Editing Linear elasticity (section)

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

Thomson's solution or Kelvin's solution is 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>
where <math>\nu</math> is Poisson's ratio, the solution may be expressed as <math display="block">u_i = G_{ik} F_k</math> where <math>F_k</math> is the force vector being applied at the point, and <math>G_{ik}</math> is a tensor [[Green's function]] which may be written in [[Cartesian coordinates]] as:
<math display="block">G_{ik} = \frac{1}{4\pi\mu r} \left[ \left(1 - \frac{1}{2b}\right) \delta_{ik} + \frac{1}{2b} \frac{x_i x_k}{r^2} \right]</math>

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>
and it may be explicitly written as:
<math display="block">G_{ik}=\frac{1}{4\pi\mu r} \begin{bmatrix}

1-\frac{1}{2b}+\frac{1}{2b}\frac{x^2}{r^2} &
  \frac{1}{2b}\frac{xy} {r^2} &
  \frac{1}{2b}\frac{xz} {r^2} \\

  \frac{1}{2b}\frac{yx} {r^2} &
1-\frac{1}{2b}+\frac{1}{2b}\frac{y^2}{r^2} &
  \frac{1}{2b}\frac{yz} {r^2} \\

  \frac{1}{2b}\frac{zx} {r^2} &
  \frac{1}{2b}\frac{zy} {r^2} &
1-\frac{1}{2b}+\frac{1}{2b}\frac{z^2}{r^2} 
\end{bmatrix}</math>

In cylindrical coordinates (<math>\rho,\phi,z\,\!</math>) it may be written as:
<math display="block">G_{ik} = \frac{1}{4\pi \mu r} \begin{bmatrix}
1 - \frac{1}{2b} \frac{z^2}{r^2} & 0 & \frac{1}{2b} \frac{\rho z}{r^2}\\
0 & 1 - \frac{1}{2b} & 0\\
\frac{1}{2b} \frac{z \rho}{r^2}& 0 & 1 - \frac{1}{2b} \frac{\rho^2}{r^2}
\end{bmatrix}</math>
where {{mvar|r}} is total distance to point.

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{\boldsymbol{\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>

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.

======Frequency domain Green's function======

Rewrite the Navier-Cauchy equations in component form<ref>{{cite web |last=Bouchbinder |first=Eran |title= Linear Elasticity I (Non‑Equilibrium Continuum Physics)|url=https://www.weizmann.ac.il/chembiophys/bouchbinder/sites/chemphys.bouchbinder/files/uploads/Courses/2021/TAs/TA4-Linear_elasticity-I.pdf |website=Weizmann Institute of Science |publisher=Department of Chemical and Biological Physics |date=5 May 2021 |format=PDF |access-date=20 May 2025}}</ref>

<math display="block">(\lambda + \mu)\partial_i \partial_j u_j +\mu\partial_j\partial_j u_i =-F_i</math>

Convert this to frequency domain, where derivative <math> \partial_i</math> maps to <math>\sqrt{-1}q_i</math>, where <math>q</math> is the wave vector
<math display="block">(\lambda + \mu)q_i q_j u_j +\mu|q|^2u_i =F_i</math>

Spatial frequency domain force to displacement Green's function is the inverse of the above

<math>G_{ij}(q) = \frac{1}{\mu}\bigg[\frac{\delta_{ij}}{|q|^2} -\frac{1}{b}\frac{q_iq_j}{|q|^4}\bigg]</math>

The stress to strain Green's function <math>\Gamma</math> is<ref>{{cite journal | last=Moulinec | first=H. | last2=Suquet | first2=P. | title=A fast numerical method for computing the linear and nonlinear mechanical properties of composites | journal=Comptes Rendus de l’Académie des Sciences II | volume=318 | pages=1417–1423 | year=1994 | url=https://lma-software-craft.cnrs.fr/wp-content/uploads/2020/11/CRAS_Moulinec_Suquet_1994.pdf | format=PDF | access-date=2025-05-17}}</ref>
<math>\Gamma_{khij} = \frac{1}{4\mu |q|^2}(\delta_{ki}q_hq_j+\delta_{hi}q_kq_j+\delta_{kj}q_hq_i+\delta_{hj}q_kq_i) -\frac{\lambda+\mu}{\mu(\lambda+2\mu)}\frac{q_iq_jq_kq_h}{|q|^4}</math>

where <math>\epsilon_{kh} = \Gamma_{khij}\sigma_{ij}</math>