A strain-softening bar with rehardening revisited

Abstract

We revisit, from the standpoint of the modern theory of phase transitions, the classical problem of stretching of a strain-softening bar with rehardening, considered earlier by Belytschko et al. We show that the classical solution, which is based on an empiric condition motivated by finite-element simulations, is not always consistent with the second law of thermodynamics. We use the model of a phase-transforming bar with a trilinear stress–strain relation and analytically consider the particular case where the stiffness of a new phase inclusion (i.e. ‘rehardening zone’) equals the stiffness of the initial phase. Instead of the empiric condition used earlier, we formulate the thermodynamic condition at the phase boundary in the most general form which respects the second law of thermodynamics. This allows us to construct a new solution, which describes the non-stationary dynamic processes in a strain-softening bar with rehardening being in agreement with the second law of thermodynamics. We also demonstrate that the solution always exists if the dissipation at the phase boundary is small enough. On the other hand, if the dissipation is strong enough, the proposed scenario of the phase transformation cannot be realized.