Fourier Variant Homogenization of the Heat Transfer Processes in Periodic Composites

The well-known parabolic Heat Transfer Equation is a simplest recognized description of phenomena related to the heat conductivity in solids with microstructure. However, it is a tool difficult to use due to the discontinuity of coefficients appearing here. The purpose of the paper is to reformulate this equation to the form that allows to represent solutions in the form of Fourier’s expansions. This equivalent re-formulation has the form of infinite number of equations with Fourier coefficients in expansion of the temperature field as the basic unknowns. The first term in Fourier representation, being an average temperature field, should satisfy the well-known parabolic heat conduction equation with Fourier coefficients as fields controlling average temperature behavior. The proposed description takes into account changes of the composite periodicity accompanying changes in the variable perpendicular to the surfaces separating components, concerning FGM - type materials and can be treated as the asymptotic version of Heat Transfer Equation obtained as a result of a certain limit passage where the cell size remains unchanged.

Visco-energetic solutions to some rate-independent systems in damage, delamination, and plasticity

This paper revolves around a newly introduced weak solvability concept for rate-independent systems, alternative to the notions of Energetic ([Formula: see text]) and Balanced Viscosity ([Formula: see text]) solutions. Visco-Energetic ([Formula: see text]) solutions have been recently obtained by passing to the time-continuous limit in a time-incremental scheme, akin to that for [Formula: see text] solutions, but perturbed by a “viscous” correction term, as in the case of [Formula: see text] solutions. However, for VE solutions this viscous correction is tuned by a fixed parameter. The resulting solution notion turns out to describe a kind of evolution in between Energetic and BV evolution. In this paper we aim to investigate the application of [Formula: see text] solutions to nonsmooth rate-independent processes in solid mechanics such as damage and plasticity at finite strains. We also address the limit passage, in the [Formula: see text] formulation, from an adhesive contact to a brittle delamination system. The analysis of these applications reveals the wide applicability of this solution concept, in particular to processes for which [Formula: see text] solutions are not available, and confirms its intermediate character between the [Formula: see text] and [Formula: see text] notions.

A gradient system with a wiggly energy and relaxed EDP-convergence

For gradient systems depending on a microstructure, it is desirable to derive a macroscopic gradient structure describing the effective behavior of the microscopic scale on the macroscopic evolution. We introduce a notion of evolutionary Gamma-convergence that relates the microscopic energy and the microscopic dissipation potential with their macroscopic limits via Gamma-convergence. This new notion generalizes the concept of EDP-convergence, which was introduced in [26], and is now called relaxed EDP-convergence. Both notions are based on De Giorgi’s energy-dissipation principle (EDP), however the special structure of the dissipation functional in terms of the primal and dual dissipation potential is, in general, not preserved under Gamma-convergence. By using suitable tiltings we study the kinetic relation directly and, thus, are able to derive a unique macroscopic dissipation potential. The wiggly-energy model of Abeyaratne-Chu-James (1996) serves as a prototypical example where this nontrivial limit passage can be fully analyzed.

Large deviations and gradient flows

In recent work we uncovered intriguing connections between Otto’s characterization of diffusion as an entropic gradient flow on the one hand and large-deviation principles describing the microscopic picture (Brownian motion) on the other. In this paper, we sketch this connection, show how it generalizes to a wider class of systems and comment on consequences and implications. Specifically, we connect macroscopic gradient flows with large-deviation principles, and point out the potential of a bigger picture emerging: we indicate that, in some non-equilibrium situations, entropies and thermodynamic free energies can be derived via large-deviation principles. The approach advocated here is different from the established hydrodynamic limit passage but extends a link that is well known in the equilibrium situation.