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m v2.05b - Bot T20 CW#61 - Fix errors for CW project (Reference before punctuation) |
add frequency ___domain Green's function for the Kelvin's solution |
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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}k_i</math>,
<math display="block">(\lambda + \mu)k_i k_j u_j +\mu|k|^2u_i =F_i</math>
Frequency ___domain Green's function is the matrix inverse of the above
<math>G_{ij}(k) = \frac{1}{\mu}\bigg[\frac{\delta_{ij}}{|k|^2} -\frac{1}{b}\frac{k_ik_j}{|k|^2}\bigg]</math>
===== Boussinesq–Cerruti solution - point force at the origin of an infinite isotropic half-space =====
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