Linear elasticity: Difference between revisions

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== Basic equations ==

Linear elastodynamics is based on three [[tensor]] equations:

* dynamic equation (an expression of [[Newtons second law#Newton.27s second law: law of acceleration|Newton's second law]])

:<math>

\partial_j \sigma_{ij} + f_i = \rho \, \partial_{tt} u_i

</math>

* [[constitutive equation]] (''anisotropic [[Hooke's law]]'')

Line 27 ⟶ 28:

The elastostatic equations are given by setting <math>\partial_t</math> to zero in the dynamic equation. The elastostatic equations are shown in their full form on the [[3-D elasticity]] entry.

== Wave equation ==

From the basic equations one gets the ''wave equation''

:<math> (\delta_{kl} \partial_{tt}-A_{kl}[\nabla]) \, u_l

= \frac{1}{\rho} f_k </math>

where

:<math> 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 ==

A ''plane wave'' has the form

:<math> \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> A_{kl}[\mathbf{k}]=\frac{1}{\rho} \, k_i \, C_{iklj} \, k_j </math>

This ''propagation condition'' may be written as

:<math>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.

== Isotropic homogeneous media ==

In [[Hooke's Law#Isotropic materials|isotropic]] media, the [[elasticity tensor]] has the form

:<math> 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 elasticity tensor is constant throughout the material), the acoustic operator becomes:

:<math>A_{ij}[\nabla]=\alpha^2 \partial_i\partial_j+\beta^2(\partial_m\partial_m\delta_{ij}-\partial_i\partial_j)\,</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 elasticity tensor is constant throughout the material), the three basic equations can be combined to form the '''elastodynamic equation''':

and the acoustic algebraic operator becomes

:<math>A_{ij}[\mathbf{k}]=\alpha^2 k_ik_j+\beta^2(k_mk_m\delta_{ij}-k_ik_j)\,</math>

:<math>

where

\mu\partial_j\partial_j u_i+\frac{\mu}{1-2\nu}\partial_i\partial_ju_j+f_i=\rho\partial_{tt}u_i

:<math> \alpha^2=\left(K+\frac{4}{3}\mu\right)/\rho \qquad \beta^2=\mu/\rho </math>

\,\,\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,\,\,

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. In the seismological literature, the corresponding plane waves are called P-waves and S-waves (see [[Seismic wave]]).

\mu\nabla^2\mathbf{u}+\frac{\mu}{1-2\nu}\nabla(\nabla\cdot\mathbf{u})+\mathbf{f}=\rho\frac{\partial^2\mathbf{u}}{\partial t^2}

</math>

and the constitutive equation may be written:

:<math>

\sigma_{ij}=\lambda\epsilon_{kk}\delta_{ij}+2\mu\epsilon_{ij}\,

</math>

==Elastostatics - the elastostatic equation ==

If we assume that a steady state has been achieved, in which there is no time dependence to any of the quantities involved, the elastodynamic equation becomes the '''elastostatic equation'''

:<math>

\mu\partial_j\partial_ju_i+\frac{\mu}{1-2\nu}\partial_i\partial_ju_j=-f_i

\,\,\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,\,\,

\mu\nabla^2\mathbf{u}+\frac{\mu}{1-2\nu}\nabla(\nabla\cdot\mathbf{u})=-\mathbf{f}

</math>

=== Thomson's solution: point force at the origin of an infinite medium ===

The most important solution of this 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 {{ref_harvard|L&LV7|Landau & Lifshitz &sect; 8|}}. Defining

:<math>a=1-2\nu\,</math>

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

where <math>/nu</math> is Poisson's ratio, the solution may be expressed as <math>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>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^3}

\right]

</math>

It may be also compactly written as:

:<math>G_{ik}=

\frac{1}{4\pi\mu}\left[\frac{\delta_{ik}}{r}-\frac{1}{2b}\frac{\partial r^2}{\partial x_i\partial x_k}\right]

</math>

and it may be explicitly written as:

:<math>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>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>

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>

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. There is also an additional &rho;-directed component which will be zero when &nu;=1.

=== Boussinesq's 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_harvard|Bou1885|Boussinesq 1885|}} and a derivation is given in {{ref_harvard|L&LV7|See Landau & Lifshitz &sect; 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 as in Cartesian coordinates as:

:<math>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_harvard|Min1885|Mindlin 1936|}}

* Contact of two elastic bodies {{ref_harvard|Hertz1882|Hertz 1882|}}

=== The biharmonic equation ===

The elastostatic equation may be written:

For a static situation (<math>\partial_t=0</math>) in isotropic materials, the wave equation becomes the elastostatic equation :

:<math>A_{ij}u_j=(\alpha^2-\beta^2)\partial_i\partial_ju_j+

Line 99 ⟶ 176:

or, in coordinate free notation <math>\nabla^4 \mathbf{u}=0</math> which is just the [[biharmonic equation]] in <math>\mathbf{u}</math>.

== Elastodynamics - The Wave equation ==

From the elastodynamic equation one gets the ''wave equation''

:<math> (\delta_{kl} \partial_{tt}-A_{kl}[\nabla]) \, u_l

= \frac{1}{\rho} f_k </math>

where

:<math> 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 [[elasticity tensor]] has the form

:<math> 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 elasticity tensor is constant throughout the material), the acoustic operator becomes:

:<math>A_{ij}[\nabla]=\alpha^2 \partial_i\partial_j+\beta^2(\partial_m\partial_m\delta_{ij}-\partial_i\partial_j)\,</math>

and the acoustic algebraic operator becomes

:<math>A_{ij}[\mathbf{k}]=\alpha^2 k_ik_j+\beta^2(k_mk_m\delta_{ij}-k_ik_j)\,</math>

where

:<math> \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. In the seismological literature, the corresponding plane waves are called P-waves and S-waves (see [[Seismic wave]]).

=== Plane waves ===

A ''plane wave'' has the form

:<math> \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> A_{kl}[\mathbf{k}]=\frac{1}{\rho} \, k_i \, C_{iklj} \, k_j </math>

This ''propagation condition'' may be written as

:<math>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==

Line 108 ⟶ 225:

== References ==

*Gurtin M. E., Introduction to Continuum Mechanics, Academic Press 1981

*L. D. Landau & E. M. Lifschitz, Theory of Elasticity, Butterworth 1986

* [http://www.pma.caltech.edu/Courses/ph136/yr2002/chap10/0210.1.pdf Elastostatics (Kip Thorne)]

* {{note_label|L&LV7||}}{{cite book |title=Theory of Elasticity |edition=3rd Edition|last=Landau |first=L.D. |authorlink=Lev Landau |coauthors=[[Evgeny Lifshitz|Lifshitz, E. M.]] |year=1986 |publisher=Butterworth Heinemann |___location=Oxford, England |isbn=0-7506-2633-X }}

* {{note_label|Bou1885||}}{{cite book |title= Application des potentiels à l'étude de l'équilibre et du mouvement des solides élastiques |last=Boussinesq|first=Joseph |authorlink=Joseph Boussinesq |year=1885 |publisher=Gauthier-Villars |___location=Paris, France |url=http://name.umdl.umich.edu/ABV5032.0001.001 }}

* {{note_label|Min1936||}}{{cite journal |last=Mindlin |first= R. D.|authorlink=Raymond D. Mindlin |year=1936|title=Force at a point in the interior of a semi-infinite solid |journal=Physics |volume=7|issue= |pages=195-202 |id= |url= |quote= }}

* {{note_label|Hertz1882||}}{{cite journal |last=Hertz |first= Heinrich|authorlink=Heinrich Hertz |year=1882|title=Contact between solid elastic bodies |journal=Journ. f'tir reine iind angewandte Math.|volume=92}}

[[Category:Solid mechanics]]