An isotropic and homogeneous material. More...
#include <IsotropicHomogeneous.hpp>
Classes | |
| struct | BulkModulus |
| struct | ShearModulus |
Public Types | |
| using | options = tmpl::list< BulkModulus, ShearModulus > |
Public Member Functions | |
| IsotropicHomogeneous (const IsotropicHomogeneous &)=default | |
| IsotropicHomogeneous & | operator= (const IsotropicHomogeneous &)=default |
| IsotropicHomogeneous (IsotropicHomogeneous &&)=default | |
| IsotropicHomogeneous & | operator= (IsotropicHomogeneous &&)=default |
| IsotropicHomogeneous (double bulk_modulus, double shear_modulus) | |
| std::unique_ptr< ConstitutiveRelation< Dim > > | get_clone () const override |
Returns a std::unique_ptr pointing to a copy of the ConstitutiveRelation. More... | |
| void | stress (gsl::not_null< tnsr::II< DataVector, Dim > * > stress, const tnsr::ii< DataVector, Dim > &strain, const tnsr::I< DataVector, Dim > &x) const override |
| The constitutive relation that characterizes the elastic properties of a material. More... | |
| double | bulk_modulus () const |
| The bulk modulus (or incompressibility) \(K\). | |
| double | shear_modulus () const |
| The shear modulus (or rigidity) \(\mu\). | |
| double | lame_parameter () const |
| The Lamé parameter \(\lambda\). | |
| double | youngs_modulus () const |
| The Young's modulus \(E\). | |
| double | poisson_ratio () const |
| The Poisson ratio \(\nu\). | |
| void | pup (PUP::er &) override |
| IsotropicHomogeneous (CkMigrateMessage *) | |
| WRAPPED_PUPable_decl_base_template (SINGLE_ARG(ConstitutiveRelation< Dim >), IsotropicHomogeneous) | |
 Public Member Functions inherited from Elasticity::ConstitutiveRelations::ConstitutiveRelation< Dim > | |
| ConstitutiveRelation (const ConstitutiveRelation &)=default | |
| ConstitutiveRelation & | operator= (const ConstitutiveRelation &)=default |
| ConstitutiveRelation (ConstitutiveRelation &&)=default | |
| ConstitutiveRelation & | operator= (ConstitutiveRelation &&)=default |
| WRAPPED_PUPable_abstract (ConstitutiveRelation) | |
| virtual std::unique_ptr< ConstitutiveRelation< Dim > > | get_clone () const =0 |
Returns a std::unique_ptr pointing to a copy of the ConstitutiveRelation. More... | |
| virtual void | stress (gsl::not_null< tnsr::II< DataVector, Dim > * > stress, const tnsr::ii< DataVector, Dim > &strain, const tnsr::I< DataVector, Dim > &x) const =0 |
| The constitutive relation that characterizes the elastic properties of a material. More... | |
Static Public Attributes | |
| static constexpr size_t | volume_dim = Dim |
| static constexpr Options::String | help |
| Static Public Attributes inherited from Elasticity::ConstitutiveRelations::ConstitutiveRelation< Dim > | |
| static constexpr size_t | volume_dim = Dim |
Detailed Description
class Elasticity::ConstitutiveRelations::IsotropicHomogeneous< Dim >
An isotropic and homogeneous material.
Details
For an isotropic and homogeneous material the linear constitutive relation \(T^{ij}=-Y^{ijkl}S_{kl}\) reduces to
\[ Y^{ijkl} = \lambda \delta^{ij}\delta^{kl} + \mu \left(\delta^{ik}\delta^{jl} + \delta^{il}\delta^{jk}\right) \\ \implies \quad T^{ij} = -\lambda \mathrm{Tr}(S) \delta^{ij} - 2\mu S^{ij} \]
with the Lamé parameter \(\lambda\) and the shear modulus (or rigidity) \(\mu\). In the parametrization chosen in this implementation we use the bulk modulus (or incompressibility)
\[ K=\lambda + \frac{2}{3}\mu \]
instead of the Lamé parameter. In this parametrization the stress-strain relation
\[ T^{ij} = -K \mathrm{Tr}(S) \delta^{ij} - 2\mu\left(S^{ij} - \frac{1}{3}\mathrm{Tr}(S)\delta^{ij}\right) \]
decomposes into a scalar and a traceless part (Eq. 11.18 in [191]). Parameters also often used in this context are the Young's modulus
\[ E=\frac{9K\mu}{3K+\mu}=\frac{\mu(3\lambda+2\mu)}{\lambda+\mu} \]
and the Poisson ratio
\[ \nu=\frac{3K-2\mu}{2(3K+\mu)}=\frac{\lambda}{2(\lambda+\mu)}\text{.} \]
Inversely, these relations read:
\[ K =\frac{E}{3(1-2\nu)}, \quad \lambda =\frac{E\nu}{(1+\nu)(1-2\nu)}, \quad \mu =\frac{E}{2(1+\nu)} \]
In two dimensions this implementation reduces to the plane-stress approximation. We assume that all stresses apply in the plane of the computational domain, which corresponds to scenarios of in-plane stretching and shearing of thin slabs of material. Since orthogonal stresses vanish as \(T^{i3}=0=T^{3i}\) we find \(\mathrm{Tr}(S)=\frac{2\mu}{\lambda + 2\mu}\mathrm{Tr}^{(2)}(S)\), where \(\mathrm{Tr}^{(2)}\) denotes that the trace only applies to the two dimensions within the plane. The constitutive relation thus reduces to
\begin{align} T^{ij}=&-\frac{2\lambda\mu}{\lambda + 2\mu}\mathrm{Tr}^{(2)}(S)\delta^{ij} - 2\mu S^{ij} \\ =&-\frac{E\nu}{1-\nu^2}\mathrm{Tr}^{(2)}(S)\delta^{ij} - \frac{E}{1+\nu}S^{ij} \end{align}
which is non-zero only in the directions of the plane. Since the stresses are also assumed to be constant along the thickness of the plane \(\partial_3T^{ij}=0\) the elasticity problem \(-\partial_i T^{ij}=F^j\) reduces to two dimensions.
Member Function Documentation
◆ get_clone()
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overridevirtual |
Returns a std::unique_ptr pointing to a copy of the ConstitutiveRelation.
Implements Elasticity::ConstitutiveRelations::ConstitutiveRelation< Dim >.
◆ stress()
|
overridevirtual |
The constitutive relation that characterizes the elastic properties of a material.
Implements Elasticity::ConstitutiveRelations::ConstitutiveRelation< Dim >.
Member Data Documentation
◆ help
|
staticconstexpr |
The documentation for this class was generated from the following file:
- src/PointwiseFunctions/Elasticity/ConstitutiveRelations/IsotropicHomogeneous.hpp