Elasticity::ConstitutiveRelations::IsotropicHomogeneous< Dim > Class Template Reference

An isotropic and homogeneous material. More...

#include <IsotropicHomogeneous.hpp>

Classes
struct	BulkModulus
struct	ShearModulus

Public Types
using	options = implementation defined

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

template<size_t Dim>
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

$$\begin{gathered} Y^{ijkl}=\lambda {\delta }^{ij}{\delta }^{kl}+\mu ({\delta }^{ik}{\delta }^{jl}+{\delta }^{il}{\delta }^{jk}) \\ ⟹T^{ij}=-\lambda \mathrm{T}\mathrm{r}(S){\delta }^{ij}-2\mu S^{ij} \end{gathered}$$

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

$$T^{ij}=-K\mathrm{T}\mathrm{r}(S){\delta }^{ij}-2\mu (S^{ij}-\frac{1}{3}\mathrm{T}\mathrm{r}(S){\delta }^{ij})$$

decomposes into a scalar and a traceless part (Eq. 11.18 in [194]). 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:

$$K=\frac{E}{3(1-2\nu )},\lambda =\frac{E\nu }{(1+\nu )(1-2\nu )},\mu =\frac{E}{2(1+\nu )}$$

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 $\mathrm{T}\mathrm{r}(S)=\frac{2\mu }{\lambda +2\mu }{\mathrm{T}\mathrm{r}}^{(2)}(S)$, where ${\mathrm{T}\mathrm{r}}^{(2)}$ denotes that the trace only applies to the two dimensions within the plane. The constitutive relation thus reduces to

$$\begin{array}{}\text{(1)}& T^{ij}=& -\frac{2\lambda \mu }{\lambda +2\mu }{\mathrm{T}\mathrm{r}}^{(2)}(S){\delta }^{ij}-2\mu S^{ij}\text{(2)}& =& -\frac{E\nu }{1-{\nu }^2}{\mathrm{T}\mathrm{r}}^{(2)}(S){\delta }^{ij}-\frac{E}{1+\nu }{S}^{ij}\end{array}$$

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 ${\partial }_3{T}^{ij}=0$ the elasticity problem $-{\partial }_i{T}^{ij}=F^j$ reduces to two dimensions.

Member Function Documentation

get_clone()

template<size_t Dim>

std::unique_ptr< ConstitutiveRelation< Dim > > Elasticity::ConstitutiveRelations::IsotropicHomogeneous< Dim >::get_clone ( ) const

overridevirtual

Returns a std::unique_ptr pointing to a copy of the ConstitutiveRelation.

Implements Elasticity::ConstitutiveRelations::ConstitutiveRelation< Dim >.

stress()

template<size_t Dim>

void Elasticity::ConstitutiveRelations::IsotropicHomogeneous< Dim >::stress ( gsl::not_null< tnsr::II< DataVector, Dim > * >  stress, const tnsr::ii< DataVector, Dim > &  strain, const tnsr::I< DataVector, Dim > &  x  ) const

overridevirtual

The constitutive relation that characterizes the elastic properties of a material.

Implements Elasticity::ConstitutiveRelations::ConstitutiveRelation< Dim >.

Member Data Documentation

help

template<size_t Dim>

constexpr Options::String Elasticity::ConstitutiveRelations::IsotropicHomogeneous< Dim >::help

staticconstexpr

Initial value:

= {
      "A constitutive relation that describes an isotropic, homogeneous "
      "material in terms of two elastic moduli. These bulk and shear moduli "
      "indicate the material's resistance to volume and shape changes, "
      "respectively. Both are measured in units of stress, typically Pascals."}

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The documentation for this class was generated from the following file:

• src/PointwiseFunctions/Elasticity/ConstitutiveRelations/IsotropicHomogeneous.hpp