In contemporary elasticity theory, the strain–energy function predominantly relies on the first invariant of the deformation tensor; a practice that has been influenced by models derived from rubber elasticity. However, this approach may not fully capture the complexities of materials exhibiting pronounced shear deformations, such as very soft biological tissues. Here, we explore the implications and potential benefits of constitutive models where the strain–energy function is exclusively a function of the second invariant, . By shifting the focus towards , we aim to address the limitations of current models in accurately describing shear-dominated behaviors and to provide a more comprehensive understanding of material responses, particularly for materials that do not conform to the assumptions underlying -centric theories. Through theoretical musings, data analysis, and automated model discovery, we investigate the feasibility of this approach and its consequences for predicting material behavior under various loading conditions. We show that the so-called “second-invariant materials” conforming to -only have interesting properties that are found in biological tissues and are fundamentally different from the traditional “first-invariant materials”.

中文翻译：

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我也是[公式省略]：一类新的仅基于第二不变量的超弹性各向同性不可压缩模型

在当代弹性理论中，应变-能量函数主要依赖于形变张量的第一个不变量；这种做法受到橡胶弹性模型的影响。然而，这种方法可能无法完全捕捉表现出明显剪切变形的材料的复杂性，例如非常柔软的生物组织。在这里，我们探讨了本构模型的含义和潜在好处，其中应变能量函数完全是第二个不变量 的函数。通过将焦点转向 ，我们的目标是解决当前模型在准确描述剪切主导行为方面的局限性，并提供对材料响应的更全面的理解，特别是对于不符合以 为中心的理论假设的材料。通过理论思考、数据分析和自动模型发现，我们研究了这种方法的可行性及其在预测各种负载条件下材料行为的结果。我们证明，符合的所谓“第二不变材料”仅具有在生物组织中发现的有趣特性，并且与传统的“第一不变材料”有根本的不同。