On universal deformations of compressible Cauchy elastic solids reinforced by inextensible fibers

Abstract

Universal deformations are those that can be maintained in the absence of body forces and with boundary tractions alone, for all materials within a given constitutive class. We study the universal deformations of compressible isotropic Cauchy elastic solids reinforced by a single family of inextensible fibers. We consider straight fibers parallel to the Cartesian $Z$-axis in the reference configuration and derive the associated universality constraints, which depend explicitly on the geometry of the deformed fibers. We study universal deformations in two cases: (i) deformed fibers are straight lines, and (ii) deformed fibers have non-vanishing curvature. For case (i), we provide a complete classification. Assuming that at least one principal invariant of the right Cauchy–Green tensor is not constant, we show that the deformed fiber direction must be an eigenvector of the Finger tensor, and the invariants depend only on the fiber arc length parameter. The universality constraints reduce to geometric restrictions on the orthogonal surfaces, which must be planes, circular cylinders, or spheres. This gives one inhomogeneous universal deformation family: the non-isochoric Family $Z_1$ of combined bending and stretching deformations. In addition, Family $0Z$ consists of homogeneous deformations that respect the inextensibility constraint. We further show that if all principal invariants are constant and deformed fibers remain straight, then only homogeneous universal deformations are possible. For case (ii), when deformed fibers have non-vanishing curvature, the universality constraints become significantly more complex. We show that the three principal invariants are functionally dependent and that the binormal to the deformed fibers is an eigenvector of the Finger tensor. The existence of universal deformations in this case remains an open problem. In particular, we demonstrate that Family 5 universal deformations of incompressible elasticity, when restricted to satisfy the inextensibility constraint, are no longer universal in fiber-reinforced solids. Finally, we prove that the universal deformations of Cauchy and hyperelastic solids with the same fiber reinforcement coincide. Our results provide the first systematic classification of universal deformations for compressible isotropic fiber-reinforced solids and include a new inhomogeneous family. These solutions may serve as benchmark problems for numerical methods.