An important problem that spans across many types of systems (e.g., mechanical and biological) is how to model the dynamics of joints or interfaces in built-up structures in such a way that the complex dynamic and energy-dissipative behavior that depends on microscale phenomena at the joint/interface is accurately captured, yet in a framework that is amenable to efficient computational analyses of the larger macroscale system of which the joint or interface is a (spatially) small part. Simulating joint behavior in finite element analysis by meshing the joint regions finely enough to capture relevant micromechanics is impractical for large-scale structural systems. A more practical approach is to devise constitutive models for the overall behavior of individual joints that accurately capture their nonlinear and energy-dissipative behavior and to locally incorporate the constitutive response into the otherwise often-linear structural model. Recent studies have successfully captured and simulated mechanical joint dynamics using computationally simple phenomenological models of combined elasticity and slip with associated friction and energy dissipation, known as Iwan models. In the present article, the author reviews the relationship, and in some cases equivalence, of one type of Iwan model to several other models of hysteretic behavior that have been used to simulate a wide range of physical phenomena. Specifically, it is shown that the “parallel-series” Iwan model has been referred to in other fields by different names, including “Maxwell resistive capacitor,” “Ishlinskii,” and “ordinary stop hysteron.” Given this, the author establishes the relationship of this Iwan model to several other hysteresis models, most significantly the classical Preisach model. Having established these relationships, it is then possible to extend analytical tools developed for a specific hysteresis model to all of the models with which it is related. Such analytical tools include experimental identification, inversion, and analysis of vibratory energy flow and dissipation. Numerical case studies of simple systems that include an Iwan-modeled joint illustrate these points.