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KASKADE 7 development version
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Convenience class for handling orthotropic materials in linear elastomechanics. More...
#include <elastoVariationalFunctionals.hh>
Convenience class for handling orthotropic materials in linear elastomechanics.
This class defines the variational formulation of the linear elasticity for orthotropic materials for vector-valued displacements \( u \), i.e. the variational functional \( u \mapsto \frac{1}{2} \epsilon : \mathcal{C}\epsilon \) with \( \epsilon = \frac{1}{2}(u_x + u_x^T) \). In an appropriately rotated coordinate system, the stiffness matrix, representing the tensor \(\mathcal{C}\) by interpreting \( \epsilon, \sigma \) as vectors, has the form
\[ C = \begin{bmatrix} * & * & \\ * & * & \\ & & * \end{bmatrix}, \quad C = \begin{bmatrix} * & * & * & & & \\ * & * & * & & & \\ * & * & * & & & \\ & & & * & & \\ & & & & * & \\ & & & & & * \end{bmatrix} \]
depending on the spatial dimension (see Wikipedia).
Definition at line 429 of file elastoVariationalFunctionals.hh.
Public Member Functions | |
| OrthotropicLameNavier (Dune::FieldMatrix< Scalar, dim, dim > const &orth_, Dune::FieldMatrix< Scalar, dim, dim > const &mat_) | |
| Constructor. More... | |
| OrthotropicLameNavier (Dune::FieldMatrix< Scalar, dim, dim > const &orth_, Dune::FieldMatrix< Scalar, dim, dim > const &mat1, Dune::FieldVector< Scalar, dim *(dim-1)/2 > const &mat2) | |
| Constructor. More... | |
| OrthotropicLameNavier (ElasticModulus const &p) | |
| Default constructor. More... | |
| OrthotropicLameNavier () | |
| Default constructor. More... | |
| Dune::FieldMatrix< Scalar, dim *(dim+1)/2, dim *(dim+1)/2 > const & | stiffnessMatrix () const |
| Returns the stiffness tensor. More... | |
| void | setLinearizationPoint (Dune::FieldMatrix< Scalar, dim, dim > const &du0_) |
| Defines the displacement derivative around which to linearize. More... | |
| Scalar | d0 () const |
| Computes the elastic energy \( \frac{1}{2} \epsilon(u_x) : \mathcal{C}\epsilon(u_x) \). More... | |
| Dune::FieldVector< Scalar, dim > | d1 (VariationalArg< Scalar, dim > const &arg) const |
| Computes the elastic energy \( \frac{1}{2} \epsilon(u_x) : \mathcal{C}\epsilon(u_x) \). More... | |
| Dune::FieldMatrix< Scalar, dim, dim > | d2 (VariationalArg< Scalar, dim > const &arg1, VariationalArg< Scalar, dim > const &arg2) const |
| Computes the local Hessian of the variational functional. For scalar functions \( \phi, \psi \), the values and derivatives at a certain point are provided in arg1 and arg 2, this computes the \( d\times d \) matrix \( A \) with \( A_{ij} = \epsilon((u_i)_x)^T \mathcal{C} \epsilon((v_j)_x) \), where \( u_i = \phi e_i, v_j = \psi e_j \) and \( e_k \) is the \( k \)-th unit vector. More... | |
| void | setLinear (bool lin) |
| Kaskade::Elastomechanics::OrthotropicLameNavier< Scalar, dim >::OrthotropicLameNavier | ( | Dune::FieldMatrix< Scalar, dim, dim > const & | orth_, |
| Dune::FieldMatrix< Scalar, dim, dim > const & | mat_ | ||
| ) |
Constructor.
| orth | The material coordinate system matrix. Has to be orthonormal. |
| mat | The material parameters. |
The columns of the orth matrix are the material coordinate directions, i.e. a multiplication by this matrix transforms from material coordinates to world coordinates.
mat is a matrix containing the material parameters in the following way:
\[ \begin{bmatrix} E_1 & G_{12} \\ \nu_{12} & E_2 \end{bmatrix}, \quad \begin{bmatrix} E_1 & G_{12} & G_{13} \\ \nu_{12} & E_2 & G_{23} \\ \nu_{13} & \nu_{23} & E_3 \end{bmatrix}, \]
where \( E_i \) is the elastic modulus in material coordinate direction \( i \), \( \nu_{ij} \) is the Poissons ratio for contraction in \( j \) direction due to strain in \( i \) direction, and \( G_{ij} \) is the shear modulus in \( ij \) plane.
| Kaskade::Elastomechanics::OrthotropicLameNavier< Scalar, dim >::OrthotropicLameNavier | ( | Dune::FieldMatrix< Scalar, dim, dim > const & | orth_, |
| Dune::FieldMatrix< Scalar, dim, dim > const & | mat1, | ||
| Dune::FieldVector< Scalar, dim *(dim-1)/2 > const & | mat2 | ||
| ) |
Constructor.
| orth | The material coordinate system matrix. Has to be orthonormal. |
| mat1 | material parameters |
| mat2 | material parameters |
The columns of the orth matrix are the material coordinate directions, i.e. a multiplication by this matrix transforms from material coordinates to world coordinates.
mat1 is a \( d \times d \) matrix containing the top left part of the stiffness matrix \( C \), and mat2 is a \( d(d-1)/2 \) vector containing the diagonal of the bottom right part.
\[ C = \begin{bmatrix} \mathrm{mat1} & \\ & \mathrm{diag}(\mathrm{mat2}) \end{bmatrix}, \]
| Kaskade::Elastomechanics::OrthotropicLameNavier< Scalar, dim >::OrthotropicLameNavier | ( | ElasticModulus const & | p | ) |
Default constructor.
This initializes the material to an isotropic material with given material parameters.
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inline |
Default constructor.
This initializes the material to an isotropic material with Lame parameters set to 1.
Definition at line 476 of file elastoVariationalFunctionals.hh.
| Scalar Kaskade::Elastomechanics::OrthotropicLameNavier< Scalar, dim >::d0 | ( | ) | const |
Computes the elastic energy \( \frac{1}{2} \epsilon(u_x) : \mathcal{C}\epsilon(u_x) \).
| Dune::FieldVector< Scalar, dim > Kaskade::Elastomechanics::OrthotropicLameNavier< Scalar, dim >::d1 | ( | VariationalArg< Scalar, dim > const & | arg | ) | const |
Computes the elastic energy \( \frac{1}{2} \epsilon(u_x) : \mathcal{C}\epsilon(u_x) \).
arg is assumed to be a scalar variational argument.
| Dune::FieldMatrix< Scalar, dim, dim > Kaskade::Elastomechanics::OrthotropicLameNavier< Scalar, dim >::d2 | ( | VariationalArg< Scalar, dim > const & | arg1, |
| VariationalArg< Scalar, dim > const & | arg2 | ||
| ) | const |
Computes the local Hessian of the variational functional. For scalar functions \( \phi, \psi \), the values and derivatives at a certain point are provided in arg1 and arg 2, this computes the \( d\times d \) matrix \( A \) with \( A_{ij} = \epsilon((u_i)_x)^T \mathcal{C} \epsilon((v_j)_x) \), where \( u_i = \phi e_i, v_j = \psi e_j \) and \( e_k \) is the \( k \)-th unit vector.
This is exactly what is to be returned from the d2 method of a a variational functional's domain cache implementing the Lame-Navier equations.
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Definition at line 520 of file elastoVariationalFunctionals.hh.
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Defines the displacement derivative around which to linearize.
Definition at line 492 of file elastoVariationalFunctionals.hh.
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Returns the stiffness tensor.
The stiffness tensor \( C \) is represented in matrix form for Voigt notation (i.e. the mapping from Voigt strain vector to Voigt stress vector). In 3D this reads
\[ \begin{bmatrix} \sigma_{11} \\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{23} \\ \sigma_{13} \\ \sigma_{12} \end{bmatrix} = C \begin{bmatrix} \epsilon_{11} \\ \epsilon_{22} \\ \epsilon_{33} \\ 2\epsilon_{23} \\ 2\epsilon_{13} \\ 2\epsilon_{12} \end{bmatrix} \]
Definition at line 487 of file elastoVariationalFunctionals.hh.
1.9.4