Elasticity::Solutions::HalfSpaceMirror Class Reference

The solution for a half-space mirror deformed by a laser beam. More...

#include <HalfSpaceMirror.hpp>

Classes
struct	AbsoluteTolerance
struct	BeamWidth
struct	IntegrationIntervals
struct	Material
struct	RelativeTolerance

Public Types
using	constitutive_relation_type = Elasticity::ConstitutiveRelations::IsotropicHomogeneous< 3 >
using	options = implementation defined

Public Member Functions
	HalfSpaceMirror (const HalfSpaceMirror &)=default
HalfSpaceMirror &	operator= (const HalfSpaceMirror &)=default
	HalfSpaceMirror (HalfSpaceMirror &&)=default
HalfSpaceMirror &	operator= (HalfSpaceMirror &&)=default
std::unique_ptr< elliptic::analytic_data::AnalyticSolution >	get_clone () const override
	HalfSpaceMirror (double beam_width, constitutive_relation_type constitutive_relation, size_t integration_intervals=350, double absolute_tolerance=1e-12, double relative_tolerance=1e-10)
double	beam_width () const
size_t	integration_intervals () const
double	absolute_tolerance () const
double	relative_tolerance () const
const constitutive_relation_type &	constitutive_relation () const
template<typename DataType , typename... RequestedTags>
tuples::TaggedTuple< RequestedTags... >	variables (const tnsr::I< DataType, 3 > &x, tmpl::list< RequestedTags... >) const
void	pup (PUP::er &p) override
	NOLINTNEXTLINE(google-runtime-references)
virtual std::unique_ptr< AnalyticSolution >	get_clone () const =0

Static Public Attributes
static constexpr Options::String	help

Detailed Description

The solution for a half-space mirror deformed by a laser beam.

Details

This solution is mapping (via the fluctuation dissipation theorem) thermal noise to an elasticity problem where a normally incident and axisymmetric laser beam with a Gaussian beam profile acts on the face of a semi-infinite mirror. Here we assume the face to be at $z=0$ and the material to extend to $+\mathrm{\infty }$ in the z-direction as well as for the mirror diameter to be comparatively large to the beam width. The mirror material is characterized by an isotropic homogeneous constitutive relation $Y^{ijkl}$ (see Elasticity::ConstitutiveRelations::IsotropicHomogeneous). In this scenario, the auxiliary elastic problem has an applied pressure distribution equal to the laser beam intensity profile $p(r)$ (see Eq. (11.94) and Eq. (11.95) in [194] with F = 1 and the time dependency dropped)

$$\begin{array}{}\text{(1)}& T^{zr}& =T^{rz}=0\text{(2)}& T^{zz}& =p(r)=\frac{e^{-\frac{r^2}{r_0^2}}}{\pi r_0^2}\text{.}\end{array}$$

in the form of a Neumann boundary condition to the face of the mirror. We find that this stress in cylinder coordinates is produced by the displacement field

$$\begin{array}{}\text{(3)}& {\xi }_r& =\frac{1}{2\mu }{\int }_0^{\mathrm{\infty }}dk{J}_1(kr)e^{(-kz)}(1-\frac{\lambda +2\mu }{\lambda +\mu }+kz)\tilde{p}(k)\text{(4)}& {\xi }_{\varphi }& =0\text{(5)}& {\xi }_z& =\frac{1}{2\mu }{\int }_0^{\mathrm{\infty }}dk{J}_0(kr)e^{(-kz)}(1+\frac{\mu }{\lambda +\mu }+kz)\tilde{p}(k)\end{array}$$

and the strain

$$\begin{array}{}\text{(6)}& \mathrm{\Theta }& =\frac{1}{2\mu }{\int }_0^{\mathrm{\infty }}dk{J}_0(kr)k{e}^{(-kz)}\left(\frac{-2\mu }{\lambda +\mu }\right)\tilde{p}(k)\text{(7)}& S_{rr}& =\mathrm{\Theta }-S_{\varphi \varphi }-S_{zz}\text{(8)}& S_{\varphi \varphi }& =\frac{{\xi }_r}{r}\text{(9)}& S_{(rz)}& =-\frac{1}{2\mu }{\int }_0^{\mathrm{\infty }}dk{J}_1(kr)k{e}^{(-kz)}\left(kz\right)\tilde{p}(k)\text{(10)}& S_{zz}& =\frac{1}{2\mu }{\int }_0^{\mathrm{\infty }}dk{J}_0(kr)k{e}^{(-kz)}(-\frac{\mu }{\lambda +\mu }-kz)\tilde{p}(k)\end{array}$$

(see Eqs. (11 a) - (11 c) and (13 a) - (13 e), with (13 c) swapped in favor of (12 c) in [134]), where $\tilde{p}(k)=\frac{1}{2\pi }{e}^{-(\frac{k{r}_0}{2}{)}^2}$ is the Hankel-Transform of the lasers intensity profile and $\mathrm{\Theta }=\mathrm{T}\mathrm{r}(S)$ the materials expansion.

Member Function Documentation

get_clone()

std::unique_ptr< elliptic::analytic_data::AnalyticSolution > Elasticity::Solutions::HalfSpaceMirror::get_clone ( ) const

inlineoverridevirtual

Implements elliptic::analytic_data::AnalyticSolution.

Member Data Documentation

help

constexpr Options::String Elasticity::Solutions::HalfSpaceMirror::help

staticconstexpr

Initial value:

{
      "A semi-infinite mirror on which a laser introduces stress perpendicular "
      "to the mirrors surface."}

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

• src/PointwiseFunctions/AnalyticSolutions/Elasticity/HalfSpaceMirror.hpp