It is common practice to characterize the constitutive law of a material indirectly. This takes place by fitting a specific stress component, which is given as a combination of response functions or derivatives of the energy function of the material. Yet, it is possible to characterize each energy derivative of the material directly. Not only that but, through a few well-designed tests, getting a set of well-distributed data that defines the evolution of the energy derivatives in the invariant space is attainable, but not for all tests. Here, each test is portrayed as an equilibrium path on the surfaces (or volumes) of the derivative of the energy function. In the framework of the homothetic tests of hyperelastic isotropic materials, we propose the definition of energetic exhaustiveness. This definition relates to the capability of a test, via its analytic formulation according to a proper set of deformation invariants, to directly provide a closed-form solution for the derivatives of the energy function. In reaching this definition and retracing the Baker–Ericksen and the empirical inequalities, an alternative form of Baker–Ericksen inequalities is presented. We demonstrate that the unequal-biaxial test alone is energetically exhaustive and that it can provide (the same and more) information on the energy compared to the uniaxial, equi-biaxial, and pure shear tests. Unequal-biaxial experiments on three rubbers are presented. The outcomes of experiments contradict the empirical inequalities and seem to suggest new hierarchical empirical inequalities. Compact and nearly exact solutions are provided to perform and design tests at a constant magnitude of distortion, thus reaching a direct and comprehensive representation of the energy.

中文翻译：

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用于直接表征超弹性各向同性材料的能量形式的能量穷举性


通常的做法是间接描述材料的本构定律。这是通过拟合特定的应力分量来实现的，该分量是响应函数或材料能量函数的导数的组合。然而，可以直接表征材料的每个能量导数。不仅如此，通过一些精心设计的测试，可以获得一组分布均匀的数据来定义不变空间中能量导数的演变，但并非所有测试都是可以实现的。在这里，每个测试都被描述为能量函数导数的表面（或体积）上的平衡路径。在超弹性各向同性材料的同构测试框架内，我们提出了能量穷举的定义。该定义与测试的能力有关，通过根据一组适当的变形不变量进行解析公式，直接为能量函数的导数提供闭式解。在得出这个定义并追溯 Baker-Ericksen 和经验不等式时，提出了 Baker-Ericksen 不等式的另一种形式。我们证明，单独的不等双轴试验在能量上是详尽的，并且与单轴、等双轴和纯剪切试验相比，它可以提供有关能量的（相同和更多）信息。介绍了三种橡胶的不等双轴实验。实验结果与经验不平等相矛盾，似乎暗示了新的等级经验不平等。提供了紧凑且近乎精确的解决方案，用于在恒定的失真幅度下执行和设计测试，从而达到能量的直接和全面的表示。