A novel asymptotic approach to the theory of non-homogeneous anisotropic plates is suggested. For the problem of linear static deformations we consider solutions, which are slowly varying in the plane of the plate in comparison to the thickness direction. A small parameter is introduced in the general equations of the theory of elasticity. According to the procedure of asymptotic splitting, the principal terms of the series expansion of the solution are determined from the conditions of solvability for the minor terms. Three-dimensional conditions of compatibility make the analysis more efficient and straightforward. We obtain the system of equations of classical Kirchhoff’s plate theory, including the balance equations, compatibility conditions, elastic relations and kinematic relations between the displacements and strain measures. Subsequent analysis of the edge layer near the contour of the plate is required in order to satisfy the remaining boundary conditions of the three-dimensional problem. Matching of the asymptotic expansions of the solution in the edge layer and inside the domain provides four classical plate boundary conditions. Additional effects, like electromechanical coupling for piezoelectric plates, can easily be incorporated into the model due to the modular structure of the analysis. The results of the paper constitute a sound basis to the equations of the theory of classical plates with piezoelectric effects, and provide a trustworthy algorithm for computation of the stressed state in the three-dimensional problem. Numerical and analytical studies of a sample electromechanical problem demonstrate the asymptotic nature of the present theory.

中文翻译：

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非均匀压电板弹性三维问题中的渐近分裂

对非均匀各向异性板的理论提出了一种新的渐近方法。对于线性静态变形问题，我们考虑解决方案，与厚度方向相比，这些解决方案在板平面内缓慢变化。在弹性理论的一般方程中引入了一个小参数。根据渐近分裂的过程，解的级数展开的主项由次项的可解性条件确定。三维兼容性条件使分析更加高效和直接。我们得到了经典基尔霍夫板理论的方程组，包括位移和应变测度之间的平衡方程、相容性条件、弹性关系和运动学关系。为了满足三维问题的剩余边界条件，需要对板轮廓附近的边缘层进行后续分析。边缘层和域内解的渐近展开匹配提供了四种经典的板边界条件。由于分析的模块化结构，附加效应，如压电板的机电耦合，可以很容易地合并到模型中。该论文的结果为具有压电效应的经典板理论方程奠定了良好的基础，并为计算三维问题中的受力状态提供了可靠的算法。样本机电问题的数值和分析研究证明了当前理论的渐进性质。