Linear elasticity

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Linear elasticity models the macroscopic mechanical properties of solids assuming "small" deformations.

1 Basic equations
2 Wave equation
3 Plane waves
4 Isotropic homogeneous media
5 The biharmonic equation
6 See also
7 References

Linear elastodynamics is based on three tensor equations:

dynamic equation

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

constitutive equation (anisotropic Hooke's law)

$\sigma_{ij} = C_{ijkl} \, \varepsilon_{kl}$

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 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 entry.

From the basic equations 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 $δ k l$ is Kronecker delta.

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 $ω 2$ 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.

In isotropic media, the elasticity tensor has the form

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

where $κ$ is incompressibility, and $μ$ is rigidity. These two parameters are also called the Lamé parameters. 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

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

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).

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

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

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 $\mathbf{u}$ .

Castigliano's method

Gurtin M. E., Introduction to Continuum Mechanics, Academic Press 1981
L. D. Landau & E. M. Lifschitz, Theory of Elasticity, Butterworth 1986
Elastostatics (Kip Thorne)

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Categories: Continuum mechanics | Structural engineering

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