Demonstrating the power of extended Masing models for hysteresis through model equivalencies and numerical investigation

Abstract

The extended Masing model (EMM) is a powerful model for hysteresis that is theoretically sound, physically meaningful and computationally efficient. Any such model is defined by specifying a virgin loading curve and is implemented for arbitrary loadings using three simple hysteresis rules. A brief history of the development of these three switching rules is given. They can be accurately and efficiently implemented using a hybrid dynamical system approach where a state event algorithm is seamlessly combined with a time-stepping algorithm for numerical solution of the equations of motion when an EMM is used for the combined restoring and damping force. It is shown why each EMM is equivalent to an Iwan distributed-element model (DEM), which generalizes a multi-linear hysteresis system (a.k.a. Maxwell slip model) that consists of a finite number of elasto-plastic elements in parallel to an infinite number of such elements (countably many or a continuum of them). This model equivalency provides a physical basis for the choice of the three EMM rules. It is also noted that each EMM is also a classical Preisach model, a class of models that is well known in the mathematical literature on hysteresis. The extended Masing model is inherently for softening hysteresis but we show that a simple modification can be used to extend it to hardening hysteresis. It is noted that the EMM can also be extended to model deteriorating hysteresis. The hysteresis behavior of the EMM is further illustrated with examples of single-degree-of-freedom and two-degrees-of-freedom systems under dynamic excitation that use for the restoring force a specific EMM model whose defining virgin loading curve has a quite general parameterized form. It is shown that if the EMM model for the restoring force in a SDOF system that is subjected to earthquake excitation is replaced by a Bouc–Wen model with the same virgin loading curve, the hysteretic response changes dramatically and exhibits substantial drifting of the hysteresis loops. This behavior of the Bouc–Wen model is the result of a physical deficiency that was first noted four decades ago.

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