A unified model of hysteresis and long-time relaxation in heterogeneous materials

A physical model of stress-strain dynamics and long-time relaxation (slow time) in structured media is proposed. The model is based on the analysis of inter-grain contacts and the resulting surface force potential with a barrier. The result is a unified description of the classical acoustic nonlinearity, stress-strain hysteresis, and logarithmic relaxation law for sound velocity (and, hence, for the frequency of nonlinear resonance in samples of structured materials). Estimates of a characteristic volume of interacting contacts give close values for the variety of consolidated materials. For weak (linear) testing waves, the logarithmic relaxation occurs if a classical quadratic nonlinearity is added to the stress-strain relation.