Preface

Our work on rate-independent systems was stimulated by our search for evolutionary material models for shape-memory alloys that are flexible enough to encompass nonlinear and nonconvex phenomena. There are two difficulties, namely (i) modeling the nonlinear material behavior and (ii) modeling the dissipative processes. Under the guidance of classical thermodynamics, most material models assume that the forces induced by dissipative effects are linear in the rates of the associated variables, such as in viscoelastic friction. However, there are certain effects, such as dry friction, plasticity, and fracture, for which the force depends on the direction of the rate but not on its magnitude. Such processes are called rate-independent and include the class of hysteresis operators. The latter may relate the material response (such as stress or magnetization) to the changes of the material state (such as strain or magnetic field, respectively). In general, there may be memory effects, and rateindependence means that any monotone temporal rescaling of the input leads to the same correspondingly rescaled output. A first mathematical formulation of such an invariance of the material response under temporal rescalings is contained in [601, § 99] in the context of hypoelastic materials. Of course, rate-independence is an idealization that is admissible only if the relevant processes of a system take place on a much slower time scale than the internal relaxation processes. In this book, we restrict our attention primarily to fully rate-independent systems, which can be justified in two ways. First, many mechanical processes can be well described by fully rate-independent models. Second, the restriction to such models allows for new mathematical tools for facilitating rigorous mathematical and numerical analysis that are not available for models with viscous dissipation. For instance, so far, the only existence result for the time-continuous evolutionary system for finite-strain elastoplasticity was obtained in the rate-independent case; see Section 4.2 and [374]. The description and discussion of rate-independent mechanical behavior has a relatively long history. Already in the eighteenth century, C.-A. Coulomb treated, besides his famous contribution to electricity, dry friction in [138], stating his Law of Friction: “Kinetic friction is independent of the sliding velocity”. In the early twentieth century, the theory of elastoplasticity became a major topic in mechanics