In continuum mechanics, plate theories are mathematical descriptions of the mechanics of flat plates that draw 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 of plates (classical plate theory) The Uflyand-Mindlin theory of plates (first-order shear plate theory)
Kirchhoff–Love theory for thin plates
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 are made in this theory: straight lines normal to the mid-surface remain straight after deformation straight lines normal to the mid-surface remain normal to the mid-surface after deformation the thickness of the plate does not change during a deformation.
Displacement field
The Kirchhoff hypothesis implies that the displacement field has the form
where and are the Cartesian coordinates on the mid-surface of the undeformed plate, is the coordinate for the thickness direction, are the in-plane displacements of the mid-surface, and is the displacement of the mid-surface in the direction.
If are the angles of rotation of the normal to the mid-surface, then in the Kirchhoff–Love theory
Strain-displacement relations
For the situation where the strains in the plate are infinitesimal and the rotations of the mid-surface normals are less than 10° the strains-displacement relations are
Therefore, the only non-zero strains are in the in-plane directions.
If the rotations of the normals to the mid-surface are in the range of 10° to 15°, the strain-displacement relations can be approximated using the von Kármán strains. Then the kinematic assumptions of Kirchhoff-Love theory lead to the following strain-displacement relations
This theory is nonlinear because of the quadratic terms in the strain-displacement relations.
Equilibrium equations
The equilibrium equations for the plate can be derived from the principle of virtual work. For the situation where the strains and rotations of the plate are small, the equilibrium equations for an unloaded plate are given by
where the stress resultants and stress moment resultants are defined as
and the thickness of the plate is . The quantities are the stresses.
If the plate is loaded by an external distributed load that is normal to the mid-surface and directed in the positive direction, the principle of virtual work then leads to the equilibrium equations
For moderate rotations, the strain-displacement relations take the von Karman form and the equilibrium equations can be expressed as
Boundary conditions
The boundary conditions that are needed to solve the equilibrium equations of plate theory can be obtained from the boundary terms in the principle of virtual work.
For small strains and small rotations, the boundary conditions are
Note that the quantity is an effective shear force.
Stress–strain relations
The stress–strain relations for a linear elastic Kirchhoff plate are given by
Since and do not appear in the equilibrium equations it is implicitly assumed that these quantities do not have any effect on the momentum balance and are neglected.
It is more convenient to work with the stress and moment resultants that enter the equilibrium equations. These are related to the displacements by
and
The extensional stiffnesses are the quantities
The bending stiffnesses (also called flexural rigidity) are the quantities
Isotropic and homogeneous Kirchhoff plate
For an isotropic and homogeneous plate, the stress–strain relations are
The moments corresponding to these stresses are
Pure bending
The displacements and are zero under pure bending conditions. For an isotropic, homogeneous plate under pure bending the governing equation is
In index notation,