Mindlin–Reissner plate theory

The Mindlin hypothesis implies that the displacements in the plate have the form

${\displaystyle {\begin{aligned}u_{\alpha }(\mathbf {x} )&=u_{\alpha }^{0}(x_{1},x_{2})-x_{3}~\varphi _{\alpha }~;~~\alpha =1,2\\u_{3}(\mathbf {x} )&=w^{0}(x_{1},x_{2})\end{aligned}}}$

where ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}}$ are the Cartesian coordinates on the mid-surface of the undeformed plate and ${\displaystyle x_{3}}$ is the coordinate for the thickness direction, ${\displaystyle u_{\alpha }^{0},~\alpha =1,2}$ are the in-plane displacements of the mid-surface, ${\displaystyle w^{0}}$ is the displacement of the mid-surface in the ${\displaystyle x_{3}}$ direction, ${\displaystyle \varphi _{1}}$ and ${\displaystyle \varphi _{2}}$ designate the angles which the normal to the mid-surface makes with the ${\displaystyle x_{3}}$ axis. Unlike Kirchhoff–Love plate theory where ${\displaystyle \varphi _{\alpha }}$ are directly related to ${\displaystyle w^{0}}$, Mindlin's theory does not require that ${\displaystyle \varphi _{1}=w_{,1}^{0}}$ and ${\displaystyle \varphi _{2}=w_{,2}^{0}}$.

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

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

${\displaystyle {\begin{aligned}\varepsilon _{\alpha \beta }&={\frac {1}{2}}(u_{\alpha ,\beta }^{0}+u_{\beta ,\alpha }^{0})-{\frac {x_{3}}{2}}~(\varphi _{\alpha ,\beta }+\varphi _{\beta ,\alpha })\\\varepsilon _{\alpha 3}&={\cfrac {1}{2}}\left(w_{,\alpha }^{0}-\varphi _{\alpha }\right)\\\varepsilon _{33}&=0\end{aligned}}}$

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 (${\displaystyle \kappa }$) is applied so that the correct amount of internal energy is predicted by the theory. Then

${\displaystyle \varepsilon _{\alpha 3}={\cfrac {1}{2}}~\kappa ~\left(w_{,\alpha }^{0}-\varphi _{\alpha }\right)}$

The equilibrium equations of a Mindlin–Reissner plate for small strains and small rotations have the form

${\displaystyle {\begin{aligned}&N_{\alpha \beta ,\alpha }=0\\&M_{\alpha \beta ,\beta }-Q_{\alpha }=0\\&Q_{\alpha ,\alpha }+q=0\end{aligned}}}$

where ${\displaystyle q}$ is an applied out-of-plane load, the in-plane stress resultants are defined as

${\displaystyle N_{\alpha \beta }:=\int _{-h}^{h}\sigma _{\alpha \beta }~dx_{3}\,,}$

the moment resultants are defined as

${\displaystyle M_{\alpha \beta }:=\int _{-h}^{h}x_{3}~\sigma _{\alpha \beta }~dx_{3}\,,}$

and the shear resultants are defined as

${\displaystyle Q_{\alpha }:=\kappa ~\int _{-h}^{h}\sigma _{\alpha 3}~dx_{3}\,.}$

Derivation of equilibrium equations

For the situation where the strains and rotations of the plate are small the virtual internal energy is given by ${\displaystyle {\begin{aligned}\delta U&=\int _{\Omega ^{0}}\int _{-h}^{h}{\boldsymbol {\sigma }}:\delta {\boldsymbol {\epsilon }}~dx_{3}~d\Omega =\int _{\Omega ^{0}}\int _{-h}^{h}\left[\sigma _{\alpha \beta }~\delta \varepsilon _{\alpha \beta }+2~\sigma _{\alpha 3}~\delta \varepsilon _{\alpha 3}\right]~dx_{3}~d\Omega \\&=\int _{\Omega ^{0}}\int _{-h}^{h}\left[{\frac {1}{2}}~\sigma _{\alpha \beta }~(\delta u_{\alpha ,\beta }^{0}+\delta u_{\beta ,\alpha }^{0})-{\frac {x_{3}}{2}}~\sigma _{\alpha \beta }~(\delta \varphi _{\alpha ,\beta }+\delta \varphi _{\beta ,\alpha })+\kappa ~\sigma _{\alpha 3}\left(\delta w_{,\alpha }^{0}-\delta \varphi _{\alpha }\right)\right]~dx_{3}~d\Omega \\&=\int _{\Omega ^{0}}\left[{\frac {1}{2}}~N_{\alpha \beta }~(\delta u_{\alpha ,\beta }^{0}+\delta u_{\beta ,\alpha }^{0})-{\frac {1}{2}}M_{\alpha \beta }~(\delta \varphi _{\alpha ,\beta }+\delta \varphi _{\beta ,\alpha })+Q_{\alpha }\left(\delta w_{,\alpha }^{0}-\delta \varphi _{\alpha }\right)\right]~d\Omega \end{aligned}}}$ where the stress resultants and stress moment resultants are defined in a way similar to that for Kirchhoff plates. The shear resultant is defined as ${\displaystyle Q_{\alpha }:=\kappa ~\int _{-h}^{h}\sigma _{\alpha 3}~dx_{3}}$ Integration by parts gives ${\displaystyle {\begin{aligned}\delta U&=\int _{\Omega ^{0}}\left[-{\frac {1}{2}}~(N_{\alpha \beta ,\beta }~\delta u_{\alpha }^{0}+N_{\alpha \beta ,\alpha }~\delta u_{\beta }^{0})+{\frac {1}{2}}(M_{\alpha \beta ,\beta }~\delta \varphi _{\alpha }+M_{\alpha \beta ,\alpha }\delta \varphi _{\beta })-Q_{\alpha ,\alpha }~\delta w^{0}-Q_{\alpha }~\delta \varphi _{\alpha }\right]~d\Omega \\&+\int _{\Gamma ^{0}}\left[{\frac {1}{2}}~(n_{\beta }~N_{\alpha \beta }~\delta u_{\alpha }^{0}+n_{\alpha }~N_{\alpha \beta }~\delta u_{\beta }^{0})-{\frac {1}{2}}(n_{\beta }~M_{\alpha \beta }~\delta \varphi _{\alpha }+n_{\alpha }M_{\alpha \beta }\delta \varphi _{\beta })+n_{\alpha }~Q_{\alpha }~\delta w^{0}\right]~d\Gamma \end{aligned}}}$ The symmetry of the stress tensor implies that ${\displaystyle N_{\alpha \beta }=N_{\beta \alpha }}$ and ${\displaystyle M_{\alpha \beta }=M_{\beta \alpha }}$. Hence, ${\displaystyle {\begin{aligned}\delta U&=\int _{\Omega ^{0}}\left[-N_{\alpha \beta ,\alpha }~\delta u_{\beta }^{0}+\left(M_{\alpha \beta ,\beta }-Q_{\alpha }\right)~\delta \varphi _{\alpha }-Q_{\alpha ,\alpha }~\delta w^{0}\right]~d\Omega \\&+\int _{\Gamma ^{0}}\left[n_{\alpha }~N_{\alpha \beta }~\delta u_{\beta }^{0}-n_{\beta }~M_{\alpha \beta }~\delta \varphi _{\alpha }+n_{\alpha }~Q_{\alpha }~\delta w^{0}\right]~d\Gamma \end{aligned}}}$ For the special case when the top surface of the plate is loaded by a force per unit area ${\displaystyle q(\mathbf {x} ^{0})}$, the virtual work done by the external forces is ${\displaystyle \delta V_{\mathrm {ext} }=\int _{\Omega ^{0}}q~\delta w^{0}~\mathrm {d} \Omega }$ Then, from the principle of virtual work, ${\displaystyle {\begin{aligned}&\int _{\Omega ^{0}}\left[N_{\alpha \beta ,\alpha }~\delta u_{\beta }^{0}-\left(M_{\alpha \beta ,\beta }-Q_{\alpha }\right)~\delta \varphi _{\alpha }+\left(Q_{\alpha ,\alpha }+q\right)~\delta w^{0}\right]~d\Omega \\&\qquad \qquad =\int _{\Gamma ^{0}}\left[n_{\alpha }~N_{\alpha \beta }~\delta u_{\beta }^{0}-n_{\beta }~M_{\alpha \beta }~\delta \varphi _{\alpha }+n_{\alpha }~Q_{\alpha }~\delta w^{0}\right]~d\Gamma \end{aligned}}}$ Using standard arguments from the calculus of variations, the equilibrium equations for a Mindlin–Reissner plate are ${\displaystyle {\begin{aligned}&N_{\alpha \beta ,\alpha }=0\\&M_{\alpha \beta ,\beta }-Q_{\alpha }=0\\&Q_{\alpha ,\alpha }+q=0\end{aligned}}}$

Bending moments and normal stresses	Torques and shear stresses

Shear resultant and shear stresses

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

${\displaystyle {\begin{aligned}n_{\alpha }~N_{\alpha \beta }&\quad \mathrm {or} \quad u_{\beta }^{0}\\n_{\alpha }~M_{\alpha \beta }&\quad \mathrm {or} \quad \varphi _{\alpha }\\n_{\alpha }~Q_{\alpha }&\quad \mathrm {or} \quad w^{0}\end{aligned}}}$

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

${\displaystyle {\begin{aligned}\sigma _{\alpha \beta }&=C_{\alpha \beta \gamma \theta }~\varepsilon _{\gamma \theta }\\\sigma _{\alpha 3}&=C_{\alpha 3\gamma \theta }~\varepsilon _{\gamma \theta }\\\sigma _{33}&=C_{33\gamma \theta }~\varepsilon _{\gamma \theta }\end{aligned}}}$

Since ${\displaystyle \sigma _{33}}$ does not appear in the equilibrium equations it is implicitly assumed that it does 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

${\displaystyle {\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{23}\\\sigma _{31}\\\sigma _{12}\end{bmatrix}}={\begin{bmatrix}C_{11}&C_{12}&0&0&0\\C_{12}&C_{22}&0&0&0\\0&0&C_{44}&0&0\\0&0&0&C_{55}&0\\0&0&0&0&C_{66}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{23}\\\varepsilon _{31}\\\varepsilon _{12}\end{bmatrix}}}$

Then

${\displaystyle {\begin{aligned}{\begin{bmatrix}N_{11}\\N_{22}\\N_{12}\end{bmatrix}}&=\int _{-h}^{h}{\begin{bmatrix}C_{11}&C_{12}&0\\C_{12}&C_{22}&0\\0&0&C_{66}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{12}\end{bmatrix}}dx_{3}\\[5pt]&=\left\{\int _{-h}^{h}{\begin{bmatrix}C_{11}&C_{12}&0\\C_{12}&C_{22}&0\\0&0&C_{66}\end{bmatrix}}~dx_{3}\right\}{\begin{bmatrix}u_{1,1}^{0}\\u_{2,2}^{0}\\{\frac {1}{2}}~(u_{1,2}^{0}+u_{2,1}^{0})\end{bmatrix}}\end{aligned}}}$

and

${\displaystyle {\begin{aligned}{\begin{bmatrix}M_{11}\\M_{22}\\M_{12}\end{bmatrix}}&=\int _{-h}^{h}x_{3}~{\begin{bmatrix}C_{11}&C_{12}&0\\C_{12}&C_{22}&0\\0&0&C_{66}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{12}\end{bmatrix}}dx_{3}\\[5pt]&=-\left\{\int _{-h}^{h}x_{3}^{2}~{\begin{bmatrix}C_{11}&C_{12}&0\\C_{12}&C_{22}&0\\0&0&C_{66}\end{bmatrix}}~dx_{3}\right\}{\begin{bmatrix}\varphi _{1,1}\\\varphi _{2,2}\\{\frac {1}{2}}(\varphi _{1,2}+\varphi _{2,1})\end{bmatrix}}\end{aligned}}}$

For the shear terms

${\displaystyle {\begin{bmatrix}Q_{1}\\Q_{2}\end{bmatrix}}=\kappa ~\int _{-h}^{h}{\begin{bmatrix}C_{55}&0\\0&C_{44}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{31}\\\varepsilon _{32}\end{bmatrix}}dx_{3}={\cfrac {\kappa }{2}}\left\{\int _{-h}^{h}{\begin{bmatrix}C_{55}&0\\0&C_{44}\end{bmatrix}}~dx_{3}\right\}{\begin{bmatrix}w_{,1}^{0}-\varphi _{1}\\w_{,2}^{0}-\varphi _{2}\end{bmatrix}}}$

The extensional stiffnesses are the quantities

${\displaystyle A_{\alpha \beta }:=\int _{-h}^{h}C_{\alpha \beta }~dx_{3}}$

The bending stiffnesses are the quantities

${\displaystyle D_{\alpha \beta }:=\int _{-h}^{h}x_{3}^{2}~C_{\alpha \beta }~dx_{3}\,.}$

The relations between the stress resultants and the generalized deformations are,

${\displaystyle {\begin{aligned}{\begin{bmatrix}N_{11}\\N_{22}\\N_{12}\end{bmatrix}}&={\cfrac {2Eh}{1-\nu ^{2}}}{\begin{bmatrix}1&\nu &0\\\nu &1&0\\0&0&1-\nu \end{bmatrix}}{\begin{bmatrix}u_{1,1}^{0}\\u_{2,2}^{0}\\{\frac {1}{2}}~(u_{1,2}^{0}+u_{2,1}^{0})\end{bmatrix}},\\[5pt]{\begin{bmatrix}M_{11}\\M_{22}\\M_{12}\end{bmatrix}}&=-{\cfrac {2Eh^{3}}{3(1-\nu ^{2})}}{\begin{bmatrix}1&\nu &0\\\nu &1&0\\0&0&1-\nu \end{bmatrix}}{\begin{bmatrix}\varphi _{1,1}\\\varphi _{2,2}\\{\frac {1}{2}}(\varphi _{1,2}+\varphi _{2,1})\end{bmatrix}},\end{aligned}}}$

and

${\displaystyle {\begin{bmatrix}Q_{1}\\Q_{2}\end{bmatrix}}=\kappa G2h{\begin{bmatrix}w_{,1}^{0}-\varphi _{1}\\w_{,2}^{0}-\varphi _{2}\end{bmatrix}}\,.}$

The bending rigidity is defined as the quantity

${\displaystyle D={\cfrac {2Eh^{3}}{3(1-\nu ^{2})}}\,.}$

For a plate of thickness ${\displaystyle h}$ (${\displaystyle h}$ of the following all indicates thickness), the bending rigidity has the form

${\displaystyle D={\cfrac {Eh^{3}}{12(1-\nu ^{2})}}\,.}$

If we ignore the in-plane extension of the plate, the governing equations are

${\displaystyle {\begin{aligned}M_{\alpha \beta ,\beta }-Q_{\alpha }&=0\\Q_{\alpha ,\alpha }+q&=0\,.\end{aligned}}}$

In terms of the generalized deformations, these equations can be written as

${\displaystyle {\begin{aligned}&\nabla ^{2}\left({\frac {\partial \varphi _{1}}{\partial x_{1}}}+{\frac {\partial \varphi _{2}}{\partial x_{2}}}\right)={\frac {q}{D}}\\&\nabla ^{2}w^{0}-{\frac {\partial \varphi _{1}}{\partial x_{1}}}-{\frac {\partial \varphi _{2}}{\partial x_{2}}}=-{\frac {q}{\kappa Gh}}\\&\nabla ^{2}\left({\frac {\partial \varphi _{1}}{\partial x_{2}}}-{\frac {\partial \varphi _{2}}{\partial x_{1}}}\right)={\frac {2\kappa Gh}{D(1-\nu )}}\left({\frac {\partial \varphi _{1}}{\partial x_{2}}}-{\frac {\partial \varphi _{2}}{\partial x_{1}}}\right)\,.\end{aligned}}}$

Derivation of equilibrium equations in terms of deformations

If we expand out the governing equations of a Mindlin plate, we have ${\displaystyle {\begin{aligned}{\frac {\partial M_{11}}{\partial x_{1}}}+{\frac {\partial M_{12}}{\partial x_{2}}}&=Q_{1}\quad \,,\quad {\frac {\partial M_{21}}{\partial x_{1}}}+{\frac {\partial M_{22}}{\partial x_{2}}}=Q_{2}\\{\frac {\partial Q_{1}}{\partial x_{1}}}+{\frac {\partial Q_{2}}{\partial x_{2}}}&=-q\,.\end{aligned}}}$ Recalling that ${\displaystyle M_{11}=-D\left({\frac {\partial \varphi _{1}}{\partial x_{1}}}+\nu {\frac {\partial \varphi _{2}}{\partial x_{2}}}\right)~,~~M_{22}=-D\left({\frac {\partial \varphi _{2}}{\partial x_{2}}}+\nu {\frac {\partial \varphi _{1}}{\partial x_{1}}}\right)~,~~M_{12}=-{\frac {D(1-\nu )}{2}}\left({\frac {\partial \varphi _{1}}{\partial x_{2}}}+{\frac {\partial \varphi _{2}}{\partial x_{1}}}\right)}$ and combining the three governing equations, we have ${\displaystyle {\frac {\partial ^{3}\varphi _{1}}{\partial x_{1}^{3}}}+{\frac {\partial ^{3}\varphi _{1}}{\partial x_{1}\,\partial x_{2}^{2}}}+{\frac {\partial ^{3}\varphi _{2}}{\partial x_{1}^{2}\,\partial x_{2}}}+{\frac {\partial ^{3}\varphi _{2}}{\partial x_{2}^{3}}}={\frac {q}{D}}\,.}$ If we define ${\displaystyle {\mathcal {M}}:=D\left({\frac {\partial \varphi _{1}}{\partial x_{1}}}+{\frac {\partial \varphi _{2}}{\partial x_{2}}}\right)}$ we can write the above equation as ${\displaystyle \nabla ^{2}{\mathcal {M}}=q\,.}$ Similarly, using the relationships between the shear force resultants and the deformations, and the equation for the balance of shear force resultants, we can show that ${\displaystyle \kappa Gh\left(\nabla ^{2}w^{0}-{\frac {\mathcal {M}}{D}}\right)=-q\,.}$ Since there are three unknowns in the problem, ${\displaystyle \varphi _{1}}$, ${\displaystyle \varphi _{2}}$, and ${\displaystyle w^{0}}$, we need a third equation which can be found by differentiating the expressions for the shear force resultants and the governing equations in terms of the moment resultants, and equating these. The resulting equation has the form ${\displaystyle \nabla ^{2}\left({\frac {\partial \varphi _{1}}{\partial x_{2}}}-{\frac {\partial \varphi _{2}}{\partial x_{1}}}\right)={\frac {2\kappa Gh}{D(1-\nu )}}\left({\frac {\partial \varphi _{1}}{\partial x_{2}}}-{\frac {\partial \varphi _{2}}{\partial x_{1}}}\right)\,.}$ Therefore, the three governing equations in terms of the deformations are ${\displaystyle {\begin{aligned}&\nabla ^{2}\left({\frac {\partial \varphi _{1}}{\partial x_{1}}}+{\frac {\partial \varphi _{2}}{\partial x_{2}}}\right)={\frac {q}{D}}\\&\nabla ^{2}w^{0}-{\frac {\partial \varphi _{1}}{\partial x_{1}}}-{\frac {\partial \varphi _{2}}{\partial x_{2}}}=-{\frac {q}{\kappa Gh}}\\&\nabla ^{2}\left({\frac {\partial \varphi _{1}}{\partial x_{2}}}-{\frac {\partial \varphi _{2}}{\partial x_{1}}}\right)={\frac {2\kappa Gh}{D(1-\nu )}}\left({\frac {\partial \varphi _{1}}{\partial x_{2}}}-{\frac {\partial \varphi _{2}}{\partial x_{1}}}\right)\,.\end{aligned}}}$

The boundary conditions along the edges of a rectangular plate are

${\displaystyle {\begin{aligned}{\text{simply supported}}\quad &\quad w^{0}=0,M_{11}=0~({\text{or}}~M_{22}=0),\varphi _{1}=0~({\text{ or }}\varphi _{2}=0)\\{\text{clamped}}\quad &\quad w^{0}=0,\varphi _{1}=0,\varphi _{2}=0\,.\end{aligned}}}$