Mem-Spring Models Combined with Hybrid Dynamical System Approach to Represent Material Behavior

Material stress-strain behaviors and system load-displacement responses characterized with the origin-crossing input-ouput feature under repetitive loading and unloading conditions can be modeled efficiently by mem-spring models. Mem-springs are from a new family of state-space hysteresis constitutive and component models recently introduced to engineering mechanics by building on analogy with the recent concepts of the memristor, memcapacitor, and meminductor in electrical engineering. In this study, the modeling capability of mem-springs was explored, first by relaxing the continuity condition in the original definition to allow switching, and second, by leveraging a hybrid dynamical system approach as a framework for the switching. State variables were employed in a systematic manner, including the absement (i.e., the time integral of strain or displacement) or momentum (i.e., the time integral of stress or restoring force), and, when needed, internal variables that capture long- and/or short-term memory. Secant rather than tangent stiffness of the input-output pair was a signature of the data analysis. In this manner, mem-springs were constructed as parsimonious and physically meaningful hysteresis models at a constitutive and component level. Mem-springs incorporated into the simplest hybrid dynamical system can render rich hysteretic behaviors and responses, which have the potential to be very useful in engineering mechanics applications. Both rate-dependent and rate-independent hysteresis can be captured by mem-spring models. A partial model for concrete and qualitative models for rubber and the Mullins effect are presented as illustrative examples.