Reference documentation for deal.II version Git c9feb145b3 20210301 21:22:52 +0100

#include <deal.II/physics/elasticity/standard_tensors.h>
Static Public Member Functions  
Scalar derivatives  
template<typename Number >  
static constexpr SymmetricTensor< 2, dim, Number >  ddet_F_dC (const Tensor< 2, dim, Number > &F) 
Tensor derivatives  
template<typename Number >  
static constexpr SymmetricTensor< 4, dim, Number >  dC_inv_dC (const Tensor< 2, dim, Number > &F) 
Static Public Attributes  
Metric tensors  
static constexpr const SymmetricTensor< 2, dim >  I = unit_symmetric_tensor<dim>() 
static constexpr const SymmetricTensor< 4, dim >  S = identity_tensor<dim>() 
static constexpr const SymmetricTensor< 4, dim >  IxI 
Projection operators  
static constexpr const SymmetricTensor< 4, dim >  dev_P = deviator_tensor<dim>() 
template<typename Number >  
static constexpr SymmetricTensor< 4, dim, Number >  Dev_P (const Tensor< 2, dim, Number > &F) 
template<typename Number >  
static constexpr SymmetricTensor< 4, dim, Number >  Dev_P_T (const Tensor< 2, dim, Number > &F) 
A collection of tensor definitions that mostly conform to notation used in standard scientific literature, in particular the book of Wriggers (2008). The citation for this reference, as well as other notation used here, can be found in the description for the Physics::Elasticity namespace.
Definition at line 46 of file standard_tensors.h.

static 
Return the fourthorder referential deviatoric tensor, as constructed from the deformation gradient tensor F
. Also known as the deviatoric operator, this tensor projects a secondorder symmetric tensor onto a deviatoric space (for which the hydrostatic component is removed).
This referential isochoric projection tensor is defined as
\[ \hat{\mathcal{P}} \dealcoloneq \frac{\partial \bar{\mathbf{C}}}{\partial \mathbf{C}} \]
with
\[ \bar{\mathbf{C}} \dealcoloneq J^{2/\textrm{dim}} \mathbf{C} \qquad \text{,} \qquad \mathbf{C} = \mathbf{F}^{T}\cdot\mathbf{F} \qquad \text{and} \qquad J = \textrm{det}\mathbf{F} \]
such that, for any secondorder (referential) symmetric tensor, the following holds:
\[ \{ \bullet \} : \hat{\mathcal{P}} \dealcoloneq J^{2/\textrm{dim}} \left[ \{ \bullet \}  \frac{1}{\textrm{dim}}\left[\mathbf{C} : \{ \bullet \}\right] \mathbf{C}^{1} \right] = \mathtt{Dev\_P} \left( \{ \bullet \} \right) \, . \]
It can therefore be readily shown that
\[ \mathtt{Dev\_P} \left( \{ \bullet \} \right) : \mathbf{C} = 0 \, . \]
\[ \mathbf{S} = 2\frac{\partial \psi \left( \bar{\mathbf{C}} \right)}{\partial \mathbf{C}} = 2\frac{\partial \psi \left( \bar{\mathbf{C}} \right)}{\partial \bar{\mathbf{C}}} : \frac{\partial \bar{\mathbf{C}}}{\partial \mathbf{C}} = \bar{\mathbf{S}} : \hat{\mathcal{P}} \equiv \hat{\mathcal{P}}^{T} : \bar{\mathbf{S}} \, . \]

static 
Return the transpose of the fourthorder referential deviatoric tensor, as constructed from the deformation gradient tensor F
. The result performs the following operation:
\[ \hat{\mathcal{P}}^{T} : \{ \bullet \} = J^{2/\textrm{dim}} \left[ \{ \bullet \}  \frac{1}{\textrm{dim}} \left[\mathbf{C}^{1} : \{ \bullet \}\right] \mathbf{C} \right] = \mathtt{Dev\_P\_T} \{ \bullet \} \]

static 
Return the derivative of the volumetric Jacobian \(J = \text{det} \mathbf{F}\) with respect to the right CauchyGreen tensor, as constructed from the deformation gradient tensor F
. The computed result is
\[ \frac{\partial J}{\partial \mathbf{C}} = \frac{1}{2} J \mathbf{C}^{1} \]
with
\[ \mathbf{C} = \mathbf{F}^{T}\cdot\mathbf{F} \, . \]

static 
Return the derivative of the inverse of the right CauchyGreen tensor with respect to the right CauchyGreen tensor itself, as constructed from the deformation gradient tensor F
. The result, accounting for symmetry, is defined in index notation as
\[ \left[ \frac{\partial \mathbf{C}^{1}}{\partial \mathbf{C}} \right]_{IJKL} \dealcoloneq \frac{1}{2}[ C^{1}_{IK}C^{1}_{JL} + C^{1}_{IL}C^{1}_{JK} ] \]

static 
The secondorder referential/spatial symmetric identity (metric) tensor \(\mathbf{I}\).
This is defined such that, for any rank2 tensor or symmetric tensor, the following holds:
\[ \mathbf{I} \cdot \{ \bullet \} = \{ \bullet \} \cdot \mathbf{I} = \{ \bullet \} \qquad \text{and} \qquad \mathbf{I} : \{ \bullet \} = \textrm{trace} \{ \bullet \} \, . \]
This definition aligns with the rank2 symmetric tensor returned by unit_symmetric_tensor(). If one is to interpret the tensor as a matrix, then this simply corresponds to the identity matrix.
Definition at line 73 of file standard_tensors.h.

static 
The fourthorder referential/spatial unit symmetric tensor \(\mathcal{S}\).
This is defined such that for a general rank2 tensor \(\{ \hat{\bullet} \}\) the following holds:
\[ \mathcal{S} : \{ \hat{\bullet} \} \dealcoloneq \dfrac{1}{2} \left[ \{ \hat{\bullet} \} + \{ \hat{\bullet} \}^T \right] \, . \]
As a corollary to this, for any secondorder symmetric tensor \(\{ \bullet \}\)
\[ \mathcal{S} : \{ \bullet \} = \{ \bullet \} : \mathcal{S} = \{ \bullet \} \, . \]
This definition aligns with the fourthorder symmetric tensor \(\mathcal{S}\) introduced in the Physics::Elasticity namespace description and that which is returned by identity_tensor().
Definition at line 106 of file standard_tensors.h.

static 
The fourthorder referential/spatial tensor \(\mathbf{I} \otimes \mathbf{I}\).
This is defined such that, for any rank2 tensor, the following holds:
\[ [\mathbf{I} \otimes \mathbf{I}] : \{ \bullet \} = \textrm{trace}\{ \bullet \} \mathbf{I} \, . \]
Definition at line 122 of file standard_tensors.h.

static 
The fourthorder spatial deviatoric tensor. Also known as the deviatoric operator, this tensor projects a secondorder symmetric tensor onto a deviatoric space (for which the hydrostatic component is removed).
This is defined as
\[ \mathcal{P} \dealcoloneq \mathcal{S}  \frac{1}{\textrm{dim}} \mathbf{I} \otimes \mathbf{I} \]
where \(\mathcal{S}\) is the fourthorder unit symmetric tensor and \(\mathbf{I}\) is the secondorder identity tensor.
For any secondorder (spatial) symmetric tensor the following holds:
\[ \mathcal{P} : \{ \bullet \} \dealcoloneq \{ \bullet \}  \frac{1}{\textrm{dim}} \left[ \{ \bullet \} : \mathbf{I} \right]\mathbf{I} = \mathcal{P}^{T} : \{ \bullet \} = \mathtt{dev\_P} \left( \{ \bullet \} \right) \]
and, therefore,
\[ \mathtt{dev\_P} \left( \{ \bullet \} \right) : \mathbf{I} = \mathrm{trace}(\mathtt{dev\_P} \left( \{ \bullet \} \right)) = 0 \, . \]
This definition aligns with the fourthorder symmetric tensor that is returned by deviator_tensor().
Definition at line 172 of file standard_tensors.h.