Reference documentation for deal.II version GIT relicensing660g9ae06c0eb4 20240517 13:10:02+00:00

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) 
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.
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 \(\frac{\partial}{\partial \mathbf{X}}\).
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 \(J = \text{det}\left(\mathbf{F}\right)\).
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 \(J = \text{det}\left(\mathbf{F}\right)\).
SymmetricTensor< 2, dim, Number > Physics::Elasticity::Kinematics::C  (  const Tensor< 2, dim, Number > &  F  ) 
Return the symmetric right CauchyGreen deformation tensor, as constructed from the deformation gradient tensor F
. The result is expressed as
\[ \mathbf{C} \dealcoloneq \mathbf{F}^{T}\cdot\mathbf{F} \, . \]
SymmetricTensor< 2, dim, Number > Physics::Elasticity::Kinematics::b  (  const Tensor< 2, dim, Number > &  F  ) 
Return the symmetric left CauchyGreen deformation tensor, as constructed from the deformation gradient tensor F
. The result is expressed as
\[ \mathbf{b} \dealcoloneq \mathbf{F}\cdot\mathbf{F}^{T} \, . \]
SymmetricTensor< 2, dim, Number > Physics::Elasticity::Kinematics::E  (  const Tensor< 2, dim, Number > &  F  ) 
Return the symmetric GreenLagrange strain tensor, as constructed from the deformation gradient tensor F
. The result is expressed as
\[ \mathbf{E} \dealcoloneq \frac{1}{2} \left[ \mathbf{F}^{T}\cdot\mathbf{F}  \mathbf{I} \right] \, . \]
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 \(\frac{\partial}{\partial \mathbf{X}}\).
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
\[ \mathbf{e} \dealcoloneq \frac{1}{2} \left[ \mathbf{I}  \mathbf{F}^{T}\cdot\mathbf{F}^{1} \right] \, . \]
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
\[ \mathbf{l} \dealcoloneq \dot{\mathbf{F}}\cdot\mathbf{F}^{1} \, . \]
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.
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.