Reference documentation for deal.II version Git 1ab60d1 2017-10-16 22:02:15 +0200
Physics::Elasticity::Kinematics Namespace Reference

Functions

Deformation tensors
template<int dim, typename Number >
Tensor< 2, dim, Number > F (const Tensor< 2, dim, Number > &Grad_u)
 
template<int dim, typename Number >
Tensor< 2, dim, Number > F_iso (const Tensor< 2, dim, Number > &F)
 
template<int dim, typename Number >
SymmetricTensor< 2, dim, Number > F_vol (const Tensor< 2, dim, Number > &F)
 
template<int dim, typename Number >
SymmetricTensor< 2, dim, Number > C (const Tensor< 2, dim, Number > &F)
 
template<int dim, typename Number >
SymmetricTensor< 2, dim, Number > b (const Tensor< 2, dim, Number > &F)
 
Strain tensors
template<int dim, typename Number >
SymmetricTensor< 2, dim, Number > E (const Tensor< 2, dim, Number > &F)
 
template<int dim, typename Number >
SymmetricTensor< 2, dim, Number > epsilon (const Tensor< 2, dim, Number > &Grad_u)
 
template<int dim, typename Number >
SymmetricTensor< 2, dim, Number > e (const Tensor< 2, dim, Number > &F)
 
Strain rate tensors
template<int dim, typename Number >
Tensor< 2, dim, Number > l (const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
 
template<int dim, typename Number >
SymmetricTensor< 2, dim, Number > d (const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
 
template<int dim, typename Number >
Tensor< 2, dim, Number > w (const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
 

Detailed Description

A collection of tensor definitions for deformation and strain measures, as well as a few special transformations, that conform to notation used in standard scientific literature, in particular the books of Holzapfel (2007) and Wriggers (2008). The citation for these references, as well as other notation used here, can be found in the description for the Physics::Elasticity namespace.

Note
These hold specifically for the codimension 0 case, where the metric tensor is the identity tensor.
Author
Jean-Paul Pelteret, Andrew McBride, 2016

Function Documentation

template<int dim, typename Number >
Tensor<2, dim, Number> Physics::Elasticity::Kinematics::F ( const Tensor< 2, dim, Number > &  Grad_u)

Returns the deformation gradient tensor, as constructed from the material displacement gradient tensor Grad_u. The result is expressed as

\[ \mathbf{F} := \nabla_{0} \boldsymbol{\varphi} \left( \mathbf{X} \right) =\mathbf{I} + \nabla_{0}\mathbf{u} \]

where \(\mathbf{u} = \mathbf{u}\left(\mathbf{X}\right)\) is the displacement at position \(\mathbf{X}\) in the referential configuration. The differential operator \(\nabla_{0}\) is defined as \(\frac{\partial}{\partial \mathbf{X}}\).

Note
For a discussion of the background of this function, see P. Wriggers: "Nonlinear finite element methods" (2008), and in particular formula (3.14) on p. 23 (or thereabouts).
For a discussion of the background of this function, see G. A. Holzapfel: "Nonlinear solid mechanics. A Continuum Approach for Engineering" (2007), and in particular formula (2.39) on p. 71 (or thereabouts).
template<int dim, typename Number >
Tensor<2, dim, Number> Physics::Elasticity::Kinematics::F_iso ( const Tensor< 2, dim, Number > &  F)

Returns the isochoric counterpart of the deformation gradient tensor F . The result is expressed as

\[ \mathbf{F}^{\text{iso}} := J^{-1/\textrm{dim}} \mathbf{F} \]

where \(J = \text{det}\left(\mathbf{F}\right)\).

Note
For a discussion of the background of this function, see P. Wriggers: "Nonlinear finite element methods" (2008), and in particular formula (3.28) on p. 29 (or thereabouts).
For a discussion of the background of this function, see G. A. Holzapfel: "Nonlinear solid mechanics. A Continuum Approach for Engineering" (2007), and in particular formula (6.79) on p. 228 (or thereabouts).
template<int dim, typename Number >
SymmetricTensor<2, dim, Number> Physics::Elasticity::Kinematics::F_vol ( const Tensor< 2, dim, Number > &  F)

Returns the volumetric counterpart of the deformation gradient tensor F . The result is expressed as

\[ \mathbf{F}^{\text{vol}} := J^{1/\textrm{dim}} \mathbf{I} \]

where \(J = \text{det}\left(\mathbf{F}\right)\).

Note
For a discussion of the background of this function, see P. Wriggers: "Nonlinear finite element methods" (2008), and in particular formula (3.28) on p. 29 (or thereabouts).
For a discussion of the background of this function, see G. A. Holzapfel: "Nonlinear solid mechanics. A Continuum Approach for Engineering" (2007), and in particular formula (6.79) on p. 228 (or thereabouts).
template<int dim, typename Number >
SymmetricTensor<2, dim, Number> Physics::Elasticity::Kinematics::C ( const Tensor< 2, dim, Number > &  F)

Returns the symmetric right Cauchy-Green deformation tensor, as constructed from the deformation gradient tensor F. The result is expressed as

\[ \mathbf{C} := \mathbf{F}^{T}\cdot\mathbf{F} \, . \]

Note
For a discussion of the background of this function, see P. Wriggers: "Nonlinear finite element methods" (2008), and in particular formula (3.15) on p. 23 (or thereabouts).
For a discussion of the background of this function, see G. A. Holzapfel: "Nonlinear solid mechanics. A Continuum Approach for Engineering" (2007), and in particular formula (2.65) on p. 78 (or thereabouts).
template<int dim, typename Number >
SymmetricTensor<2, dim, Number> Physics::Elasticity::Kinematics::b ( const Tensor< 2, dim, Number > &  F)

Returns the symmetric left Cauchy-Green deformation tensor, as constructed from the deformation gradient tensor F. The result is expressed as

\[ \mathbf{b} := \mathbf{F}\cdot\mathbf{F}^{T} \, . \]

Note
For a discussion of the background of this function, see P. Wriggers: "Nonlinear finite element methods" (2008), and in particular formula (3.25) on p. 28 (or thereabouts).
For a discussion of the background of this function, see G. A. Holzapfel: "Nonlinear solid mechanics. A Continuum Approach for Engineering" (2007), and in particular formula (2.79) on p. 81 (or thereabouts).
template<int dim, typename Number >
SymmetricTensor<2, dim, Number> Physics::Elasticity::Kinematics::E ( const Tensor< 2, dim, Number > &  F)

Returns the symmetric Green-Lagrange strain tensor, as constructed from the deformation gradient tensor F. The result is expressed as

\[ \mathbf{E} := \frac{1}{2}[\mathbf{F}^{T}\cdot\mathbf{F} - \mathbf{I}] \, . \]

Note
For a discussion of the background of this function, see P. Wriggers: "Nonlinear finite element methods" (2008), and in particular formula (3.15) on p. 23 (or thereabouts).
For a discussion of the background of this function, see G. A. Holzapfel: "Nonlinear solid mechanics. A Continuum Approach for Engineering" (2007), and in particular formula (6.29) on p. 79 (or thereabouts).
template<int dim, typename Number >
SymmetricTensor<2, dim, Number> Physics::Elasticity::Kinematics::epsilon ( const Tensor< 2, dim, Number > &  Grad_u)

Returns the symmetric small strain tensor, as constructed from the displacement gradient tensor Grad_u. The result is expressed as

\[ \boldsymbol{\varepsilon} := \frac{1}{2} \left[ \nabla_{0}\mathbf{u} + [\nabla_{0}\mathbf{u}]^{T} \right] \, . \]

where \(\mathbf{u} = \mathbf{u}(\mathbf{X})\) is the displacement at position \(\mathbf{X}\) in the referential configuration. The differential operator \(\nabla_{0}\) is defined as \(\frac{\partial}{\partial \mathbf{X}}\).

Note
For a discussion of the background of this function, see P. Wriggers: "Nonlinear finite element methods" (2008), and in particular formula (3.17) on p. 24 (or thereabouts).
template<int dim, typename Number >
SymmetricTensor<2, dim, Number> Physics::Elasticity::Kinematics::e ( const Tensor< 2, dim, Number > &  F)

Returns the symmetric Almansi strain tensor, as constructed from the deformation gradient tensor F. The result is expressed as

\[ \mathbf{e} := \frac{1}{2} \left[ \mathbf{I} - \mathbf{F}^{-T}\cdot\mathbf{F}^{-1} \right] \, . \]

Note
For a discussion of the background of this function, see P. Wriggers: "Nonlinear finite element methods" (2008), and in particular formula (3.35) on p. 30 (or thereabouts).
For a discussion of the background of this function, see G. A. Holzapfel: "Nonlinear solid mechanics. A Continuum Approach for Engineering" (2007), and in particular formula (2.83) on p. 81 (or thereabouts).
template<int dim, typename Number >
Tensor<2, dim, Number> Physics::Elasticity::Kinematics::l ( const Tensor< 2, dim, Number > &  F,
const Tensor< 2, dim, Number > &  dF_dt 
)

Returns the spatial velocity gradient tensor, as constructed from the deformation gradient tensor F and its material time derivative dF_dt (the material velocity gradient). The result is expressed as

\[ \mathbf{l} := \dot{\mathbf{F}}\cdot\mathbf{F}^{-1} \, . \]

Note
For a discussion of the background of this function, see P. Wriggers: "Nonlinear finite element methods" (2008), and in particular formula (3.47) on p. 32 (or thereabouts).
For a discussion of the background of this function, see G. A. Holzapfel: "Nonlinear solid mechanics. A Continuum Approach for Engineering" (2007), and in particular formula (2.141) on p. 96 (or thereabouts).
template<int dim, typename Number >
SymmetricTensor<2, dim, Number> Physics::Elasticity::Kinematics::d ( const Tensor< 2, dim, Number > &  F,
const Tensor< 2, dim, Number > &  dF_dt 
)

Returns the rate of deformation tensor (also known as the rate of strain tensor), as constructed from the deformation gradient tensor F and its material time derivative dF_dt (the material velocity gradient). The result is expressed as

\[ \mathbf{d} := \frac{1}{2} \left[ \mathbf{l} + \mathbf{l}^{T} \right] \]

where

\[ \mathbf{l} = \dot{\mathbf{F}}\cdot\mathbf{F}^{-1} \]

is the spatial velocity gradient tensor.

Note
For a discussion of the background of this function, see P. Wriggers: "Nonlinear finite element methods" (2008), and in particular formula (3.49) on p. 32 (or thereabouts).
For a discussion of the background of this function, see G. A. Holzapfel: "Nonlinear solid mechanics. A Continuum Approach for Engineering" (2007), and in particular formula (2.148) on p. 97 (or thereabouts).
template<int dim, typename Number >
Tensor<2, dim, Number> Physics::Elasticity::Kinematics::w ( const Tensor< 2, dim, Number > &  F,
const Tensor< 2, dim, Number > &  dF_dt 
)

Returns the rate of rotation tensor (also known as the vorticity tensor), as constructed from the deformation gradient tensor F and its material time derivative dF_dt (the material velocity gradient). The result is expressed as

\[ \mathbf{w} := \frac{1}{2} \left[ \mathbf{l} - \mathbf{l}^{T} \right] \]

where

\[ \mathbf{l} = \dot{\mathbf{F}}\cdot\mathbf{F}^{-1} \]

is the spatial velocity gradient tensor.

Note
For a discussion of the background of this function, see G. A. Holzapfel: "Nonlinear solid mechanics. A Continuum Approach for Engineering" (2007), and in particular formula (2.149) on p. 97 (or thereabouts).