The equilibrium equations and the traction boundary conditions are evaluated on the basis of the condition of the stationarity of the Lagrangian for coupled strain gradient elasticity. The quadratic form of strain energy can be written as a function of the strain and the second gradient of displacement and contains a fourth-, a fifth- and a sixth-order stiffness tensor \({\mathbb {C}}_4\), \({\mathbb {C}}_5\) and \({\mathbb {C}}_6\), respectively. Assuming invariance under rigid body motions the balance of linear and angular momentum is obtained. The uniqueness theorem (Kirchhoff) for the mixed boundary value problem is proved for the case of the coupled linear strain gradient elasticity (novel). To this end, the total potential energy is altered to be presented as an uncoupled quadratic form of the strain and the modified second gradient of displacement vector. Such a transformation leads to a decoupling of the equation of the potential energy density. The uniqueness of the solution is proved in the standard manner by considering the difference between two solutions.

平衡方程和牵引边界条件是基于耦合应变梯度弹性的拉格朗日方程的平稳性条件来评估的。应变能的二次形式可以写成应变和位移的第二梯度的函数，并包含四阶、五阶和六阶刚度张量\({\mathbb {C}}_4\)，\({\mathbb {C}}_5\)和\({\mathbb {C}}_6\)， 分别。假设在刚体运动下具有不变性，则获得线性动量和角动量的平衡。在耦合线性应变梯度弹性的情况下证明了混合边值问题的唯一性定理（Kirchhoff）（新颖）。为此，总势能被更改为应变的非耦合二次形式和位移矢量的修改后的第二梯度。这种转换导致势能密度方程的解耦。通过考虑两个解之间的差异，以标准方式证明解的唯一性。