18 research outputs found
Singular Behavior of Electric Field of High Contrast Concentrated Composites
A heterogeneous medium of constituents with vastly different mechanical
properties, whose inhomogeneities are in close proximity to each other, is
considered. The gradient of the solution to the corresponding problem exhibits
singular behavior (blow up) with respect to the distance between
inhomogeneities. This paper introduces a concise procedure for capturing the
leading term of gradient's asymptotics precisely. This procedure is based on a
thorough study of the system's energy. The developed methodology allows for
straightforward generalization to heterogeneous media with a nonlinear
constitutive description
Explicit corrector in homogenization of monotone operators and its application to nonlinear dielectric elastomer composites
This paper concerns the rigorous periodic homogenization for a weakly coupled
electroelastic system of a nonlinear electrostatic equation with an elastic
equation enriched with electrostriction. Such coupling is employed to describe
dielectric elastomers or deformable (elastic) dielectrics. It is shown that the
effective response of the system consists of a homogeneous dielectric elastomer
described by a nonlinear weakly coupled system of PDEs whose coefficients
depend on the coefficients of the original heterogeneous material, the geometry
of the composite and the periodicity of the original microstructure. The
approach developed here for this nonlinear problem allows obtaining an explicit
corrector result for the homogenization of monotone operators with minimal
regularity assumptions. Two gradient estimates for elastic systems with
discontinuous coefficients are also obtained.Comment: We provide a new proof to extend the explicit first-order corrector
result in the first version of this paper. The new explicit corrector result
(cf. Theorem 1) holds globally and unifies the previous classical correct
results in homogenization of the divergence equation (both linear and
nonlinear). New references are added. Comments are welcome
Asymptotic analysis of an array of closely spaced absolutely conductive inclusions
We consider the conductivity problem in an array structure with
square closely spaced absolutely conductive inclusions of the high concentra-
tion, i.e. the concentration of inclusions is assumed to be close to 1. The
problem depends on two small parameters: ", the ratio of the period of the
micro-structure to the characteristic macroscopic size, and  , the ratio of the
thickness of the strips of the array structure and the period of the micro-
structure. The complete asymptotic expansion of the solution to problem is
constructed and justified