Hyperelasticity with volumetric-isochoric split

Models nonlinear deformation of a (nearly) incompressible elastic solid with a mixed displacement-pressure formulation. More...

Description

Hyperelastic model with volumetric-isochoric split.

Physical model

This model solves the finite-strain quasi-static balance laws for a hyperelastic solid using a mixed displacement-pressure formulation. The split between volumetric and isochoric response is handled by an independent pressure-like variable \( p_s \), which prevents volumetric locking for (nearly-)incompressible materials.

Kinematics

The deformation map \( \chi : \Omega_0 \to \Omega \) maps material points \( \mathbf{X} \) to spatial points \( \mathbf{x} = \chi(\mathbf{X}) \). The displacement field is \( \mathbf{d} = \mathbf{x} - \mathbf{X} \). The deformation gradient and its Jacobian are

\[ \mathbf{F} = \mathbf{I} + \nabla_X \mathbf{d}, \quad J = \det\mathbf{F}. \]

Governing equations

Momentum subproblem (displacement \( \mathbf{d} \)):

\[ -\nabla_X \cdot \mathbf{P} = \mathbf{f}, \quad \mathbf{P} = \mathbf{P}_\text{iso}(\mathbf{F}) + p_s J \mathbf{F}^{-\top}, \]

where the problem supplies the stress without the volumetric part via isochoricFirstPiolaKirchhoffStressTensor(F, elemDisc, ipGlobal) and the body force \( \mathbf{f} \) via source; the model adds \( p_s J \mathbf{F}^{-\top} \).

Pressure subproblem (pressure \( p_s \)):

\[ p_s^{\mathrm{eq}}(J) - p_s = 0, \]

where the equilibrium pressure \( p_s^{\mathrm{eq}}(J) \) is supplied by the pressure problem via problem.volumetricPressure(J).

The type tags HyperelasticVolIsoMomentumModel and HyperelasticVolIsoPressureModel are coupled via HyperelasticVolIsoCouplingManager.

Classes
class	Dumux::HyperelasticVolIsoCouplingManager< MDTraits >
	Coupling manager for the mixed u–p hyperelastic model. More...
class	Dumux::HyperelasticVolIsoMomentumLocalResidual< TypeTag >
	Momentum local residual for the mixed u–p hyperelastic model. More...
class	Dumux::HyperelasticVolIsoPressureLocalResidual< TypeTag >
	Pressure local residual for the mixed u–p hyperelastic model. More...
struct	Dumux::HyperelasticVolIsoMomentumIndices< dim >
	Index set for the momentum subdomain. More...
struct	Dumux::HyperelasticVolIsoMomentumModelTraits< dim >
	Model traits for the momentum subdomain ( \( \dim \) displacement DOFs per node). More...
struct	Dumux::HyperelasticVolIsoPressureIndices
	Index set for the pressure subdomain. More...
struct	Dumux::HyperelasticVolIsoPressureModelTraits
	Model traits for the pressure subdomain (1 pressure DOF per node). More...
struct	Dumux::Properties::TTag::HyperelasticVolIsoMomentumModel
	Type tag for the momentum subdomain of the mixed u–p model. More...
struct	Dumux::Properties::TTag::HyperelasticVolIsoPressureModel
	Type tag for the pressure subdomain of the mixed u–p model. More...
class	Dumux::HyperelasticVolIsoMomentumVariables< Traits >
	Volume variables for the momentum subdomain (displacement). More...
class	Dumux::HyperelasticVolIsoPressureVariables< Traits >
	Volume variables for the pressure subdomain (bulk pressure). More...

Files
file	couplingmanager.hh
	Coupling manager for the mixed u–p hyperelastic model.
file	localresidual.hh
	Local residuals for the volumetric-isochoric split hyperelastic mixed u–p model.
file	model.hh
	Hyperelastic model with volumetric-isochoric split.
file	variables.hh
	Volume variables for the volumetric-isochoric split hyperelastic mixed u–p model.