Dissipative Response Due to Discontinuous Strains in Bars of Unstable Elastic Material

摘要

Some elastic materials are capable of sustaining finite equilibrium deformations with discontinuous strains. Boundary-value problems for such unstable elastic materials often possess an infinity 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 elastic materials can be used to model macroscopic behavior similar to that associated with viscoplasticity.