Linear elasticity: Difference between revisions

Revision as of 23:46, 12 May 2022

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====Solutions for elastostatic cases====

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!===== Thomson's solution - point force in an infinite isotropic medium =====

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|The most important solution of the Navier–Cauchy or elastostatic equation is for that of a force acting at a point in an infinite isotropic medium. This solution was found by [[William Thomson, 1st Baron Kelvin|William Thomson]] (later Lord Kelvin) in 1848 (Thomson 1848). This solution is the analog of [[Coulomb's law]] in [[electrostatics]]. A derivation is given in Landau & Lifshitz.<ref name=LL>{{cite book |title=Theory of Elasticity |edition=3rd|last=Landau |first=L.D. |author-link=Lev Landau |author2=Lifshitz, E. M. |author-link2=Evgeny Lifshitz |year=1986 |publisher=Butterworth Heinemann |___location=Oxford, England |isbn=0-7506-2633-X }}</ref>{{rp|§8}} Defining

<math display="block">a = 1-2\nu</math>

<math display="block">b = 2(1-\nu) = a+1</math>

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It may be also compactly written as:

<math display="block">G_{ik} = \frac{1}{4\pi\mu} \left[\frac{\delta_{ik}}{r} - \frac{1}{2b} \frac{\partial^2 r}{\partial x_i \partial x_k}\right]</math>

<math display="block">G_{ik}=

\frac{1}{4\pi\mu} \left[\frac{\delta_{ik}}{r} - \frac{1}{2b} \frac{\partial^2 r}{\partial x_i \partial x_k}\right]</math>

and it may be explicitly written as:

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It is particularly helpful to write the displacement in cylindrical coordinates for a point force <math>F_z</math> directed along the z-axis. Defining <math>\hat{\mathbf{\rho}}</math> and <math>\hat{\mathbf{z}}</math> as unit vectors in the <math>\rho</math> and <math>z</math> directions respectively yields:

<math display="block">

\mathbf{u} = \frac{F_z}{4\pi\mu r} \left[\frac{1}{4(1-\nu)} \, \frac{\rho z}{r^2} \hat{\boldsymbol{\rho}} + \left(1-\frac{1}{4(1-\nu)}\,\frac{\rho^2}{r^2}\right)\hat{\mathbf{z}}\right]</math>

\,\!</math>

It can be seen that there is a component of the displacement in the direction of the force, which diminishes, as is the case for the potential in electrostatics, as 1/''r'' for large ''r''. There is also an additional ρ-directed component.

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!===== Boussinesq–Cerruti solution - point force at the origin of an infinite isotropic half-space =====

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|Another useful solution is that of a point force acting on the surface of an infinite half-space. It was derived by Boussinesq<ref>{{cite book |title= Application des potentiels à l'étude de l'équilibre et du mouvement des solides élastiques |last=Boussinesq|first=Joseph |author-link=Joseph Boussinesq |year=1885 |publisher=Gauthier-Villars |___location=Paris, France |url=http://name.umdl.umich.edu/ABV5032.0001.001 }}</ref> for the normal force and Cerruti for the tangential force and a derivation is given in Landau & Lifshitz.<ref name=LL/>{{rp|§8}} In this case, the solution is again written as a Green's tensor which goes to zero at infinity, and the component of the stress tensor normal to the surface vanishes. This solution may be written in Cartesian coordinates as [note: a=(1-2ν) and b=2(1-ν), ν= = Poissons ratio]:

!Boussinesq–Cerruti solution - point force at the origin of an infinite isotropic half-space

|-

|Another useful solution is that of a point force acting on the surface of an infinite half-space. It was derived by Boussinesq<ref>{{cite book |title= Application des potentiels à l'étude de l'équilibre et du mouvement des solides élastiques |last=Boussinesq|first=Joseph |author-link=Joseph Boussinesq |year=1885 |publisher=Gauthier-Villars |___location=Paris, France |url=http://name.umdl.umich.edu/ABV5032.0001.001 }}</ref> for the normal force and Cerruti for the tangential force and a derivation is given in Landau & Lifshitz.<ref name=LL/>{{rp|§8}} In this case, the solution is again written as a Green's tensor which goes to zero at infinity, and the component of the stress tensor normal to the surface vanishes. This solution may be written in Cartesian coordinates as [note: a=(1-2ν) and b=2(1-ν), ν== Poissons ratio]:

:<math display="block">G_{ik} = \frac{1}{4\pi\mu} \begin{bmatrix}

\frac{b}{r}+\frac{x^2}{r^3}-\frac{ax^2}{r(r+z)^2}-\frac{az}{r(r+z)} &

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\frac{b}{r}+\frac{z^2}{r^3}

\end{bmatrix}

\,\!</math>

===== Other solutions: =====

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Other solutions:

* Point force inside an infinite isotropic half-space.<ref>{{cite journal |last=Mindlin |first= R. D.|author-link=Raymond D. Mindlin |year=1936|title=Force at a point in the interior of a semi-infinite solid |journal=Physics |volume=7|issue= 5|pages=195–202 |url= http://www.dtic.mil/get-tr-doc/pdf?AD=AD0012375|doi=10.1063/1.1745385 |bibcode = 1936Physi...7..195M }}</ref>

* Point force on a surface of an isotropic half-space.<ref name="tribonet"/>