The elastic metric: A review of elasticity with large distortions

The mechanical behavior of soft matter is characterized by large shape changes, often accompanied by small changes of elastic energy; non-linear elasticity, with large, inhomogeneous, and anisotropic distortions, that may evolve in time, proved to be an effective modeling tool for many of such soft materials. Here, we deal with the definition of an appropriate strain measure, called the elastic metric, upon which the elastic energy density can be defined. Moreover, we discuss two key issues about distortions: one deals with the notion of compatible distortions, that is, distortion fields yielding a global configuration without any change of the elastic energy; the other concerns the symmetries of the material responses. We also present few selected examples of non-linear, anisotropic, elastic response where distortions play a key role.