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.

Un-jamming due to energetic instability: statics to dynamics

“…the first term couples elastic and total strain together, while the second term is relaxing towards zero 33 -in case of finite T g , with rate 1 T g . The third term can be positive (or negative, e.g., at strain reversal 34 ), and thus works against (or with) the relaxation term.…”

“…On the other hand, at fixed confining pressure, P, a jammed system, at finite, but small T g (tapping) will develop to a state such that the elastic pressure, P = P − P T ≈ P , remains constant; relaxation of then corresponds to an increase of density, i.e., compaction. 34 After large strain, one has a positive product, e * ij̇ * ij > 0 , but at strain reversal the same term will be negative, for a while, until the elastic deviatoric strain reverts direction.…”

“…In order to describe solid-like granular matter on the macroscopic scale, concepts from elasto-and visco-plastic theories were used [15,16,21,[26][27][28][29][30][31][32][33][34][35] including instabilities, yield and failure [29,[36][37][38][39][40][41][42][43][44][45]. Recently, statistical mechanics/ physics concepts [19,46], helped to better understand the probabilities for plastic deformations [17-21, 34, 47-49], force network change/growth [50,51], stress-based meso ensembles [52], or stress-relaxation [11,[22][23][24].…”

“…In fact, encoding the transition in the elastic energy certainly affects the flowing state as well. The mechanism for yield is here related to elastic energy (irrespective whether the pressure or the shear stress is too large, or the density too small), as traditionally encompassed by concepts like plastic potentials, yield functions, or flow rules [21,30,32,34,106,122], see Fig. 2 and textbooks like Ref.…”

“…This could be confusing, as it is not related to typical plasticity formulations, and neither is it connected to concepts of plastic potentials or flow functions (see Refs. [32,34,122]). The term plastic strain rate is more appropriate for the other dissipative contributions discussed in the next two sections, on transient elasticity and granular media.…”