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
email: trusk@aem.umn.edu
