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{{Continuum mechanics|solid}}
'''Linear elasticity''' is a mathematical model of how solid objects [[deformation (physics)|deform]] and become internally [[stress (mechanics)|stressed]]
The fundamental
==Mathematical formulation==
Equations governing a linear elastic [[boundary value problem]] are based on three [[tensor]] [[partial differential equation]]s for the [[conservation of momentum|balance of linear momentum]] and six [[infinitesimal strain]]-[[displacement field (mechanics)|displacement]] relations. The system of differential equations is completed by a set of [[linear equation|linear]] algebraic [[
=== Direct tensor form ===
In direct [[tensor]] form that is independent of the choice of coordinate system, these governing equations are:<ref name="Slau">{{Cite book |last=Slaughter
* [[Cauchy momentum equation]], which is an expression of [[Newton's laws of motion#Newton's second law|Newton's second law]]. In convective form it is written as: <math display="block">\boldsymbol{\nabla} \cdot \boldsymbol{\sigma} + \mathbf{F} = \rho \ddot{\mathbf{u}} </math>
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* [[Constitutive equations]]. For elastic materials, [[Hooke's law]] represents the material behavior and relates the unknown stresses and strains. The general equation for Hooke's law is <math display="block"> \boldsymbol{\sigma} = \mathsf{C}:\boldsymbol{\varepsilon},</math>
where <math>\boldsymbol{\sigma}</math> is the [[Cauchy stress tensor]], <math>\boldsymbol{\varepsilon}</math> is the [[infinitesimal strain]] tensor, <math>\mathbf{u}</math> is the [[
=== Cartesian coordinate form ===
{{Einstein_summation_convention}}
Expressed in terms of components with respect to a rectangular [[Cartesian coordinate]] system, the governing equations of linear elasticity are:<ref name="Slau" />
* [[Cauchy momentum equation|Equation of motion]]: <math display="block"> \sigma_{ji,j} + F_i = \rho \partial_{tt} u_i</math> where the <math>{(\bullet)}_{,j}</math> subscript is a shorthand for <math>\partial{(\bullet)} / \partial x_j</math> and <math>\partial_{tt}</math> indicates <math>\partial^2 / \partial t^2</math>, <math> \sigma_{ij} = \sigma_{ji}</math> is the Cauchy [[Stress (physics)|stress]] tensor, <math> F_i</math> is the body force density, <math> \rho</math> is the mass density, and <math> u_i</math> is the displacement.{{pb}}These are 3 [[System of linear equations#Independence|independent]] equations with 6 independent unknowns (stresses).{{pb}} In engineering notation, they are: <math display="block">\begin{align}
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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===
In cylindrical coordinates (<math>r,\theta,z</math>) the equations of motion are<ref name="Slau" />
<math display="block">\begin{align}
& \frac{\partial \sigma_{rr}}{\partial r} + \frac{1}{r}\frac{\partial \sigma_{r\theta}}{\partial \theta} + \frac{\partial \sigma_{rz}}{\partial z} + \cfrac{1}{r}(\sigma_{rr}-\sigma_{\theta\theta}) + F_r = \rho~\frac{\partial^2 u_r}{\partial t^2} \\
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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 ===
In spherical coordinates (<math>r,\theta,\phi</math>) the equations of motion are<ref name="Slau" />
<math display="block">\begin{align}
& \frac{\partial \sigma_{rr}}{\partial r} + \cfrac{1}{r}\frac{\partial \sigma_{r\theta}}{\partial \theta} + \cfrac{1}{r\sin\theta}\frac{\partial \sigma_{r\phi}}{\partial \phi} + \cfrac{1}{r} (2\sigma_{rr}-\sigma_{\theta\theta}-\sigma_{\phi\phi}+\sigma_{r\theta}\cot\theta) + F_r = \rho~\frac{\partial^2 u_r}{\partial t^2} \\
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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
<math display="block"> \sigma_{ij} = \lambda \delta_{ij} \varepsilon_{kk}+2\mu\varepsilon_{ij}</math>
where λ is [[
<math display="block">\varepsilon_{ij} = \frac{1}{9K} \delta_{ij} \sigma_{kk} + \frac{1}{2\mu} \left(\sigma_{ij} - \tfrac{1}{3} \delta_{ij} \sigma_{kk}\right)</math>
which is again, a scalar part on the left and a traceless shear part on the right. More simply:
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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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These last 3 equations are the steady Navier–Cauchy equations, which can be also expressed in vector notation as
<math display="block">(\lambda+\mu) \nabla(\nabla \cdot \mathbf{u}) + \mu \nabla^2\mathbf{u} + \mathbf{F} = \boldsymbol{0}</math>
}}
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<math display="block"> (1+\nu)\sigma_{ij,kk}+\sigma_{kk,ij}=0.</math>
A necessary, but insufficient, condition for compatibility under this situation is <math>\boldsymbol{\nabla}^4\boldsymbol{\sigma} = \boldsymbol{0}</math> or <math>\sigma_{ij,kk\ell\ell} = 0</math>.<ref name="Slau" />
These constraints, along with the equilibrium equation (or equation of motion for elastodynamics) allow the calculation of the stress tensor field. Once the stress field has been calculated from these equations, the strains can be obtained from the constitutive equations, and the displacement field from the strain-displacement equations.
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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
* 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]].
=== Elastodynamics in terms of displacements ===
{{Expand section|more principles, a brief explanation to each type of wave|discuss=Talk:Linear elasticity#New section needed|date=September 2010}}
Elastodynamics is the study of '''elastic waves''' and involves linear elasticity with variation in time. An '''elastic wave''' is a type of [[mechanical wave]] that propagates in elastic or [[
The linear momentum equation is simply the equilibrium equation with an additional inertial term:
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<math display="block">\left( C_{ijkl} u_{(k},_{l)}\right) ,_{j}+F_{i}=\rho \ddot{u}_{i}.</math>
If the material is isotropic and homogeneous, one obtains the (general, or transient) '''Navier–Cauchy equation''':
<math display="block">
\mu u_{i,jj} + (\mu+\lambda)u_{j,ij}+F_i=\rho\partial_{tt}u_i
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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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=== Elastodynamics in terms of stresses ===
Elimination of displacements and strains from the governing equations leads to the '''Ignaczak equation of elastodynamics'''<ref
<math display="block">\left( \rho ^{-1} \sigma _{(ik},_{k}\right) ,_{j)} - S_{ijkl} \ddot{\sigma}_{kl} + \left( \rho ^{-1} F_{(i}\right) ,_{j)} = 0.</math>
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*[[Castigliano's method]]
*[[Cauchy momentum equation]]
*[[Clapeyron's theorem
*[[Contact mechanics]]
*[[Deformation (mechanics)|Deformation]]
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== References ==
{{
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