Kirchhoff-Love plate

Models bending of a thin plate using the Kirchhoff-Love theory. More...

Description

Kirchhoff-Love plate model.

In the Kirchhoff-Love model, the plate is very thin and the rotation angles \( \boldsymbol{\theta} \) are identified with the gradient of the vertical deformation \( w \):

\[ \boldsymbol{\theta} = \nabla w. \]

The equilibrium equation reads

\[ \nabla\cdot(\nabla\cdot\mathbf{M}) = F, \]

where \( F \) is the out-of-plane load and the moment resultant for an isotropic material is

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

with the bending modulus \( D = Et^3/(12(1-\nu^2)) \), Young's modulus \( E \), Poisson ratio \( \nu \), and plate thickness \( t \). This is a fourth-order PDE in \( w \), which is not directly amenable to standard lowest-order finite volume discretization.

Mixed form using Helmholtz decomposition

To reduce the problem to a system of second-order equations, we define the shear resultant vector \( \mathbf{q} := \nabla\cdot\mathbf{M}(\boldsymbol{\theta}) \) and apply the Helmholtz decomposition

\[ \mathbf{q} = \nabla\varphi + \mathbf{J}\nabla\psi, \]

where \( \varphi \) is the gradient (irrotational) potential, \( \psi \) is the curl (solenoidal) potential, and \( \mathbf{J} = \begin{bmatrix}0&1\\-1&0\end{bmatrix} \) is the 90° rotation matrix (so that \(\mathbf{J}\nabla\psi\) is the 2D curl of \(\psi\), i.e. \((-\partial_y\psi,\,\partial_x\psi)^T\)). Substituting into the equilibrium equation and the constraint \( \nabla w - \boldsymbol{\theta} = \mathbf{0} \) (taking its divergence and curl respectively), the system reads [25]

\begin{align} \nabla\cdot\nabla\varphi &= F,\\ -\nabla\cdot(\nabla w - \boldsymbol{\theta}) &= 0,\\ -\nabla\cdot(\mathbf{J}\boldsymbol{\theta}) &= 0,\\ -\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 in the implemented order \( (\varphi, w, \psi) \). In particular, equations (1) and (2) are scalar second-order equations in \( \varphi \) and \( w \), while equation (3) is a scalar constraint equation. 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 current implementation solves only the static (equilibrium) problem.

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

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

Classes
class	Dumux::KirchhoffLovePlateLocalResidualDeformation< TypeTag >
	Local residual for the Kirchhoff-Love model (deformation and potentials) More...
class	Dumux::KirchhoffLovePlateLocalResidualRotation< TypeTag >
	Local residual for the Kirchhoff-Love model (rotations) More...
struct	Dumux::KirchhoffLovePlateTraits
	KirchhoffLovePlateTraits. More...
struct	Dumux::KirchhoffLovePlateRotationModelTraits
	KirchhoffLovePlateRotationModelTraits. More...
class	Dumux::KirchhoffLovePlateDeformationVolumeVariables< Traits >
	Volume variables for the Kirchhoff-Love model (deformation, potentials) More...
class	Dumux::KirchhoffLovePlateRotationVolumeVariables< Traits >
	Volume variables for the rotations. More...

Files
file	solidmechanics/plate/kirchhoff_love/couplingmanager.hh
	Coupling manager for the Kirchhoff-Love model.
file	solidmechanics/plate/kirchhoff_love/localresidual.hh
	Local residual for the Kirchhoff-Love model.
file	solidmechanics/plate/kirchhoff_love/model.hh
	Kirchhoff-Love plate model.
file	solidmechanics/plate/kirchhoff_love/volumevariables.hh
	Volume variables for the Kirchhoff-Love model.