Physics::Elasticity::Kinematics Namespace Reference

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

F()

template<int dim, typename Number >

Tensor<2, dim, Number> Physics::Elasticity::Kinematics::F

(

const Tensor< 2, dim, Number > &

Grad_u

)

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

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

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).

F_iso()

template<int dim, typename Number >

Tensor<2, dim, Number> Physics::Elasticity::Kinematics::F_iso

(

const Tensor< 2, dim, Number > &

F

)

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

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

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).

F_vol()

template<int dim, typename Number >

SymmetricTensor<2, dim, Number> Physics::Elasticity::Kinematics::F_vol

(

const Tensor< 2, dim, Number > &

F

)

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

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

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).

C()

template<int dim, typename Number >

SymmetricTensor<2, dim, Number> Physics::Elasticity::Kinematics::C

(

const Tensor< 2, dim, Number > &

F

)

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

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

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).

b()

template<int dim, typename Number >

SymmetricTensor<2, dim, Number> Physics::Elasticity::Kinematics::b

(

const Tensor< 2, dim, Number > &

F

)

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

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

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).

E()

template<int dim, typename Number >

SymmetricTensor<2, dim, Number> Physics::Elasticity::Kinematics::E

(

const Tensor< 2, dim, Number > &

F

)

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

\[ \mathbf{E} \dealcoloneq \frac{1}{2} \left[ \mathbf{F}^{T}\cdot\mathbf{F} - \mathbf{I} \right] \, . \]

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).

epsilon()

template<int dim, typename Number >

SymmetricTensor<2, dim, Number> Physics::Elasticity::Kinematics::epsilon

(

const Tensor< 2, dim, Number > &

Grad_u

)

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

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

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).

e()

template<int dim, typename Number >

SymmetricTensor<2, dim, Number> Physics::Elasticity::Kinematics::e

(

const Tensor< 2, dim, Number > &

F

)

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

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

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).

l()

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
	)

Return 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

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

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).

d()

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
	)

Return 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

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

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).

w()

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
	)

Return 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

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

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).