Variational dimension reduction in nonlinear elasticity: a Young measure approach

Starting form 3D elasticity, we deduce the variational limit of the string and of the membrane on the space of one and two-dimensional gradient Young measures, respectively. The physical requirement that the energy becomes infinite when the volume locally vanishes is taken into account in the string model. The rate at which the energy density blows up characterizes the effective domain of the limit energy. The limit problem uniquely determines the energy density of the thin structure.