Valery P. Smyshlyaev

Multi-scale modeling of polycrystal plasticity: a workshop report

Abstract The workshop on multi-scale modeling of polycrystal plasticity was held on April 9–11, 1997 at the Institute for Mechanics and Materials at the University of California, San Diego in La Jolla, CA. This workshop addressed length-scale issues associated with developing a predictive capability in the modeling of the plastic deformation of polycrystals by the incorporation of more physically based information in the models. The goals of the workshop were to: (1) establish a model system that is well suited to the multi-scale modeling methodology; (2) explore a set of discrete simulation methods at the continuum-scale, meso-scale, micro-scale, and atomic-scale; and (3) identify critical links connecting the length scales which will allow information to be passed among scales and allow the end goal of predictive models at the continuum scale. This paper presents the technical summary of the topics covered by the speakers at the workshop and a discussion of critical issues at each length scale.

Homogenization of spectral problems in bounded domains with doubly high contrasts

Homogenization of a spectral problem in a bounded domain with a high contrast in both stiffness and density is considered. For a special critical scaling, two-scale asymptotic expansions for eigenvalues and eigenfunctions are constructed. Two-scale limit equations are derived and relate to certain non-standard self-adjoint operators. In particular they explicitly display the first two terms in the asymptotic expansion for the eigenvalues, with a surprising bound for the error of order

MEMORY EFFECT IN HOMOGENIZATION OF A VISCOELASTIC KELVIN-VOIGT MODEL WITH TIME-DEPENDENT COEFFICIENTS

\varepsilon^{5/4}

Homogenization of a thermo-chemo-viscoelastic Kelvin-Voigt model

proved.

Non-local homogenized limits for composite media with highly anisotropic periodic fibres

This paper is motivated by modeling the procedure of formation of a composite material constituted of solid fibers and of a solidifying matrix. The solidification process for the matrix depends on the temperature and on the reticulation rate which thereby influence the mechanical properties of the matrix. The mechanical properties are described by a viscoelastic medium equation of Kelvin-Voigt type with rapidly oscillating periodic coefficients depending on the temperature and the reticulation rate. That is modeled as an initial boundary value problem with time-dependent elasticity and viscosity tensors to account for the solidification, and the mechanical and/or thermal forcing. First we prove the existence and uniqueness of the solution for the problem and obtain a priori estimates. Then we derive the homogenized problem, characterize its coefficients including explicit memory terms, and prove that it admits a unique solution. Finally, we prove error bounds for the asymptotic solution, and establish some related regularity properties of the homogenized solution.

Propagation and localization of elastic waves in highly anisotropic periodic composites via two-scale homogenization

The paper is devoted to a model for the procedure of formation of a composite material constituted of solid fibers and of a solidifying matrix. The solidification process for the matrix depends on the temperature and on the degree of cure, which are used for the modeling of the mechanical properties of the matrix. Namely, the mechanical properties are described by Kelvin-Voigt viscoelastic equation with rapidly oscillating periodic coefficients depending on the temperature and the degree of cure. The latter are in turn solutions of a thermo-chemical problem with rapidly varying coefficients. We prove an error estimate for approximation of the viscoelastic problem by the same equation but with the coefficients depending on solution to the homogenized thermo-chemical problem. This estimate, in combination with our recent estimates for the viscoelastic (with time-dependent coefficients) and thermo-chemical homogenization problems, generates the overall error bound for the asymptotic solution to the full coup...