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Reference documentation for deal.II version 9.6.0
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#include <deal.II/physics/elasticity/standard_tensors.h>
Static Public Member Functions | |
Scalar derivatives | |
template<typename Number > | |
static DEAL_II_HOST constexpr SymmetricTensor< 2, dim, Number > | ddet_F_dC (const Tensor< 2, dim, Number > &F) |
Tensor derivatives | |
template<typename Number > | |
static DEAL_II_HOST 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 DEAL_II_HOST constexpr SymmetricTensor< 4, dim, Number > | Dev_P (const Tensor< 2, dim, Number > &F) |
template<typename Number > | |
static DEAL_II_HOST 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 44 of file standard_tensors.h.
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staticconstexpr |
Return the fourth-order referential deviatoric tensor, as constructed from the deformation gradient tensor F
. Also known as the deviatoric operator, this tensor projects a second-order 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 second-order (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
\[ \mathtt{Dev\_P} \left( \{ \bullet \} \right) : \mathbf{C} = 0 \, . \]
\[ \mathbf{S} = 2\frac{\partial \psi \left( \bar{\mathbf{C}} \right)}{\partial \mathbf{C}} = 2\frac{\partial \psi \left( \bar{\mathbf{C}} \right)}{\partial \bar{\mathbf{C}}} : \frac{\partial \bar{\mathbf{C}}}{\partial \mathbf{C}} = \bar{\mathbf{S}} : \hat{\mathcal{P}} \equiv \hat{\mathcal{P}}^{T} : \bar{\mathbf{S}} \, . \]
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staticconstexpr |
Return the transpose of the fourth-order referential deviatoric tensor, as constructed from the deformation gradient tensor F
. The result performs the following operation:
\[ \hat{\mathcal{P}}^{T} : \{ \bullet \} = J^{-2/\textrm{dim}} \left[ \{ \bullet \} - \frac{1}{\textrm{dim}} \left[\mathbf{C}^{-1} : \{ \bullet \}\right] \mathbf{C} \right] = \mathtt{Dev\_P\_T} \{ \bullet \} \]
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staticconstexpr |
Return the derivative of the volumetric Jacobian \(J = \text{det} \mathbf{F}\) with respect to the right Cauchy-Green 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
\[ \mathbf{C} = \mathbf{F}^{T}\cdot\mathbf{F} \, . \]
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staticconstexpr |
Return the derivative of the inverse of the right Cauchy-Green tensor with respect to the right Cauchy-Green tensor itself, as constructed from the deformation gradient tensor F
. The result, accounting for symmetry, is defined in index notation as
\[ \left[ \frac{\partial \mathbf{C}^{-1}}{\partial \mathbf{C}} \right]_{IJKL} \dealcoloneq -\frac{1}{2}[ C^{-1}_{IK}C^{-1}_{JL} + C^{-1}_{IL}C^{-1}_{JK} ] \]
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staticconstexpr |
The second-order referential/spatial symmetric identity (metric) tensor \(\mathbf{I}\).
This is defined such that, for any rank-2 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 rank-2 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 69 of file standard_tensors.h.
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staticconstexpr |
The fourth-order referential/spatial unit symmetric tensor \(\mathcal{S}\).
This is defined such that for a general rank-2 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 second-order symmetric tensor \(\{ \bullet \}\)
\[ \mathcal{S} : \{ \bullet \} = \{ \bullet \} : \mathcal{S} = \{ \bullet \} \, . \]
This definition aligns with the fourth-order symmetric tensor \(\mathcal{S}\) introduced in the Physics::Elasticity namespace description and that which is returned by identity_tensor().
Definition at line 102 of file standard_tensors.h.
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staticconstexpr |
The fourth-order referential/spatial tensor \(\mathbf{I} \otimes \mathbf{I}\).
This is defined such that, for any rank-2 tensor, the following holds:
\[ [\mathbf{I} \otimes \mathbf{I}] : \{ \bullet \} = \textrm{trace}\{ \bullet \} \mathbf{I} \, . \]
Definition at line 118 of file standard_tensors.h.
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staticconstexpr |
The fourth-order spatial deviatoric tensor. Also known as the deviatoric operator, this tensor projects a second-order 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 fourth-order unit symmetric tensor and \(\mathbf{I}\) is the second-order identity tensor.
For any second-order (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 fourth-order symmetric tensor that is returned by deviator_tensor().
Definition at line 168 of file standard_tensors.h.