Linear elasticity: Difference between revisions

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Revision as of 15:52, 10 May 2022

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== (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:

<math display="block"> C_{ijkl}

:<math> C_{ijkl}

= K \, \delta_{ij}\, \delta_{kl}

+\mu\, (\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}-\textstyle{\frac{2}{3}}\, \delta_{ij}\,\delta_{kl})

\,\!</math>{{citation needed|date=June 2012}}

where <math>\delta_{ij}\,\!</math> is the [[Kronecker delta]], ''K''&nbsp; 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 (\varepsilon_{ij} - \tfrac{1}{3} \delta_{ij} \varepsilon_{kk}).</math>

:<math> \sigma_{ij}

=K\delta_{ij}\varepsilon_{kk}+2\mu(\varepsilon_{ij}-\textstyle{\frac{1}{3}}\delta_{ij}\varepsilon_{kk}).

\,\!</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 name=aki>{{cite book |title= Quantitative Seismology |last1=Aki|first1=Keiiti |last2=Richards|first2= Paul G. | author-link1=Keiiti Aki |author2-link=Paul G. richards |year=2002 | edition= 2| publisher=University Science Books |___location=Sausalito, California}}</ref>

<math display="block"> \sigma_{ij}

=\lambda \delta_{ij} \varepsilon_{kk}+2\mu\varepsilon_{ij}</math><ref>Continuum Mechanics for Engineers 2001 Mase, Eq. 5.12-2</ref>

:<math> \sigma_{ij}

=\lambda \delta_{ij} \varepsilon_{kk}+2\mu\varepsilon_{ij}

\,\!</math><ref>Continuum Mechanics for Engineers 2001 Mase, Eq. 5.12-2</ref>

where λ is [[Lamé parameters|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 name=sommerfeld>{{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>

:<math>\varepsilon_{ij}

= \frac{1}{9K}\delta_{ij}\sigma_{kk} + \frac{1}{2\mu}\left(\sigma_{ij}-\textstyle{\frac{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>

:<math>\varepsilon_{ij}

where <math>\nu</math> is [[Poisson's ratio]] and <math>E</math> is [[Young's modulus]].

=\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>

:<math>

\sigma_{ji,j}+ F_i = 0.

\,\!</math>

:{| class="collapsible collapsed" width="30%" style="text-align:left"

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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">\begin{align}

\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). \\

\end{align}</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>

:<math>

\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 '''Navier–Cauchy equations''' given below.

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| title = Derivation of 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}\sigma_xleft( 2\mu\frac{\partial u_x}{\partial x}+ \lambda \left(\frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\tau_partial y}+ \frac{yx\partial u_z}{\partial z}\right)\right) + \mu\frac{\partial}{\partial y} \left(\frac{\partial u_x}{\partial y}+ \frac{\partial u_y}{\tau_partial x}\right)+ \mu\frac{zx}\partial}{\partial z} \left(\frac{\partial u_z}{\partial x} + F_x\frac{\partial =u_x}{\partial z}\right) +F_x=0</math>

:<math>\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 yz} \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_yu_z}{\partial x^2}+ \frac{\partial^2 u_yu_z}{\partial y^2}+ \frac{\partial^2 u_yu_z}{\partial z^2}\right)+F_yF_z=0\,\!</math>

:<math>\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 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} = 0</math>

}}

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===== The biharmonic equation =====

The elastostatic equation may be written:

:<math display="block">(\alpha^2-\beta^2) u_{j,ij}+

\beta^2u_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>

:<math>(\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^2u_{j,iij} = 0</math>

:<math>\alpha^2u_{j,iij} = 0\,\!</math>

from which we conclude that:

<math display="block">u_{j,iij} = 0.</math>

:<math>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>

:<math>(\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>

:<math>\beta^2u_{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>.

:<math>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====

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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>

{| class="collapsible collapsed" width="30%" style="text-align:left"

:<math>\varepsilon_{ij,km}+\varepsilon_{km,ij}-\varepsilon_{ik,jm}-\varepsilon_{jm,ik}=0.\,\!</math>

:{| class="collapsible collapsed" width="30%" style="text-align:left"

!Engineering notation

|-

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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>

:<math>\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

:<math display="block"> (1+\nu)\sigma_{ij,kk}+\sigma_{kk,ij}=0.\,\!</math><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>

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/>

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

:{| class="toccolours collapsible collapsed" style="max-width: 90%; text-align:left"

!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">ab = 2(1-2\nu\,\!) = a+1</math>

where <math>\nu</math> is Poisson's ratio, the solution may be expressed as

:<math>b=2(1-\nu)=a+1\,\!</math>

<math display="block">u_i = G_{ik} F_k</math>

where <math>\nu\F_k</math> is the force vector being applied at the point,\! and <math>G_{ik}</math> is Poissona tensor [[Green's ratio,function]] the solutionwhich may be expressedwritten in [[Cartesian coordinates]] as:

<math display="block">G_{ik}=

\frac{1}{4\pi\mu r} \left[

:<math>u_i=G_{ik}F_k\,\!</math>

\left(1 - \frac{1}{2b}\right) \delta_{ik} + \frac{1}{2b} \frac{x_i x_k}{r^2}

\right]</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>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>

:<math>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}

:<math>G_{ik}=\frac{1}{4\pi\mu r}\begin{bmatrix}

1-\frac{1}{2b}+\frac{1}{2b}\frac{x^2}{r^2} &

Line 298 ⟶ 258:

\frac{1}{2b}\frac{zy} {r^2} &

1-\frac{1}{2b}+\frac{1}{2b}\frac{z^2}{r^2}

\end{bmatrix}</math>

\,\!</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{\mathbf{\rho}}</math> and <math>\hat{\mathbf{z}}</math> as unit vectors in the <math>\rho</math> and <math>z</math> directions respectively yields:

:<math>G_{ik}=\frac{1}{4\pi \mu r}\begin{bmatrix}

<math display="block">

1-\frac{1}{2b}\frac{z^2}{r^2}&0&\frac{1}{2b}\frac{\rho z}{r^2}\\

\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]

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 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{\mathbf{\rho}}\,\!</math> and <math>\hat{\mathbf{z}}\,\!</math> as unit vectors in the <math>\rho\,\!</math> and <math>z\,\!</math> directions respectively yields:

:<math>

\mathbf{u}=\frac{F_z}{4\pi\mu r}\left[\frac{1}{4(1-\nu)}\,\frac{\rho z}{r^2}\hat{\mathbf{\rho}} + \left(1-\frac{1}{4(1-\nu)}\,\frac{\rho^2}{r^2}\right)\hat{\mathbf{z}}\right]

\,\!</math>

Line 326 ⟶ 281:

|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 [note: a=(1-2ν) and b=2(1-ν), ν== Poissons 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)} &

Line 347 ⟶ 302:

* 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|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 ===

Line 354 ⟶ 309:

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>

:<math>

\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>

:<math>

\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 '''Navier–Cauchy equation''':

<math display="block">

\mu u_{i,jj} + (\mu+\lambda)u_{j,ij}+F_i=\rho\partial_{tt}u_i

:<math>

\quad \text{or} \quad

\mu u_{i,jj}+(\mu+\lambda)u_{j,ij}+F_i=\rho\partial_{tt}u_i

\mu \nabla^2\mathbf{u} + (\mu+\lambda)\nabla(\nabla\cdot\mathbf{u}) + \mathbf{F}=\rho\frac{\partial^2\mathbf{u}}{\partial t^2}.</math>

\quad \mathrm{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_ik_jk_i k_j + \beta^2(k_mk_mk_m k_m \delta_{ij}-k_ik_jk_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 name=OS>[[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>

:<math>

\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

:<math>

_{ij}\right) +\left( \rho ^{-1}\sigma _F_{(ik},_{ki}\right) ,_{j)}-\frac{1}{2\mu }\left(= 0. </math>

\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.