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'''Linear elasticity''' is a mathematical model of how solid objects [[deformation (physics)|deform]] and become internally [[stress (mechanics)|stressed]] by prescribed loading conditions. It is a simplification of the more general [[Finite strain theory|nonlinear theory of elasticity]] and a branch of [[continuum mechanics]].
The fundamental
==Mathematical formulation==
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* [[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 |last1=Belen'kii |last2= Salaev|date= 1988|title= Deformation effects in layer crystals|journal= Uspekhi Fizicheskikh Nauk|volume= 155|issue= 5|pages= 89–127|doi= 10.3367/UFNr.0155.198805c.0089|doi-access= free}}</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).
===Cylindrical coordinate form===
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\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
=== Spherical coordinate form ===
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== (An)isotropic (in)homogeneous media ==
In [[Hooke's
= K \, \delta_{ij}\, \delta_{kl}
+ \mu\, (\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}- \tfrac{2}{3}\, \delta_{ij}\,\delta_{kl})
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<math display="block"> \sigma_{ij} = K \delta_{ij} \varepsilon_{kk} + 2\mu \left(\varepsilon_{ij} - \tfrac{1}{3} \delta_{ij} \varepsilon_{kk}\right).</math>
This expression separates the stress into a scalar part on the left which may be associated with a scalar pressure, and a traceless part on the right which may be associated with shear forces. A simpler expression is:<ref>{{Cite book |
<math display="block"> \sigma_{ij} = \lambda \delta_{ij} \varepsilon_{kk}+2\mu\varepsilon_{ij}</math>
where λ is [[Lamé's first parameter]]. Since the constitutive equation is simply a set of linear equations, the strain may be expressed as a function of the stresses as:<ref>{{cite book |title= Mechanics of Deformable Bodies |last=Sommerfeld|first=Arnold |author-link=Arnold Sommerfeld|year=1964 |publisher=Academic Press |___location=New York}}</ref>
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====Displacement formulation====
In this case, the displacements are prescribed everywhere in the boundary. In this approach, the strains and stresses are eliminated from the formulation, leaving the displacements as the unknowns to be solved for in the governing equations.
First, the strain-displacement equations are substituted into the constitutive equations (Hooke's
<math display="block">\sigma_{ij} = \lambda \delta_{ij} \varepsilon_{kk}+2\mu\varepsilon_{ij}
= \lambda\delta_{ij}u_{k,k}+\mu\left(u_{i,j}+u_{j,i}\right).
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===== Thomson's solution - point force in an infinite isotropic medium =====
<math display="block">a = 1-2\nu</math>
<math display="block">b = 2(1-\nu) = a+1</math>
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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======
===== 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 [recall: <math>a=(1-2\nu)</math> and <math>b=2(1-\nu)</math>, <math>\nu</math> = Poisson's ratio]:▼
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 |access-date=20 May 2025}}</ref>
<math display="block">G_{ik} = \frac{1}{4\pi\mu} \begin{bmatrix}▼
<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 | last1=Moulinec | first1=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, Série 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 | 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>
▲===== 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.<ref name="tribonet" /> It was derived by Boussinesq<ref>{{cite book |title=
\begin{bmatrix}
\frac{xy}{r^3}-\frac{axy}{r(r+z)^2}&▼
\frac{
\frac{(2 r (\nu r + z) + z^2) x y}{r^2 (r + z)^2} &
\frac{
\frac{b
\frac{
\frac{
\frac{
\end{bmatrix}
</math>
===== 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|archive-url= https://web.archive.org/web/20170923074956/http://www.dtic.mil/get-tr-doc/pdf?AD=AD0012375|url-status= dead|archive-date= September 23, 2017|doi=10.1063/1.1745385 |bibcode = 1936Physi...7..195M|url-access=subscription}}</ref>
* Contact of two elastic bodies: the Hertz solution (see [http://www.tribonet.org/cmdownloads/hertz-contact-calculator/ Matlab code]).<ref>{{cite journal |last=Hertz |first= Heinrich|author-link=Heinrich Hertz |year=1882 |title=Contact between solid elastic bodies |journal=Journal für die reine und angewandte Mathematik|volume=92}}</ref> See also the page on [[Contact mechanics]].
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is the ''acoustic differential operator'', and <math> \delta_{kl}</math> is [[Kronecker delta]].
In [[Hooke's
<math display="block"> C_{ijkl}
= K \, \delta_{ij}\, \delta_{kl}
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