SERVE-Boundary Conditions
299
A SERVE is defined as the smallest microstructural domain with the following
characteristics:
1. Distribution functions of morphological parameters in the SERVE should be
statistically equivalent to those for the overall microstructure. This is classified
as a microstructure-based SERVE or M-SERVE [20–22].
2. Effective material properties and response functions for the SERVE should be
equivalent to those of the entire microstructure. This is classified as a propertybased SERVE or P-SERVE [20–22].
3. The SERVE should not depend on the location in the microstructure, e.g., A,B,C,
and D in Fig. 1c, or the applied loading. The size of the SERVE should be
optimal in terms of representing deformation mechanisms and overall response.
A smaller than necessary SERVE size may not include all possible deformation
mechanisms and lead to erroneous estimation of effective properties, while
a larger than required size can require exorbitant computational resources. A
variety of methods have been developed for estimating RVE sizes of random
media, where direct numerical simulations are performed with different boundary conditions [23–25]. Various statistical descriptors have been used to estimate
the RVE size to be sampled for obtaining the bulk material response [23, 26–29].
These descriptors include distributions of the local fiber volume fraction, nearest
neighbor distance, radial basis functions, and n-point correlation functions, e.g.,
in [30–34].
While methods of RVE estimation have generally focused only on the effective
microstructural domain and its volume, little consideration has been given to the
appropriateness of the boundary conditions. Three types of boundary conditions
have been conventionally applied on the RVE for solving the micromechanics
problem. These are:
1. Affine transformation-based displacement boundary condition (ATDBC),
expressed as:
u
A
i =
0
ij x j on RV E
Here, 0
ij is a constant applied far-field strain, and x j are the boundary coordinates
measured with respect to the geometrical centroid of the RVE. This condition
provides the lower or Voigt bound of the solution.
2. Uniform traction boundary condition (UTBC) given as:
T i = σ
0
ij n j on RV E
T i is the applied traction on the RVE boundary resulting in a constant stress σ 0
ij ,
where n j is the unit normal to the RVE boundary. This condition provides the
upper or Reuss bound of the solution.
299
A SERVE is defined as the smallest microstructural domain with the following
characteristics:
1. Distribution functions of morphological parameters in the SERVE should be
statistically equivalent to those for the overall microstructure. This is classified
as a microstructure-based SERVE or M-SERVE [20–22].
2. Effective material properties and response functions for the SERVE should be
equivalent to those of the entire microstructure. This is classified as a propertybased SERVE or P-SERVE [20–22].
3. The SERVE should not depend on the location in the microstructure, e.g., A,B,C,
and D in Fig. 1c, or the applied loading. The size of the SERVE should be
optimal in terms of representing deformation mechanisms and overall response.
A smaller than necessary SERVE size may not include all possible deformation
mechanisms and lead to erroneous estimation of effective properties, while
a larger than required size can require exorbitant computational resources. A
variety of methods have been developed for estimating RVE sizes of random
media, where direct numerical simulations are performed with different boundary conditions [23–25]. Various statistical descriptors have been used to estimate
the RVE size to be sampled for obtaining the bulk material response [23, 26–29].
These descriptors include distributions of the local fiber volume fraction, nearest
neighbor distance, radial basis functions, and n-point correlation functions, e.g.,
in [30–34].
While methods of RVE estimation have generally focused only on the effective
microstructural domain and its volume, little consideration has been given to the
appropriateness of the boundary conditions. Three types of boundary conditions
have been conventionally applied on the RVE for solving the micromechanics
problem. These are:
1. Affine transformation-based displacement boundary condition (ATDBC),
expressed as:
u
A
i =
0
ij x j on RV E
Here, 0
ij is a constant applied far-field strain, and x j are the boundary coordinates
measured with respect to the geometrical centroid of the RVE. This condition
provides the lower or Voigt bound of the solution.
2. Uniform traction boundary condition (UTBC) given as:
T i = σ
0
ij n j on RV E
T i is the applied traction on the RVE boundary resulting in a constant stress σ 0
ij ,
where n j is the unit normal to the RVE boundary. This condition provides the
upper or Reuss bound of the solution.