Editing Linear elasticity (section)

======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>