324
S. Ghosh et al.
6 Summary and Conclusions
This chapter discusses the development of the exterior statistics-based boundary
conditions or ESBCs, for statistically equivalent RVEs of elastic composites with
a nonuniform distribution of fibers [35, 38, 39]. Boundary conditions are complementary to the micromechanical SERVE domain for micromechanical simulations.
The ESBCs overcome deficiencies with conventionally applied boundary conditions, such as the affine transformation-based displacement boundary conditions
(ATDBCs) or periodic boundary conditions (PBCs) in evaluating homogenized
material properties. These deficiencies arise from overlooking the actual statistics
of heterogeneities in nonuniform microstructures, where the effect of the exterior
microstructure on the SERVE can be significant. The SERVE-ESBC model is
capable of resulting in an optimal SERVE domain due to the representation of
realistic boundary conditions. This results in significant computational efficiency.
Development of the ESBCs for the large exterior region needs characterization for
statistical analysis, which is efficiently accomplished for most material systems.
Validation results clearly show the significant advantage and potential of this
method, both in terms of the volume to be modeled for determining effective
mechanical properties and the number of iterations. In the case of high fiber volume
fractions, higher-order correlation functions such as S 3 , convoluted with the proper
three-body polarization tensor, may be helpful. In conclusion, the ESBCs are very
effective boundary conditions when modeling linear elastic heterogeneous materials
with nonuniform distributions of heterogeneities. Extension to nonlinear materials
will however require a different formulation due to the use of superposition
methods in this approach. Alternative approaches are in consideration for nonlinear
heterogeneous materials.
Appendix: Eshelby Tensors for Circular Cylindrical Fibers
For a cylindrical fiber of circular cross-section with a radius a and centroid at x I ,
let r = x − x I , x being a generic field point. Let ρ =
a
r with r = |x − x I | and
θ = (x − x I ). Then the interior and exterior Eshelby tensors S ij kl and ˆ
G ij kl
x, x I
have been given in [41] as:
S ij kl = {α}
T
{ ij kl }(θ ) and ˆ
G ij kl
x, x
I
= {β}
T (r){ ij kl }(θ )
(32)
The material-dependent vectors {α} and {β} are:
S. Ghosh et al.
6 Summary and Conclusions
This chapter discusses the development of the exterior statistics-based boundary
conditions or ESBCs, for statistically equivalent RVEs of elastic composites with
a nonuniform distribution of fibers [35, 38, 39]. Boundary conditions are complementary to the micromechanical SERVE domain for micromechanical simulations.
The ESBCs overcome deficiencies with conventionally applied boundary conditions, such as the affine transformation-based displacement boundary conditions
(ATDBCs) or periodic boundary conditions (PBCs) in evaluating homogenized
material properties. These deficiencies arise from overlooking the actual statistics
of heterogeneities in nonuniform microstructures, where the effect of the exterior
microstructure on the SERVE can be significant. The SERVE-ESBC model is
capable of resulting in an optimal SERVE domain due to the representation of
realistic boundary conditions. This results in significant computational efficiency.
Development of the ESBCs for the large exterior region needs characterization for
statistical analysis, which is efficiently accomplished for most material systems.
Validation results clearly show the significant advantage and potential of this
method, both in terms of the volume to be modeled for determining effective
mechanical properties and the number of iterations. In the case of high fiber volume
fractions, higher-order correlation functions such as S 3 , convoluted with the proper
three-body polarization tensor, may be helpful. In conclusion, the ESBCs are very
effective boundary conditions when modeling linear elastic heterogeneous materials
with nonuniform distributions of heterogeneities. Extension to nonlinear materials
will however require a different formulation due to the use of superposition
methods in this approach. Alternative approaches are in consideration for nonlinear
heterogeneous materials.
Appendix: Eshelby Tensors for Circular Cylindrical Fibers
For a cylindrical fiber of circular cross-section with a radius a and centroid at x I ,
let r = x − x I , x being a generic field point. Let ρ =
a
r with r = |x − x I | and
θ = (x − x I ). Then the interior and exterior Eshelby tensors S ij kl and ˆ
G ij kl
x, x I
have been given in [41] as:
S ij kl = {α}
T
{ ij kl }(θ ) and ˆ
G ij kl
x, x
I
= {β}
T (r){ ij kl }(θ )
(32)
The material-dependent vectors {α} and {β} are: