deal.II version GIT relicensing-1989-gd7a2c90e4e 2024-10-14 01:50:00+00:00
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Namespaces | |
namespace | Contravariant |
namespace | Covariant |
namespace | Piola |
namespace | Rotations |
Functions | |
Special operations | |
template<int dim, typename Number > | |
Tensor< 1, dim, Number > | nansons_formula (const Tensor< 1, dim, Number > &N, const Tensor< 2, dim, Number > &F) |
Basis transformations | |
template<int dim, typename Number > | |
Tensor< 1, dim, Number > | basis_transformation (const Tensor< 1, dim, Number > &V, const Tensor< 2, dim, Number > &B) |
template<int dim, typename Number > | |
Tensor< 2, dim, Number > | basis_transformation (const Tensor< 2, dim, Number > &T, const Tensor< 2, dim, Number > &B) |
template<int dim, typename Number > | |
SymmetricTensor< 2, dim, Number > | basis_transformation (const SymmetricTensor< 2, dim, Number > &T, const Tensor< 2, dim, Number > &B) |
template<int dim, typename Number > | |
Tensor< 4, dim, Number > | basis_transformation (const Tensor< 4, dim, Number > &H, const Tensor< 2, dim, Number > &B) |
template<int dim, typename Number > | |
SymmetricTensor< 4, dim, Number > | basis_transformation (const SymmetricTensor< 4, dim, Number > &H, const Tensor< 2, dim, Number > &B) |
A collection of operations to assist in the transformation of tensor quantities from the reference to spatial configuration, and vice versa. These types of transformation are typically used to re-express quantities measured or computed in one configuration in terms of a second configuration.
We will use the same notation for the coordinates \(\mathbf{X}, \mathbf{x}\), transformations \(\varphi\), differential operator \(\nabla_{0}\) and deformation gradient \(\mathbf{F}\) as discussed for namespace Physics::Elasticity.
As a further point on notation, we will follow Holzapfel (2007) and denote the push forward transformation as \(\chi\left(\bullet\right)\) and the pull back transformation as \(\chi^{-1}\left(\bullet\right)\). We will also use the annotation \(\left(\bullet\right)^{\sharp}\) to indicate that a tensor \(\left(\bullet\right)\) is a contravariant tensor, and \(\left(\bullet\right)^{\flat}\) that it is covariant. In other words, these indices do not actually change the tensor, they just indicate the kind of object a particular tensor is.
Tensor< 1, dim, Number > Physics::Transformations::nansons_formula | ( | const Tensor< 1, dim, Number > & | N, |
const Tensor< 2, dim, Number > & | F | ||
) |
Return the result of applying Nanson's formula for the transformation of the material surface area element \(d\mathbf{A}\) to the current surfaces area element \(d\mathbf{a}\) under the nonlinear transformation map \(\mathbf{x} = \boldsymbol{\varphi} \left( \mathbf{X} \right)\).
The returned result is the spatial normal scaled by the ratio of areas between the reference and spatial surface elements, i.e.
\[ \mathbf{n} \frac{da}{dA} \dealcoloneq \textrm{det} \mathbf{F} \, \mathbf{F}^{-T} \cdot \mathbf{N} = \textrm{cof} \mathbf{F} \cdot \mathbf{N} \, . \]
[in] | N | The referential normal unit vector \(\mathbf{N}\) |
[in] | F | The deformation gradient tensor \(\mathbf{F} \left( \mathbf{X} \right)\) |
Tensor< 1, dim, Number > Physics::Transformations::basis_transformation | ( | const Tensor< 1, dim, Number > & | V, |
const Tensor< 2, dim, Number > & | B | ||
) |
Return a vector with a changed basis, i.e.
\[ \mathbf{V}^{\prime} \dealcoloneq \mathbf{B} \cdot \mathbf{V} \]
[in] | V | The vector to be transformed \(\mathbf{V}\) |
[in] | B | The transformation matrix \(\mathbf{B}\) |
Tensor< 2, dim, Number > Physics::Transformations::basis_transformation | ( | const Tensor< 2, dim, Number > & | T, |
const Tensor< 2, dim, Number > & | B | ||
) |
Return a rank-2 tensor with a changed basis, i.e.
\[ \mathbf{T}^{\prime} \dealcoloneq \mathbf{B} \cdot \mathbf{T} \cdot \mathbf{B}^{T} \]
[in] | T | The tensor to be transformed \(\mathbf{T}\) |
[in] | B | The transformation matrix \(\mathbf{B}\) |
SymmetricTensor< 2, dim, Number > Physics::Transformations::basis_transformation | ( | const SymmetricTensor< 2, dim, Number > & | T, |
const Tensor< 2, dim, Number > & | B | ||
) |
Return a symmetric rank-2 tensor with a changed basis, i.e.
\[ \mathbf{T}^{\prime} \dealcoloneq \mathbf{B} \cdot \mathbf{T} \cdot \mathbf{B}^{T} \]
[in] | T | The tensor to be transformed \(\mathbf{T}\) |
[in] | B | The transformation matrix \(\mathbf{B}\) |
Tensor< 4, dim, Number > Physics::Transformations::basis_transformation | ( | const Tensor< 4, dim, Number > & | H, |
const Tensor< 2, dim, Number > & | B | ||
) |
Return a rank-4 tensor with a changed basis, i.e. (in index notation):
\[ H_{ijkl}^{\prime} \dealcoloneq B_{iI} B_{jJ} H_{IJKL} B_{kK} B_{lL} \]
[in] | H | The tensor to be transformed \(\mathbf{T}\) |
[in] | B | The transformation matrix \(\mathbf{B}\) |
SymmetricTensor< 4, dim, Number > Physics::Transformations::basis_transformation | ( | const SymmetricTensor< 4, dim, Number > & | H, |
const Tensor< 2, dim, Number > & | B | ||
) |
Return a symmetric rank-4 tensor with a changed basis, i.e. (in index notation):
\[ H_{ijkl}^{\prime} \dealcoloneq B_{iI} B_{jJ} H_{IJKL} B_{kK} B_{lL} \]
[in] | H | The tensor to be transformed \(\mathbf{T}\) |
[in] | B | The transformation matrix \(\mathbf{B}\) |