The linear theory of coupled gradient elasticity has been considered for hemitropic second gradient materials, specifically the positive definiteness of the strain and strain gradient energy density, which is assumed to be a quadratic form of the strain and of the second gradient of the displacement. The existence of the mixed, fifth-rank coupling term significantly complicates the problem. To obtain inequalities for the positive definiteness including the coupling term, a diagonalization in terms of block matrices is given, such that the potential energy density is obtained in an uncoupled quadratic form of a modified strain and the second gradient of displacement. Using orthonormal bases for the second-rank strain tensor and third-rank strain gradient tensor results in matrix representations for the modified fourth-rank and the sixth-rank tensors, such that Sylvester’s formula and eigenvalue criteria can be applied to yield conditions for positive definiteness. Both criteria result in the same constraints on the constitutive parameters. A comparison with results available in the literature was possible only for the special case that the coupling term vanishes. These coincide with our results.

对于半各向异性的第二梯度材料，已经考虑了耦合梯度弹性的线性理论，特别是应变和应变梯度能量密度的正定性，这被假定为应变和位移第二梯度的二次形式。混合的第五级耦合项的存在使问题变得更加复杂。为了获得包括耦合项的正定性的不等式，给出了基于块矩阵的对角线化，从而以修正的应变和第二位移梯度的非耦合二次形式获得势能密度。对第二级应变张量和第三级应变梯度张量使用正交基可以得到修改后的第四级和第六级张量的矩阵表示，从而可以将Sylvester公式和特征值标准应用于产生正定性的条件。这两个标准导致对本构参数的相同约束。仅在耦合项消失的特殊情况下，才有可能与文献中的结果进行比较。这些与我们的结果一致。