Arash Yavari

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 and by applying only boundary tractions. Universal deformations play a crucial role
in nonlinear elasticity. In this talk, we first discuss the same problem for homogeneous 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. We next extend Ericksen's analysis of
universal deformations to inhomogeneous compressible and incompressible isotropic and
anisotropic solids. We show that a necessary condition for the known universal deformations of
homogeneous isotropic solids to be universal for inhomogeneous solids is that inhomogeneities
respect the symmetries of the deformations. Symmetries of a deformation are encoded in the
symmetries of its pulled-back metric (or its right Cauchy-Green strain). We show that this necessary
condition is sufficient as well for all the known families of universal deformations except for Family
5. Finally we consider both compressible and incompressible inhomogeneous transversely isotropic,
orthotropic, and monoclinic solids. We show that the universality constraints for inhomogeneous
anisotropic solids include those of the corresponding 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 (position dependence of the
energy function) 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 analytically study functionally-graded
fiber-reinforced elastic solids.