij =u(i,j), we have

SCALING IN THERMOELASTICITY OF RANDOM MATERIALS 273
0 8 16 24 32 40
1.72
1.74
1.76
1.78
δ (L/d)
Tr(αij)/2 x10−5 /oC
Inclusion(40%) in a Matrix
Property Mismatches:
Ei
__
Em=3, νi
__
νm=1, αi
__
αm=2, cp
i
__
cp
m=2.
← 〈αt〉
0 8 16 24 32 40
1.48
1.5
1.52
1.54
1.56
1.58
δ (L/d)
Tr(αij)/2 x10−5 /oC
Inclusion(40%) in a Matrix
Property Mismatches:
Ei
__
Em=3, νi
__
νm=1, αi
__
αm= ,
1
__
5
cp
i
__
cp
m=2.
← 〈αt〉
(a) (b)
0 8 16 24 32 40
−2.9
−2.8
−2.7
−2.6
−2.5
δ (L/d)
Tr(Γij)/2 x106 Pa/oC
Inclusion(40%) in a Matrix
Property Mismatches:
Ei
__
Em=3, νi
__
νm=1, αi
__
αm=2, cp
i
__
cp
m=2.
← 〈Γd〉
0 8 16 24 32 40
−2.21
−2.2
−2.19
−2.18
δ (L/d)
Tr(Γij)/2 x106 Pa/oC
Inclusion(40%) in a Matrix
Property Mismatches:
Ei
__
Em=3, νi
__
νm=1, αi
__
αm= ,
1
__
5
cp
i
__
cp
m=2.
← 〈Γd〉
(c) (d)
0 8 16 24 32 40
−2.85
−2.75
−2.65
−2.55
−2.45
δ (L/d)
Tr(Γij)/2 x106 Pa/oC
Inclusion(40%) in a Matrix
Property Mismatches:
Ei
__
Em=3, νi
__
νm=1, αi
__
αm=2, cp
i
__
cp
m=2.
〈Γd〉
〈Γt〉
0 8 16 24 32 40
−2.25
−2.23
−2.21
−2.19
−2.17
δ (L/d)
Tr(Γij)/2 x106 Pa/oC
Inclusion(40%) in a Matrix
Property Mismatches:
Ei
__
Em=3, νi
__
νm=1, αi
__
αm= ,
1
__
5
cp
i
__
cp
m=2.
〈Γd〉
〈Γt〉
(e) (f)
Figure 7. Hierarchies of upper and lower bounds for two compos-
ite models (#1 and #2) of Section 4.2.2: (a) and (b) scale effects on
thermal strain coefficients under zero traction boundary condition;
(c) and (d) scale effects on thermal stress coefficients under zero
displacement boundary condition; (e) and (f) mesoscale bounds
(upper and lower) on thermal stress coefficients.
σ′
ij ε′
ij =1
VZV
(σij −σij )(εij −εij)dV
=1
VZVn[(σij −σij)(ui−ui)],j −(σij,j −σij,j )(ui−ui)odV
=1
VZ∂V
[(σij −σij )(ui−ui)] njdS
=1
VZ∂V
[(ti−σij nj)(ui−εijxj)] dS.
(A.2)

274 XIANGDONG DU AND MARTIN OSTOJA-STARZEWSKI
In (A.2) the second line follows from the integration by parts. Next, the second term
in the the second part of the integrand may be dropped assuming the absence of
body and inertia forces. The third line follows from the Green-Gauss theorem, and
the fourth from the Cauchy’s stress concept and the property of affine displacement
fields.
Acknowledgements. The work reported herein has been made possible through
support by the NSERC and the Canada Research Chairs program.
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Received November 2005; revised February 2006.
E-mail address:xdu1@po-box.mcgill.ca
E-mail address:martinos@uiuc.edu