A hyperelastic material law formulated in terms of the invariants \( I_1, I_2, I_3 \) of the Cauchy-Green strain tensor \( C \). More...

#include <concepts.hh>

Detailed Description

A hyperelastic material law formulated in terms of the invariants \( I_1, I_2, I_3 \) of the Cauchy-Green strain tensor \( C \).

See also: Kaskade::InvariantsMaterialLaw

Definition at line 703 of file concepts.hh.

Public Types
using	Scalar = unspecified
	The scalar field type (usually double). More...

using	Invariants = Dune::FieldVector< Scalar, dim >
	A dim-dimensional vector type. More...

Public Member Functions
void	InvariantsMaterialConcept (Invariants const &i,...)
	Constructs the material at shifted invariants \( i \). More...

Scalar	d0 () const
	Evaluates the stored energy density \( W(I) \). More...

Scalar	d1 (Invariants const &i1) const
	Evaluates the first directional derivative \( W'(I)\tilde I \). More...

Scalar	d2 (Invariants const &i1, Invariants const &i2) const
	Evaluates the second directional derivative \( W''(\epsilon)\epsilon_1\epsilon_2 \). More...

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

Member Typedef Documentation

◆ Invariants

using InvariantsMaterialConcept::Invariants = Dune::FieldVector<Scalar,dim>

A dim-dimensional vector type.

Definition at line 719 of file concepts.hh.

◆ Scalar

using InvariantsMaterialConcept::Scalar = unspecified

The scalar field type (usually double).

Definition at line 709 of file concepts.hh.

Constructor & Destructor Documentation

◆ InvariantsMaterialConcept()

void InvariantsMaterialConcept::InvariantsMaterialConcept	(	Invariants const &	i,
			...
	)

Constructs the material at shifted invariants \( i \).

The invariants of \( A \) are not given by their actual value \( I_k \), but in the shifted invariants \( i_k \) such that \( i_k(I) = 0 \). This allows a numerically stable evaluation of the material laws in the vicinity of the reference configuration.

For \( d=2 \) we have \( i_1 = I_1-2 = \mathrm{tr}(A)-2 \) and \( i_2 = I_2-1 = |A| - 1 \). For \( d=3 \) we have \( i_1 = I_1-3 = \mathrm{tr}(A)-3 \), \( i_2 = I_2-3 = \frac{1}{2}(\mathrm{tr}(A)^2 - \mathrm{tr}(A^2)) - 3\), and \( i_3 = I_3-1 = |A| - 1 \).

Parameters

i	the shifted invariants

Member Function Documentation

◆ d0()

Scalar InvariantsMaterialConcept::d0 ( ) const

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

◆ d1()

Scalar InvariantsMaterialConcept::d1 ( Invariants const & i1 ) const

Evaluates the first directional derivative \( W'(I)\tilde I \).

◆ d2()

Scalar InvariantsMaterialConcept::d2	(	Invariants const &	i1,
		Invariants const &	i2
	)		const

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

Member Data Documentation

◆ dim

int const InvariantsMaterialConcept::dim = unspecified

static

The spatial dimension (usually 2 or 3).

Definition at line 714 of file concepts.hh.

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

concepts.hh