Solid mechanics linear elasticity

Models linear elastic deformation of a solid. More...

Description

Deformation of a solid body using the theory of linear elasticity (small deformations).

This model describes the deformation of a solid body using the theory of linear elasticity. The momentum balance equation of the solid body can be expressed by

\[\nabla\cdot\boldsymbol{\sigma} + \rho \mathbf{g} + \mathbf{f} = \rho\ddot{\mathbf{u}}, \]

where \( \boldsymbol{\sigma} \) is the stress tensor, \( \rho \) is the density of the solid, \( \mathbf{f} \) in \( \mathrm{N/m^3} \) is the external force acting on the body per unit volume (e.g. magnetism), and \( \mathbf{u} = \mathbf{x} - \mathbf{x}_{\mathrm{initial}} \) is the displacement, defined as the difference in material points \( \mathbf{x} \) and \( \mathbf{x}_{\mathrm{initial}} \) in the deformed and undeformed (initial) state, respectively. The model assumes quasi-static conditions, that is, the above momentum balance equation is solved under the assumption that the acceleration term \( \rho\ddot{\mathbf{u}} \approx 0\).

Per default, Hookes' Law is used for expressing the stress tensor \( \boldsymbol{\sigma} \) as a function of the displacement:

\[\boldsymbol{\sigma} = \lambda\mathrm{tr}(\boldsymbol{\varepsilon}) \mathbf{I} + 2G \boldsymbol{\varepsilon}, \]

with

\[\boldsymbol{\varepsilon} = \frac{1}{2} \left[ \nabla\mathbf{u} + (\nabla\mathbf{u})^{\mathrm{T}} \right]. \]

Primary variables are the displacements in each direction \( \mathbf{u} \). Gravity can be enabled or disabled via a runtime parameter.

Classes
class	Dumux::FVElasticSpatialParams< GridGeometry, Scalar, Implementation >
	The base class for spatial parameters of linear elastic geomechanical problems. More...
struct	Dumux::ElasticIndices
	The indices for the linear elasticity model. More...
struct	Dumux::LameParams< Scalar >
	Structure encapsulating the lame parameters. More...
class	Dumux::ElasticLocalResidual< TypeTag >
	Element-wise calculation of the local residual for problems using the elastic model considering linear elasticity. More...
struct	Dumux::ElasticModelTraits< dim, numSolidComp >
	Specifies a number properties of the elastic model. More...
struct	Dumux::ElasticVolumeVariablesTraits< PV, DV, MT, SST, SSY >
	Traits class for the volume variables of the elastic model. More...
class	Dumux::StressVariablesCache< Scalar, GridGeometry, DiscretizationMethod >
	The stress variables cache classes for models involving solid mechanics. Store data required for stress calculation. More...
class	Dumux::ElasticVolumeVariables< Traits >
	Contains the quantities which are constant within a finite volume in the elastic model. More...

Files
file	fvspatialparams.hh
	The base class for spatial parameters of linear elastic geomechanical problems.
file	indices.hh
	Defines the indices for the elastic model.
file	lameparams.hh
	Structure encapsulating the lame parameters.
file	localresidual.hh
	Element-wise calculation of the local residual for problems using the elastic model considering linear elasticity.
file	model.hh
	Deformation of a solid body using the theory of linear elasticity (small deformations).
file	stressvariablescache.hh
	Base class for the stress variables cache.
file	volumevariables.hh
	Quantities required by the elastic model defined on a sub-control volume.