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Linear elasticity is a mathematical model of how solid objectsdeform and become internallystressed by prescribed loading conditions. It is a simplification of the more generalnonlinear theory of elasticity and a branch ofcontinuum mechanics.

The fundamental assumptions of linear elasticity areinfinitesimal strains — meaning, "small" deformations — and linear relationships between the components of stress and strain — hence the "linear" in its name. Linear elasticity is valid only for stress states that do not produceyielding. Its assumptions are reasonable for many engineering materials and engineering design scenarios. Linear elasticity is therefore used extensively instructural analysis and engineering design, often with the aid offinite element analysis.

Equations governing a linear elasticboundary value problem are based on threetensorpartial differential equations for thebalance of linear momentum and sixinfinitesimal strain-displacement relations. The system of differential equations is completed by a set oflinear algebraicconstitutive relations.

In directtensor form that is independent of the choice of coordinate system, these governing equations are:^[1]

• Cauchy momentum equation, which is an expression ofNewton's second law. In convective form it is written as:$${\displaystyle \mathbf{\nabla }\cdot \mathit{\sigma }+\mathbf{F}=\rho \ddot{\mathbf{u}}}$$
• Strain-displacement equations:$${\displaystyle \mathit{\epsilon }=\frac{1}{2}\left[\mathbf{\nabla }\mathbf{u}+(\mathbf{\nabla }\mathbf{u}{)}^{\mathrm{T}}\right]}$$
• 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$${\displaystyle \mathit{\sigma }=\mathsf{C}:\mathit{\epsilon },}$$

where${\displaystyle \mathit{\sigma }}$ is theCauchy stress tensor,${\displaystyle \mathit{\epsilon }}$ is theinfinitesimal strain tensor,${\displaystyle \mathbf{u}}$ is thedisplacement vector,${\displaystyle \mathsf{C}}$ is the fourth-orderstiffness tensor,${\displaystyle \mathbf{F}}$ is the body force per unit volume,${\displaystyle \rho }$ is the mass density,${\displaystyle \mathbf{\nabla }}$ represents thenabla operator,${\displaystyle (\bullet {)}^{\mathrm{T}}}$ represents atranspose,${\displaystyle \ddot{(\bullet )}}$ represents the secondmaterial derivative with respect to time, and${\displaystyle \mathsf{A}:\mathsf{B}=A_{ij}{B}_{ij}}$ is the inner product of two second-order tensors (summation over repeated indices is implied).

Expressed in terms of components with respect to a rectangularCartesian coordinate system, the governing equations of linear elasticity are:^[1]

• Equation of motion:$${\displaystyle {\sigma }_{ji,j}+F_i=\rho {\mathrm{\partial }}_{tt}{u}_i}$$ where the${\displaystyle {(\bullet )}_{,j}}$ subscript is a shorthand for${\displaystyle \mathrm{\partial }(\bullet )/\mathrm{\partial }{x}_j}$ and${\displaystyle {\mathrm{\partial }}_{tt}}$ indicates${\displaystyle {\mathrm{\partial }}^2/\mathrm{\partial }{t}^2}$,${\displaystyle {\sigma }_{ij}={\sigma }_{ji}}$ is the Cauchystress tensor,${\displaystyle F_i}$ is the body force density,${\displaystyle \rho }$ is the mass density, and${\displaystyle u_i}$ is the displacement.
  These are 3independent equations with 6 independent unknowns (stresses).
  In engineering notation, they are:$${\displaystyle \begin{array}{r}\frac{\mathrm{\partial }{\sigma }_x}{\mathrm{\partial }x}+\frac{\mathrm{\partial }{\tau }_{yx}}{\mathrm{\partial }y}+\frac{\mathrm{\partial }{\tau }_{zx}}{\mathrm{\partial }z}+F_x=\rho \frac{{\mathrm{\partial }}^2{u}_x}{\mathrm{\partial }{t}^2}\\ \frac{\mathrm{\partial }{\tau }_{xy}}{\mathrm{\partial }x}+\frac{\mathrm{\partial }{\sigma }_y}{\mathrm{\partial }y}+\frac{\mathrm{\partial }{\tau }_{zy}}{\mathrm{\partial }z}+F_y=\rho \frac{{\mathrm{\partial }}^2{u}_y}{\mathrm{\partial }{t}^2}\\ \frac{\mathrm{\partial }{\tau }_{xz}}{\mathrm{\partial }x}+\frac{\mathrm{\partial }{\tau }_{yz}}{\mathrm{\partial }y}+\frac{\mathrm{\partial }{\sigma }_z}{\mathrm{\partial }z}+F_z=\rho \frac{{\mathrm{\partial }}^2{u}_z}{\mathrm{\partial }{t}^2}\end{array}}$$
• Strain-displacement equations:$${\displaystyle {\epsilon }_{ij}=\frac{1}{2}(u_{j,i}+u_{i,j})}$$ where${\displaystyle {\epsilon }_{ij}={\epsilon }_{ji}}$ is the strain. These are 6 independent equations relating strains and displacements with 9 independent unknowns (strains and displacements).
  In engineering notation, they are:$${\displaystyle \begin{array}{r}{ϵ}_x=\frac{\mathrm{\partial }{u}_x}{\mathrm{\partial }x}\\ {ϵ}_y=\frac{\mathrm{\partial }{u}_y}{\mathrm{\partial }y}\\ {ϵ}_z=\frac{\mathrm{\partial }{u}_z}{\mathrm{\partial }z}\end{array}\begin{array}{r}{\gamma }_{xy}=\frac{\mathrm{\partial }{u}_x}{\mathrm{\partial }y}+\frac{\mathrm{\partial }{u}_y}{\mathrm{\partial }x}\\ {\gamma }_{yz}=\frac{\mathrm{\partial }{u}_y}{\mathrm{\partial }z}+\frac{\mathrm{\partial }{u}_z}{\mathrm{\partial }y}\\ {\gamma }_{zx}=\frac{\mathrm{\partial }{u}_z}{\mathrm{\partial }x}+\frac{\mathrm{\partial }{u}_x}{\mathrm{\partial }z}\end{array}}$$
• Constitutive equations. The equation for Hooke's law is:$${\displaystyle {\sigma }_{ij}=C_{ijkl}{\epsilon }_{kl}}$$ where${\displaystyle C_{ijkl}}$ 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^[2]${\displaystyle C_{ijkl}=C_{klij}=C_{jikl}=C_{ijlk}}$.

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). By specifying the boundary conditions, the boundary value problem is fully defined. To solve the system two approaches can be taken according to boundary conditions of the boundary value problem: adisplacement formulation, and astress formulation.

In cylindrical coordinates (${\displaystyle r,\theta ,z}$) the equations of motion are^[1]The strain-displacement relations areand the constitutive relations are the same as in Cartesian coordinates, except that the indices 1,2,3 now stand for${\displaystyle r}$,${\displaystyle \theta }$,${\displaystyle z}$, respectively.

In spherical coordinates (${\displaystyle r,\theta ,\varphi }$) the equations of motion are^[1]

The strain tensor in spherical coordinates is

Inisotropic 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]}$${\displaystyle C_{ijkl}=K{\delta }_{ij}{\delta }_{kl}+\mu ({\delta }_{ik}{\delta }_{jl}+{\delta }_{il}{\delta }_{jk}-\frac{2}{3}{\delta }_{ij}{\delta }_{kl})}$$ where${\displaystyle {\delta }_{ij}}$ is theKronecker delta,K is thebulk modulus (or incompressibility), and${\displaystyle \mu }$ is theshear modulus (or rigidity), twoelastic 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 ishomogeneous, then the elastic moduli will be independent of the position in the medium. The constitutive equation may now be written as:$${\displaystyle {\sigma }_{ij}=K{\delta }_{ij}{\epsilon }_{kk}+2\mu \left({\epsilon }_{ij}-\frac{1}{3}{\delta }_{ij}{\epsilon }_{kk}\right).}$$

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:^[3][4]$${\displaystyle {\sigma }_{ij}=\lambda {\delta }_{ij}{\epsilon }_{kk}+2\mu {\epsilon }_{ij}}$$where λ isLamé'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:^[5]$${\displaystyle {\epsilon }_{ij}=\frac{1}{9K}{\delta }_{ij}{\sigma }_{kk}+\frac{1}{2\mu }\left({\sigma }_{ij}-\frac{1}{3}{\delta }_{ij}{\sigma }_{kk}\right)}$$which is again, a scalar part on the left and a traceless shear part on the right. More simply:$${\displaystyle {\epsilon }_{ij}=\frac{1}{2\mu }{\sigma }_{ij}-\frac{\nu }{E}{\delta }_{ij}{\sigma }_{kk}=\frac{1}{E}[(1+\nu ){\sigma }_{ij}-\nu {\delta }_{ij}{\sigma }_{kk}]}$$where${\displaystyle \nu }$ isPoisson's ratio and${\displaystyle E}$ isYoung's modulus.

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. Theequilibrium equations are then$${\displaystyle {\sigma }_{ji,j}+F_i=0.}$$In engineering notation (with tau asshear stress),

• ${\displaystyle \frac{\mathrm{\partial }{\sigma }_x}{\mathrm{\partial }x}+\frac{\mathrm{\partial }{\tau }_{yx}}{\mathrm{\partial }y}+\frac{\mathrm{\partial }{\tau }_{zx}}{\mathrm{\partial }z}+F_x=0}$
• ${\displaystyle \frac{\mathrm{\partial }{\tau }_{xy}}{\mathrm{\partial }x}+\frac{\mathrm{\partial }{\sigma }_y}{\mathrm{\partial }y}+\frac{\mathrm{\partial }{\tau }_{zy}}{\mathrm{\partial }z}+F_y=0}$
• ${\displaystyle \frac{\mathrm{\partial }{\tau }_{xz}}{\mathrm{\partial }x}+\frac{\mathrm{\partial }{\tau }_{yz}}{\mathrm{\partial }y}+\frac{\mathrm{\partial }{\sigma }_z}{\mathrm{\partial }z}+F_z=0}$

This section will discuss only the isotropic homogeneous case.

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:$${\displaystyle {\sigma }_{ij}=\lambda {\delta }_{ij}{\epsilon }_{kk}+2\mu {\epsilon }_{ij}=\lambda {\delta }_{ij}{u}_{k,k}+\mu \left(u_{i,j}+u_{j,i}\right).}$$Differentiating (assuming${\displaystyle \lambda }$ and${\displaystyle \mu }$ are spatially uniform) yields:$${\displaystyle {\sigma }_{ij,j}=\lambda u_{k,ki}+\mu \left(u_{i,jj}+u_{j,ij}\right).}$$Substituting into the equilibrium equation yields:$${\displaystyle \lambda u_{k,ki}+\mu \left(u_{i,jj}+u_{j,ij}\right)+F_i=0}$$or (replacing double (dummy) (=summation) indices k,k by j,j and interchanging indices, ij to, ji after the, by virtue ofSchwarz' theorem)$${\displaystyle \mu u_{i,jj}+(\mu +\lambda )u_{j,ji}+F_i=0}$$where${\displaystyle \lambda }$ and${\displaystyle \mu }$ areLamé 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 theelastostatic equations, the special case of the steadyNavier–Cauchy equations given below.

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 elastostatic equation may be written:$${\displaystyle ({\alpha }^2-{\beta }^2)u_{j,ij}+{\beta }^2{u}_{i,mm}=-F_i.}$$

Taking thedivergence of both sides of the elastostatic equation and assuming the body forces has zero divergence (homogeneous in domain) (${\displaystyle F_{i,i}=0}$) we have$${\displaystyle ({\alpha }^2-{\beta }^2)u_{j,iij}+{\beta }^2{u}_{i,imm}=0.}$$

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:$${\displaystyle {\alpha }^2{u}_{j,iij}=0}$$ from which we conclude that:$${\displaystyle u_{j,iij}=0.}$$

Taking theLaplacian of both sides of the elastostatic equation, and assuming in addition${\displaystyle F_{i,kk}=0}$, we have$${\displaystyle ({\alpha }^2-{\beta }^2)u_{j,kkij}+{\beta }^2{u}_{i,kkmm}=0.}$$

From the divergence equation, the first term on the left is zero (Note: again, the summed indices need not match) and we have:$${\displaystyle {\beta }^2{u}_{i,kkmm}=0}$$from which we conclude that:$${\displaystyle u_{i,kkmm}=0}$$or, in coordinate free notation${\displaystyle {\mathrm{\nabla }}^4\mathbf{u}=0}$ which is just thebiharmonic equation in${\displaystyle \mathbf{u}}$.

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 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:$${\displaystyle {\epsilon }_{ij,km}+{\epsilon }_{km,ij}-{\epsilon }_{ik,jm}-{\epsilon }_{jm,ik}=0.}$$In engineering notation, they are:

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 theBeltrami-Michell equations of compatibility:$${\displaystyle {\sigma }_{ij,kk}+\frac{1}{1+\nu }{\sigma }_{kk,ij}+F_{i,j}+F_{j,i}+\frac{\nu }{1-\nu }{\delta }_{i,j}{F}_{k,k}=0.}$$In the special situation where the body force is homogeneous, the above equations reduce to^[6]$${\displaystyle (1+\nu ){\sigma }_{ij,kk}+{\sigma }_{kk,ij}=0.}$$

A necessary, but insufficient, condition for compatibility under this situation is${\displaystyle {\mathbf{\nabla }}^4\mathit{\sigma }=\mathbf{0}}$ or${\displaystyle {\sigma }_{ij,kk\ell \ell }=0}$.^[1]

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 ofstress 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.

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 byWilliam Thomson (later Lord Kelvin) in 1848 (Thomson 1848). This solution is the analog ofCoulomb's law inelectrostatics. A derivation is given in Landau & Lifshitz.^[7]: §8 Defining$${\displaystyle a=1-2\nu }$$$${\displaystyle b=2(1-\nu )=a+1}$$where${\displaystyle \nu }$ is Poisson's ratio, the solution may be expressed as$${\displaystyle u_i=G_{ik}{F}_k}$$ where${\displaystyle F_k}$ is the force vector being applied at the point, and${\displaystyle G_{ik}}$ is a tensorGreen's function which may be written inCartesian coordinates as:$${\displaystyle G_{ik}=\frac{1}{4\pi \mu r}\left[\left(1-\frac{1}{2b}\right){\delta }_{ik}+\frac{1}{2b}\frac{x_i{x}_k}{r^2}\right]}$$

It may be also compactly written as:$${\displaystyle G_{ik}=\frac{1}{4\pi \mu }\left[\frac{{\delta }_{ik}}{r}-\frac{1}{2b}\frac{{\mathrm{\partial }}^{2}r}{\mathrm{\partial }{x}_i\mathrm{\partial }{x}_k}\right]}$$and it may be explicitly written as:

In cylindrical coordinates (${\displaystyle \rho ,\varphi ,z}$) it may be written as:wherer is total distance to point.

It is particularly helpful to write the displacement in cylindrical coordinates for a point force${\displaystyle F_z}$ directed along the z-axis. Defining${\displaystyle \hat{\mathit{\rho }}}$ and${\displaystyle \hat{\mathbf{z}}}$ as unit vectors in the${\displaystyle \rho }$ and${\displaystyle z}$ directions respectively yields:$${\displaystyle \mathbf{u}=\frac{F_z}{4\pi \mu r}\left[\frac{1}{4(1-\nu )}\frac{\rho z}{r^2}\hat{\mathit{\rho }}+\left(1-\frac{1}{4(1-\nu )}\frac{{\rho }^2}{r^2}\right)\hat{\mathbf{z}}\right]}$$

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 larger. There is also an additional ρ-directed component.

Another useful solution is that of a point force acting on the surface of an infinite half-space. It was derived by Boussinesq^[8] for the normal force and Cerruti for the tangential force and a derivation is given in Landau & Lifshitz.^[7]: §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:${\displaystyle a=(1-2\nu )}$ and${\displaystyle b=2(1-\nu )}$,${\displaystyle \nu }$ = Poisson's ratio]:

• Point force inside an infinite isotropic half-space.^[9]
• Point force on a surface of an isotropic half-space.^[6]
• Contact of two elastic bodies: the Hertz solution (seeMatlab code).^[10] See also the page onContact mechanics.

This sectionneeds expansion with: more principles, a brief explanation to each type of wave. You can help byadding to it. (talk)(September 2010)

Elastodynamics is the study ofelastic waves and involves linear elasticity with variation in time. Anelastic wave is a type ofmechanical wave that propagates in elastic orviscoelastic materials. The elasticity of the material provides the restoringforce of the wave. When they occur in theEarth as the result of anearthquake or other disturbance, elastic waves are usually calledseismic waves.

The linear momentum equation is simply the equilibrium equation with an additional inertial term:$${\displaystyle {\sigma }_{ji,j}+F_i=\rho {\ddot{u}}_i=\rho {\mathrm{\partial }}_{tt}{u}_i.}$$

If the material is governed by anisotropic Hooke's law (with the stiffness tensor homogeneous throughout the material), one obtains thedisplacement equation of elastodynamics:$${\displaystyle \left(C_{ijkl}{u}_{(k}{,}_{l)}\right){,}_j+F_i=\rho {\ddot{u}}_i.}$$

If the material is isotropic and homogeneous, one obtains the (general, or transient)Navier–Cauchy equation:$${\displaystyle \mu u_{i,jj}+(\mu +\lambda )u_{j,ij}+F_i=\rho {\mathrm{\partial }}_{tt}{u}_i\text{or}\mu {\mathrm{\nabla }}^2\mathbf{u}+(\mu +\lambda )\mathrm{\nabla }(\mathrm{\nabla }\cdot \mathbf{u})+\mathbf{F}=\rho \frac{{\mathrm{\partial }}^2\mathbf{u}}{\mathrm{\partial }{t}^2}.}$$

The elastodynamic wave equation can also be expressed as$${\displaystyle \left({\delta }_{kl}{\mathrm{\partial }}_{tt}-A_{kl}[\mathrm{\nabla }]\right)u_l=\frac{1}{\rho }{F}_k}$$where$${\displaystyle A_{kl}[\mathrm{\nabla }]=\frac{1}{\rho }{\mathrm{\partial }}_i{C}_{iklj}{\mathrm{\partial }}_j}$$is theacoustic differential operator, and${\displaystyle {\delta }_{kl}}$ isKronecker delta.

Inisotropic media, the stiffness tensor has the form$${\displaystyle C_{ijkl}=K{\delta }_{ij}{\delta }_{kl}+\mu ({\delta }_{ik}{\delta }_{jl}+{\delta }_{il}{\delta }_{jk}-\frac{2}{3}{\delta }_{ij}{\delta }_{kl})}$$where${\displaystyle K}$ is thebulk modulus (or incompressibility), and${\displaystyle \mu }$ is theshear modulus (or rigidity), twoelastic moduli. If the material is homogeneous (i.e. the stiffness tensor is constant throughout the material), the acoustic operator becomes:$${\displaystyle A_{ij}[\mathrm{\nabla }]={\alpha }^2{\mathrm{\partial }}_i{\mathrm{\partial }}_j+{\beta }^2({\mathrm{\partial }}_m{\mathrm{\partial }}_m{\delta }_{ij}-{\mathrm{\partial }}_i{\mathrm{\partial }}_j)}$$

Forplane waves, the above differential operator becomes theacoustic algebraic operator:$${\displaystyle A_{ij}[\mathbf{k}]={\alpha }^2{k}_i{k}_j+{\beta }^2(k_m{k}_m{\delta }_{ij}-k_i{k}_j)}$$where$${\displaystyle {\alpha }^2=\left(K+\frac{4}{3}\mu \right)/\rho {\beta }^2=\mu /\rho }$$are theeigenvalues of${\displaystyle A[\hat{\mathbf{k}}]}$ witheigenvectors${\displaystyle \hat{\mathbf{u}}}$ parallel and orthogonal to the propagation direction${\displaystyle \hat{\mathbf{k}}}$, respectively. The associated waves are calledlongitudinal andshear elastic waves. In the seismological literature, the corresponding plane waves are called P-waves and S-waves (seeSeismic wave).

Elimination of displacements and strains from the governing equations leads to theIgnaczak equation of elastodynamics^[11]$${\displaystyle \left({\rho }^{-1}{\sigma }_{(ik}{,}_k\right){,}_{j)}-S_{ijkl}{\ddot{\sigma }}_{kl}+\left({\rho }^{-1}{F}_{(i}\right){,}_{j)}=0.}$$

In the case of local isotropy, this reduces to$${\displaystyle \left({\rho }^{-1}{\sigma }_{(ik}{,}_k\right){,}_{j)}-\frac{1}{2\mu }\left({\ddot{\sigma }}_{ij}-\frac{\lambda }{3\lambda +2\mu }{\ddot{\sigma }}_{kk}{\delta }_{ij}\right)+\left({\rho }^{-1}{F}_{(i}\right){,}_{j)}=0.}$$

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.

For anisotropic media, the stiffness tensor${\displaystyle C_{ijkl}}$ is more complicated. The symmetry of the stress tensor${\displaystyle {\sigma }_{ij}}$ means that there are at most 6 different elements of stress. Similarly, there are at most 6 different elements of the strain tensor${\displaystyle {\epsilon }_{ij}}$. Hence the fourth-order stiffness tensor${\displaystyle C_{ijkl}}$ may be written as a matrix${\displaystyle C_{\alpha \beta }}$ (a tensor of second order).Voigt notation is the standard mapping for tensor indices,

With this notation, one can write the elasticity matrix for any linearly elastic medium as:

As shown, the matrix${\displaystyle C_{\alpha \beta }}$ is symmetric, this is a result of the existence of a strain energy density function which satisfies${\displaystyle {\sigma }_{ij}=\frac{\mathrm{\partial }W}{\mathrm{\partial }{\epsilon }_{ij}}}$. Hence, there are at most 21 different elements of${\displaystyle C_{\alpha \beta }}$.

The isotropic special case has 2 independent elements:

The simplest anisotropic case, that of cubic symmetry has 3 independent elements:

The case oftransverse isotropy, also called polar anisotropy, (with a single axis (the 3-axis) of symmetry) has 5 independent elements: