The universal program of linear elasticity

Universal displacements are those displacements that can be maintained, in the absence of body forces, by applying only boundary tractions for any material in a given class of materials. Therefore, equilibrium equations must be satisfied for arbitrary elastic moduli for a given anisotropy class. These conditions can be expressed as a set of partial differential equations for the displacement field that we call universality constraints. The classification of universal displacements in homogeneous linear elasticity has been completed for all the eight anisotropy classes. Here, we extend our previous work by studying universal displacements in inhomogeneous anisotropic linear elasticity assuming that the directions of anisotropy are known. We show that universality constraints of inhomogeneous linear elasticity include those of homogeneous linear elasticity. For each class and for its known universal displacements, we find the most general inhomogeneous elastic moduli that are consistent with the universality constrains. It is known that the larger the symmetry group, the larger the space of universal displacements. We show that the larger the symmetry group, the more severe the universality constraints are on the inhomogeneities of the elastic moduli. In particular, we show that inhomogeneous isotropic and inhomogeneous cubic linear elastic solids do not admit universal displacements and we completely characterize the universal inhomogeneities for the other six anisotropy classes.

Automated summary

This study extends previous work by investigating universal displacements in inhomogeneous anisotropic linear elasticity, assuming known anisotropy directions. It shows that universality constraints in inhomogeneous linear elasticity include those in homogeneous linear elasticity. Additionally, the larger the symmetry group, the more stringent the universality constraints are on the inhomogeneities of the elastic moduli. It concludes that inhomogeneous isotropic and inhomogeneous cubic linear elastic solids do not allow for universal displacements, and characterizes the universal inhomogeneities for the other six anisotropy classes.