Formulation of problems in the general Kirchhoff--Love theory of inhomogeneous anisotropic plates

V. I. Gorbachev ; L. A. Kabanova

Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 3 (2018), pp. 43-50

Résumé

In this paper we study the procedure of reducing the three-dimensional problem of elasticity theory for a thin inhomogeneous anisotropic plate to a two-dimensional problem in the median plane. The plate is in equilibrium under the action of bulk and surface forces of general form. A notion of internal force factors is introduced. Equations for force factors (equilibrium equations in the median plane) are obtained from the thickness-averaged three-dimensional equations of elasticity theory. In order to establish the relation between the internal force factors and the characteristics of the deformed middle surface, we use some prior assumptions on the distribution of displacements along the thickness of the plate. To arrange these assumptions in order, the displacements of plate points are expanded into Taylor series in the transverse coordinate with consideration of the physical hypotheses on the deformation of a material fiber that is originally perpendicular to the median plane. The well-known Kirchhoff–Love hypothesis is considered in detail. A closed system of equations for the theory of inhomogeneous anisotropic plates is obtained on the basis of the Kirchhoff–Love hypothesis. The boundary conditions are formulated from the Lagrange variation principle.