In this paper, we propose a generalized strain energy density function based on invariants of stretch tensor with arbitrary exponents. We employ polynomial, logarithmic and exponential functions of these invariants to develop the strain energy functions. We also study characteristics and applications of the proposed model for isotropy, transverse isotropy, orthotropy with a special focus on initial/residual stress symmetry. The proposed invariants generalize the existing strain energy potentials constructed with invariants of Cauchy stretch. We construct generalized strain energy functions for initial stress problems using initial stress symmetries. We obtain objectivity of strain energy for initially stressed transversely isotropic solids to derive the invariants and study resulting symmetry. By extending Dunford–Taylor integration based approach and tensor diagonalization approach, we obtain stress through differentiation of anisotropic scalar invariants with respect to a tensor. These approaches are usually applicable for derivative of isotropic tensor functions with respect to tensors. Using initial stress compatibility, we derive constraint equations for material parameters by evaluating the limits of Cauchy stress in the reference configuration. In order to apply the proposed model for initial stress problems, we further investigate bending and unbending of hyperelastic structures. We study bending of a rectangle to a cylinder and unbending of a cylinder to a rectangle in presence of initial/residual stress and observe both magnifying and moderating effects of initial stress for stress distribution and flexural characteristics. Using experimental data we substantiate the proposed model for seat foam, bovine pericardium and rabbit skin which represent compressible isotropic and incompressible orthotropic materials. For demonstration, we use both polynomial and exponential functions of these invariants. Furthermore, we develop a nonlinear finite element computational model for thermo-hyperelastic structure where thermal stress represents the initial stress. We corroborate this stress–strain data with the proposed model for initial stress symmetry and observe nice agreements.

在本文中，我们提出了基于具有任意指数的拉伸张量不变性的广义应变能密度函数。我们使用这些不变量的多项式，对数和指数函数来开发应变能函数。我们还研究了所提出的各向同性，横向各向同性，正交各向异性模型的特性和应用，特别关注初始/残余应力对称性。提出的不变量概括了用柯西拉伸的不变量构造的现有应变能势。我们使用初始应力对称性构造了用于初始应力问题的广义应变能函数。我们获得初始受力的横观各向同性固体的应变能客观性，以得出不变量并研究对称性。通过扩展基于Dunford–Taylor积分的方法和张量对角化方法，我们可以通过对张量进行各向异性标量不变量的区分来获得应力。这些方法通常适用于关于张量的各向同性张量函数的导数。使用初始应力兼容性，我们通过评估参考构型中柯西应力的极限来导出材料参数的约束方程式。为了将建议的模型应用于初始应力问题，我们进一步研究了超弹性结构的弯曲和未弯曲。我们研究了在存在初始/残余应力的情况下，矩形向圆柱体的弯曲以及圆柱体向矩形的未弯曲，并观察了初始应力对应力分布和弯曲特性的放大和缓和作用。使用实验数据，我们证实了座椅泡沫，牛心包和兔皮的提议模型，它们代表了可压缩的各向同性和不可压缩的正交异性材料。为了演示，我们使用这些不变量的多项式和指数函数。此外，我们为热-超弹性结构开发了非线性有限元计算模型，其中热应力代表初始应力。我们用建议的初始应力对称性模型来验证该应力-应变数据，并观察到很好的一致性。我们为热-超弹性结构开发了一个非线性有限元计算模型，其中热应力代表初始应力。我们用建议的初始应力对称性模型来验证该应力-应变数据，并观察到很好的一致性。我们为热-超弹性结构开发了一个非线性有限元计算模型，其中热应力代表初始应力。我们用建议的初始应力对称性模型来验证该应力-应变数据，并观察到很好的一致性。