### The Universal Program of Nonlinear Hyperelasticity

Fluids and Materials Seminar

27th October 2022, 2:00 pm – 3:00 pm

Online talk, Zoom

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.

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