Two-Dimensional Model of a Plate, Made of Material with the General Anisotropy

A new two-dimensional linear model of the second order accuracy describing deformations of an anisotropic heterogeneous in the thickness direction plate is proposed. The case of the general anisotropy with 21 elastic modules is studied. The asymptotic expansions of solutions of 3D equations of the theory of elasticity in power series with small thickness parameter are used. The zero asymptotic approximation was constructed earlier and it is similar to the Kirchhoff–Love model. Also earlier the models of the second order accuracy were built for an isotropic material and for partial cases of anisotropy (for transversely isotropic and for monoclinic materials). In this work the general case is studied. A peculiarity of the proposed model is that the model includes the zero, the first, and the second approximations in contrary to the more simple models where summands of the first asymptotic order are absent. The proposed model may be applied to multi-layered and to functionally graded plates. The model may be used to solve various static and vibration problems. A 2D system of three PDE with the constant coefficients is obtained. The harmonic solution is investigated more detailed, and in this case the problem is reduced to a linear algebraic system.