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.

本文基于新型框架提出的应变能密度函数，对不可压缩的横向各向同性材料的力学行为进行了建模。就物理学而言，假设应变能密度取决于变形似乎是合理的。正确的柯西-格林张量是描述实体元素变形的基本工具。实体元素变形描述的常用方法之一是确定三个方向的拉伸以及其直角角度的变化。此处，描述实体元素的变形的想法是确定其纵向拉伸以及其三个面的面积变化。在这方面，根据固体（连续）单元的纵向拉伸和面积拉伸，提出了应变能密度函数。应用应变能密度形式的基本要求，获得了一个通用框架，以确保对横向各向同性材料进行建模的正定性和多凸性条件。在此框架的基础上，提出了适合于应变能密度函数的数学形式，以符合相应线性化理论和应变加劲条件以及零变形构型下的零应力和应变能密度的要求。在下文中，提出了具有很少参数的用于应变能函数的多凸简单数学形式。而且，这种提议的形式可以在材料参数和线性理论的三个独立的弹性模量之间建立合理的联系。为了评估所提出的应变能密度函数的性能，采用了具有纯均质变形的不可压缩横向材料的一些测试数据。发现在测试数据和从所提出的模型获得的结果之间存在很强的兼容性。