The Universal Program of Nonlinear Hyperelasticity

For a given class of materials, universal deformations are those that can be maintained in the absence of body forces by applying only boundary tractions.  Universal deformations play a crucial role in nonlinear elasticity. To date, their classification has been accomplished for homogeneous isotropic solids following Ericksen's seminal work, and  homogeneous anisotropic solids and inhomogeneous isotropic solids in our recent works. In this paper we study universal deformations for inhomogeneous anisotropic solids defined as materials whose energy function depends on position. We consider both compressible and incompressible transversely isotropic, orthotropic, and monoclinic solids. We show that the universality constraints---the constraints that are dictated by the equilibrium equations and the arbitrariness of the energy function---for inhomogeneous anisotropic solids include those of inhomogeneous isotropic and homogeneous anisotropic solids. For compressible solids, universal deformations are homogeneous and the material preferred directions are uniform. For each of the three classes of anisotropic solids we find the corresponding universal inhomogeneities---those inhomogeneities  that are consistent with the universality constraints. For incompressible anisotropic solids we find the universal inhomogeneities for each of the six known families of universal deformations.

This work provides a systematic approach to study analytically  functionally-graded fiber-reinforced elastic solids.