On the Convergence of the Energy, Stress Tensors, and Eigenvalues in Homogenization Problems of Elasticity

Abstract

In this paper we study the convergence of energy integrals, stress tensors and frequencies of free vibrations for non‐homogeneous and porous elastic bodies with a periodic structure of period ε as ε → 0. The results are based on the estimates for solutions of the boundary value problem of the elasticity system with rapidly oscillating periodic coefficients, obtained in [4], [11].

An Introduction to the Homogenization Modeling of Non-Newtonian and Electrokinetic Flows in Porous Media

“…The equivalence between the norms ∥D(w)∥ L q (Ω ε ) 9 and ∥∇w∥ L q (Ω ε ) 9 is due to Korn's inequality in porous media 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 [65] and references therein) For each w ∈ W 1,q 0 (Ω ε ) 3 , 1 < q < +∞, we have the inequality…”

Hashin–Shtrikman bounds of periodic linear elastic media with cubic symmetry

“…Tensor p true_ true_ is called the polarization tensor. The problem is linear in p true_ true_ and has a unique solution [14, 15] u true_ . Therefore, its deformation tensor e true_ true_ ( u true_ ) is a bounded linear function of p true_ true_ , denoted e true_ true_ ( u true_ ) = P i p true_ true_ .…”

A discontinuous Poisson–Boltzmann equation with interfacial jump: homogenisation and residual error estimate

“…Indeed, since with an arbitrary constant solves also ( 47 ), the zero average condition is sufficient (and necessary) to ensure the uniqueness of the solution, see e.g. [ 21 ]. Finally, the solution is smooth locally in .…”

“…The system ( 47 ) is essential to determine the efficient coefficient matrix of the macroscopic model averaged over . In fact, following the lines of [ 21 , 23 ], we shall establish an orthogonal decomposition of Helmholtz type for the oscillating coefficients .…”

“…For these purposes, we develop a variational technique based on orthogonal Helmholtz decomposition following the lines of [ 21 , 23 ]. In a periodic cell, we decompose oscillating coefficients (describing the electric permittivity) by using the nontrivial kernel in the space of vector valued periodic functions, which is represented by sums of constant and divergence free (and thus, skew symmetric) vector fields (cf.…”