View source for Plate theory

{{short description|Mathematical model of the stresses within flat plates under loading}}
{{For|the theory of geological plates|Plate tectonics}}
{{Continuum mechanics|cTopic=[[Solid mechanics]]}}
[[Image:ESPIvibration.jpg|thumb|250px | right| Vibration mode of a clamped square plate]]

In [[continuum mechanics]], '''plate theories''' are mathematical descriptions of the [[mechanics]] of flat [[Plate (structure)|plates]] that draw on the [[bending|theory of beams]].  Plates are defined as plane [[List of structural elements|structural elements]] with a small thickness compared to the planar dimensions.<ref name=timo>Timoshenko, S. and Woinowsky-Krieger, S. "Theory of plates and shells". McGraw–Hill New York, 1959.</ref>  The typical thickness to width ratio of a plate structure is less than 0.1.{{Citation needed|date=March 2020}}  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 (mechanics)|deformation]] and [[stress (mechanics)|stress]]es in a plate subjected to [[Structural load|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 [[Gustav Kirchhoff|Kirchhoff]]–[[Augustus Edward Hough Love|Love]] theory of plates (classical plate theory)
* The [[Reissner-Mindlin plate theory|Reissner-Mindlin]] theory of plates (first-order shear plate theory)

== Kirchhoff–Love theory for thin plates ==
{{main|Kirchhoff–Love plate theory}}
{{Einstein_summation_convention}}

[[Image:Plaque mince deplacement element matiere.svg|thumb|250px | Deformation of a thin plate highlighting the displacement, the mid-surface (red) and the normal to the mid-surface (blue)]]
The [[Gustav Kirchhoff|Kirchhoff]]–[[Augustus Edward Hough Love|Love]] theory is an extension of [[beam theory|Euler–Bernoulli beam theory]] to thin plates. The theory was developed in 1888 by Love<ref>A. E. H. Love, ''On the small free vibrations and deformations of elastic shells'', Philosophical trans. of the Royal Society (London), 1888, Vol. série A, N° 17 p. 491–549.</ref> 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:<ref name=Reddy>Reddy, J. N., 2007, '''Theory and analysis of elastic plates and shells''', CRC Press, Taylor and Francis.</ref>
* 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 (vector)|displacement]] field has the form
{{Equation box 1  |indent =:| equation=
:<math>
   \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}
</math>
| cellpadding | border|border colour=#50C878 |background colour = #ECFCF4}}
where <math>x_1</math> and <math>x_2</math> are the Cartesian coordinates on the mid-surface of the undeformed plate, <math>x_3</math> is the coordinate for the thickness direction, <math>u^0_1, u^0_2</math> are the in-plane displacements of the mid-surface, and <math>w^0</math> is the displacement of the mid-surface in the <math>x_3</math> direction.

If <math>\varphi_\alpha</math> are the angles of rotation of the [[Surface normal|normal]] to the mid-surface, then in the Kirchhoff–Love theory
<math>
   \varphi_\alpha = w^0_{,\alpha} \,.
 </math>
{|
|[[Image:Plaque mince deplacement rotation fibre neutre new.svg|thumb|600px|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 [[infinitesimal strain theory|strains-displacement]] relations are
:<math>
  \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}
</math>
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 [[Theodore von Kármán|von Kármán]] strains.  Then the kinematic assumptions of Kirchhoff-Love theory lead to the following strain-displacement relations
:<math>
  \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}
</math>
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
:<math>
   \begin{align}
     N_{\alpha\beta,\alpha} & = 0 \\
     M_{\alpha\beta,\alpha\beta} & = 0
   \end{align}
</math>
where the stress resultants and stress moment resultants are defined as
:<math>
   N_{\alpha\beta} := \int_{-h}^h \sigma_{\alpha\beta}~dx_3 ~;~~
   M_{\alpha\beta} := \int_{-h}^h x_3~\sigma_{\alpha\beta}~dx_3
</math>
and the thickness of the plate is <math>2h</math>.  The quantities <math>\sigma_{\alpha\beta}</math> are the stresses.

If the plate is loaded by an external distributed load <math>q(x)</math> that is normal to the mid-surface and directed in the positive <math>x_3</math> direction, the principle of virtual work then leads to the equilibrium equations
{{Equation box 1  |indent =:| equation=
<math>
   \begin{align}
     N_{\alpha\beta,\alpha} & = 0 \\
     M_{\alpha\beta,\alpha\beta} - q & = 0
   \end{align}
</math>
| cellpadding | border|border colour=#50C878 |background colour = #ECFCF4}}

For moderate rotations, the strain-displacement relations take the von Karman form and the equilibrium equations can be expressed as
:<math>
   \begin{align}
     N_{\alpha\beta,\alpha} & = 0 \\
     M_{\alpha\beta,\alpha\beta} + [N_{\alpha\beta}~w^0_{,\beta}]_{,\alpha} - q & = 0
   \end{align}
</math>

=== 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
:<math>
   \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}
</math>
Note that the quantity <math> n_\alpha~M_{\alpha\beta,\beta}</math> is an effective shear force.

=== Stress–strain relations ===
The stress–strain relations for a linear elastic Kirchhoff plate are given by
:<math>
   \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}
</math>
Since <math>\sigma_{\alpha 3}</math> and <math>\sigma_{33}</math> 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
:<math>
   \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}
</math>
and
:<math>
   \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} \,.
</math>
The ''' extensional stiffnesses''' are the quantities
:<math>
   A_{\alpha\beta} := \int_{-h}^h C_{\alpha\beta}~dx_3
</math>
The ''' bending stiffnesses''' (also called '''flexural rigidity''') are the quantities
:<math>
   D_{\alpha\beta} := \int_{-h}^h x_3^2~C_{\alpha\beta}~dx_3
</math>

== Isotropic and homogeneous Kirchhoff plate ==
{{main|Kirchhoff–Love plate theory}}
For an isotropic and  homogeneous plate, the stress–strain relations are
:<math>
   \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} \,.
 </math>
The moments corresponding to these stresses are
:<math>
   \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}
</math>

=== Pure bending ===
The displacements <math>u^0_1</math> and <math>u^0_2</math> are zero under [[pure bending]] conditions.  For an isotropic, homogeneous plate under pure bending the governing equation is
:<math>
   \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\,.
 </math>
In index notation,
:<math>
   w^0_{,1111} + 2~w^0_{,1212} +  w^0_{,2222} = 0 \,.
 </math>
In direct tensor notation, the governing equation is
{{Equation box 1  |indent =:| equation=
<math>
   \nabla^2\nabla^2 w = 0 \,.
 </math>
| cellpadding | border|border colour=#50C878 |background colour = #ECFCF4}}

=== Transverse loading ===
For a transversely loaded plate without axial deformations, the governing equation has the form
:<math>
   \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}
 </math>
where
:<math>
   D := \cfrac{2h^3E}{3(1-\nu^2)} \,.
 </math>
for a plate with thickness <math>2h</math>.
In index notation,
:<math>
   w^0_{,1111} + 2\,w^0_{,1212} + w^0_{,2222} = -\frac{q}{D}
 </math>
and in direct notation

{{Equation box 1  |indent =:| equation=
:<math>
  \nabla^2\nabla^2 w = -\frac{q}{D} \,.
 </math>
| cellpadding | border|border colour=#50C878 |background colour = #ECFCF4}}

In cylindrical coordinates <math>(r, \theta, z)</math>, the governing equation is
:<math>
   \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}\,.
</math>

== Orthotropic and homogeneous Kirchhoff plate ==
For an [[orthotropic material|orthotropic]] plate
:<math>
   \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}
    \,.
 </math>
Therefore,
:<math>
   \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}
 </math>
and
:<math>
   \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}
    \,.
 </math>

=== Transverse loading ===
The governing equation of an orthotropic Kirchhoff plate loaded transversely by a distributed load <math>q</math> per unit area is
:<math>
   D_x w^0_{,1111} + 2 D_{xy} w^0_{,1122} + D_y w^0_{,2222} = -q
 </math>
where
:<math>
   \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}
 </math>

== Dynamics of thin Kirchhoff plates ==
{{main|Kirchhoff–Love plate theory}}
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
{{Equation box 1  |indent =:| equation=
:<math>
   \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}
</math>
| cellpadding | border|border colour=#50C878 |background colour = #ECFCF4}}
where, for a plate with density <math>\rho = \rho(x)</math>,
:<math>
   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
</math>
and
:<math>
   \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}
</math>

The figures below show some vibrational modes of a circular plate.
<gallery widths="250px">
Image:Drum vibration mode01.gif|mode ''k'' = 0, ''p'' = 1
Image:Drum vibration mode12.gif|mode ''k'' = 1, ''p'' = 2
</gallery>

=== 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
:<math>
   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} \,.
 </math>
where <math>D</math> is the bending stiffness of the plate.  For a uniform plate of thickness <math>2h</math>,
:<math>
   D := \cfrac{2h^3E}{3(1-\nu^2)} \,.
 </math>
In direct notation
{{Equation box 1  |indent =:| equation=
:<math>
   D\,\nabla^2\nabla^2 w^0 = -q(x, y, t) - 2\rho h \, \ddot{w}^0 \,.
 </math>
| cellpadding | border|border colour=#50C878 |background colour = #ECFCF4}}

== Reissner-Mindlin theory for thick plates ==
{{main|Mindlin–Reissner plate theory}}
{{Einstein_summation_convention}}
In the theory of thick plates, contributed to by [[Eric Reissner]],<ref>E. Reissner, 1945, The effect of transverse shear deformation on the bending of elastic plates, ASME Journal of Applied Mechanics, Vol. 12, pp. A68–77.</ref> [[Raymond Mindlin]],<ref> R. D. Mindlin, ''Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates'', Journal of Applied Mechanics, 1951, Vol. 18 p. 31–38.</ref> and Yakov S. Uflyand,<ref>Uflyand, Ya. S.,1948, Wave Propagation by Transverse Vibrations of Beams and Plates, PMM: Journal of Applied Mathematics and Mechanics, Vol. 12, 287-300 (in Russian)</ref><ref>Elishakoff ,I.,2020, '''Handbook on Timoshenko-Ehrenfest Beam and Uflyand-Mindlin Plate Theories''', World Scientific, Singapore, {{ISBN|978-981-3236-51-6}}</ref> the normal to the mid-surface remains straight but not necessarily perpendicular to the mid-surface.  If <math>\varphi_1</math> and <math>\varphi_2</math> designate the angles which the mid-surface makes with the <math>x_3</math> axis then
:<math>
    \varphi_1 \ne w_{,1} ~;~~ \varphi_2 \ne w_{,2}
 </math>

Then the Mindlin–Reissner hypothesis implies that
{{Equation box 1  |indent =:| equation=
:<math>
   \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}
</math>
| cellpadding | border|border colour=#50C878 |background colour = #ECFCF4}}

=== Strain-displacement relations ===
Depending on the amount of rotation of the plate normals two different approximations for the strains can be derived from the basic kinematic assumptions.

For small strains and small rotations the strain-displacement relations for Mindlin–Reissner plates are
:<math>
  \begin{align}
    \varepsilon_{\alpha\beta} & = \frac{1}{2}(u^0_{\alpha,\beta}+u^0_{\beta,\alpha})
      - \frac{x_3}{2}~(\varphi_{\alpha,\beta} + \varphi_{\beta,\alpha})\\
    \varepsilon_{\alpha 3} & = \cfrac{1}{2}\left(w^0_{,\alpha}- \varphi_\alpha\right) \\
    \varepsilon_{33} & = 0
  \end{align}
</math>
The shear strain, and hence the [[shear stress]], across the thickness of the plate is not neglected in this theory.  However, the shear strain is constant across the thickness of the plate.  This cannot be accurate since the shear stress is known to be parabolic even for simple plate geometries.  To account for the inaccuracy in the shear strain, a '''shear correction factor''' (<math>\kappa</math>) is applied so that the correct amount of internal energy is predicted by the theory.  Then
:<math>
   \varepsilon_{\alpha 3} = \cfrac{1}{2}~\kappa~\left(w^0_{,\alpha}- \varphi_\alpha\right)
 </math>

=== Equilibrium equations ===
The equilibrium equations have slightly different forms depending on the amount of bending expected in the plate.  For the situation where the strains and rotations of the plate are small the equilibrium equations for a Mindlin–Reissner plate are
{{Equation box 1  |indent =:| equation=
:<math>
   \begin{align}
     & N_{\alpha\beta,\alpha} = 0 \\
     & M_{\alpha\beta,\beta}-Q_\alpha = 0 \\
     & Q_{\alpha,\alpha}+q = 0 \,.
   \end{align}
 </math>
| cellpadding | border|border colour=#50C878 |background colour = #ECFCF4}}
The resultant shear forces in the above equations are defined as
:<math>
   Q_\alpha := \kappa~\int_{-h}^h \sigma_{\alpha 3}~dx_3 \,.
</math>

=== Boundary conditions ===
The boundary conditions are indicated by the boundary terms in the principle of virtual work.

If the only external force is a vertical force on the top surface of the plate, the boundary conditions are
:<math>
   \begin{align}
      n_\alpha~N_{\alpha\beta} & \quad \mathrm{or} \quad u^0_\beta \\
      n_\alpha~M_{\alpha\beta} & \quad \mathrm{or} \quad \varphi_\alpha \\
      n_\alpha~Q_\alpha & \quad \mathrm{or} \quad w^0
   \end{align}
</math>

=== Constitutive relations ===
The stress–strain relations for a linear elastic Mindlin–Reissner plate are given by
:<math>
   \begin{align}
     \sigma_{\alpha\beta} & = C_{\alpha\beta\gamma\theta}~\varepsilon_{\gamma\theta} \\
     \sigma_{\alpha 3} & = C_{\alpha 3\gamma\theta}~\varepsilon_{\gamma\theta} \\
     \sigma_{33} & = C_{33\gamma\theta}~\varepsilon_{\gamma\theta}
   \end{align}
</math>
Since <math>\sigma_{33}</math> does not appear in the equilibrium equations it is implicitly assumed that it do not have any effect on the momentum balance and is neglected.  This assumption is also called the '''plane stress''' assumption.  The remaining stress–strain relations for an [[orthotropic material]], in matrix form, can be written as
:<math>
   \begin{bmatrix}\sigma_{11} \\ \sigma_{22} \\ \sigma_{23} \\ \sigma_{31} \\ \sigma_{12} \end{bmatrix} = 
   \begin{bmatrix} C_{11} & C_{12} & 0 & 0 & 0 \\ C_{12} & C_{22} & 0 & 0 & 0 \\
                   0 & 0 & C_{44} & 0 & 0 \\
                   0 & 0 & 0 & C_{55} & 0 \\ 0 & 0 & 0 & 0 & C_{66}\end{bmatrix}
   \begin{bmatrix}\varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{23} \\ \varepsilon_{31} \\ \varepsilon_{12}\end{bmatrix}
</math>
Then,
:<math>
   \begin{bmatrix}N_{11} \\ N_{22} \\ N_{12} \end{bmatrix} = 
   \left\{
   \int_{-h}^h \begin{bmatrix} C_{11} & C_{12} & 0 \\ C_{12} & C_{22} & 0 \\
                   0 & 0 & C_{66} \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}
</math>
and
:<math>
   \begin{bmatrix}M_{11} \\ M_{22} \\ M_{12} \end{bmatrix} = -\left\{
   \int_{-h}^h x_3^2~\begin{bmatrix} C_{11} & C_{12} & 0 \\ C_{12} & C_{22} & 0 \\
                   0 & 0 & C_{66} \end{bmatrix}~dx_3 \right\}
   \begin{bmatrix} \varphi_{1,1} \\ \varphi_{2,2} \\ \frac{1}{2}~(\varphi_{1,2}+\varphi_{2,1}) \end{bmatrix}
</math>
For the shear terms
:<math>
   \begin{bmatrix}Q_1 \\ Q_2 \end{bmatrix} = \cfrac{\kappa}{2}\left\{
   \int_{-h}^h \begin{bmatrix} C_{55} & 0 \\ 0 & C_{44}  \end{bmatrix}~dx_3 \right\}
   \begin{bmatrix} w^0_{,1} - \varphi_1 \\ w^0_{,2} - \varphi_2 \end{bmatrix}
</math>
The ''' extensional stiffnesses''' are the quantities
:<math>
   A_{\alpha\beta} := \int_{-h}^h C_{\alpha\beta}~dx_3
</math>
The ''' bending stiffnesses''' are the quantities
:<math>
   D_{\alpha\beta} := \int_{-h}^h x_3^2~C_{\alpha\beta}~dx_3
</math>

== Isotropic and homogeneous Reissner-Mindlin plates ==
{{main|Mindlin–Reissner plate theory}}
For uniformly thick, homogeneous, and isotropic plates, the stress–strain relations in the plane of the plate are
:<math>
   \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} \,.
</math>
where <math>E</math> is the Young's modulus, <math>\nu</math> is the Poisson's ratio, and <math>\varepsilon_{\alpha\beta}</math> are the in-plane strains.  The through-the-thickness shear stresses and strains are related by
:<math>
   \sigma_{31} = 2G\varepsilon_{31} \quad \text{and} \quad
   \sigma_{32} = 2G\varepsilon_{32}
</math>
where <math>G = E/(2(1+\nu))</math> is the [[shear modulus]].

=== Constitutive relations ===
The relations between the stress resultants and the generalized displacements for an isotropic Mindlin–Reissner plate are:
:<math>
   \begin{bmatrix}N_{11} \\ N_{22} \\ N_{12} \end{bmatrix} =
    \cfrac{2Eh}{1-\nu^2} \begin{bmatrix} 1 & \nu & 0 \\ \nu & 1 & 0 \\
                   0 & 0 & 1-\nu \end{bmatrix}
   \begin{bmatrix} u^0_{1,1} \\ u^0_{2,2} \\ \frac{1}{2}~(u^0_{1,2}+u^0_{2,1}) \end{bmatrix} \,,
</math>
:<math>
   \begin{bmatrix}M_{11} \\ M_{22} \\ M_{12} \end{bmatrix} =
    -\cfrac{2Eh^3}{3(1-\nu^2)} \begin{bmatrix} 1 & \nu & 0 \\ \nu & 1 & 0 \\
                   0 & 0 & 1-\nu \end{bmatrix}
   \begin{bmatrix} \varphi_{1,1} \\ \varphi_{2,2} \\ \frac{1}{2}(\varphi_{1,2}+\varphi_{2,1}) \end{bmatrix} \,,
</math>
and
:<math>
   \begin{bmatrix}Q_1 \\ Q_2 \end{bmatrix} = \kappa G h
   \begin{bmatrix} w^0_{,1} - \varphi_1 \\ w^0_{,2} - \varphi_2 \end{bmatrix} \,.
</math>
The [[bending rigidity]] is defined as the quantity
:<math>
   D = \cfrac{2Eh^3}{3(1-\nu^2)}  \,.
</math>
For a plate of thickness <math>H</math>, the bending rigidity has the form
:<math>
   D = \cfrac{EH^3}{12(1-\nu^2)}  \,.

</math>

where 
<math>
h=\frac{H}{2}
</math>

=== Governing equations ===
If we ignore the in-plane extension of the plate, the governing equations are
:<math>
   \begin{align}
     M_{\alpha\beta,\beta}-Q_\alpha & = 0 \\
     Q_{\alpha,\alpha}+q & = 0 \,.
   \end{align}
</math>
In terms of the generalized deformations <math>w^0, \varphi_1, \varphi_2</math>, the three governing equations are
{{Equation box 1  |indent =:| equation=
:<math>
   \begin{align}
    &\nabla^2 \left(\frac{\partial \varphi_1}{\partial x_1} + \frac{\partial \varphi_2}{\partial x_2}\right) = -\frac{q}{D} \\
   &\nabla^2 w^0 - \frac{\partial \varphi_1}{\partial x_1} - \frac{\partial \varphi_2}{\partial x_2} = -\frac{q}{\kappa G h} \\
   &\nabla^2 \left(\frac{\partial \varphi_1}{\partial x_2} - \frac{\partial \varphi_2}{\partial x_1}\right) = -\frac{2\kappa G h}{D(1-\nu)}\left(\frac{\partial \varphi_1}{\partial x_2} - \frac{\partial \varphi_2}{\partial x_1}\right) \,.
   \end{align}
</math>
| cellpadding | border|border colour=#50C878 |background colour = #ECFCF4}}
The boundary conditions along the edges of a rectangular plate are
:<math>
  \begin{align}
    \text{simply supported} \quad & \quad w^0 = 0, M_{11} = 0 ~(\text{or}~M_{22} = 0),
    \varphi_1 = 0 ~(\text{or}~\varphi_2 = 0) \\
    \text{clamped} \quad & \quad w^0 = 0, \varphi_1 = 0,  \varphi_{2} = 0  \,.
  \end{align}
</math>

== Reissner–Stein static theory for isotropic cantilever plates ==
In general, exact solutions for cantilever plates using plate theory are quite involved and few exact solutions can be found in the literature.  Reissner and Stein<ref name=Reissner51>E. Reissner and M. Stein. Torsion and transverse bending of cantilever plates. Technical Note 2369, National Advisory Committee for Aeronautics,Washington, 1951.</ref> provide a simplified theory for cantilever plates that is an improvement over older theories such as Saint-Venant plate theory.

The Reissner-Stein theory assumes a transverse displacement field of the form
:<math>
  w(x,y) = w_x(x) + y\,\theta_x(x) \,.
</math>
The governing equations for the plate then reduce to two coupled ordinary differential equations:
{{Equation box 1  |indent =:| equation=
:<math>
  \begin{align}
  & bD \frac{\mathrm{d}^4w_x}{\mathrm{d}x^4}
   = q_1(x) - n_1(x)\cfrac{d^2 w_x}{d x^2} - \cfrac{d n_1}{d x}\,\cfrac{d w_x}{d x}
     - \frac{1}{2}\cfrac{d n_2}{d x}\,\cfrac{d \theta_x}{d x} - \frac{n_2(x)}{2}\cfrac{d^2 \theta_x}{d x^2} \\
  &\frac{b^3D}{12}\,\frac{\mathrm{d}^4\theta_x}{\mathrm{d}x^4} - 2bD(1-\nu)\cfrac{d^2 \theta_x}{d x^2}
   = q_2(x) - n_3(x)\cfrac{d^2 \theta_x}{d x^2} - \cfrac{d n_3}{d x}\,\cfrac{d \theta_x}{d x}
    - \frac{n_2(x)}{2}\,\cfrac{d^2 w_x}{d x^2} - \frac{1}{2}\cfrac{d n_2}{d x}\,\cfrac{d w_x}{d x}
  \end{align}
</math>
| cellpadding | border|border colour=#50C878 |background colour = #ECFCF4}}
where
:<math>
  \begin{align}
    q_1(x) & = \int_{-b/2}^{b/2}q(x,y)\,\text{d}y ~,~~ q_2(x) = \int_{-b/2}^{b/2}y\,q(x,y)\,\text{d}y~,~~
    n_1(x) = \int_{-b/2}^{b/2}n_x(x,y)\,\text{d}y \\
    n_2(x) & = \int_{-b/2}^{b/2}y\,n_x(x,y)\,\text{d}y ~,~~ n_3(x)  = \int_{-b/2}^{b/2}y^2\,n_x(x,y)\,\text{d}y  \,.
  \end{align}
</math>
At <math>x = 0</math>, since the beam is clamped, the boundary conditions are
:<math>
  w(0,y) = \cfrac{d w}{d x}\Bigr|_{x=0} = 0 \qquad \implies \qquad
  w_x(0) = \cfrac{d w_x}{d x}\Bigr|_{x=0} = \theta_x(0) = \cfrac{d \theta_x}{d x}\Bigr|_{x=0} = 0 \,.
</math>
The boundary conditions at <math>x = a</math> are
:<math>
  \begin{align}
  & bD\cfrac{d^3 w_x}{d x^3} + n_1(x)\cfrac{d w_x}{d x} + n_2(x)\cfrac{d \theta_x}{d x} + q_{x1} = 0 \\
  & \frac{b^3D}{12}\cfrac{d^3 \theta_x}{d x^3} + \left[n_3(x) -2bD(1-\nu)\right]\cfrac{d \theta_x}{d x}
    + n_2(x)\cfrac{d w_x}{d x} + t = 0 \\
  & bD\cfrac{d^2 w_x}{d x^2} + m_1 = 0 \quad,\quad  \frac{b^3D}{12}\cfrac{d^2 \theta_x}{d x^2} + m_2 = 0
  \end{align}
</math>
where
:<math>
  \begin{align}
    m_1 & = \int_{-b/2}^{b/2}m_x(y)\,\text{d}y ~,~~ m_2 = \int_{-b/2}^{b/2}y\,m_x(y)\,\text{d}y ~,~~
    q_{x1} = \int_{-b/2}^{b/2}q_x(y)\,\text{d}y \\
    t  & = q_{x2} + m_3 = \int_{-b/2}^{b/2}y\,q_x(y)\,\text{d}y + \int_{-b/2}^{b/2}m_{xy}(y)\,\text{d}y  \,.
  \end{align}
</math>
:{| class="toccolours collapsible collapsed" width="60%" style="text-align:left"
!Derivation of Reissner–Stein cantilever plate equations
|-
|The strain energy of bending of a thin rectangular plate of uniform thickness <math>h</math> is given by
:<math>
   U = \frac{1}{2} \int_0^a \int_{-b/2}^{b/2}D\left\{\left(\frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2}\right)^2 +
     2(1-\nu)\left[\left(\frac{\partial^2 w}{\partial x \partial y}\right)^2 - \frac{\partial^2 w}{\partial x^2}\frac{\partial^2 w}{\partial y^2}\right]
     \right\}\text{d}x\text{d}y
</math>
where <math>w</math> is the transverse displacement, <math>a</math> is the length, <math>b</math> is the width, <math>\nu</math> is the Poisson's
ratio, <math>E</math> is the Young's modulus, and
:<math>
  D = \frac{Eh^3}{12(1-\nu)}.
</math>
The potential energy of transverse loads <math>q(x,y)</math> (per unit length) is
:<math>
  P_q = \int_0^a \int_{-b/2}^{b/2}q(x,y)\, w(x,y)\,\text{d}x\text{d}y \,.
</math>
The potential energy of in-plane loads <math>n_x(x,y)</math> (per unit width) is
:<math>
  P_n = \frac{1}{2} \int_0^a \int_{-b/2}^{b/2}n_x(x,y)\,\left(\frac{\partial w}{\partial x}\right)^2\,\text{d}x\text{d}y \,.
</math>
The potential energy of tip forces <math>q_x(y)</math> (per unit width), and bending moments <math>m_x(y)</math> and <math>m_{xy}(y)</math>
(per unit width) is
:<math>
  P_t = \int_{-b/2}^{b/2}\left(q_x(y)\,w(x,y) - m_x(y)\,\frac{\partial w}{\partial x} + m_{xy}(y)\,\frac{\partial w}{\partial y}\right)\text{d}x\text{d}y \,.
</math>
A balance of energy requires that the total energy is
:<math>
  W = U - (P_q + P_n + P_t) \,.
</math>
With the Reissener–Stein assumption for the displacement, we have
:<math>
  U = \int_0^a\frac{bD}{24}\left[12\left(\cfrac{d^2 w_x}{d x^2}\right)^2 +
       b^2\left(\cfrac{d^2 \theta_x}{d x^2}\right)^2 + 24(1-\nu)\left(\cfrac{d \theta_x}{d x}\right)^2\right]\,\text{d}x\,,
</math>
:<math>
  P_q = \int_0^a\left[\left(\int_{-b/2}^{b/2}q(x,y)\,\text{d}y\right)w_x + \left(\int_{-b/2}^{b/2}yq(x,y)\,\text{d}y\right)\theta_x\right]\,dx \,,
</math>
:<math>
  \begin{align}
  P_n & = \frac{1}{2}\int_0^a\left[\left(\int_{-b/2}^{b/2}n_x(x,y)\,\text{d}y\right)\left(\cfrac{d w_x}{d x}\right)^2 +
    \left(\int_{-b/2}^{b/2}y n_x(x,y)\,\text{d}y\right)\cfrac{d w_x}{d x}\,\cfrac{d \theta_x}{d x} \right.\\
    & \left. \qquad\qquad +\left(\int_{-b/2}^{b/2}y^2 n_x(x,y)\,\text{d}y\right)\left(\cfrac{d \theta_x}{d x}\right)^2\right]\text{d}x\,,
  \end{align}
</math>
and
:<math>
  \begin{align}
  P_t & = \left(\int_{-b/2}^{b/2}q_x(y)\,\text{d}y\right)w_x -
        \left(\int_{-b/2}^{b/2}m_x(y)\,\text{d}y\right)\cfrac{d w_x}{d x} +
        \left[\int_{-b/2}^{b/2}\left(y q_x(y) + m_{xy}(y)\right)\,\text{d}y\right]\theta_x  \\
      & \qquad \qquad  -\left(\int_{-b/2}^{b/2}y m_x(y)\,\text{d}y\right)\cfrac{d \theta_x}{d x} \,.
  \end{align}
</math>
Taking the first variation of <math>W</math> with respect to <math>(w_x, \theta_x, x)</math> and
setting it to zero gives us the Euler equations
:<math> \text{(1)} \qquad
  bD \frac{\mathrm{d}^4w_x}{\mathrm{d}x^4}
   = q_1(x) - n_1(x)\cfrac{d^2 w_x}{d x^2} - \cfrac{d n_1}{d x}\,\cfrac{d w_x}{d x}
     - \frac{1}{2}\cfrac{d n_2}{d x}\,\cfrac{d \theta_x}{d x} - \frac{n_2(x)}{2}\cfrac{d^2 \theta_x}{d x^2}
</math>
and
:<math> \text{(2)} \qquad
  \frac{b^3D}{12}\,\frac{\mathrm{d}^4\theta_x}{\mathrm{d}x^4} - 2bD(1-\nu)\cfrac{d^2 \theta_x}{d x^2}
   = q_2(x) - n_3(x)\cfrac{d^2 \theta_x}{d x^2} - \cfrac{d n_3}{d x}\,\cfrac{d \theta_x}{d x}
    - \frac{n_2(x)}{2}\,\cfrac{d^2 w_x}{d x^2} - \frac{1}{2}\cfrac{d n_2}{d x}\,\cfrac{d w_x}{d x}
</math>
where
:<math>
  \begin{align}
    q_1(x) & = \int_{-b/2}^{b/2}q(x,y)\,\text{d}y ~,~~ q_2(x) = \int_{-b/2}^{b/2}y\,q(x,y)\,\text{d}y~,~~
    n_1(x) = \int_{-b/2}^{b/2}n_x(x,y)\,\text{d}y \\
    n_2(x) & = \int_{-b/2}^{b/2}y\,n_x(x,y)\,\text{d}y ~,~~ n_3(x)  = \int_{-b/2}^{b/2}y^2\,n_x(x,y)\,\text{d}y.
  \end{align}
</math>
Since the beam is clamped at <math>x = 0</math>, we have
:<math>
  w(0,y) = \cfrac{d w}{d x}\Bigr|_{x=0} = 0 \qquad \implies \qquad
  w_x(0) = \cfrac{d w_x}{d x}\Bigr|_{x=0} = \theta_x(0) = \cfrac{d \theta_x}{d x}\Bigr|_{x=0} = 0 \,.
</math>
The boundary conditions at <math>x = a</math> can be found by integration by parts:
:<math>
  \begin{align}
  & bD\cfrac{d^3 w_x}{d x^3} + n_1(x)\cfrac{d w_x}{d x} + n_2(x)\cfrac{d \theta_x}{d x} + q_{x1} = 0 \\
  & \frac{b^3D}{12}\cfrac{d^3 \theta_x}{d x^3} + \left[n_3(x) -2bD(1-\nu)\right]\cfrac{d \theta_x}{d x}
    + n_2(x)\cfrac{d w_x}{d x} + t = 0 \\
  & bD\cfrac{d^2 w_x}{d x^2} + m_1 = 0 \quad,\quad  \frac{b^3D}{12}\cfrac{d^2 \theta_x}{d x^2} + m_2 = 0
  \end{align}
</math>
where
:<math>
  \begin{align}
    m_1 & = \int_{-b/2}^{b/2}m_x(y)\,\text{d}y ~,~~ m_2 = \int_{-b/2}^{b/2}y\,m_x(y)\,\text{d}y ~,~~
    q_{x1} = \int_{-b/2}^{b/2}q_x(y)\,\text{d}y \\
    t  & = q_{x2} + m_3 = \int_{-b/2}^{b/2}y\,q_x(y)\,\text{d}y + \int_{-b/2}^{b/2}m_{xy}(y)\,\text{d}y.
  \end{align}
</math>
|}

== See also ==
* [[Bending of plates]]
* [[Vibration of plates]]
* [[Infinitesimal strain theory]]
* [[Membrane theory of shells]]
*[[Finite strain theory]]
*[[Stress (mechanics)]]
*[[Stress resultants]]
*[[Linear elasticity]]
*[[Bending]]
*[[Föppl–von Kármán equations]]
*[[Euler–Bernoulli beam equation]]
*[[Timoshenko beam theory]]

== References ==
<references/>

{{Structural engineering topics}}
{{DEFAULTSORT:Plate Theory}}
[[Category:Continuum mechanics]]