Universal Deformations in Anisotropic Nonlinear Elastic Solids

Universal deformations of an elastic solid are deformations that can be achieved for all possible strain-energy density functions and suitable boundary conditions. They play a central role in nonlinear elasticity and their classification has been mostly accomplished for isotropic solids following Ericksen's seminal work. Here, we address the same problem for transversely isotropic, orthotropic, and monoclinic solids. In this case, there are no general solutions unless universal material preferred directions are also specified. First, we show that for compressible transversely isotropic, orthotropic, and monoclinic solids universal deformations are homogeneous and that the material preferred directions are uniform. Second, for incompressible transversely isotropic, orthotropic, and monoclinic solids we derive the corresponding universality constraints. These are constraints that are imposed by equilibrium equations and the arbitrariness of the energy function. We show that these constraints include those of incompressible isotropic solids. Hence, we consider the known universal deformations for each of the six known families of universal deformations for isotropic solids and find the corresponding universal material preferred directions for transversely isotropic, orthotropic, and monoclinic solids. This work provides a systematic way to study analytically fiber-reinforced elastic solids.