Linear elasticity: Difference between revisions

Revision as of 21:57, 9 May 2022

Revision as of 15:25, 10 May 2022

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In direct [[tensor]] form that is independent of the choice of coordinate system, these governing equations are:<ref name=Slau>Slaughter, W. S., (2002), ''The linearized theory of elasticity'', Birkhauser.</ref>

* [[Momentum#Linear momentum for a system|Equation of motion]], which is an expression of [[Newton's laws of motion#Newton's second law|Newton's second law]]: <math display="block">\boldsymbol{\nabla} \cdot \boldsymbol{\sigma} + \mathbf{F} = \rho \ddot{\mathbf{u}} </math>

* [[Infinitesimal strain theory|Strain-displacement]] equations: <math display="block">\boldsymbol{\varepsilon} = \tfrac{1}{2} \left[\boldsymbol{\nabla}\mathbf{u} + (\boldsymbol{\nabla}\mathbf{u})^\mathrm{T}\right]\,\!</math>

* [[Constitutive equations]]. For elastic materials, [[Hooke's law]] represents the material behavior and relates the unknown stresses and strains. The general equation for Hooke's law is <math display="block"> \boldsymbol{\sigma} = \mathsf{C}:\boldsymbol{\varepsilon},</math>

where <math>\boldsymbol{\sigma}</math> is the [[Cauchy stress tensor]], <math>\boldsymbol{\varepsilon}</math> is the [[infinitesimal strain]] tensor, <math>\mathbf{u}</math> is the [[Displacement (vector)|displacement vector]], <math>\mathsf{C}</math> is the fourth-order [[stiffness tensor]], <math>\mathbf{F}</math> is the body force per unit volume, <math>\rho</math> is the mass density, <math>\boldsymbol{\nabla}</math> represents the [[nabla operator]], <math>(\bullet)^\mathrm{T}</math> represents a [[transpose]], <math>\ddot{(\bullet)}</math> represents the second derivative with respect to time, and <math>\mathsf{A}:\mathsf{B} = A_{ij}B_{ij}</math> is the inner product of two second-order tensors (summation over repeated indices is implied).

=== Cartesian coordinate form ===

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Expressed in terms of components with respect to a rectangular [[Cartesian coordinate]] system, the governing equations of linear elasticity are:<ref name=Slau/>

* [[Cauchy momentum equation|Equation of motion]]: <math display="block"> \sigma_{ji,j} + F_i = \rho \partial_{tt} u_i</math>

:<math>\sigma_{ji,j}+ F_i = \rho \partial_{tt} u_i\,\!</math>

:{| class="collapsible collapsed" width="30%" style="text-align:left"

!Engineering notation

|-

|<math>\frac{\partial \sigma_x}{\partial x} + \frac{\partial \tau_{yx}}{\partial y} + \frac{\partial \tau_{zx}}{\partial z} + F_x = \rho \frac{\partial^2 u_x}{\partial t^2}\,\!</math>

<math>\frac{\partial \tau_{xy}}{\partial x} + \frac{\partial \sigma_y}{\partial y} + \frac{\partial \tau_{zy}}{\partial z} + F_y = \rho \frac{\partial^2 u_y}{\partial t^2}\,\!</math>

<math>\frac{\partial \tau_{xz}}{\partial x} + \frac{\partial \tau_{yz}}{\partial y} + \frac{\partial \sigma_z}{\partial z} + F_z = \rho \frac{\partial^2 u_z}{\partial t^2}\,\!</math>

|}

:where the <math>{(\bullet)}_{,j}</math> subscript is a shorthand for <math>\partial{(\bullet)} / \partial x_j</math> and <math>\partial_{tt}</math> indicates <math>\partial^2 / \partial t^2</math>, <math> \sigma_{ij} = \sigma_{ji}\,\!</math> is the Cauchy [[Stress (physics)|stress]] tensor, <math> F_i\,\!</math> is the body force density, <math> \rho\,\!</math> is the mass density, and <math> u_i\,\!</math> is the displacement.

:These are 3 [[System of linear equations#Independence|independent]] equations with 6 independent unknowns (stresses).

* [[Deformation (mechanics)#Infinitesimal strain|Strain-displacement]] equations: <math display="block">\varepsilon_{ij} =\frac{1}{2} (u_{j,i} + u_{i,j})</math>

:<math>\varepsilon_{ij} =\frac{1}{2} (u_{j,i}+u_{i,j})\,\!</math>

:{| class="collapsible collapsed" width="30%" style="text-align:left"

!Engineering notation

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|<math>\gamma_{zx}=\frac{\partial u_z}{\partial x}+\frac{\partial u_x}{\partial z}\,\!</math>

|}

:where <math> \varepsilon_{ij}=\varepsilon_{ji}\,\!</math> is the strain. These are 6 independent equations relating strains and displacements with 9 independent unknowns (strains and displacements).

* [[Constitutive equations]]. The equation for Hooke's law is: <math display="block">

:\sigma_{ij} = C_{ijkl} \, \varepsilon_{kl} </math> where <math>C_{ijkl}</math> is the stiffness tensor. These are 6 independent equations relating stresses and strains. The requirement of the symmetry of the stress and strain tensors lead to equality of many of the elastic constants, reducing the number of different elements to 21<ref>{{cite journal |last=Belen'kii |last2= Salaev|date= 1988|title= Deformation effects in layer crystals|journal= Uspekhi Fizicheskikh Nauk|volume= 155|pages= 89|doi= 10.3367/UFNr.0155.198805c.0089}}</ref> <math> C_{ijkl} = C_{klij} = C_{jikl} = C_{ijlk}</math>.

:<math>

\sigma_{ij} = C_{ijkl} \, \varepsilon_{kl}

\,\!</math>

:where <math>C_{ijkl}</math> is the stiffness tensor. These are 6 independent equations relating stresses and strains. The requirement of the symmetry of the stress and strain tensors lead to equality of many of the elastic constants, reducing the number of different elements to 21<ref>{{cite journal |last=Belen'kii |last2= Salaev|date= 1988|title= Deformation effects in layer crystals|journal= Uspekhi Fizicheskikh Nauk|volume= 155|pages= 89|doi= 10.3367/UFNr.0155.198805c.0089}}</ref> <math> C_{ijkl}=C_{klij}=C_{jikl}=C_{ijlk}</math>.

An elastostatic boundary value problem for an isotropic-homogeneous media is a system of 15 independent equations and equal number of unknowns (3 equilibrium equations, 6 strain-displacement equations, and 6 constitutive equations). Specifying the boundary conditions, the boundary value problem is completely defined. To solve the system two approaches can be taken according to boundary conditions of the boundary value problem: a '''displacement formulation''', and a '''stress formulation'''.

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===Cylindrical coordinate form===

In cylindrical coordinates (<math>r,\theta,z</math>) the equations of motion are<ref name=Slau/>

<math display="block">\begin{align}

:<math>

& \frac{\partial \sigma_{rzrr}}{\partial r} + \cfracfrac{1}{r}\frac{\partial \sigma_{r\theta z}}{\partial \theta} + \frac{\partial \sigma_{zzrz}}{\partial z} + \cfrac{1}{r}(\sigma_{rzrr}-\sigma_{\theta\theta}) + F_zF_r = \rho~\frac{\partial^2 u_zu_r}{\partial t^2} \\

\begin{align}

& \frac{\partial \sigma_{rrr\theta}}{\partial r} + \cfracfrac{1}{r} \frac{\partial \sigma_{r\theta\theta}}{\partial \theta} + \frac{\partial \sigma_{rz\theta z}}{\partial z} + \cfracfrac{12}{r}(\sigma_{rr}-\sigma_{\thetar\theta}) + F_rF_\theta = \rho~\frac{\partial^2 u_ru_\theta}{\partial t^2} \\

& \frac{\partial \sigma_{r\thetarz}}{\partial r} + \cfracfrac{1}{r}\frac{\partial \sigma_{\theta\theta z}}{\partial \theta} + \frac{\partial \sigma_{\theta zzz}}{\partial z} + \cfracfrac{21}{r} \sigma_{r\thetarz} + F_\thetaF_z = \rho~\frac{\partial^2 u_\thetau_z}{\partial t^2} \\

\,\!end{align}</math>

& \frac{\partial \sigma_{rz}}{\partial r} + \cfrac{1}{r}\frac{\partial \sigma_{\theta z}}{\partial \theta} + \frac{\partial \sigma_{zz}}{\partial z} + \cfrac{1}{r}\sigma_{rz} + F_z = \rho~\frac{\partial^2 u_z}{\partial t^2}

\end{align}

</math>

The strain-displacement relations are

<math display="block">\begin{align}

:<math>

\varepsilon_{rr} & = \frac{\partial u_r}{\partial r} ~;~~

\begin{align}

\varepsilon_{rr\theta\theta} & = \frac{1}{r} \left(\cfrac{\partial u_ru_\theta}{\partial r\theta} + u_r\right) ~;~~

\varepsilon_{\theta\thetazz} = \cfrac{1}{r}\left(\cfracfrac{\partial u_\thetau_z}{\partial \thetaz} + u_r\right) ~;~~\

\varepsilon_{zzr\theta} & = \frac{1}{2} \left(\cfrac{1}{r}\cfrac{\partial u_zu_r}{\partial z\theta} + \cfrac{\partial u_\theta}{\partial r}- \cfrac{u_\theta}{r}\right) ~;~~

\varepsilon_{r\theta z} & = \cfrac{1}{2} \left(\cfrac{1}{r}\cfrac{\partial u_ru_\theta}{\partial \thetaz} + \cfrac{1}{r}\cfrac{\partial u_\thetau_z}{\partial r}- \cfrac{u_\theta}{r}\right) ~;~~

\varepsilon_{\theta zzr} = \cfrac{1}{2} \left(\cfrac{\partial u_\thetau_r}{\partial z} + \cfrac{1}{r}\cfrac{\partial u_z}{\partial \thetar}\right) ~;~~

\end{align}</math>

\varepsilon_{zr} = \cfrac{1}{2}\left(\cfrac{\partial u_r}{\partial z} + \cfrac{\partial u_z}{\partial r}\right)

\end{align}

</math>

and the constitutive relations are the same as in Cartesian coordinates, except that the indices <math>1</math>,<math>2</math>,<math>3</math> now stand for <math>r</math>,<math>\theta</math>,<math>z</math>, respectively.

=== Spherical coordinate form ===

In spherical coordinates (<math>r,\theta,\phi</math>) the equations of motion are<ref name=Slau/>

<math display="block">\begin{align}

:<math>

& \frac{\partial \sigma_{rr}}{\partial r} + \cfrac{1}{r}\frac{\partial \sigma_{r\theta}}{\partial \theta} + \cfrac{1}{r\sin\theta}\frac{\partial \sigma_{r\phi}}{\partial \phi} + \cfrac{1}{r} (2\sigma_{rr}-\sigma_{\theta\theta}-\sigma_{\phi\phi}+\sigma_{r\theta}\cot\theta) + F_r = \rho~\frac{\partial^2 u_r}{\partial t^2} \\

\begin{align}

& \frac{\partial \sigma_{rr}}{\partial r} + \cfrac{1}{r}\frac{\partial \sigma_{r\theta}}{\partial \theta} + \cfrac{1}{r\sin\theta}\frac{\partial \sigma_{r\phi}}{\partial \phi} + \cfrac{1}{r}(2\sigma_{rr}-\sigma_{\theta\theta}-\sigma_{\phi\phi}+\sigma_{r\theta}\cot\theta) + F_r = \rho~\frac{\partial^2 u_r}{\partial t^2} \\

& \frac{\partial \sigma_{r\theta}}{\partial r} + \cfrac{1}{r}\frac{\partial \sigma_{\theta\theta}}{\partial \theta} + \cfrac{1}{r\sin\theta}\frac{\partial \sigma_{\theta \phi}}{\partial \phi} + \cfrac{1}{r}[(\sigma_{\theta\theta}-\sigma_{\phi\phi})\cot\theta + 3\sigma_{r\theta}] + F_\theta = \rho~\frac{\partial^2 u_\theta}{\partial t^2} \\

& \frac{\partial \sigma_{r\phi}}{\partial r} + \cfrac{1}{r}\frac{\partial \sigma_{\theta \phi}}{\partial \theta} + \cfrac{1}{r\sin\theta}\frac{\partial \sigma_{\phi\phi}}{\partial \phi} + \cfrac{1}{r}(2\sigma_{\theta\phi}\cot\theta+3\sigma_{r\phi}) + F_\phi = \rho~\frac{\partial^2 u_\phi}{\partial t^2}

\end{align}</math>

[[File:3D Spherical.svg|thumb|240px|right|Spherical coordinates (''r'', ''&theta; θ'', ''&phi;φ'') as commonly used in ''physics'': radial distance ''r'', polar angle ''&theta;θ'' ([[theta]]), and azimuthal angle ''&phi;φ'' ([[phi]]). The symbol ''&rho;ρ'' ([[rho]]) is often used instead of ''r''.]]

</math>

[[File:3D Spherical.svg|thumb|240px|right|Spherical coordinates (''r'', ''&theta;'', ''&phi;'') as commonly used in ''physics'': radial distance ''r'', polar angle ''&theta;'' ([[theta]]), and azimuthal angle ''&phi;'' ([[phi]]). The symbol ''&rho;'' ([[rho]]) is often used instead of ''r''.]]

The strain tensor in spherical coordinates is

<math display="block">\begin{align}

:<math>

\varepsilon_{rr} & = \frac{\partial u_r}{\partial r}\\

\begin{align}

\varepsilon_{rr\theta\theta} & = \frac{1}{r} \left(\frac{\partial u_ru_\theta}{\partial r\theta} + u_r\right)\\

\varepsilon_{\thetaphi\thetaphi} & = \frac{1}{r\sin\theta} \left(\frac{\partial u_\thetaphi}{\partial \thetaphi} + u_r\sin\theta + u_\theta\cos\theta\right)\\

\varepsilon_{r\phi\phitheta} & = \frac{1}{r\sin\theta2} \left(\frac{1}{r} \frac{\partial u_\phiu_r}{\partial \phitheta} + u_r\sinfrac{\theta +partial u_\theta}{\cospartial r} - \frac{u_\theta}{r}\right) \\

\varepsilon_{r\theta \phi} & = \frac{1}{22r} \left([\frac{1}{r\sin\theta}\frac{\partial u_ru_\theta}{\partial \thetaphi} + \left(\frac{\partial u_\thetaphi}{\partial r\theta} - \frac{u_\phi \cot\theta}{r}\right) \right]\\

\varepsilon_{\thetar \phi} & = \frac{1}{2r2} \left[(\frac{1}{r \sin \theta} \frac{\partial u_\thetau_r}{\partial \phi} +\left( \frac{\partial u_\phi}{\partial \thetar} - \frac{u_\phi \cot\theta}{r}\right)\right]\\.

\end{align}</math>

\varepsilon_{r \phi} & = \frac{1}{2} \left(\frac{1}{r \sin \theta} \frac{\partial u_r}{\partial \phi} + \frac{\partial u_\phi}{\partial r} - \frac{u_\phi}{r}\right).

\end{align}

</math>

== (An)isotropic (in)homogeneous media ==