Elasticity::ConstitutiveRelations::CubicCrystal Class Reference

A cubic crystalline material. More...

#include <CubicCrystal.hpp>

Classes
struct	C_11
struct	C_12
struct	C_44

Public Types
using	options = implementation defined

Public Member Functions
	CubicCrystal (const CubicCrystal &)=default
CubicCrystal &	operator= (const CubicCrystal &)=default
	CubicCrystal (CubicCrystal &&)=default
CubicCrystal &	operator= (CubicCrystal &&)=default
	CubicCrystal (double c_11, double c_12, double c_44)
std::unique_ptr< ConstitutiveRelation< 3 > >	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, 3 > * > stress, const tnsr::ii< DataVector, 3 > &strain, const tnsr::I< DataVector, 3 > &x) const override
	The constitutive relation that characterizes the elastic properties of a material.
double	c_11 () const
	The 1st group parameter $c_{11}=\frac{1-\nu }{\nu }\lambda$.
double	c_12 () const
	The 2nd group parameter; the $\mathrm{L}\mathrm{a}\mathrm{m}\acute{\mathrm{e}}$ parameter $c_{12}=\lambda$.
double	c_44 () const
	The 3rd group parameter; the shear modulus (rigidity) $c_{44}=\mu$.
double	youngs_modulus () const
	The Young's modulus $E$.
double	poisson_ratio () const
	The Poisson ratio $\nu$.
void	pup (PUP::er &) override
	CubicCrystal (CkMigrateMessage *)
	WRAPPED_PUPable_decl_base_template (SINGLE_ARG(ConstitutiveRelation< 3 >), CubicCrystal)
Public Member Functions inherited from Elasticity::ConstitutiveRelations::ConstitutiveRelation< 3 >
	ConstitutiveRelation (const ConstitutiveRelation &)=default
	ConstitutiveRelation (ConstitutiveRelation &&)=default
ConstitutiveRelation &	operator= (const 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.

Static Public Attributes
static constexpr size_t	volume_dim = 3
static constexpr Options::String	help
Static Public Attributes inherited from Elasticity::ConstitutiveRelations::ConstitutiveRelation< 3 >
static constexpr size_t	volume_dim

Detailed Description

A cubic crystalline material.

Details

For a cubic crystalline material the Elasticity tensor in the linear constitutive relation $T^{ij}=-Y^{ijkl}{S}_{kl}$ reduces to

$$Y^{ijkl}=\{\begin{array}{ll}{c}_{11}& \mathrm{f}\mathrm{o}\mathrm{r}i=j=k=l\\ c_{12}& \mathrm{f}\mathrm{o}\mathrm{r}i=j,k=l,i\ne k\\ c_{44}& \mathrm{f}\mathrm{o}\mathrm{r}i=k,j=l,i\ne j\mathrm{o}\mathrm{r}i=l,j=k,i\ne j\end{array}$$

with the three independent parameters: the $\mathrm{L}\mathrm{a}\mathrm{m}\acute{\mathrm{e}}$ parameter $\lambda$, the Shear modulus $\mu$ and the Poisson ratio $\nu$. In the parametrization chosen in this implementation we use the experimental group parameters $c_{11}$, $c_{12}$ and $c_{44}$, related by;

$$c_{11}=\frac{1-\nu }{\nu }\lambda =\frac{(1-\nu )E}{(1+\nu )(1-2\nu )},c_{12}=\lambda =\frac{E\nu }{(1+\nu )(1-2\nu )},c_{44}=\mu$$

and inversely;

$$E=\frac{(c_{11}+2c_{12})(c_{11}-c_{12})}{c_{11}+c_{12}},\nu ={(1+\frac{c_{11}}{c_{12}})}^{-1},\mu =c_{44}$$

The stress-strain relation then reduces to

$$T^{ij}=\{\begin{array}{ll}-(c_{11}-c_{12})S^{ij}-c_{12}\mathrm{T}\mathrm{r}(S)& \mathrm{f}\mathrm{o}\mathrm{r}i=j\\ -2c_{44}{S}^{ij}& \mathrm{f}\mathrm{o}\mathrm{r}i\ne j\end{array}$$

In the case where the shear modulus satisfies $c_{44}=\frac{c_{11}-c_{12}}{2}$ the constitutive relation is that of an isotropic material (see Elasticity::ConstitutiveRelations::IsotropicHomogeneous).

Member Function Documentation

get_clone()

std::unique_ptr< ConstitutiveRelation< 3 > > Elasticity::ConstitutiveRelations::CubicCrystal::get_clone ( ) const

overridevirtual

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

Implements Elasticity::ConstitutiveRelations::ConstitutiveRelation< 3 >.

Member Data Documentation

help

constexpr Options::String Elasticity::ConstitutiveRelations::CubicCrystal::help

staticconstexpr

Initial value:

= {
      "A constitutive relation that describes a cubic, crystalline material in "
      "terms of the three independent group paremeters. The parameters "
      "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/CubicCrystal.hpp