The inverse Hooke's law and complementary strain energy density has been examined in the context of the theory of coupled gradient elasticity for second gradient materials. To this end, it was assumed that the potential energy density is a quadratic form of the strain and of the second gradient of displacement. Existence of the coupling term significantly complicates the problem. To avoid this complication the equation for the potential energy density was transformed in order to present it as an uncoupled quadratic form of a modified strain and the second gradient of displacement or of the strain and a modified second gradient of displacement. These transformations, which is in essence a block matrix diagonalization, lead to a decoupling of strains and strain gradient in the potential energy density and makes it possible to determine tensorial relations for the compliance tensors of fourth-, fifth- , and sixth-rank. Both modifications result in the same compliance tensors and are valid for an arbitrary material symmetry class. In the case of hemitropic materials, the compliance tensors have the same symmetry and the same form as the stiffness tensors and are characterized by eight independent constants, namely the two classical isotropic constants, five constants in the strain gradient part and one constant in the coupling term. Explicit expressions for these eight parameters are obtained from the tensorial relations for the compliance tensors and are compared with the direct solution of a linear system for the compliance's. All three solutions are identical, what we consider as a verification of the presented results.

中文翻译：

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耦合应变梯度弹性中的逆胡克定律和互补应变能

已在第二梯度材料的耦合梯度弹性理论的背景下检查了逆虎克定律和互补应变能量密度。为此，假设势能密度是应变和位移的第二梯度的二次形式。耦合项的存在使问题显着复杂化。为了避免这种复杂性，势能密度方程被转换，以便将其表示为修正应变和第二位移梯度或应变和修正第二位移梯度的非耦合二次形式。这些变换本质上是块矩阵对角化，导致势能密度中应变和应变梯度的解耦，并使确定第四、第五和第六阶柔量张量的张量关系成为可能。两种修改都会产生相同的柔量张量，并且对任意材料对称类都有效。在半向性材料的情况下，柔量张量与刚度张量具有相同的对称性和相同的形式，并具有八个独立的常数，即两个经典各向同性常数，五个常数在应变梯度部分和一个常数在耦合学期。这八个参数的显式表达式是从柔量张量的张量关系中获得的，并与柔量线性系统的直接解进行比较。