Reissner-Mindlin plate theory

Mindlin theory

Mindlin's theory was originally derived for isotropic plates using equilibrium considerations. A more general version of the theory based on energy considerations is discussed here.

Assumed displacement field

The Mindlin hypothesis implies that the displacements in the plate have the form : \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} where x_1 and x_2 are the Cartesian coordinates on the mid-surface of the undeformed plate and x_3 is the coordinate for the thickness direction, u^0_\alpha,~ \alpha=1,2 are the in-plane displacements of the mid-surface, w^0 is the displacement of the mid-surface in the x_3 direction, \varphi_1 and \varphi_2 designate the angles which the normal to the mid-surface makes with the x_3 axis. Unlike Kirchhoff–Love plate theory where \varphi_\alpha are directly related to w^0, Mindlin's theory does not require that \varphi_1 = w^0_{,1} and \varphi_2 = w^0_{,2}.

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 : \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} 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 (\kappa) is applied so that the correct amount of internal energy is predicted by the theory. Then : \varepsilon_{\alpha 3} = \cfrac{1}{2}\kappa~\left(w^0_{,\alpha}- \varphi_\alpha\right)

Equilibrium equations

The equilibrium equations of a Mindlin–Reissner plate for small strains and small rotations have the form : \begin{align} & N_{\alpha\beta,\alpha} = 0 \ & M_{\alpha\beta,\beta}-Q_\alpha = 0 \ & Q_{\alpha,\alpha}+q = 0 \end{align} where q is an applied out-of-plane load, the in-plane stress resultants are defined as : N_{\alpha\beta} := \int_{-h}^h \sigma_{\alpha\beta}dx_3 ,, the moment resultants are defined as : M_{\alpha\beta} := \int_{-h}^h x_3\sigma_{\alpha\beta}dx_3 ,, and the shear resultants are defined as : Q_\alpha := \kappa\int_{-h}^h \sigma_{\alpha 3}dx_3 ,. :{| class="toccolours collapsible collapsed" width="60%" style="text-align:left" !Derivation of equilibrium equations |- | For the situation where the strains and rotations of the plate are small the virtual internal energy is given by : \begin{align} \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^0_{\alpha,\beta}+\delta u^0_{\beta,\alpha}) - \frac{x_3}{2}\sigma_{\alpha\beta}(\delta \varphi_{\alpha,\beta}+\delta\varphi_{\beta,\alpha}) + \kappa~\sigma_{\alpha 3}\left(\delta w^0_{,\alpha} - \delta \varphi_\alpha\right)\right]dx_3~d\Omega \ & = \int_{\Omega^0} \left[\frac{1}{2}N_{\alpha\beta}(\delta u^0_{\alpha,\beta}+\delta u^0_{\beta,\alpha}) - \frac{1}{2}M_{\alpha\beta}(\delta \varphi_{\alpha,\beta}+\delta\varphi_{\beta,\alpha}) + Q_\alpha\left(\delta w^0_{,\alpha} - \delta \varphi_\alpha\right)\right]d\Omega \end{align} 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 : Q_\alpha := \kappa\int_{-h}^h \sigma_{\alpha 3}dx_3 Integration by parts gives : \begin{align} \delta U & = \int_{\Omega^0} \left[-\frac{1}{2}(N_{\alpha\beta,\beta}\delta u^0_{\alpha}+N_{\alpha\beta,\alpha}\delta u^0_{\beta})

• \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^0_\alpha+n_\alpha~N_{\alpha\beta}\delta u^0_{\beta})

• \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{align} The symmetry of the stress tensor implies that N_{\alpha\beta} = N_{\beta\alpha} and M_{\alpha\beta} = M_{\beta\alpha}. Hence, : \begin{align} \delta U & = \int_{\Omega^0} \left[-N_{\alpha\beta,\alpha}\delta u^0_{\beta}

• \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^0_{\beta}

• n_\beta~M_{\alpha\beta}\delta \varphi_{\alpha} + n_\alpha~Q_\alpha\delta w^0\right]d\Gamma \end{align} For the special case when the top surface of the plate is loaded by a force per unit area q(\mathbf{x}^0), the virtual work done by the external forces is : \delta V_{\mathrm{ext}} = \int_{\Omega^0} q\delta w^0~\mathrm{d}\Omega Then, from the principle of virtual work, : \begin{align} & \int_{\Omega^0} \left[N_{\alpha\beta,\alpha}~\delta u^0_{\beta}
• \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^0_{\beta}
• n_\beta~M_{\alpha\beta}\delta \varphi_{\alpha} + n_\alpha~Q_\alpha\delta w^0\right]~d\Gamma \end{align} Using standard arguments from the calculus of variations, the equilibrium equations for a Mindlin–Reissner plate are : \begin{align} & N_{\alpha\beta,\alpha} = 0 \ & M_{\alpha\beta,\beta}-Q_\alpha = 0 \ & Q_{\alpha,\alpha}+q = 0 \end{align} |}

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 : \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}

Stress–strain relations

The stress–strain relations for a linear elastic Mindlin–Reissner plate are given by : \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} Since \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 : \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 : \begin{align} \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^0_{1,1} \ u^0_{2,2} \ \frac{1}{2}(u^0_{1,2}+u^0_{2,1}) \end{bmatrix} \end{align} and : \begin{align} \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{align} For the shear terms : \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^0_{,1} - \varphi_1 \ w^0_{,2} - \varphi_2 \end{bmatrix} The ** extensional stiffnesses** are the quantities : A_{\alpha\beta} := \int_{-h}^h C_{\alpha\beta}~dx_3 The ** bending stiffnesses** are the quantities : D_{\alpha\beta} := \int_{-h}^h x_3^2~C_{\alpha\beta}~dx_3 ,.

Mindlin theory for isotropic plates

For uniformly thick, homogeneous, and isotropic plates, the stress–strain relations in the plane of the plate 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 & \cfrac{1-\nu}{2} \end{bmatrix} \begin{bmatrix}\varepsilon_{11} \ \varepsilon_{22} \ 2\varepsilon_{12} \end{bmatrix} ,. where E is the Young's modulus, \nu is the Poisson's ratio, and \varepsilon_{\alpha\beta} are the in-plane strains. The through-the-thickness shear stresses and strains are related by : \sigma_{31} = 2G\varepsilon_{31} \quad \text{and} \quad \sigma_{32} = 2G\varepsilon_{32} where G = E/(2(1+\nu)) is the shear modulus.

Constitutive relations

The relations between the stress resultants and the generalized deformations are, : \begin{align} \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}, \[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{align} and : \begin{bmatrix}Q_1 \ Q_2 \end{bmatrix} = \kappa G 2 h \begin{bmatrix} w^0_{,1} - \varphi_1 \ w^0_{,2} - \varphi_2 \end{bmatrix} ,.

In the above, : D = \cfrac{2Eh^3}{3(1-\nu^2)} ,. is referred to as the bending rigidity (or bending modulus).

For a plate of thickness \tilde{h}=2h, the bending rigidity has the form : D = \cfrac{E\tilde{h}^3}{12(1-\nu^2)} ,. from now on, in all the equations below, we will refer to h as the total thickness of the plate, and as not the semi-thickness (as in the above equations).

Governing equations

If we ignore the in-plane extension of the plate, the governing equations are : \begin{align} M_{\alpha\beta,\beta}-Q_\alpha & = 0 \ Q_{\alpha,\alpha}+q & = 0 ,. \end{align} In terms of the generalized deformations, these equations can be written as : \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} :{| class="toccolours collapsible collapsed" width="60%" style="text-align:left" !Derivation of equilibrium equations in terms of deformations |- | If we expand out the governing equations of a Mindlin plate, we have : \begin{align} \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{align} Recalling that : 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 : \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 : \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 : \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 : \kappa G h \left(\nabla^2 w^0 - \frac{\mathcal{M}}{D}\right) = -q ,. Since there are three unknowns in the problem, \varphi_1, \varphi_2, and 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 : \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) ,. Therefore, the three governing equations in terms of the deformations are : \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} |} The boundary conditions along the edges of a rectangular plate are : \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}

Relationship to Reissner's theory

The canonical constitutive relations for shear deformation theories of isotropic plates can be expressed asLim, G. T. and Reddy, J. N., 2003, On canonical bending relationships for plates, International Journal of Solids and Structures, vol. 40, pp. 3039–3067.These equations use a slightly different sign convention than the preceding discussion. : \begin{align} M_{11} & = D\left[\mathcal{A}\left(\frac{\partial \varphi_1}{\partial x_1}+\nu\frac{\partial \varphi_2}{\partial x_2}\right)

• (1-\mathcal{A})\left(\frac{\partial^2 w^0}{\partial x_1^2} + \nu\frac{\partial^2 w^0}{\partial x_2^2}\right)\right]

• \frac{q}{1-\nu},\mathcal{B}\[5pt] M_{22} & = D\left[\mathcal{A}\left(\frac{\partial \varphi_2}{\partial x_2}+\nu\frac{\partial \varphi_1}{\partial x_1}\right)

• (1-\mathcal{A})\left(\frac{\partial^2 w^0}{\partial x_2^2} + \nu\frac{\partial^2 w^0}{\partial x_1^2}\right)\right]

• \frac{q}{1-\nu},\mathcal{B}\[5pt] M_{12} & = \frac{D(1-\nu)}{2}\left[\mathcal{A}\left(\frac{\partial \varphi_1}{\partial x_2}+\frac{\partial \varphi_2}{\partial x_1}\right)

• 2(1-\mathcal{A}),\frac{\partial^2 w^0}{\partial x_1 \partial x_2}\right] \ Q_1 & = \mathcal{A} \kappa G h\left(\varphi_1 + \frac{\partial w^0}{\partial x_1}\right) \[5pt] Q_2 & = \mathcal{A} \kappa G h\left(\varphi_2 + \frac{\partial w^0}{\partial x_2}\right) ,. \end{align} Note that the plate thickness is h (and not 2h) in the above equations and D = Eh^3/[12(1-\nu^2)]. If we define a Marcus moment, : \mathcal{M} = D\left[\mathcal{A}\left(\frac{\partial \varphi_1}{\partial x_1} + \frac{\partial \varphi_2}{\partial x_2}\right)
• (1-\mathcal{A})\nabla^2 w^0\right] + \frac{2q}{1-\nu^2}\mathcal{B} we can express the shear resultants as : \begin{align} Q_1 & = \frac{\partial \mathcal{M}}{\partial x_1}

• \frac{D(1-\nu)}{2}\left[\mathcal{A}\frac{\partial }{\partial x_2}\left(\frac{\partial \varphi_1}{\partial x_2} -\frac{\partial \varphi_2}{\partial x_1}\right)\right] - \frac{\mathcal{B}}{1+\nu}\frac{\partial q}{\partial x_1} \[5pt] Q_2 & = \frac{\partial \mathcal{M}}{\partial x_2}

• \frac{D(1-\nu)}{2}\left[\mathcal{A}\frac{\partial }{\partial x_1}\left(\frac{\partial \varphi_1}{\partial x_2} -\frac{\partial \varphi_2}{\partial x_1}\right)\right] - \frac{\mathcal{B}}{1+\nu}\frac{\partial q}{\partial x_2},. \end{align} These relations and the governing equations of equilibrium, when combined, lead to the following canonical equilibrium equations in terms of the generalized displacements. : \begin{align} & \nabla^2 \left(\mathcal{M} - \frac{\mathcal{B}}{1+\nu},q\right) = -q \ & \kappa G h\left(\nabla^2 w^0 + \frac{\mathcal{M}}{D}\right) = -\left(1 - \cfrac{\mathcal{B} c^2}{1+\nu}\right)q \ & \nabla^2 \left(\frac{\partial \varphi_1}{\partial x_2} - \frac{\partial \varphi_2}{\partial x_1}\right) = c^2\left(\frac{\partial \varphi_1}{\partial x_2} - \frac{\partial \varphi_2}{\partial x_1}\right) \end{align} where : c^2 = \frac{2\kappa G h}{D(1-\nu)} ,. In Mindlin's theory, w^0 is the transverse displacement of the mid-surface of the plate and the quantities \varphi_1 and \varphi_2 are the rotations of the mid-surface normal about the x_2 and x_1-axes, respectively. The canonical parameters for this theory are \mathcal{A} = 1 and \mathcal{B} = 0. The shear correction factor \kappa usually has the value 5/6.

On the other hand, in Reissner's theory, w^0 is the weighted average transverse deflection while \varphi_1 and \varphi_2 are equivalent rotations which are not identical to those in Mindlin's theory.

Relationship to Kirchhoff–Love theory

If we define the moment sum for Kirchhoff–Love theory as : \mathcal{M}^K := -D\nabla^2 w^K we can show that : \mathcal{M} = \mathcal{M}^K + \frac{\mathcal{B}}{1+\nu},q + D \nabla^2 \Phi where \Phi is a biharmonic function such that \nabla^2 \nabla^2 \Phi = 0. We can also show that, if w^K is the displacement predicted for a Kirchhoff–Love plate, : w^0 = w^K + \frac{\mathcal{M}^K}{\kappa G h}\left(1 - \frac{\mathcal{B} c^2}{2}\right)

• \Phi + \Psi where \Psi is a function that satisfies the Laplace equation, \nabla^2 \Psi = 0. The rotations of the normal are related to the displacements of a Kirchhoff–Love plate by : \begin{align} \varphi_1 = - \frac{\partial w^K}{\partial x_1}
• \frac{1}{\kappa G h}\left(1 - \frac{1}{\mathcal{A}} - \frac{\mathcal{B} c^2}{2}\right)Q_1^K

• \frac{\partial }{\partial x_1}\left(\frac{D}{\kappa G h \mathcal{A}}\nabla^2 \Phi + \Phi - \Psi\right)
• \frac{1}{c^2}\frac{\partial \Omega}{\partial x_2} \ \varphi_2 = - \frac{\partial w^K}{\partial x_2}

• \frac{1}{\kappa G h}\left(1 - \frac{1}{\mathcal{A}} - \frac{\mathcal{B} c^2}{2}\right)Q_2^K

• \frac{\partial }{\partial x_2}\left(\frac{D}{\kappa G h \mathcal{A}}\nabla^2 \Phi + \Phi - \Psi\right)
• \frac{1}{c^2}\frac{\partial \Omega}{\partial x_1} \end{align} where : Q_1^K = -D\frac{\partial }{\partial x_1}\left(\nabla^2 w^K\right) ~,~~ Q_2^K = -D\frac{\partial }{\partial x_2}\left(\nabla^2 w^K\right) ~,~~ \Omega := \frac{\partial \varphi_1}{\partial x_2} - \frac{\partial \varphi_2}{\partial x_1} ,.

References

References

1. R. D. Mindlin, 1951, ''Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates'', ASME Journal of Applied Mechanics, Vol. 18 pp. 31–38.
2. 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.
3. Wang, C. M., Lim, G. T., Reddy, J. N, Lee, K. H., 2001, Relationships between bending solutions of Reissner and Mindlin plate theories, Engineering Structures, vol. 23, pp. 838–849.
4. Reddy, J. N., 1999, '''Theory and analysis of elastic plates''', Taylor and Francis, Philadelphia.