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# Linear elasticity

Linear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions. Linear elasticity relies upon the continuum hypothesis and is applicable at macroscopic (and sometimes microscopic) length scales. Linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of continuum mechanics. The fundamental "linearizing" assumptions of linear elasticity are: "small" deformations (or strains) and linear relationships between the components of stress and strain. In addition linear elasticity is only valid for stress states that do not produce 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 through the aid of finite element analysis. This article presents a summary of some of the basic equations used to describe linear elasticity mathematically in tensor notation. For an alternative presentation using engineering notation, see the article on 3-D elasticity.

## Basic equations

Linear elastodynamics is based on three tensor equations:

• dynamic equation (an expression of Newton's second law)
$\partial_j \sigma_{ij}+ f_i = \rho \, \partial_{tt} u_i$
$\sigma_{ij} = C_{ijkl} \, \varepsilon_{kl}$
• kinematic equation
$\varepsilon_{ij} =\frac{1}{2} (\partial_i u_j+\partial_j u_i)$

where:

• $\sigma_{ij}=\sigma_{ji} \,$ is the Cauchy stress
• $f_i \,$ is the body force
• $\rho \,$ is the mass density
• $u_i \,$ is the displacement
• $C_{ijkl}=C_{klij}=C_{jikl}=C_{ijlk} \,$ is the elasticity tensor
• $\varepsilon_{ij}=\varepsilon_{ji} \,$ is the strain
• $\partial_i$ is the partial derivative $\partial/\partial x_i$ and $\partial_t$ is $\partial/\partial t$.

The basic elastostatic equations are given by setting $\partial_t$ to zero in the dynamic equation. The elastostatic equations are shown in their full form on the 3-D elasticity article.

Just as a spring which is compressed or expanded holds potential energy, so a strained material will possess an energy density due to the deformation. The energy density due to deformation is given by:

$F=\frac{1}{2}\sigma_{ij}\,\varepsilon_{ij}$

## Isotropic homogeneous media

In isotropic media, the elasticity tensor has the form

$C_{ijkl} = K \, \delta_{ij}\, \delta_{kl} +\mu\, (\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}-\frac{2}{3}\, \delta_{ij}\,\delta_{kl})$

where K is the bulk modulus (or incompressibility), and μ 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:

$\mu\partial_j\partial_j u_i+\frac{\mu}{1-2\nu}\partial_i\partial_ju_j+f_i=\rho\partial_{tt}u_i \,\,\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,\,\, \mu\nabla^2\mathbf{u}+\frac{\mu}{1-2\nu}\nabla(\nabla\cdot\mathbf{u})+\mathbf{f}=\rho\frac{\partial^2\mathbf{u}}{\partial t^2}$

and the constitutive equation may be written:

$\sigma_{ij}=\frac{2\nu}{1-2\nu}\mu\epsilon_{kk}\delta_{ij}+2\mu\epsilon_{ij}\,$

## 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

$\mu\partial_j\partial_ju_i+\frac{\mu}{1-2\nu}\partial_i\partial_ju_j=-f_i \,\,\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,\,\, \mu\nabla^2\mathbf{u}+\frac{\mu}{1-2\nu}\nabla(\nabla\cdot\mathbf{u})=-\mathbf{f}$

### 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 (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 § 8). Defining

$a=1-2\nu\,$
$b=2(1-\nu)=a+1\,$

where / nu is Poisson's ratio, the solution may be expressed as ui = Gikfk where fk is the force vector being applied at the point, and Gik is a tensor Green's function which may be written in Cartesian coordinates as:

$G_{ik}= \frac{1}{4\pi\mu r}\left[ \left(1-\frac{1}{2b}\right)\delta_{ik}+\frac{1}{2b}\frac{x_i x_k}{r^3} \right]$

It may be also compactly written as:

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

and it may be explicitly written as:

$G_{ik}=\frac{1}{4\pi\mu r}\begin{bmatrix} 1-\frac{1}{2b}+\frac{1}{2b}\frac{x^2}{r^2} & \frac{1}{2b}\frac{xy} {r^2} & \frac{1}{2b}\frac{xz} {r^2} \\ \frac{1}{2b}\frac{yx} {r^2} & 1-\frac{1}{2b}+\frac{1}{2b}\frac{y^2}{r^2} & \frac{1}{2b}\frac{yz} {r^2} \\ \frac{1}{2b}\frac{zx} {r^2} & \frac{1}{2b}\frac{zy} {r^2} & 1-\frac{1}{2b}+\frac{1}{2b}\frac{z^2}{r^2} \end{bmatrix}$

In cylindrical coordinates (ρ,φ,z) it may be written as:

$G_{ik}=\frac{1}{4\pi \mu r}\begin{bmatrix} 1-\frac{1}{2b}\frac{z^2}{r^2}&0&\frac{1}{2b}\frac{\rho z}{r^2}\\ 0&1-\frac{1}{2b}&0\\ \frac{1}{2b}\frac{z \rho}{r^2}&0&1-\frac{1}{2b}\frac{\rho^2}{r^2} \end{bmatrix}$

It is particularly helpful to write the displacement in cylindrical coordinates for a point force Fz directed along the z-axis. Defining $\hat{\mathbf{\rho}}$ and $\hat{\mathbf{z}}$ as unit vectors in the ρ and z directions respectively yields:

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

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 ρ-directed component.

### 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(Boussinesq 1885) and a derivation is given in (See Landau & Lifshitz § 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:

$G_{ik}=\frac{1}{4\pi\mu}\begin{bmatrix} \frac{b}{r}+\frac{x^2}{r^3}-\frac{ax^2}{r(r+z)^2}+\frac{az}{r(r+z)} & \frac{xy}{r^3}-\frac{axy}{r(r+z)^2}& \frac{xz}{r^3}-\frac{ax}{r(r+z)}\\ \frac{yx}{r^3} -\frac{ayx}{r(r+z)^2}& \frac{b}{r}+\frac{y^2}{r^3}-\frac{ay^2}{r(r+z)^2}+\frac{az}{r(r+z)} & \frac{yz}{r^3} -\frac{ay}{r(r+z)}\\ \frac{zx}{r^3}+\frac{ax}{r(r+z)}& \frac{zy}{r^3}+\frac{ay}{r(r+z)}& \frac{b}{r}+\frac{z^2}{r^3} \end{bmatrix}$

Other solutions:

### The biharmonic equation

The elastostatic equation may be written:

$A_{ij}u_j=(\alpha^2-\beta^2)\partial_i\partial_ju_j+ \beta^2\partial_m\partial_mu_i=-f_i/\rho$

Taking the divergence of both sides of the elastostatic equation and assuming a conservative force, ($\partial_i f_i=0$) we have

$\partial_i A_{ij}u_j = (\alpha^2-\beta^2)\partial_i\partial_i\partial_ju_j+\beta^2\partial_i\partial_m\partial_mu_i = 0$

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:

$\partial_i A_{ij}u_j = \alpha^2\partial_i\partial_i\partial_ju_j = 0$

from which we conclude that:

$\partial_i\partial_i\partial_ju_j = 0$

Taking the Laplacian of both sides of the elastostatic equation, a conservative force will give $\partial_k\partial_kf_i=0$ and we have

$\partial_k\partial_kA_{ij}u_j = (\alpha^2-\beta^2)\partial_k\partial_k\partial_i\partial_ju_j+\beta^2\partial_k\partial_k\partial_m\partial_mu_i=0$

From the divergence equation, the first term on the right is zero (Note: again, the summed indices need not match) and we have:

$\partial_k\partial_kA_{ij}u_j = \beta^2\partial_k\partial_k\partial_m\partial_mu_i=0$

from which we conclude that:

$\partial_k\partial_k\partial_m\partial_mu_i=0$

or, in coordinate free notation $\nabla^4 \mathbf{u}=0$ which is just the biharmonic equation in $\mathbf{u}$.

## Elastodynamics - The Wave equation

From the elastodynamic equation one gets the wave equation

$(\delta_{kl} \partial_{tt}-A_{kl}[\nabla]) \, u_l = \frac{1}{\rho} f_k$

where

$A_{kl}[\nabla]=\frac{1}{\rho} \, \partial_i \, C_{iklj} \, \partial_j$

is the acoustic differential operator, and δkl is Kronecker delta.

In isotropic media, the elasticity tensor has the form

$C_{ijkl} = K \, \delta_{ij}\, \delta_{kl} +\mu\, (\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}-\frac{2}{3}\, \delta_{ij}\,\delta_{kl})$

where K is the bulk modulus (or incompressibility), and μ 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:

$A_{ij}[\nabla]=\alpha^2 \partial_i\partial_j+\beta^2(\partial_m\partial_m\delta_{ij}-\partial_i\partial_j)\,$

and the acoustic algebraic operator becomes

$A_{ij}[\mathbf{k}]=\alpha^2 k_ik_j+\beta^2(k_mk_m\delta_{ij}-k_ik_j)\,$

where

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

are the eigenvalues of $A[\hat{\mathbf{k}}]$ with eigenvectors $\hat{\mathbf{u}}$ parallel and orthogonal to the propagation direction $\hat{\mathbf{k}}$, 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

$\mathbf{u}[\mathbf{x}, \, t] = U[\mathbf{k} \cdot \mathbf{x} - \omega \, t] \, \hat{\mathbf{u}}$

with $\hat{\mathbf{u}}$ of unit length. It is a solution of the wave equation with zero forcing, if and only if ω2 and $\hat{\mathbf{u}}$ constitute an eigenvalue/eigenvector pair of the acoustic algebraic operator

$A_{kl}[\mathbf{k}]=\frac{1}{\rho} \, k_i \, C_{iklj} \, k_j$

This propagation condition may be written as

$A[\hat{\mathbf{k}}] \, \hat{\mathbf{u}}=c^2 \, \hat{\mathbf{u}}$

where $\hat{\mathbf{k}} = \mathbf{k} / \sqrt{\mathbf{k}\cdot\mathbf{k}}$ denotes propagation direction and $c=\omega/\sqrt{\mathbf{k}\cdot\mathbf{k}}$ is phase velocity.