Universal displacements in linear elasticity

In nonlinear elasticity, universal deformations are the deformations that exist for arbitrary strain-energy density functions and suitable tractions at the boundaries. Here, we discuss the equivalent problem for linear elasticity. We characterize the universal displacements of linear elasticity: those displacement fields that can be maintained by applying boundary tractions in the absence of body forces for any linear elastic solid in a given anisotropy class. We show that the universal displacements for compressible isotropic linear elastic solids are constant-divergence harmonic vector fields. We note that any divergence-free displacement field is a universal displacement for incompressible linear elastic solids. Further, we characterize the universal displacement fields for all the anisotropy classes, namely triclinic, monoclinic, tetragonal, trigonal, orthotropic, transversely isotropic, and cubic solids. As expected, universal displacements explicitly depend on the anisotropy class: the smaller the symmetry group, the smaller the space of universal displacements. In the extreme case of triclinic material where the symmetry group only contains the identity and minus identity, the only possible universal displacements are linear homogeneous functions.