Detailed Description

Transformation of tensors that are defined in terms of a set of contravariant bases. Rank-1 and rank-2 contravariant tensors \(\left(\bullet\right)^{\sharp} = \mathbf{T}\) (and its spatial counterpart \(\mathbf{t}\)) typically satisfy the relation

\[ \int_{V_{0}} \nabla_{0} \cdot \mathbf{T} \; dV = \int_{\partial V_{0}} \mathbf{T} \cdot \mathbf{N} \; dA = \int_{\partial V_{t}} \mathbf{T} \cdot \mathbf{n} \; da = \int_{V_{t}} \nabla \cdot \mathbf{t} \; dv \]

where \(V_{0}\) and \(V_{t}\) are respectively control volumes in the reference and spatial configurations, and their surfaces \(\partial V_{0}\) and \(\partial V_{t}\) have the outwards facing normals \(\mathbf{N}\) and \(\mathbf{n}\).

Author: Jean-Paul Pelteret, Andrew McBride, 2016

Function Documentation

◆ push_forward() [1/5]

template<int dim, typename Number >

Tensor<1, dim, Number> Physics::Transformations::Contravariant::push_forward	(	const Tensor< 1, dim, Number > &	V,
		const Tensor< 2, dim, Number > &	F
	)

Return the result of the push forward transformation on a contravariant vector, i.e.

\[ \chi\left(\bullet\right)^{\sharp} \dealcoloneq \mathbf{F} \cdot \left(\bullet\right)^{\sharp} \]

Parameters

[in]	V	The (referential) vector to be operated on
[in]	F	The deformation gradient tensor \(\mathbf{F} \left( \mathbf{X} \right)\)

Returns: \(\chi\left( \mathbf{V} \right)\)

◆ push_forward() [2/5]

template<int dim, typename Number >

Tensor<2, dim, Number> Physics::Transformations::Contravariant::push_forward	(	const Tensor< 2, dim, Number > &	T,
		const Tensor< 2, dim, Number > &	F
	)

Return the result of the push forward transformation on a rank-2 contravariant tensor, i.e.

\[ \chi\left(\bullet\right)^{\sharp} \dealcoloneq \mathbf{F} \cdot \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{T} \]

Parameters

[in]	T	The (referential) rank-2 tensor to be operated on
[in]	F	The deformation gradient tensor \(\mathbf{F} \left( \mathbf{X} \right)\)

Returns: \(\chi\left( \mathbf{T} \right)\)

◆ push_forward() [3/5]

template<int dim, typename Number >

SymmetricTensor<2, dim, Number> Physics::Transformations::Contravariant::push_forward	(	const SymmetricTensor< 2, dim, Number > &	T,
		const Tensor< 2, dim, Number > &	F
	)

Return the result of the push forward transformation on a rank-2 contravariant symmetric tensor, i.e.

\[ \chi\left(\bullet\right)^{\sharp} \dealcoloneq \mathbf{F} \cdot \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{T} \]

Parameters

[in]	T	The (referential) rank-2 symmetric tensor to be operated on
[in]	F	The deformation gradient tensor \(\mathbf{F} \left( \mathbf{X} \right)\)

Returns: \(\chi\left( \mathbf{T} \right)\)

◆ push_forward() [4/5]

template<int dim, typename Number >

Tensor<4, dim, Number> Physics::Transformations::Contravariant::push_forward	(	const Tensor< 4, dim, Number > &	H,
		const Tensor< 2, dim, Number > &	F
	)

Return the result of the push forward transformation on a rank-4 contravariant tensor, i.e. (in index notation)

\[ \left[ \chi\left(\bullet\right)^{\sharp} \right]_{ijkl} \dealcoloneq F_{iI} F_{jJ} \left(\bullet\right)^{\sharp}_{IJKL} F_{kK} F_{lL} \]

Parameters

[in]	H	The (referential) rank-4 tensor to be operated on
[in]	F	The deformation gradient tensor \(\mathbf{F} \left( \mathbf{X} \right)\)

Returns: \(\chi\left( \mathbf{H} \right)\)

◆ push_forward() [5/5]

template<int dim, typename Number >

SymmetricTensor<4, dim, Number> Physics::Transformations::Contravariant::push_forward	(	const SymmetricTensor< 4, dim, Number > &	H,
		const Tensor< 2, dim, Number > &	F
	)

Return the result of the push forward transformation on a rank-4 contravariant symmetric tensor, i.e. (in index notation)

\[ \left[ \chi\left(\bullet\right)^{\sharp} \right]_{ijkl} \dealcoloneq F_{iI} F_{jJ} \left(\bullet\right)^{\sharp}_{IJKL} F_{kK} F_{lL} \]

Parameters

[in]	H	The (referential) rank-4 symmetric tensor to be operated on
[in]	F	The deformation gradient tensor \(\mathbf{F} \left( \mathbf{X} \right)\)

Returns: \(\chi\left( \mathbf{H} \right)\)

◆ pull_back() [1/5]

template<int dim, typename Number >

Tensor<1, dim, Number> Physics::Transformations::Contravariant::pull_back	(	const Tensor< 1, dim, Number > &	v,
		const Tensor< 2, dim, Number > &	F
	)

Return the result of the pull back transformation on a contravariant vector, i.e.

\[ \chi^{-1}\left(\bullet\right)^{\sharp} \dealcoloneq \mathbf{F}^{-1} \cdot \left(\bullet\right)^{\sharp} \]

Parameters

[in]	v	The (spatial) vector to be operated on
[in]	F	The deformation gradient tensor \(\mathbf{F} \left( \mathbf{X} \right)\)

Returns: \(\chi^{-1}\left( \mathbf{v} \right)\)

◆ pull_back() [2/5]

template<int dim, typename Number >

Tensor<2, dim, Number> Physics::Transformations::Contravariant::pull_back	(	const Tensor< 2, dim, Number > &	t,
		const Tensor< 2, dim, Number > &	F
	)

Return the result of the pull back transformation on a rank-2 contravariant tensor, i.e.

\[ \chi^{-1}\left(\bullet\right)^{\sharp} \dealcoloneq \mathbf{F}^{-1} \cdot \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{-T} \]

Parameters

[in]	t	The (spatial) tensor to be operated on
[in]	F	The deformation gradient tensor \(\mathbf{F} \left( \mathbf{X} \right)\)

Returns: \(\chi^{-1}\left( \mathbf{t} \right)\)

◆ pull_back() [3/5]

template<int dim, typename Number >

SymmetricTensor<2, dim, Number> Physics::Transformations::Contravariant::pull_back	(	const SymmetricTensor< 2, dim, Number > &	t,
		const Tensor< 2, dim, Number > &	F
	)

Return the result of the pull back transformation on a rank-2 contravariant symmetric tensor, i.e.

\[ \chi^{-1}\left(\bullet\right)^{\sharp} \dealcoloneq \mathbf{F}^{-1} \cdot \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{-T} \]

Parameters

[in]	t	The (spatial) symmetric tensor to be operated on
[in]	F	The deformation gradient tensor \(\mathbf{F} \left( \mathbf{X} \right)\)

Returns: \(\chi^{-1}\left( \mathbf{t} \right)\)

◆ pull_back() [4/5]

template<int dim, typename Number >

Tensor<4, dim, Number> Physics::Transformations::Contravariant::pull_back	(	const Tensor< 4, dim, Number > &	h,
		const Tensor< 2, dim, Number > &	F
	)

Return the result of the pull back transformation on a rank-4 contravariant tensor, i.e. (in index notation)

\[ \left[ \chi^{-1}\left(\bullet\right)^{\sharp} \right]_{IJKL} \dealcoloneq F^{-1}_{Ii} F^{-1}_{Jj} \left(\bullet\right)^{\sharp}_{ijkl} F^{-1}_{Kk} F^{-1}_{Ll} \]

Parameters

[in]	h	The (spatial) tensor to be operated on
[in]	F	The deformation gradient tensor \(\mathbf{F} \left( \mathbf{X} \right)\)

Returns: \(\chi^{-1}\left( \mathbf{h} \right)\)

◆ pull_back() [5/5]

template<int dim, typename Number >

SymmetricTensor<4, dim, Number> Physics::Transformations::Contravariant::pull_back	(	const SymmetricTensor< 4, dim, Number > &	h,
		const Tensor< 2, dim, Number > &	F
	)

Return the result of the pull back transformation on a rank-4 contravariant symmetric tensor, i.e. (in index notation)

\[ \left[ \chi^{-1}\left(\bullet\right)^{\sharp} \right]_{IJKL} \dealcoloneq F^{-1}_{Ii} F^{-1}_{Jj} \left(\bullet\right)^{\sharp}_{ijkl} F^{-1}_{Kk} F^{-1}_{Ll} \]

Parameters

[in]	h	The (spatial) symmetric tensor to be operated on
[in]	F	The deformation gradient tensor \(\mathbf{F} \left( \mathbf{X} \right)\)

Returns: \(\chi^{-1}\left( \mathbf{h} \right)\)