HyperelasticMaterialLaw Class Reference

A hyperelastic material law. More...

#include <concepts.hh>

Detailed Description

A hyperelastic material law.

See also

Kaskade::MaterialLawBase

Definition at line 620 of file concepts.hh.

Public Types
using	Scalar = unspecified
	The scalar field type (usually double). More...
using	Tensor = unspecified
	The type of stress and strain tensors (usually Dune::FieldMatrix<Scalar,dim,dim>). More...
using	VoigtTensor = unspecified
	The type of stress and strain tensors (usually Dune::FieldMatrix<Scalar,dim*(dim+1)/2,dim*(dim+1)/2>). More...

Public Member Functions
void	setLinearizationPoint (Tensor const &e)
	Defines a new evaluation/linearization point. More...
Scalar	d0 () const
	Evaluates the stored energy density \( W(\epsilon) \). More...
Scalar	d1 (Tensor const &e1) const
	Evaluates the first directional derivative \( W'(\epsilon)\epsilon_1 \). More...
Scalar	d2 (Tensor const &e1, Tensor const &e2) const
	Evaluates the second directional derivative \( W''(\epsilon)\epsilon_1\epsilon_2 \). More...
Tensor	stress () const
	Returns the 2nd Piola-Kirchhoff stress tensor \( \sigma = W'(E) \) corresponding to the current strain \( E \). More...
VoigtTensor	strainToStressMatrix () const
	Returns the tangent stiffness tensor \( C(E) = W''(E)\) mapping strain tensor variations to stress tensor (2nd Piola-Kirchhoff) variations. More...

Static Public Attributes
static int const	dim = unspecified
	The spatial dimension (usually 2 or 3). More...

Member Typedef Documentation

Scalar

using HyperelasticMaterialLaw::Scalar = unspecified

The scalar field type (usually double).

Definition at line 626 of file concepts.hh.

Tensor

using HyperelasticMaterialLaw::Tensor = unspecified

The type of stress and strain tensors (usually Dune::FieldMatrix<Scalar,dim,dim>).

Definition at line 636 of file concepts.hh.

VoigtTensor

using HyperelasticMaterialLaw::VoigtTensor = unspecified

The type of stress and strain tensors (usually Dune::FieldMatrix<Scalar,dim*(dim+1)/2,dim*(dim+1)/2>).

Definition at line 641 of file concepts.hh.

Member Function Documentation

d0()

Scalar HyperelasticMaterialLaw::d0

(

)

const

Evaluates the stored energy density \( W(\epsilon) \).

d1()

Scalar HyperelasticMaterialLaw::d1

(

Tensor const &

e1

)

const

Evaluates the first directional derivative \( W'(\epsilon)\epsilon_1 \).

d2()

Scalar HyperelasticMaterialLaw::d2	(	Tensor const &	e1,
		Tensor const &	e2
	)		const

Evaluates the second directional derivative \( W''(\epsilon)\epsilon_1\epsilon_2 \).

The second derivative is symmetric, i.e. d2(e1,e2)==d2(e2,e1) holds for any e1, e2.

setLinearizationPoint()

void HyperelasticMaterialLaw::setLinearizationPoint

(

Tensor const &

e

)

Defines a new evaluation/linearization point.

Parameters

e	the strain tensor

strainToStressMatrix()

VoigtTensor HyperelasticMaterialLaw::strainToStressMatrix

(

)

const

Returns the tangent stiffness tensor \( C(E) = W''(E)\) mapping strain tensor variations to stress tensor (2nd Piola-Kirchhoff) variations.

In this context, the symmetric matrices \( \sigma \) and \( \epsilon \) are interpreted as \(d(d+1)/2\)-vectors in Voigt notation as returned by Kaskade::pack. Note that strain tensor off-diagonal entries appear with factor two in the vector.

Now, a matrix \( C\in\mathbb{R}^{(d+1)d/2\times(d+1)d/2} \) is returned such that for infinitesimal changes \( \delta\epsilon \) the corresponding change in \( \sigma(\epsilon+\delta\epsilon) = \sigma(\epsilon)+\delta\sigma\) is given as \( \delta\sigma = C \delta\epsilon \). Note that due to the symmetry of the stored energy Hessian \( W''(E) \), the Voigt representation of the stiffness tensor is a symmetric matrix.

The following invariant holds for any strain tensors eps1 and eps2:

d2(eps1,eps2) == pack(eps1)*(strainToStressMatrix()*pack(eps2))

The Drucker/Hill stability criterion states that the returned matrix shall be positive semidefinite.

stress()

Tensor HyperelasticMaterialLaw::stress

(

)

const

Returns the 2nd Piola-Kirchhoff stress tensor \( \sigma = W'(E) \) corresponding to the current strain \( E \).

The following invariant holds for any strain tensor eps:

d1(eps) == contraction(stress(),eps)

Member Data Documentation

dim

int const HyperelasticMaterialLaw::dim = unspecified

static

The spatial dimension (usually 2 or 3).

Definition at line 631 of file concepts.hh.

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The documentation for this class was generated from the following file:

• concepts.hh