View source for Linear elasticity

{{Short description|Mathematical model of how solid objects deform}}
{{More footnotes|date=September 2010}}
{{Continuum mechanics|solid}}

'''Linear elasticity''' is a mathematical model as to 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 "linearizing" assumptions of linear elasticity are: [[Infinitesimal strain theory|infinitesimal strains]] or "small" deformations (or strains) and linear relationships between the components of stress and strain. In addition linear elasticity is valid only for stress states that do not produce [[Yield (engineering)|yielding]].

These assumptions are reasonable for many engineering materials and engineering design scenarios. Linear elasticity is therefore used extensively in [[structural analysis]] and engineering design, often with the aid of [[finite element analysis]].

==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 [[constitutive relations]].

=== 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 |first=William S. |url=http://link.springer.com/10.1007/978-1-4612-0093-2 |title=The Linearized Theory of Elasticity |date=2002 |publisher=Birkhäuser Boston |isbn=978-1-4612-6608-2 |location=Boston, MA |language=en |doi=10.1007/978-1-4612-0093-2}}</ref>

* [[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>
* [[Infinitesimal strain theory|Strain-displacement]] equations: <math display="block">\boldsymbol{\varepsilon} = \tfrac{1}{2} \left[\boldsymbol{\nabla}\mathbf{u} + (\boldsymbol{\nabla}\mathbf{u})^\mathrm{T}\right]</math>
* [[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 [[displacement vector]], <math>\mathsf{C}</math> is the fourth-order [[stiffness tensor]], <math>\mathbf{F}</math> is the body force per unit volume, <math>\rho</math> is the mass density, <math>\boldsymbol{\nabla}</math> represents the [[nabla operator]], <math>(\bullet)^\mathrm{T}</math> represents a [[transpose]], <math>\ddot{(\bullet)}</math> represents the second [[material derivative]] with respect to time, and <math>\mathsf{A}:\mathsf{B} = A_{ij}B_{ij}</math> is the inner product of two second-order tensors (summation over repeated indices is implied).

=== 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}
\frac{\partial \sigma_x}{\partial x} + \frac{\partial \tau_{yx}}{\partial y} + \frac{\partial \tau_{zx}}{\partial z} + F_x = \rho \frac{\partial^2 u_x}{\partial t^2} \\
\frac{\partial \tau_{xy}}{\partial x} + \frac{\partial \sigma_y}{\partial y} + \frac{\partial \tau_{zy}}{\partial z} + F_y = \rho \frac{\partial^2 u_y}{\partial t^2} \\
\frac{\partial \tau_{xz}}{\partial x} + \frac{\partial \tau_{yz}}{\partial y} + \frac{\partial \sigma_z}{\partial z} + F_z = \rho \frac{\partial^2 u_z}{\partial t^2}
\end{align}</math>
* [[Deformation (mechanics)#Strain|Strain-displacement]] equations: <math display="block">\varepsilon_{ij} =\frac{1}{2} (u_{j,i} + u_{i,j})</math> where <math> \varepsilon_{ij}=\varepsilon_{ji}\,\!</math> is the strain.  These are 6 independent equations relating strains and displacements with 9 independent unknowns (strains and displacements).{{pb}} In engineering notation, they are: <math display="block">\begin{align}
\epsilon_x=\frac{\partial u_x}{\partial x} \\
\epsilon_y=\frac{\partial u_y}{\partial y} \\
\epsilon_z=\frac{\partial u_z}{\partial z}
\end{align}
\qquad
\begin{align}
\gamma_{xy}=\frac{\partial u_x}{\partial y}+\frac{\partial u_y}{\partial x} \\
\gamma_{yz}=\frac{\partial u_y}{\partial z}+\frac{\partial u_z}{\partial y} \\
\gamma_{zx}=\frac{\partial u_z}{\partial x}+\frac{\partial u_x}{\partial z}
\end{align}</math>
* [[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). Specifying the boundary conditions, the boundary value problem is completely defined. To solve the system two approaches can be taken according to boundary conditions of the boundary value problem: a '''displacement formulation''', and a '''stress formulation'''.

===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} \\
  & \frac{\partial \sigma_{r\theta}}{\partial r} + \frac{1}{r} \frac{\partial \sigma_{\theta\theta}}{\partial \theta} + \frac{\partial \sigma_{\theta z}}{\partial z} + \frac{2}{r}\sigma_{r\theta} + F_\theta = \rho~\frac{\partial^2 u_\theta}{\partial t^2} \\
  & \frac{\partial \sigma_{rz}}{\partial r} + \frac{1}{r}\frac{\partial \sigma_{\theta z}}{\partial \theta} + \frac{\partial \sigma_{zz}}{\partial z} + \frac{1}{r} \sigma_{rz} + F_z = \rho~\frac{\partial^2 u_z}{\partial t^2}
\end{align}</math>
The strain-displacement relations are
<math display="block">\begin{align}
  \varepsilon_{rr} & = \frac{\partial u_r}{\partial r} ~;~~
  \varepsilon_{\theta\theta}  = \frac{1}{r} \left(\cfrac{\partial u_\theta}{\partial \theta} + u_r\right) ~;~~
  \varepsilon_{zz}  = \frac{\partial u_z}{\partial z} \\
  \varepsilon_{r\theta} & = \frac{1}{2} \left(\cfrac{1}{r}\cfrac{\partial u_r}{\partial \theta} + \cfrac{\partial u_\theta}{\partial r}- \cfrac{u_\theta}{r}\right) ~;~~
  \varepsilon_{\theta z}  = \cfrac{1}{2} \left(\cfrac{\partial u_\theta}{\partial z} + \cfrac{1}{r}\cfrac{\partial u_z}{\partial \theta}\right) ~;~~
  \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 1,2,3 now stand for <math>r</math>,<math>\theta</math>,<math>z</math>, respectively.

=== 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} \\
    & \frac{\partial \sigma_{r\theta}}{\partial r} + \cfrac{1}{r}\frac{\partial \sigma_{\theta\theta}}{\partial \theta} + \cfrac{1}{r\sin\theta}\frac{\partial \sigma_{\theta \phi}}{\partial \phi} + \cfrac{1}{r}[(\sigma_{\theta\theta}-\sigma_{\phi\phi})\cot\theta + 3\sigma_{r\theta}] + F_\theta = \rho~\frac{\partial^2 u_\theta}{\partial t^2} \\
    & \frac{\partial \sigma_{r\phi}}{\partial r} + \cfrac{1}{r}\frac{\partial \sigma_{\theta \phi}}{\partial \theta} + \cfrac{1}{r\sin\theta}\frac{\partial \sigma_{\phi\phi}}{\partial \phi} + \cfrac{1}{r}(2\sigma_{\theta\phi}\cot\theta+3\sigma_{r\phi}) + F_\phi = \rho~\frac{\partial^2 u_\phi}{\partial t^2}
\end{align}</math>
[[File:3D Spherical.svg|thumb|240px|right|Spherical coordinates (''r'', '' θ'', ''φ'') as commonly used in ''physics'': radial distance ''r'', polar angle ''θ'' ([[theta]]), and azimuthal angle ''φ'' ([[phi]]). The symbol ''ρ'' ([[rho]]) is often used instead of ''r''.]]
The strain tensor in spherical coordinates is
<math display="block">\begin{align}
  \varepsilon_{rr} & = \frac{\partial u_r}{\partial r}\\
  \varepsilon_{\theta\theta}& = \frac{1}{r} \left(\frac{\partial u_\theta}{\partial \theta} + u_r\right)\\
 \varepsilon_{\phi\phi} & = \frac{1}{r\sin\theta} \left(\frac{\partial u_\phi}{\partial \phi} + u_r\sin\theta + u_\theta\cos\theta\right)\\
  \varepsilon_{r\theta} & = \frac{1}{2} \left(\frac{1}{r} \frac{\partial u_r}{\partial \theta} + \frac{\partial u_\theta}{\partial r} - \frac{u_\theta}{r}\right) \\
  \varepsilon_{\theta \phi}  & = \frac{1}{2r} \left[\frac{1}{\sin\theta}\frac{\partial u_\theta}{\partial \phi} +\left(\frac{\partial u_\phi}{\partial \theta} - u_\phi \cot\theta\right)\right]\\
  \varepsilon_{r \phi} & = \frac{1}{2} \left(\frac{1}{r \sin \theta} \frac{\partial u_r}{\partial \phi} + \frac{\partial u_\phi}{\partial r} - \frac{u_\phi}{r}\right).
\end{align}</math>

== (An)isotropic (in)homogeneous media ==
In [[Hooke's Law#Isotropic materials|isotropic]] media, the stiffness tensor gives the relationship between the stresses (resulting internal stresses) and the strains (resulting deformations). For an isotropic medium, the stiffness tensor has no preferred direction: an applied force will give the same displacements (relative to the direction of the force) no matter the direction in which the force is applied. In the isotropic case, the stiffness tensor may be written:{{citation needed|date=June 2012}} <math display="block"> C_{ijkl}
=  K \, \delta_{ij}\, \delta_{kl}
+ \mu\, (\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}- \tfrac{2}{3}\, \delta_{ij}\,\delta_{kl})
</math> where <math>\delta_{ij}</math> is the [[Kronecker delta]], ''K'' is the [[bulk modulus]] (or incompressibility), and <math>\mu</math> is the [[shear modulus]] (or rigidity), two [[elastic moduli]].  If the medium is inhomogeneous, the isotropic model is sensible if either the medium is piecewise-constant or weakly inhomogeneous; in the strongly inhomogeneous smooth model, anisotropy has to be accounted for. If the medium is [[Homogeneous (chemistry)|homogeneous]], then the elastic moduli will be independent of the position in the medium. The constitutive equation may now be written as:
<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 |last=Aki |first=Keiiti |author-link=Keiiti Aki |title=Quantitative seismology |last2=Richards |first2=Paul G. |author-link2=Paul G. richards |publisher=University Science Books |year=2002 |isbn=978-1-891389-63-4 |edition=2 |location=Mill Valley, California}}</ref><ref>Continuum Mechanics for Engineers 2001 Mase, Eq. 5.12-2</ref>
<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>
<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:
<math display="block">\varepsilon_{ij}
= \frac{1}{2\mu}\sigma_{ij} - \frac{\nu}{E} \delta_{ij}\sigma_{kk} = \frac{1}{E} [(1+\nu) \sigma_{ij}-\nu\delta_{ij}\sigma_{kk}]</math>
where <math>\nu</math> is [[Poisson's ratio]] and <math>E</math> is [[Young's modulus]].

===Elastostatics===
Elastostatics is the study of linear elasticity under the conditions of equilibrium, in which all forces on the elastic body sum to zero, and the displacements are not a function of time. The [[Momentum#Linear momentum for a system|equilibrium equations]] are then <math display="block"> \sigma_{ji,j} + F_i = 0.</math>
In engineering notation (with tau as [[shear stress]]),
* <math>\frac{\partial \sigma_x}{\partial x} + \frac{\partial \tau_{yx}}{\partial y} + \frac{\partial \tau_{zx}}{\partial z} + F_x = 0</math>
*<math>\frac{\partial \tau_{xy}}{\partial x} + \frac{\partial \sigma_y}{\partial y} + \frac{\partial \tau_{zy}}{\partial z} + F_y = 0</math>
*<math>\frac{\partial \tau_{xz}}{\partial x} + \frac{\partial \tau_{yz}}{\partial y} + \frac{\partial \sigma_z}{\partial z} + F_z = 0</math>
This section will discuss only the isotropic homogeneous case.

====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 Law), eliminating the strains as unknowns:
<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).
</math>
Differentiating (assuming <math>\lambda</math> and <math>\mu</math> are spatially uniform) yields:
<math display="block">\sigma_{ij,j} = \lambda u_{k,ki}+\mu\left(u_{i,jj}+u_{j,ij}\right).</math>
Substituting into the equilibrium equation yields:
<math display="block">\lambda u_{k,ki}+\mu\left(u_{i,jj} + u_{j,ij}\right) + F_i = 0</math>
or (replacing double (dummy) (=summation) indices k,k by j,j and interchanging indices, ij to, ji after the, by virtue of [[Symmetry of second derivatives|Schwarz' theorem]])
<math display="block">\mu u_{i,jj} + (\mu+\lambda) u_{j,ji} + F_i = 0</math>
where <math>\lambda</math> and <math>\mu</math> are [[Lamé parameters]].
In this way, the only unknowns left are the displacements, hence the name for this formulation. The governing equations obtained in this manner are called the ''elastostatic equations'', the special case of the steady '''Navier–Cauchy equations''' given below.

{{math proof
| title = Derivation of steady Navier–Cauchy equations in Engineering notation
| proof = First, the <math>x</math>-direction will be considered. Substituting the strain-displacement equations into the equilibrium equation in the <math>x</math>-direction we have
<math display="block">\sigma_x = 2 \mu \varepsilon_x + \lambda(\varepsilon_x + \varepsilon_y +\varepsilon_z) = 2 \mu \frac{\partial u_x}{\partial x} + \lambda \left(\frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} + \frac{\partial u_z}{\partial z}\right)</math>
<math display="block">\tau_{xy} = \mu\gamma_{xy} = \mu\left(\frac{\partial u_x}{\partial y} + \frac{\partial u_y}{\partial x}\right)</math>
<math display="block">\tau_{xz} = \mu\gamma_{zx} = \mu\left(\frac{\partial u_z}{\partial x} + \frac{\partial u_x}{\partial z}\right)</math>

Then substituting these equations into the equilibrium equation in the <math>x\,\!</math>-direction we have
<math display="block">\frac{\partial \sigma_x}{\partial x} + \frac{\partial \tau_{yx}}{\partial y} + \frac{\partial \tau_{zx}}{\partial z} + F_x = 0</math>
<math display="block">\frac{\partial}{\partial x}\left( 2\mu\frac{\partial u_x}{\partial x}+ \lambda \left(\frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y}+ \frac{\partial u_z}{\partial z}\right)\right) + \mu\frac{\partial}{\partial y} \left(\frac{\partial u_x}{\partial y}+ \frac{\partial u_y}{\partial x}\right)+ \mu\frac{\partial}{\partial z} \left(\frac{\partial u_z}{\partial x} + \frac{\partial u_x}{\partial z}\right) +F_x=0</math>

Using the assumption that <math>\mu</math> and <math>\lambda</math> are constant we can rearrange and get:
<math display="block">\left(\lambda+\mu\right)\frac{\partial}{\partial x} \left(\frac{\partial u_x}{\partial x} +\frac{\partial u_y}{\partial y} + \frac{\partial u_z}{\partial z}\right)+\mu \left(\frac{\partial^2 u_x}{\partial x^2} + \frac{\partial^2 u_x}{\partial y^2}+ \frac{\partial^2 u_x}{\partial z^2}\right) + F_x= 0</math>

Following the same procedure for the <math>y\,\!</math>-direction and <math>z\,\!</math>-direction we have
<math display="block">\left(\lambda + \mu\right) \frac{\partial}{\partial y} \left(\frac{\partial u_x}{\partial x} +\frac{\partial u_y}{\partial y} +\frac{\partial u_z}{\partial z}\right)+\mu\left(\frac{\partial^2 u_y}{\partial x^2} + \frac{\partial^2 u_y}{\partial y^2} + \frac{\partial^2 u_y}{\partial z^2}\right) + F_y = 0</math>
<math display="block">\left(\lambda+\mu\right) \frac{\partial}{\partial z} \left(\frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} + \frac{\partial u_z}{\partial z}\right)+ \mu \left(\frac{\partial^2 u_z}{\partial x^2} + \frac{\partial^2 u_z}{\partial y^2} + \frac{\partial^2 u_z}{\partial z^2}\right) + F_z=0</math>

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>
}}

Once the displacement field has been calculated, the displacements can be replaced into the strain-displacement equations to solve for strains, which later are used in the constitutive equations to solve for stresses.

===== The biharmonic equation =====
The elastostatic equation may be written:
<math display="block">(\alpha^2-\beta^2) u_{j,ij} + \beta^2 u_{i,mm} = -F_i.</math>

Taking the [[divergence]] of both sides of the elastostatic equation and assuming the body forces has zero divergence (homogeneous in domain) (<math>F_{i,i}=0\,\!</math>) we have
<math display="block">(\alpha^2-\beta^2) u_{j,iij} + \beta^2u_{i,imm} = 0.</math>

Noting that summed indices need not match, and that the partial derivatives commute, the two differential terms are seen to be the same and we have: <math display="block">\alpha^2 u_{j,iij} = 0</math> from which we conclude that: <math display="block">u_{j,iij} = 0.</math>

Taking the [[Laplacian]] of both sides of the elastostatic equation, and assuming in addition <math>F_{i,kk}=0\,\!</math>, we have
<math display="block">(\alpha^2-\beta^2) u_{j,kkij} + \beta^2u_{i,kkmm} = 0.</math>

From the divergence equation, the first term on the left is zero (Note: again, the summed indices need not match) and we have:
<math display="block">\beta^2 u_{i,kkmm} = 0</math>
from which we conclude that:
<math display="block">u_{i,kkmm} = 0</math>
or, in coordinate free notation <math>\nabla^4 \mathbf{u} = 0</math> which is just the [[biharmonic equation]] in <math>\mathbf{u}\,\!</math>.

====Stress formulation====
In this case, the surface tractions are prescribed everywhere on the surface boundary. In this approach, the strains and displacements are eliminated leaving the stresses as the unknowns to be solved for in the governing equations. Once the stress field is found, the strains are then found using the constitutive equations.

There are six independent components of the stress tensor which need to be determined, yet in the displacement formulation, there are only three components of the displacement vector which need to be determined. This means that there are some constraints which must be placed upon the stress tensor, to reduce the number of degrees of freedom to three. Using the constitutive equations, these constraints are derived directly from corresponding constraints which must hold for the strain tensor, which also has six independent components. The constraints on the strain tensor are derivable directly from the definition of the strain tensor as a function of the displacement vector field, which means that these constraints introduce no new concepts or information. It is the constraints on the strain tensor that are most easily understood. If the elastic medium is visualized as a set of infinitesimal cubes in the unstrained state, then after the medium is strained, an arbitrary strain tensor must yield a situation in which the distorted cubes still fit together without overlapping. In other words, for a given strain, there must exist a continuous vector field (the displacement) from which that strain tensor can be derived. The constraints on the strain tensor that are required to assure that this is the case were discovered by Saint Venant, and are called the "[[Saint-Venant's compatibility condition|Saint Venant compatibility equations]]". These are 81 equations, 6 of which are independent non-trivial equations, which relate the different strain components. These are expressed in index notation as:
<math display="block">\varepsilon_{ij,km}+\varepsilon_{km,ij}-\varepsilon_{ik,jm}-\varepsilon_{jm,ik}=0.</math>
In engineering notation, they are: <math display="block">\begin{align}
&\frac{\partial^2 \epsilon_x}{\partial y^2} + \frac{\partial^2 \epsilon_y}{\partial x^2} = 2 \frac{\partial^2 \epsilon_{xy}}{\partial x \partial y} \\
&\frac{\partial^2 \epsilon_y}{\partial z^2} + \frac{\partial^2 \epsilon_z}{\partial y^2} = 2 \frac{\partial^2 \epsilon_{yz}}{\partial y \partial z} \\
&\frac{\partial^2 \epsilon_x}{\partial z^2} + \frac{\partial^2 \epsilon_z}{\partial x^2} = 2 \frac{\partial^2 \epsilon_{zx}}{\partial z \partial x} \\
&\frac{\partial^2 \epsilon_x}{\partial y \partial z} = \frac{\partial}{\partial x} \left ( -\frac{\partial \epsilon_{yz}}{\partial x} + \frac{\partial \epsilon_{zx}}{\partial y} + \frac{\partial \epsilon_{xy}}{\partial z}\right) \\
&\frac{\partial^2 \epsilon_y}{\partial z \partial x} = \frac{\partial}{\partial y} \left ( \frac{\partial \epsilon_{yz}}{\partial x} - \frac{\partial \epsilon_{zx}}{\partial y} + \frac{\partial \epsilon_{xy}}{\partial z}\right) \\
&\frac{\partial^2 \epsilon_z}{\partial x \partial y} = \frac{\partial}{\partial z} \left ( \frac{\partial \epsilon_{yz}}{\partial x} + \frac{\partial \epsilon_{zx}}{\partial y} - \frac{\partial \epsilon_{xy}}{\partial z}\right)
\end{align}</math>

The strains in this equation are then expressed in terms of the stresses using the constitutive equations, which yields the corresponding constraints on the stress tensor. These constraints on the stress tensor are known as the ''Beltrami-Michell'' equations of compatibility:
<math display="block">\sigma_{ij,kk} + \frac{1}{1+\nu}\sigma_{kk,ij} + F_{i,j} + F_{j,i} + \frac{\nu}{1-\nu}\delta_{i,j} F_{k,k} = 0.</math>
In the special situation where the body force is homogeneous, the above equations reduce to<ref name="tribonet">{{Cite news| url=http://www.tribonet.org/wiki/elastic-deformation/ |title=Elastic Deformation|last=tribonet|date=2017-02-16 | newspaper=Tribology |access-date=2017-02-16 | language=en-US}}</ref>
<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.

An alternative solution technique is to express the stress tensor in terms of [[stress functions]] which automatically yield a solution to the equilibrium equation. The stress functions then obey a single differential equation which corresponds to the compatibility equations.

====Solutions for elastostatic cases====

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

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.

===== 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 |archive-date=2024-09-03 |access-date=2007-12-19 |archive-url=https://web.archive.org/web/20240903234441/https://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABV5032.0001.001 |url-status=live }}</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]:

<math display="block">G_{ik} = \frac{1}{4\pi\mu} \begin{bmatrix}

\frac{b}{r}+\frac{x^2}{r^3}-\frac{ax^2}{r(r+z)^2}-\frac{az}{r(r+z)} &
\frac{xy}{r^3}-\frac{axy}{r(r+z)^2}&
\frac{xz}{r^3}-\frac{ax}{r(r+z)}\\

\frac{yx}{r^3} -\frac{ayx}{r(r+z)^2}&
\frac{b}{r}+\frac{y^2}{r^3}-\frac{ay^2}{r(r+z)^2}-\frac{az}{r(r+z)} &
\frac{yz}{r^3} -\frac{ay}{r(r+z)}\\

\frac{zx}{r^3}-\frac{ax}{r(r+z)}&
\frac{zy}{r^3}-\frac{ay}{r(r+z)}&
\frac{b}{r}+\frac{z^2}{r^3}
\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}}</ref>
* Point force on a surface of an isotropic half-space.<ref name="tribonet" />
* 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 [[viscoelastic]] materials.  The elasticity of the material provides the restoring [[force]] of the wave.  When they occur in the [[Earth]] as the result of an [[earthquake]] or other disturbance, elastic waves are usually called [[seismic wave]]s.

The linear momentum equation is simply the equilibrium equation with an additional inertial term:
<math display="block"> \sigma_{ji,j}+ F_i = \rho\,\ddot{u}_i = \rho \, \partial_{tt} u_i.</math>

If the material is governed by anisotropic Hooke's law (with the stiffness tensor homogeneous throughout the material), one obtains the '''displacement equation of elastodynamics''':
<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
\quad \text{or} \quad
\mu \nabla^2\mathbf{u} + (\mu+\lambda)\nabla(\nabla\cdot\mathbf{u}) + \mathbf{F}=\rho\frac{\partial^2\mathbf{u}}{\partial t^2}.</math>

The elastodynamic wave equation can also be expressed as
<math display="block"> \left(\delta_{kl} \partial_{tt} - A_{kl}[\nabla]\right) u_l = \frac{1}{\rho} F_k</math>
where
<math display="block"> A_{kl}[\nabla]=\frac{1}{\rho} \, \partial_i \, C_{iklj} \, \partial_j</math>
is the ''acoustic differential operator'', and <math> \delta_{kl}</math> is [[Kronecker delta]].

In [[Hooke's Law#Isotropic materials|isotropic]] media, the stiffness tensor has the form
<math display="block"> C_{ijkl}
= K \, \delta_{ij}\, \delta_{kl}
+ \mu\, (\delta_{ik}\delta_{jl} + \delta_{il} \delta_{jk} - \frac{2}{3}\, \delta_{ij}\, \delta_{kl})</math>
where
<math>K</math> is the [[bulk modulus]] (or incompressibility), and <math>\mu</math> is the [[shear modulus]] (or rigidity), two [[elastic moduli]]. If the material is homogeneous (i.e. the stiffness tensor is constant throughout the material), the acoustic operator becomes:
<math display="block">A_{ij}[\nabla] = \alpha^2 \partial_i \partial_j + \beta^2 (\partial_m \partial_m \delta_{ij} - \partial_i \partial_j)</math>

For [[plane waves]], the above differential operator becomes the ''acoustic algebraic operator'': 
<math display="block">A_{ij}[\mathbf{k}] = \alpha^2 k_i k_j + \beta^2(k_m k_m \delta_{ij}-k_i k_j)</math>
where
<math display="block"> \alpha^2 = \left(K+\frac{4}{3}\mu\right)/\rho \qquad \beta^2 = \mu / \rho</math>
are the [[eigenvalue]]s of <math>A[\hat{\mathbf{k}}]</math> with [[eigenvector]]s <math>\hat{\mathbf{u}}</math> parallel and orthogonal to the propagation direction <math>\hat{\mathbf{k}}\,\!</math>, respectively. The associated waves are called ''longitudinal'' and ''shear'' elastic waves. In the seismological literature, the corresponding plane waves are called P-waves and S-waves (see [[Seismic wave]]).

=== Elastodynamics in terms of stresses ===
Elimination of displacements and strains from the governing equations leads to the '''Ignaczak equation of elastodynamics'''<ref>[[Ostoja-Starzewski, M.]], (2018), ''Ignaczak equation of elastodynamics'', Mathematics and Mechanics of Solids. {{doi|10.1177/1081286518757284}}</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>

In the case of local isotropy, this reduces to
<math display="block">\left( \rho ^{-1} \sigma _{(ik},_{k}\right) ,_{j)} - \frac{1}{2\mu } \left( 
\ddot{\sigma}_{ij} - \frac{\lambda }{3 \lambda +2\mu }\ddot{\sigma}_{kk}\delta
_{ij}\right) +\left( \rho ^{-1} F_{(i}\right) ,_{j)} = 0. </math>

The principal characteristics of this formulation include: (1) avoids gradients of compliance but introduces gradients of mass density; (2) it is derivable from a variational principle; (3) it is advantageous for handling traction initial-boundary value problems, (4) allows a tensorial classification of elastic waves, (5) offers a range of applications in elastic wave propagation problems; (6) can be extended to dynamics of classical or micropolar solids with interacting fields of diverse types (thermoelastic, fluid-saturated porous, piezoelectro-elastic...) as well as nonlinear media.

== Anisotropic homogeneous media ==

{{Main|Hooke's law}}

For anisotropic media, the stiffness tensor <math> C_{ijkl}</math> is more complicated.  The symmetry of the stress tensor <math>\sigma_{ij}</math> means that there are at most 6 different elements of stress. Similarly, there are at most 6 different elements of the strain tensor <math>\varepsilon_{ij}\,\!</math>.  Hence the fourth-order stiffness tensor <math> C_{ijkl}</math> may be written as a matrix <math>C_{\alpha \beta}</math> (a tensor of second order). [[Voigt notation]] is the standard mapping for tensor indices,
<math display="block"> 
\begin{matrix}
ij & =\\
\Downarrow & \\
\alpha  & =
\end{matrix}

 \begin{matrix}
11 & 22 & 33 & 23,32 & 13,31 & 12,21 \\
\Downarrow & \Downarrow & \Downarrow & \Downarrow & \Downarrow & \Downarrow & \\
1  &2 &  3 &  4 &  5 & 6
\end{matrix}</math>

With this notation, one can write the elasticity matrix for any linearly elastic medium as:
<math display="block"> C_{ijkl}  \Rightarrow C_{\alpha \beta} = \begin{bmatrix}
 C_{11} & C_{12} & C_{13} & C_{14} & C_{15} & C_{16} \\
 C_{12} & C_{22} & C_{23} & C_{24} & C_{25} & C_{26} \\
 C_{13} & C_{23} & C_{33} & C_{34} & C_{35} & C_{36} \\
 C_{14} & C_{24} & C_{34} & C_{44} & C_{45} & C_{46} \\
 C_{15} & C_{25} & C_{35} & C_{45} & C_{55} & C_{56} \\
 C_{16} & C_{26} & C_{36} & C_{46} & C_{56} & C_{66} 
\end{bmatrix}.</math>

As shown, the matrix <math> C_{\alpha \beta}</math> is symmetric, this is a result of the existence of a strain energy density function which satisfies <math>\sigma_{ij} = \frac{\partial W}{\partial\varepsilon_{ij}}</math>.  Hence, there are at most 21 different elements of <math> C_{\alpha \beta}\,\!</math>.

The isotropic special case has 2 independent elements:
<math display="block"> C_{\alpha \beta} = \begin{bmatrix}
 K+4 \mu\ /3  & K-2 \mu\ /3 & K-2 \mu\ /3 & 0 & 0  & 0 \\
 K-2 \mu\ /3  & K+4 \mu\ /3 &  K-2 \mu\ /3 & 0 & 0  & 0 \\
 K-2 \mu\ /3  & K-2 \mu\ /3 & K+4 \mu\ /3 & 0 & 0  & 0 \\
 0  & 0 & 0 & \mu\ & 0  & 0 \\
 0  & 0 & 0 & 0 & \mu\  & 0 \\
 0  & 0 & 0 & 0 & 0  & \mu\ 
\end{bmatrix}.</math>

The simplest anisotropic case, that of cubic symmetry has 3 independent elements:
<math display="block"> C_{\alpha \beta} = \begin{bmatrix}
 C_{11}  &  C_{12} &  C_{12} & 0 & 0  & 0 \\
 C_{12}  &  C_{11} &  C_{12} & 0 & 0  & 0 \\
 C_{12}  & C_{12}  &  C_{11} & 0 & 0  & 0 \\
 0  & 0 & 0 & C_{44} & 0  & 0 \\
 0  & 0 & 0 & 0 & C_{44}  & 0 \\
 0  & 0 & 0 & 0 & 0  & C_{44} 
\end{bmatrix}.</math>

The case of [[transverse isotropy]], also called polar anisotropy, (with a single axis (the 3-axis) of symmetry) has 5 independent elements:
<math display="block"> C_{\alpha \beta} = \begin{bmatrix}
  C_{11}  &  C_{11}-2C_{66} &  C_{13} & 0 & 0  & 0 \\
 C_{11}-2C_{66}  &  C_{11} &  C_{13} & 0 & 0  & 0 \\
  C_{13}  & C_{13}  &  C_{33} & 0 & 0  & 0 \\
 0  & 0 & 0 & C_{44} & 0  & 0 \\
 0  & 0 & 0 & 0 & C_{44}  & 0 \\
 0  & 0 & 0 & 0 & 0  & C_{66} 
\end{bmatrix}.</math>

When the transverse isotropy is weak (i.e. close to isotropy), an alternative parametrization utilizing [[Thomsen parameters]], is convenient for the formulas for wave speeds.

The case of orthotropy (the symmetry of a brick) has 9 independent elements:
<math display="block"> C_{\alpha \beta} = \begin{bmatrix}
 C_{11} & C_{12} & C_{13} & 0 & 0 & 0 \\
 C_{12} & C_{22} & C_{23} & 0 & 0 & 0 \\
 C_{13} & C_{23} & C_{33} & 0 & 0 & 0 \\
 0  & 0 & 0 & C_{44} & 0 & 0 \\
 0  & 0 & 0 & 0 & C_{55} & 0 \\
 0  & 0 & 0 & 0 & 0 & C_{66} 
\end{bmatrix}.</math>

=== Elastodynamics ===
The elastodynamic wave equation for anisotropic media can be expressed as
<math display="block"> (\delta_{kl} \partial_{tt} - A_{kl}[\nabla])\, u_l = \frac{1}{\rho} F_k</math>
where
<math display="block"> A_{kl}[\nabla]=\frac{1}{\rho} \, \partial_i \, C_{iklj} \, \partial_j</math>
is the ''acoustic differential operator'', and <math> \delta_{kl}</math> is [[Kronecker delta]].

==== Plane waves and Christoffel equation ====
A ''plane wave'' has the form
<math display="block"> \mathbf{u}[\mathbf{x}, \, t] = U[\mathbf{k} \cdot  \mathbf{x} - \omega \, t] \, \hat{\mathbf{u}}</math>
with <math>\hat{\mathbf{u}}\,\!</math> of unit length.
It is a solution of the wave equation with zero forcing, if and only if <math> \omega^2 </math> and <math>\hat{\mathbf{u}}</math> constitute an eigenvalue/eigenvector pair of the ''acoustic algebraic operator''
<math display="block"> A_{kl}[\mathbf{k}]=\frac{1}{\rho} \, k_i \, C_{iklj} \, k_j.</math>
This ''propagation condition'' (also known as the '''Christoffel equation''') may be written as
<math display="block">A[\hat{\mathbf{k}}] \, \hat{\mathbf{u}} = c^2 \, \hat{\mathbf{u}}</math>
where
<math>\hat{\mathbf{k}} = \mathbf{k} / \sqrt{\mathbf{k}\cdot\mathbf{k}}</math>
denotes propagation direction and <math>c = \omega / \sqrt{\mathbf{k} \cdot \mathbf{k}}</math> is phase velocity.

==See also==
{{Continuum mechanics|cTopic=[[Solid mechanics]]}}
{{div col|colwidth=20em}}
*[[Castigliano's method]]
*[[Cauchy momentum equation]]
*[[Clapeyron's theorem]]
*[[Contact mechanics]]
*[[Deformation (mechanics)|Deformation]]
*[[Elasticity (physics)]]
*[[GRADELA]]
*[[Hooke's law]]
*[[Infinitesimal strain theory]]
*[[Michell solution]]
*[[Plasticity (physics)]]
*[[Signorini problem]]
*[[Spring system]]
*[[Stress (mechanics)]]
*[[Stress functions]]
{{div col end}}

== References ==
{{reflist}}

{{Authority control}}

{{DEFAULTSORT:Linear Elasticity}}
[[Category:Elasticity (physics)]]
[[Category:Solid mechanics]]
[[Category:Sound]]