One Dimensional Nonlinear Wave Propagation in a Rate Independent Pinched Hysteretic Material

Theoretical and numerical studies on wave propagation in symmetric hysteretic material models such as Preisach–Mayergoyz (P-M), Hodgdon, Power Law, and Bouc-Wen, etc. by researchers like McCall, Guyer, Van Den Abeele, Johnson, Meurer, Nazarov, Radostin, Zhan, Ghodake, and others (1994–2021) demonstrated generation of only odd harmonics. In this study, nonlinear wave propagation in a 1D symmetric pinched rate-independent hysteretic material is discussed. A 1D space is discretized as a long chain of spring-mass elements by adding hysteretic elements in parallel. Hysteretic elements are modeled as a rate-independent pinched hysteresis model proposed by Biswas (2016), which is an improvement to Reid’s and Muravskii’s models. The system of equations is solved numerically. Results show that propagation of a single frequency wave generates only odd harmonics. Very nice evolving symmetric pinched hysteretic loops are observed for a Gaussian input pulse. Due to one-way two-wave mixing, sum and difference frequencies are observed along with the odd harmonics of the input frequencies. In mixing also symmetric pinched hysteretic curves are observed corresponding to input frequencies and their higher harmonics. Instead of minor loops as seen in Bouc-Wen, Two-States (Ghodake 2020), and Asymmetric Bouc-Wen (Ghodake 2021) models, small symmetric pinched loops are observed due to mixing.