CNA 2001 Summer School




Some open problems in solid mechanics: from phase transitions and fracture to heat conductivity and biomechanics

Lev Truskinovsky

ABSTRACT: The goal of this series of four lectures is to attract attention of mathematicians to several intriguing and challenging problems originating from the recent research in solid mechanics. Most of the problems group around cases of non-separability of length scales due to severe nonlinearity, associated with multi-stability of mechanical system at a micro-level. The main issues include: relation between local and global minima, compatibility of discrete and continuum descriptions, internal deficiency of pure mechanical picture and the necessity of thermodynamic approach. More specifically, we'll discuss the dichotomies between the formation of a single crack and multiple cracking, between the motion of a single interface and the coherent evolution of a microstructure, between acoustic radiation by a moving defect and the closely related issue of heat release, between the hysteresis in shape memory alloys and the ideology of energy minimization. The biomechanical part of the lectures will contain discussion of some recent work on active elasticity of muscles modeled as bi-stable systems living in somewhat counterintuitive Brownian environment. In every case, the lectures will contain a review of the known facts and the insights concerning the unsolved problem. The common nature of the problems will be emphasized.

Lev Truskinovsky
University of Minnesota
Aerospace Engineering & Mechanics