Mindlin-Reissner plate

Models bending of a thick plate using the Mindlin-Reissner theory. More...

Description

Mindlin-Reissner plate model.

Mindlin-Reissner plate model for small strains and small rotations. The stress resultants are defined by integration over the plate thickness \( [-\tfrac{t}{2}, \tfrac{t}{2}] \):

\[ N_{\alpha\beta} := \int_{-t/2}^{t/2}\sigma_{\alpha\beta}\,dx_3,\quad M_{\alpha\beta} := \int_{-t/2}^{t/2} x_3\,\sigma_{\alpha\beta}\,dx_3,\quad Q_\alpha := \int_{-t/2}^{t/2}\sigma_{\alpha 3}\,dx_3. \]

Ignoring in-plane stresses, the equilibrium equations are

\begin{align} \nabla\cdot\mathbf{Q} &= F,\\ -\nabla\cdot\mathbf{M} + \mathbf{Q} &= \mathbf{0}, \end{align}

where \( F \) is the out-of-plane load. Using a plane-stress constitutive law, the moment resultants for an isotropic material are

\[ \mathbf{M}(\boldsymbol{\theta}) = -D\left\{(1-\nu)\,\boldsymbol{\varepsilon}(\boldsymbol{\theta}) + \nu\operatorname{tr}(\boldsymbol{\varepsilon}(\boldsymbol{\theta}))\,\mathbf{I}\right\}, \]

with \( \boldsymbol{\varepsilon}(\boldsymbol{\theta}) = \frac{1}{2}(\nabla\boldsymbol{\theta}+(\nabla\boldsymbol{\theta})^T) \), bending modulus \( D = Et^3/(12(1-\nu^2)) \), and the shear resultants

\[ \mathbf{Q}(\boldsymbol{\theta}, w) = \kappa G t\,(\nabla w - \boldsymbol{\theta}), \]

where \( G = E/(2(1+\nu)) \) is the shear modulus and \( \kappa \) the shear correction factor.

Mixed form using potentials

The shear resultants admit the Helmholtz decomposition \( \mathbf{Q} = \nabla\varphi + \mathbf{J}\nabla\psi \) in terms of scalar potentials \( \varphi \) and \( \psi \). With the rotation matrix \( \mathbf{J} = \begin{bmatrix}0&1\\-1&0\end{bmatrix} \), the system becomes

\begin{align} \nabla\cdot\nabla\varphi &= F,\\ -\nabla\cdot(\nabla w - \boldsymbol{\theta}) &= -\nabla\cdot((\kappa Gt)^{-1}\nabla\varphi),\\ -\nabla\cdot(\mathbf{J}\boldsymbol{\theta}) &= \nabla\cdot((\kappa Gt)^{-1}\mathbf{J} \mathbf{J}\nabla\psi),\\ -\nabla\cdot(\mathbf{M}(\boldsymbol{\theta}) - \mathbf{I}\varphi - \mathbf{J}\psi) &= \mathbf{0}. \end{align}

Equations (1)-(3) are the deformation-and-potentials sub-problem according to the implemented order for the primary variables \( (\varphi, w, \psi) \). All three are scalar second-order equations in \( \varphi \), \( w \), and \( \psi \), respectively. Equation (4) is a vector second-order equation for the rotation field \( \boldsymbol{\theta} \).

Primary variables

The deformation sub-problem has three primary variables per DOF:

• shear gradient potential \( \varphi \)
• vertical deformation \( w \)
• shear curl potential \( \psi \)

The rotation sub-problem has two primary variables per DOF:

• rotation component \( \theta_x \)
• rotation component \( \theta_y \)

Classes
class	Dumux::MindlinReissnerPlateLocalResidualDeformation< TypeTag >
	Local residual for the Mindlin-Reissner model (deformation and potentials) More...
class	Dumux::MindlinReissnerPlateLocalResidualRotation< TypeTag >
	Local residual for the Mindlin-Reissner model (rotations) More...

Files
file	solidmechanics/plate/mindlin_reissner/localresidual.hh
	Local residual for the Mindlin-Reissner model.
file	solidmechanics/plate/mindlin_reissner/model.hh
	Mindlin-Reissner plate model.