In this paper, a new approach for stress-softening of an isotropic, incompressible, hyperelastic and rectangular beam that undergoes cyclic bending-unbending deformation, is presented. Employing an exponential softening function, damage response of the hyperelastic beam due to cyclic finite bending is investigated. The stress-softening phenomenon occurs in elastomeric materials when they deform for the first time. Under the same deformation, the stress required in reloading is smaller than the initial loading stage. This is known as the Mullins effect. To verify the accuracy of the proposed solution, finite element analysis of the same problem is carried out. In this study, a principal stretch-based strain energy function i.e., Ogden model and an invariant-based function such as a newly introduced Exp–Exp model are used for all bending, unbending and re-bending procedures. The proposed method needs a much shorter time compared to FEM simulations. Thus, in design and optimization of the structures under bending that requires a large number of analyses, the proposed semi-analytical solution can be considered as an efficient tool for studying the effects of different material and geometrical parameters.

中文翻译：

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循环弯曲下超弹性材料应力软化的有限应变解析解

在本文中，提出了一种对经历循环弯曲-非弯曲变形的各向同性、不可压缩、超弹性和矩形梁进行应力软化的新方法。采用指数软化函数，研究了由于循环有限弯曲引起的超弹性梁的损伤响应。弹性材料在第一次变形时会出现应力软化现象。在相同变形下，再加载所需的应力小于初始加载阶段。这被称为穆林斯效应。为了验证所提出解决方案的准确性，对同一问题进行了有限元分析。在这项研究中，基于拉伸的主要应变能函数（即 Ogden 模型）和基于不变量的函数（例如新引入的 Exp-Exp 模型）用于所有弯曲，伸直和重新弯曲程序。与 FEM 模拟相比，所提出的方法需要更短的时间。因此，在需要大量分析的弯曲结构的设计和优化中，所提出的半解析解决方案可以被视为研究不同材料和几何参数影响的有效工具。