Dissipation potentials from elastic collapse

Abstract

Generalizing Maxwell's (Maxwell 1867 IV. Phil. Trans. R. Soc. Lond. 157 , 49–88 ( doi:10.1098/rstl.1867.0004 )) classical formula, this paper shows how the dissipation potentials for a dissipative system can be derived from the elastic potential of an elastic system undergoing continual failure and recovery. Hence, stored elastic energy gives way to dissipated elastic energy. This continuum-level response is attributed broadly to dissipative microscopic transitions over a multi-well potential energy landscape of a type studied in several previous works, dating from Prandtl's (Prandtl 1928 Ein Gedankenmodell zur kinetischen Theorie der festen Körper. ZAMM 8 , 85–106) model of plasticity. Such transitions are assumed to take place on a characteristic time scale T , with a nonlinear viscous response that becomes a plastic response for T → 0 . We consider both discrete mechanical systems and their continuum mechanical analogues, showing how the Reiner–Rivlin fluid arises from nonlinear isotropic elasticity. A brief discussion is given in the conclusions of the possible extensions to other dissipative processes.