Plate theory

In continuum mechanics, plate theories are mathematical descriptions of the mechanics of flat plates that draws on the theory of beams. Plates are defined as plane structural elements with a small thickness compared to the planar dimensions. The typical thickness to width ratio of a plate structure is less than 0.1. A plate theory takes advantage of this disparity in length scale to reduce the full three-dimensional solid mechanics problem to a two-dimensional problem. The aim of plate theory is to calculate the deformation and stresses in a plate subjected to loads.

Of the numerous plate theories that have been developed since the late 19th century, two are widely accepted and used in engineering. These are

The Kirchhoff–Love theory is an extension of Euler–Bernoulli beam theory to thin plates. The theory was developed in 1888 by Love using assumptions proposed by Kirchhoff. It is assumed that a mid-surface plane can be used to represent the three-dimensional plate in two-dimensional form.

The following kinematic assumptions that are made in this theory:

The Kirchhoff hypothesis implies that the displacement field has the form

where ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}}$ are the Cartesian coordinates on the mid-surface of the undeformed plate, ${\displaystyle x_{3}}$ is the coordinate for the thickness direction, ${\displaystyle u_{1}^{0},u_{2}^{0}}$ are the in-plane displacements of the mid-surface, and ${\displaystyle w^{0}}$ is the displacement of the mid-surface in the ${\displaystyle x_{3}}$ direction.

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