solidmechanics/hyperelastic/model.hh File Reference

Hyperelastic model. More...

#include <dumux/common/properties.hh>
#include <dumux/common/properties/model.hh>
#include "spatialparams.hh"
#include "localresidual.hh"
#include "volumevariables.hh"

Go to the source code of this file.

Description

Deformation of a solid body using the theory of (nonlinear) elasticity (large deformations)

This model describes the deformation of a solid body using the finite strain theory. The displacement vector of a material point following the deformation function $\mathbf{x}=\chi (\mathbf{X})$ is given by the difference of current position and position in the reference configuration:

$$\mathbf{d}=\mathbf{x}-\mathbf{X}.$$

An important kinematic quantity characterizing the deformation is the deformation gradient

$$\mathbf{F}=\frac{\partial \mathbf{x}}{\partial \mathbf{X}}=\mathbf{I}+\mathrm{\nabla }\mathbf{d}.$$

The equilibrium equation (nonlinear elastostatics) in the reference frame is given by

$$-\mathrm{\nabla }\cdot \mathbf{P}=\mathbf{f}$$

where $\mathbf{f}$ in $\mathrm{N}/{\mathrm{m}}^3$ is the external force acting on the body per unit volume and $\mathbf{P}=\mathbf{P}(\mathbf{F})$ is the first Piola-Kirchhoff stress tensor which relates the stress in the current and the reference configuration. It is related to the Cauchy stress, $\mathbf{P}=J\mathit{\sigma }{\mathbf{F}}^{-T}$, where $J=\det \mathbf{F}$.

A suitable constitutive law for $\mathbf{P}=\mathbf{P}(\mathbf{F})$ completes the model. All hyperelastic materials can be described in terms of a strain energy density function $\psi (\mathbf{F})$ [66] and the first Piola-Kirchhoff stress tensor can be computed as

$$\mathbf{P}=\frac{\partial \psi }{\partial \mathbf{F}}=\mathbf{F}2\frac{\partial \psi }{\partial \mathbf{C}},$$

where $\mathbf{C}={\mathbf{F}}^T\mathbf{F}$ is the right Cauchy-Green tensor and the term $\mathbf{S}=2(\partial \psi /\partial \mathbf{C})$ is the second Piola-Kirchhoff stress tensor. This model expects the user problem implementation to provide a function firstPiolaKirchhoffStressTensor(F) implementing the constitutive law.

Classes
struct	Dumux::HyperelasticVolumeVariablesTraits< PV, MT >
struct	Dumux::HyperelasticIndices
struct	Dumux::HyperelasticModelTraits< dim >
	HyperelasticModelTraits. More...
struct	Dumux::Properties::TTag::Hyperelastic
struct	Dumux::Properties::ModelTraits< TypeTag, TTag::Hyperelastic >
struct	Dumux::Properties::LocalResidual< TypeTag, TTag::Hyperelastic >
struct	Dumux::Properties::VolumeVariables< TypeTag, TTag::Hyperelastic >
	Set the volume variables property. More...
struct	Dumux::Properties::SpatialParams< TypeTag, TTag::Hyperelastic >

Namespaces
namespace	Dumux
namespace	Dumux::Properties
	The energy balance equation for a porous solid.
namespace	Dumux::Properties::TTag
	Type tag for numeric models.

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