Universal Relations for Fiber-Reinforced Elastic Materials

Two general universal relations for both compressible and incompressible, isotropic elastic materials reinforced by a single field of inextensible fibers are derived as components of an axial vector condition. The constitutive equations comprise a system of six scalar equations linear in four constraint and material response functions. It is known from the manifold method applied to this linear system that, in general, at least two universal relations exist. Hence, depending on the rank of a coefficient matrix, the two axial vector component equations comprise the complete set of universal relations for the fiber-reinforced material. These equations are valid for all deformations, and they hold independently of the balance equations and boundary conditions. The results are illustrated for a homogeneous simple shear with triaxial stretch of a material having various single fiber arrangements. For the homogeneous shear and two kinds of nonhomogeneous deformations, the balance equations and/or boundary conditions lead to an additional equation relating the constraint reaction and material response functions. Use of this condition together with the scalar constitutive equations has the effect of reducing the rank of the coefficient matrix, and hence additional universal relations are known to exist. Thus, besides the aforementioned pair of general universal relations, for the homogeneous shear and two specific nonhomogeneous deformations studied here, we deduce from the constitutive equations additional new universal relations.