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S. Ghosh et al.
mve
σ
mve
ij (x)δδ
mve
ij (x)dd = 0
( 4 )
subject to the affine transformation-based displacement boundary conditions
u
A
i (x
∞ ) =
0
ij x
∞
j on
∞
(5)
Here, δδ ij is the virtual strain, and x ∞
j are the coordinates of a point on the MVE
boundary ∞ relative to a reference point, such as the centroid of mve . Since
the MVE typically consists of a large population of heterogeneities, the solution of
the weak form (4) is computationally prohibitive. To avert this, only a statistically
equivalent subset of the MVE domain with explicit representation of dispersed
heterogeneities is identified as the SERVE for detailed micromechanical analyses. A
candidate SERVE is highlighted in Fig. 2a. This domain should be optimally small
to make it computationally tractable. Thus, the ratio of the length scales of the MVE
(L mve ) to that of the SERVE (L serve ) should be sufficiently large, i.e.,
L mve
L serve >> 1.
For reducing the MVE boundary value problem in equation (4) to that of the
SERVE, the MVE domain mve is partitioned into two complementary domains,
i.e., a SERVE domain serve and its exterior domain ext , such that mve =
ext ∪ serve . The effect of the exterior domain ext is manifested through equivalent
conditions on the SERVE boundary serve , adequately reflecting the interaction of
ext with serve . It should result in the same invariant strain energy for the SERVE
as for the entire MVE with the applied affine displacement conditions on ∞ . To
achieve this, the equation of principle of virtual work (4) is written as the sum of the
respective virtual work terms in the complementary domains of Fig. 2a as:
ext
σ
ext
ij (x) δδ
ext
ij (x) dd +
serve
σ
serve
ij
(x) δδ
serve
ij
(x) dd = 0
( 6 )
Applying the divergence theorem to the first term containing the integral over ext ,
incorporating equilibrium conditions in the absence of body forces, i.e., σ ij,j (x) =
0, and with ¯
mve
ij
= 0
ik on ∞ , the principle of virtual work (6) reduces to that of
the SERVE as:
serve
σ
serve
ij
(x) δδ
serve
ij
(x) dd −
serve
T
ext
i (x) δu
ext
i (x) dd = 0
( 7 )
T ext
i (x) is the traction on serve resulting from the stresses in the domain ext
exterior to the SERVE. The second term in equation (7) will drop out if an effective
displacement field can be prescribed on serve , since δu ext
i = 0 on serve . This can be
incorporated through the augmentation of the affine transformation-based boundary
displacement field u A
i (x serve ) = 0
il x serve by an additional perturbation term, which
represents the effects of heterogeneities in ext on serve . Since the solution process
will not involve an explicit numerical solution of the exterior domain problem in
ext , a special analytical solution that involves the statistics of the exterior domain
S. Ghosh et al.
mve
σ
mve
ij (x)δδ
mve
ij (x)dd = 0
( 4 )
subject to the affine transformation-based displacement boundary conditions
u
A
i (x
∞ ) =
0
ij x
∞
j on
∞
(5)
Here, δδ ij is the virtual strain, and x ∞
j are the coordinates of a point on the MVE
boundary ∞ relative to a reference point, such as the centroid of mve . Since
the MVE typically consists of a large population of heterogeneities, the solution of
the weak form (4) is computationally prohibitive. To avert this, only a statistically
equivalent subset of the MVE domain with explicit representation of dispersed
heterogeneities is identified as the SERVE for detailed micromechanical analyses. A
candidate SERVE is highlighted in Fig. 2a. This domain should be optimally small
to make it computationally tractable. Thus, the ratio of the length scales of the MVE
(L mve ) to that of the SERVE (L serve ) should be sufficiently large, i.e.,
L mve
L serve >> 1.
For reducing the MVE boundary value problem in equation (4) to that of the
SERVE, the MVE domain mve is partitioned into two complementary domains,
i.e., a SERVE domain serve and its exterior domain ext , such that mve =
ext ∪ serve . The effect of the exterior domain ext is manifested through equivalent
conditions on the SERVE boundary serve , adequately reflecting the interaction of
ext with serve . It should result in the same invariant strain energy for the SERVE
as for the entire MVE with the applied affine displacement conditions on ∞ . To
achieve this, the equation of principle of virtual work (4) is written as the sum of the
respective virtual work terms in the complementary domains of Fig. 2a as:
ext
σ
ext
ij (x) δδ
ext
ij (x) dd +
serve
σ
serve
ij
(x) δδ
serve
ij
(x) dd = 0
( 6 )
Applying the divergence theorem to the first term containing the integral over ext ,
incorporating equilibrium conditions in the absence of body forces, i.e., σ ij,j (x) =
0, and with ¯
mve
ij
= 0
ik on ∞ , the principle of virtual work (6) reduces to that of
the SERVE as:
serve
σ
serve
ij
(x) δδ
serve
ij
(x) dd −
serve
T
ext
i (x) δu
ext
i (x) dd = 0
( 7 )
T ext
i (x) is the traction on serve resulting from the stresses in the domain ext
exterior to the SERVE. The second term in equation (7) will drop out if an effective
displacement field can be prescribed on serve , since δu ext
i = 0 on serve . This can be
incorporated through the augmentation of the affine transformation-based boundary
displacement field u A
i (x serve ) = 0
il x serve by an additional perturbation term, which
represents the effects of heterogeneities in ext on serve . Since the solution process
will not involve an explicit numerical solution of the exterior domain problem in
ext , a special analytical solution that involves the statistics of the exterior domain