Kirchhoff–Love plate theory

The theory assumes that a mid-surface plane can be used to represent a three-dimensional plate in two-dimensional form.

The following kinematic assumptions that are made in this theory:[2] Let the position vector of a point in the undeformed plate be

form a Cartesian basis with origin on the mid-surface of the plate,

We can write the in-plane displacement of the mid-surface as Note that the index

are the angles of rotation of the normal to the mid-surface, then in the Kirchhoff-Love theory Note that we can think of the expression for

as the first order Taylor series expansion of the displacement around the mid-surface.

The original theory developed by Love was valid for infinitesimal strains and rotations.

The theory was extended by von Kármán to situations where moderate rotations could be expected.

For the situation where the strains in the plate are infinitesimal and the rotations of the mid-surface normals are less than 10° the strain-displacement relations are where

The equilibrium equations for the plate can be derived from the principle of virtual work.

Hence, Another integration by parts gives For the case where there are no prescribed external forces, the principle of virtual work implies that

then leads to the equilibrium equations 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.

The stress-strain relations for a linear elastic Kirchhoff plate are given by Since

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.

The remaining stress-strain relations, in matrix form, can be written as Then, and The extensional stiffnesses are the quantities The bending stiffnesses (also called flexural rigidity) are the quantities The Kirchhoff-Love constitutive assumptions lead to zero shear forces.

For isotropic plates, these equations lead to Alternatively, these shear forces can be expressed as where If the rotations of the normals to the mid-surface are in the range of 10

If the strain-displacement relations take the von Karman form, the equilibrium equations can be expressed as For an isotropic and homogeneous plate, the stress-strain relations are where

, the stresses are For an isotropic and homogeneous plate under pure bending, the governing equations reduce to Here we have assumed that the in-plane displacements do not vary with

Hence In direct tensor notation, the governing equation of the plate is where we have assumed that the displacements

In rectangular Cartesian coordinates, the governing equation is and in cylindrical coordinates it takes the form Solutions of this equation for various geometries and boundary conditions can be found in the article on bending of plates.

, and we have Under certain loading conditions a flat plate can be bent into the shape of the surface of a cylinder.

In that case and and the governing equations become[3] The dynamic theory of thin plates determines the propagation of waves in the plates, and the study of standing waves and vibration modes.

Hence, after switching the sequence of integration, we have Integration by parts over the mid-surface gives Again, since the variations are zero at the beginning and the end of the time interval under consideration, we have For the dynamic case, the variation in the internal energy is given by Integration by parts and invoking zero variation at the boundary of the mid-surface gives If there is an external distributed force

acting normal to the surface of the plate, the virtual external work done is From the principle of virtual work, or more precisely, Hamilton's principle for a deformable body, we have

The figures below show some vibrational modes of a circular plate.

The governing equations simplify considerably for isotropic and homogeneous plates for which the in-plane deformations can be neglected.

In that case we are left with one equation of the following form (in rectangular Cartesian coordinates): where

, In direct notation For free vibrations, the governing equation becomes For an isotropic and homogeneous plate, the stress-strain relations are where

The strain-displacement relations for Kirchhoff-Love plates are Therefore, the resultant moments corresponding to these stresses are The governing equation for an isotropic and homogeneous plate of uniform thickness

in the absence of in-plane displacements is Differentiation of the expressions for the moment resultants gives us Plugging into the governing equations leads to Since the order of differentiation is irrelevant we have