Dynamic Models for Yielding and Friction Hysteresis

In a system with hysteresis, the system response lags the excitation [16, 21] — for any excitation [10]. Hysteretic behavior is commonly depicted as loops in graphs (2D plots) of periodic (oscillatory) output vs. periodic input. In rate-dependent hysteresis, the size and shape of the hysteresis loop changes with the rate or frequency of the input. If the loop collapses to a function (e.g., a curved line) for any input (e.g., quasi-static), then the system is not hysteretic [10]. Hysteresis implies a non-linear relationship between inputs and outputs: differential equation models for hysteresis must be nonlinear; convolution models for hysteresis must be nonhomogeneous. Linear visco-elastic materials are rate-dependent but are not hysteretic because forces and displacements are proportional in the limit of quasi-static loading. This document describes members of the Duhem [9, 15, 17, 20] class of nonlinear ordinary differential equations