On the dissipative response due to discontinuous strains in bars of unstable elastic material

Some elastic materials are capable of sustaining finite equilibrium deformations with discontinuous strains. Boundary-value problems for such “unstable” elastic materials often possess an infinite number of solutions, suggesting that the theory suffers from a constitutive deficiency. In the setting of the one-dimensional theory of bars in tension, the present paper explores the consequences of supplementing the theory with further constitutive information. This additional information pertains to the surface of strain discontinuity and consists of a ‘kinetic relation’ and a criterion for the “initiation” of such a surface. We show that the quasi-static response of the bar to a prescribed force history is then fully determined. In particular, we observe how unstable clastic materials can he used to model macroscopic behavior similar to that associated with viscoplasticity