On the quasi-analytic treatment of hysteretic nonlinear response in elastic wave propagation

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Abstract
Microscopic features and their hysteretic behavior can be used to predict the macroscopic response of materials in dynamic experiments. Preisach modeling of hysteresis provides a refined procedure to obtain the stress–strain relation under arbitrary conditions, depending on the pressure history of the material. For hysteretic materials, the modulus is discontinuous at each stress–strain reversal which leads to difficulties in obtaining an analytic solution to the wave equation. Numerical implementation of the integral Preisach formulation is complicated as well. Under certain conditions an analytic expression of the modulus can be deduced from the Preisach model and an elementary description of elastic wave propagation in the presence of hysteresis can be obtained. This approach results in a second-order partial differential equation with discontinuous coefficients. Classical nonlinear representations used in acoustics can be found as limiting cases. The differential equation is solved in the frequency domain by application of Green’s function theory and perturbation methods. Limitations of this quasi-analytic approach are discussed in detail. Model examples are provided illustrating the influence of hysteresis on wave propagation and are compared to simulations derived from classical nonlinear theory. Special attention is given to the role of hysteresis in nonlinear attenuation. In addition guidance is provided for inverting a set of experimental data that fall within the validity region of this theory. This work will lead to a new type of NDT characterization of materials using their nonlinear response.