For a given class of materials, universal displacements are those displacements that can be maintained for any member of the class by applying only boundary tractions. In this paper we study universal displacements in compressible anisotropic linear elastic solids reinforced by a family of inextensible fibers. For each symmetry class and for a uniform distribution of straight fibers respecting the corresponding symmetry we characterize the respective universal displacements. A goal of this paper is to investigate how an internal constraint affects the set of universal displacements. We have observed that other than the triclinic and cubic solids in the other five classes (a fiber-reinforced solid with straight fibers cannot be isotropic) the presence of inextensible fibers enlarges the set of universal displacements.

In linear elasticity, universal displacements for a given symmetry class are those displacements that can be maintained by only applying boundary tractions (no body forces) and for arbitrary elastic constants in the symmetry class. In a previous work, we showed that the larger the symmetry group, the larger the space of universal displacements. Here, we generalize these ideas to the case of anelasticity. In linear anelasticity, the total strain is additively decomposed into elastic strain and anelastic strain, often referred to as an eigenstrain. We show that the universality constraints (equilibrium equations and arbitrariness of the elastic constants) completely specify the universal elastic strains for each of the eight anisotropy symmetry classes. The corresponding universal eigenstrains are the set of solutions to a system of second-order linear PDEs that ensure compatibility of the total strains. We show that for three symmetry classes, namely triclinic, monoclinic, and trigonal, only compatible (impotent) eigenstrains are universal. For the remaining five classes universal eigenstrains (up to the impotent ones) are the set of solutions to a system of linear second-order PDEs with certain arbitrary forcing terms that depend on the symmetry class.

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.

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.

The elastic Ericksen's problem consists of finding deformations in isotropic hyperelastic solids that can be maintained for arbitrary strain-energy density functions. In the compressible case, Ericksen showed that only homogeneous deformations are possible. Here, we solve the anelastic version of the same problem, that is we determine both the deformations and the eigenstrains such that a solution to the anelastic problem exists for arbitrary strain-energy density functions. Anelasticity is described by finite eigenstrains. In a nonlinear solid, these eigenstrains can be modeled by a Riemannian material manifold whose metric depends on their distribution. In this framework, we show that the natural generalization of the concept of homogeneous deformations is the notion of covariantly homogeneous deformations — deformations with covariantly constant deformation gradients. We prove that these deformations are the only universal deformations and that they put severe restrictions on possible universal eigenstrains. We show that, in a simply-connected body, for any distribution of universal eigenstrains the material manifold is a symmetric Riemannian manifold and that in dimensions two and three the universal eigenstrains are zero-stress.