The 3D dynamic equations in elasticity for a thin transversely inhomogeneous plate are subject to asymptotic analysis over the low-frequency range. The leading and first order approximations are derived. The former is given by a biharmonic equation on the mid-plane generalizing the classical Kirchhoff equation for plate bending. A simple explicit formula for the effective bending stiffness is presented. The refined first order equation involves the same biharmonic operator, as the leading order one, along with corrections expressed through Laplacians. However, the constant coefficients at these corrections take the form of sophisticated repeated integrals across the plate thickness. The formulae for the transverse variations of the displacement and stress components, especially relevant for FGM structures, are also obtained. The scope for comparison of the developed asymptotic results and the existing ad hoc considerations on the subject seems to be limited, in contrast to the homogeneous setup, due to a more substantial deviation between the predictions offered by these two approaches.

横向非均匀薄板弹性的 3D 动力学方程在低频范围内进行渐近分析。导出了前导和一阶近似值。前者由中平面上的双调和方程给出，该方程推广了用于板弯曲的经典基尔霍夫方程。给出了有效弯曲刚度的一个简单的显式公式。细化的一阶方程包含与前导阶相同的双谐波算子，以及通过拉普拉斯算子表达的修正。然而，这些修正中的常数系数采用跨板厚度的复杂重复积分的形式。还获得了位移和应力分量的横向变化公式，特别是与 FGM 结构相关的公式。