Abstract

The Duhem model has been widely used to describe hysteretic behaviours, particularly those observed in piezoelectric materials. We explore the thermomechanical basis of the Duhem model. In a conservative system, the time rate of the dependent variable (e.g., stress) is related to the time rate of the independent variable (e.g., strain) through the second derivative of the Helmholtz free energy. To account for hysteresis in a nonconservative system, Duhem augmented the expression of the time rate of the dependent variable for a conservative system by adding a term featuring a piecewise continuous and differentiable function representing dissipation and leading to permanent changes in the state variables. We call this Duhem’s irreversibility function or simply Duhem’s function. As an example of application, we show how the Duhem model is equivalent to classical elastoplasticity with isotropic and kinematic hardening, with a judicious choice of Duhem’s function. To illustrate this example, we numerically simulate the cyclic loading of a nonlinear elastic material with linear hardening. This work shows how, after more than a century from its conception and without knowledge of the specific system (e.g., the decomposition into elastic and plastic strain), the Duhem model constitutes a viable phenomenological approach to the modelling of hysteresis.

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