Models of physical systems often contain implicitly defined surfaces separating different phases, materials, etc. which may evolve over time. Phase field and level set techniques can be used to represent such interfaces without explicitly parameterizing them. When the evolution of the free surfaces can be determined from ``Eulerian'' quantities (such as the curvature, flow velocity, etc.) phase field/level set representations have lead to very general robust algorithms and numerical codes to simulate these complex physical systems. In this talk I will discuss how such algorithms may be extended to model free surfaces, such as elastic membranes, whose evolution depends additionally upon ``Lagrangian'' quantities, e.g. strain.