Hyperelastic solid mechanics

Models nonlinear deformation of an elastic solid. More...

Description

Hyperelastic model.

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

\[\mathbf{F} = \frac{\partial \mathbf{x}}{\partial \mathbf{X}} = \mathbf{I} + \nabla \mathbf{d}. \]

The balance of momentum in the reference frame is given by

\[\varrho_0 \ddot{\mathbf{d}} - \nabla\cdot\mathbf{P} = \mathbf{f} \]

with the density \( \varrho_0 \) in the reference configuration, where the inertia term is only present for dynamic problems, and where \( \mathbf{f} \) in \( \mathrm{N/m^3} \) 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, \( \mathbf{P} = J \boldsymbol{\sigma} \mathbf{F}^{-T} \), where \( J = \operatorname{det}{\mathbf{F}} \).

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 \( \psi(\mathbf{F}) \) [73] 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. The user problem implements the constitutive law, for example as firstPiolaKirchhoffStressTensor(F), and for dynamic problems the acceleration and velocity of the degrees of freedom; see Dumux::HyperelasticLocalResidual for the interface.

Topics
	Hyperelasticity with volumetric-isochoric split
	Models nonlinear deformation of a (nearly) incompressible elastic solid with a mixed displacement-pressure formulation.

Classes
class	Dumux::HyperelasticLocalResidual< TypeTag >
	Local residual of the hyperelastic model. More...
struct	Dumux::HyperelasticModelTraits< dim >
	HyperelasticModelTraits. More...
class	Dumux::HyperelasticSpatialParams< GridGeometry, Scalar >
	Spatial parameters of the hyperelastic model for problems that define the material parameters of their constitutive law themselves. More...
class	Dumux::HyperelasticVariables< Traits >
	Variables of the hyperelastic model: the displacement at a local dof. More...

Files
file	localresidual.hh
	Local residual of the hyperelastic model.
file	model.hh
	Hyperelastic model.
file	spatialparams.hh
	Default implementation of the spatial params.
file	variables.hh
	Variables of the hyperelastic model.