A novel phenomenological viewpoint for transversely isotropic hyperelastic materials; a new strain energy density function

Abstract

In this paper, the mechanical behavior of incompressible transversely isotropic materials is modeled based on the strain energy density function proposed based on a novel framework. As far as physics is concerned, it seems plausible to assume that the strain energy density depends on the deformation. The right Cauchy–Green tensor is a fundamental tool for the deformation description of the solid elements. One of the common approaches for the deformation description of a solid element is to determining the stretches of three directions and the change in the angle of its right corners. Here, the idea of deformation description of a solid element is to determine its longitudinal stretches and the change in the area of its three faces. In this regard, the strain energy density function is proposed in terms of the longitudinal stretches and the areal stretches of a solid (continuum) element. Applying the essential requirements on the form of the strain energy density, a general framework is acquired to ensure the positive definiteness and polyconvexity conditions for modeling transversely isotropic materials. Based on this framework, the suitable mathematical forms for strain energy density function are suggested to match the requirements of the corresponding linearized theory and strain stiffening condition as well as the zero stress and strain energy density in the undeformed configuration. In the following, a polyconvex simple mathematical form for the strain energy function is proposed having few parameters. Moreover, this proposed form can establish a sensible connection between the material parameters and three independent elastic moduli of the linear theory. For evaluating the performance of the proposed strain energy density function, some test data of incompressible transversely materials with pure homogeneous deformations are employed. It is found that there exists strong compatibility between the test data and the results obtained from the proposed model.