Plate theory

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Vibration mode of a clamped square plate

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.^[1] 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 Mindlin–Reissner theory of plates (first-order shear plate theory)

Kirchhoff–Love theory for thin plates

Note: the Einstein summation convention of summing on repeated indices is used below.

Deformation of a thin plate highlighting the displacement, the mid-surface (red) and the normal to the mid-surface (blue)

The Kirchhoff–Love theory is an extension of Euler–Bernoulli beam theory to thin plates. The theory was developed in 1888 by Love^[2] 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:^[3]

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

\begin{align} u_\alpha(\mathbf{x}) & = u^0_\alpha(x_1,x_2) - x_3~\frac{\partial w^0}{\partial x_\alpha} = u^0_\alpha - x_3~w^0_{,\alpha} ~;~~\alpha=1,2 \\ u_3(\mathbf{x}) & = w^0(x_1, x_2) \end{align}

where $x_{1}$ and $x_{2}$ are the Cartesian coordinates on the mid-surface of the undeformed plate, $x_{3}$ is the coordinate for the thickness direction, $u^0_1, u^0_2$ are the in-plane displacements of the mid-surface, and $w^0$ is the displacement of the mid-surface in the $x_{3}$ direction.

If $\varphi _{\alpha }$ are the angles of rotation of the normal to the mid-surface, then in the Kirchhoff–Love theory $\varphi_\alpha = w^0_{,\alpha} \,.$

Displacement of the mid-surface (left) and of a normal (right)

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

\begin{align} \varepsilon_{\alpha\beta} & = \tfrac{1}{2}(u^0_{\alpha,\beta}+u^0_{\beta,\alpha}) - x_3~w^0_{,\alpha\beta} \\ \varepsilon_{\alpha 3} & = - w^0_{,\alpha} + w^0_{,\alpha} = 0 \\ \varepsilon_{33} & = 0 \end{align}

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

\begin{align} \varepsilon_{\alpha\beta} & = \frac{1}{2}(u^0_{\alpha,\beta}+u^0_{\beta,\alpha}+w^0_{,\alpha}~w^0_{,\beta}) - x_3~w^0_{,\alpha\beta} \\ \varepsilon_{\alpha 3} & = - w^0_{,\alpha} + w^0_{,\alpha} = 0 \\ \varepsilon_{33} & = 0 \end{align}

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

\begin{align} N_{\alpha\beta,\alpha} & = 0 \\ M_{\alpha\beta,\alpha\beta} & = 0 \end{align}

where the stress resultants and stress moment resultants are defined as

N_{\alpha\beta} := \int_{-h}^h \sigma_{\alpha\beta}~dx_3 ~;~~ M_{\alpha\beta} := \int_{-h}^h x_3~\sigma_{\alpha\beta}~dx_3

and the thickness of the plate is $2h$ . The quantities $\sigma_{\alpha\beta}$ are the stresses.

If the plate is loaded by an external distributed load $q(x)$ that is normal to the mid-surface and directed in the positive $x_{3}$ direction, the principle of virtual work then leads to the equilibrium equations

$\begin{align} N_{\alpha\beta,\alpha} & = 0 \\ M_{\alpha\beta,\alpha\beta} - q & = 0 \end{align}$

For moderate rotations, the strain-displacement relations take the von Karman form and the equilibrium equations can be expressed as

\begin{align} N_{\alpha\beta,\alpha} & = 0 \\ M_{\alpha\beta,\alpha\beta} + [N_{\alpha\beta}~w^0_{,\beta}]_{,\alpha} - q & = 0 \end{align}

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

\begin{align} n_\alpha~N_{\alpha\beta} & \quad \mathrm{or} \quad u^0_\beta \\ n_\alpha~M_{\alpha\beta,\beta} & \quad \mathrm{or} \quad w^0 \\ n_\beta~M_{\alpha\beta} & \quad \mathrm{or} \quad w^0_{,\alpha} \end{align}

Note that the quantity $n_\alpha~M_{\alpha\beta,\beta}$ is an effective shear force.

Stress–strain relations

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

\begin{bmatrix}\sigma_{11} \\ \sigma_{22} \\ \sigma_{12} \end{bmatrix} = \begin{bmatrix} C_{11} & C_{12} & C_{13} \\ C_{12} & C_{22} & C_{23} \\ C_{13} & C_{23} & C_{33} \end{bmatrix} \begin{bmatrix}\varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{12} \end{bmatrix}

Since $\sigma_{\alpha 3}$ and $\sigma_{33}$ 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

\begin{bmatrix}N_{11} \\ N_{22} \\ N_{12} \end{bmatrix} = \left\{ \int_{-h}^h \begin{bmatrix} C_{11} & C_{12} & C_{13} \\ C_{12} & C_{22} & C_{23} \\ C_{13} & C_{23} & C_{33} \end{bmatrix}~dx_3 \right\} \begin{bmatrix} u^0_{1,1} \\ u^0_{2,2} \\ \frac{1}{2}~(u^0_{1,2}+u^0_{2,1}) \end{bmatrix}

and

\begin{bmatrix}M_{11} \\ M_{22} \\ M_{12} \end{bmatrix} = -\left\{ \int_{-h}^h x_3^2~\begin{bmatrix} C_{11} & C_{12} & C_{13} \\ C_{12} & C_{22} & C_{23} \\ C_{13} & C_{23} & C_{33} \end{bmatrix}~dx_3 \right\} \begin{bmatrix} w^0_{,11} \\ w^0_{,22} \\ w^0_{,12} \end{bmatrix} \,.

The extensional stiffnesses are the quantities

A_{\alpha\beta} := \int_{-h}^h C_{\alpha\beta}~dx_3

The bending stiffnesses (also called flexural rigidity) are the quantities

D_{\alpha\beta} := \int_{-h}^h x_3^2~C_{\alpha\beta}~dx_3

Isotropic and homogeneous Kirchhoff plate

For an isotropic and homogeneous plate, the stress–strain relations are

\begin{bmatrix}\sigma_{11} \\ \sigma_{22} \\ \sigma_{12} \end{bmatrix} = \cfrac{E}{1-\nu^2} \begin{bmatrix} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & 1-\nu \end{bmatrix} \begin{bmatrix}\varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{12} \end{bmatrix} \,.

The moments corresponding to these stresses are

\begin{bmatrix}M_{11} \\ M_{22} \\ M_{12} \end{bmatrix} = -\cfrac{2h^3E}{3(1-\nu^2)}~\begin{bmatrix} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & 1-\nu \end{bmatrix} \begin{bmatrix} w^0_{,11} \\ w^0_{,22} \\ w^0_{,12} \end{bmatrix}

Pure bending

The displacements $u^0_1$ and $u^0_2$ are zero under pure bending conditions. For an isotropic, homogeneous plate under pure bending the governing equation is

\frac{\partial^4 w}{\partial x_1^4} + 2 \frac{\partial^4 w}{\partial x_1^2 \partial x_2^2} + \frac{\partial^4 w}{\partial x_2^4} = 0 \quad \text{where} \quad w := w^0\,.

In index notation,

w^0_{,1111} + 2~w^0_{,1212} + w^0_{,2222} = 0 \,.

In direct tensor notation, the governing equation is

$\nabla^2\nabla^2 w = 0 \,.$

Transverse loading

For a transversely loaded plate without axial deformations, the governing equation has the form

\frac{\partial^4 w}{\partial x_1^4} + 2 \frac{\partial^4 w}{\partial x_1^2 \partial x_2^2} + \frac{\partial^4 w}{\partial x_2^4} = -\frac{q}{D}

where

D := \cfrac{2h^3E}{3(1-\nu^2)} \,.

In index notation,

w^0_{,1111} + 2\,w^0_{,1212} + w^0_{,2222} = -\frac{q}{D}

and in direct notation

\nabla^2\nabla^2 w = -\frac{q}{D} \,.

In cylindrical coordinates $(r, \theta, z)$ , the governing equation is

\frac{1}{r}\cfrac{d }{d r}\left[r \cfrac{d }{d r}\left\{\frac{1}{r}\cfrac{d }{d r}\left(r \cfrac{d w}{d r}\right)\right\}\right] = - \frac{q}{D}\,.

Orthotropic and homogeneous Kirchhoff plate

For an orthotropic plate

\begin{bmatrix} C_{11} & C_{12} & C_{13} \\ C_{12} & C_{22} & C_{23} \\ C_{13} & C_{23} & C_{33} \end{bmatrix} = \cfrac{1}{1-\nu_{12}\nu_{21}} \begin{bmatrix} E_1 & \nu_{12}E_2 & 0 \\ \nu_{21}E_1 & E_2 & 0 \\ 0 & 0 & 2G_{12}(1-\nu_{12}\nu_{21}) \end{bmatrix} \,.

Therefore,

\begin{bmatrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{bmatrix} = \cfrac{2h}{1-\nu_{12}\nu_{21}} \begin{bmatrix} E_1 & \nu_{12}E_2 & 0 \\ \nu_{21}E_1 & E_2 & 0 \\ 0 & 0 & 2G_{12}(1-\nu_{12}\nu_{21}) \end{bmatrix}

and

\begin{bmatrix} D_{11} & D_{12} & D_{13} \\ D_{21} & D_{22} & D_{23} \\ D_{31} & D_{32} & D_{33} \end{bmatrix} = \cfrac{2h^3}{3(1-\nu_{12}\nu_{21})} \begin{bmatrix} E_1 & \nu_{12}E_2 & 0 \\ \nu_{21}E_1 & E_2 & 0 \\ 0 & 0 & 2G_{12}(1-\nu_{12}\nu_{21}) \end{bmatrix} \,.

Transverse loading

The governing equation of an orthotropic Kirchhoff plate loaded transversely by a distributed load $q$ per unit area is

D_x w^0_{,1111} + 2 D_{xy} w^0_{,1122} + D_y w^0_{,2222} = -q

where

\begin{align} D_x & = D_{11} = \frac{2h^3 E_1}{3(1 - \nu_{12}\nu_{21})} \\ D_y & = D_{22} = \frac{2h^3 E_2}{3(1 - \nu_{12}\nu_{21})} \\ D_{xy} & = D_{33} + \tfrac{1}{2}(\nu_{21} D_{11} + \nu_{12} D_{22}) = D_{33} + \nu_{21} D_{11} = \frac{4h^3 G_{12}}{3} + \frac{2h^3 \nu_{21} E_1}{3(1 - \nu_{12}\nu_{21})} \,. \end{align}

Dynamics of thin Kirchhoff plates

The dynamic theory of plates determines the propagation of waves in the plates, and the study of standing waves and vibration modes.

Governing equations

The governing equations for the dynamics of a Kirchhoff–Love plate are

\begin{align} N_{\alpha\beta,\beta} & = J_1~\ddot{u}^0_\alpha \\ M_{\alpha\beta,\alpha\beta} - q(x,t) & = J_1~\ddot{w}^0 - J_3~\ddot{w}^0_{,\alpha\alpha} \end{align}

where, for a plate with density $\rho = \rho(x)$ ,

J_1 := \int_{-h}^h \rho~dx_3 = 2~\rho~h ~;~~ J_3 := \int_{-h}^h x_3^2~\rho~dx_3 = \frac{2}{3}~\rho~h^3

and

\dot{u}_i = \frac{\partial u_i}{\partial t} ~;~~ \ddot{u}_i = \frac{\partial^2 u_i}{\partial t^2} ~;~~ u_{i,\alpha} = \frac{\partial u_i}{\partial x_\alpha} ~;~~ u_{i,\alpha\beta} = \frac{\partial^2 u_i}{\partial x_\alpha \partial x_\beta}

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

mode k = 0, p = 1
mode k = 1, p = 2

Isotropic plates

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

D\,\left(\frac{\partial^4 w^0}{\partial x_1^4} + 2\frac{\partial^4 w^0}{\partial x_1^2\partial x_2^2} + \frac{\partial^4 w^0}{\partial x_2^4}\right) = -q(x_1, x_2, t) - 2\rho h\, \frac{\partial^2 w^0}{\partial t^2} \,.

where $D$ is the bending stiffness of the plate. For a uniform plate of thickness $2h$ ,

D := \cfrac{2h^3E}{3(1-\nu^2)} \,.

In direct notation

D\,\nabla^2\nabla^2 w^0 = -q(x, y, t) - 2\rho h \, \ddot{w}^0 \,.

Mindlin–Reissner theory for thick plates

Note: the Einstein summation convention of summing on repeated indices is used below.

In the theory of thick plates, or theory of Raymond Mindlin^[4] and Eric Reissner, the normal to the mid-surface remains straight but not necessarily perpendicular to the mid-surface. If $\varphi _{1}$ and $\varphi _{2}$ designate the angles which the mid-surface makes with the $x_{3}$ axis then

\varphi_1 \ne w_{,1} ~;~~ \varphi_2 \ne w_{,2}

Then the Mindlin–Reissner hypothesis implies that

\begin{align} u_\alpha(\mathbf{x}) & = u^0_\alpha(x_1,x_2) - x_3~\varphi_\alpha ~;~~\alpha=1,2 \\ u_3(\mathbf{x}) & = w^0(x_1, x_2) \end{align}