Linear elasticity

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

Basic equations

Linear elastodynamics is based on three tensor equations:

• dynamic equation

${\displaystyle \partial _{j}\sigma _{ij}+f_{i}=\rho \,\partial _{tt}u_{i}}$ 

• constitutive equation (anisotropic Hooke's law)

${\displaystyle \sigma _{ij}=C_{ijkl}\,\varepsilon _{kl}}$ 

• kinematic equation

${\displaystyle \varepsilon _{ij}={\frac {1}{2}}(\partial _{i}u_{j}+\partial _{j}u_{i})}$ 

where:

• ${\displaystyle \sigma _{ij}=\sigma _{ji}\,}$  is the Cauchy stress
• ${\displaystyle f_{i}\,}$  is the body force
• ${\displaystyle \rho \,}$  is the mass density
• ${\displaystyle u_{i}\,}$  is the displacement
• ${\displaystyle C_{ijkl}=C_{klij}=C_{jikl}=C_{ijlk}\,}$  is the elasticity tensor
• ${\displaystyle \varepsilon _{ij}=\varepsilon _{ji}\,}$  is the strain
• ${\displaystyle \partial _{i}}$  is the partial derivative ${\displaystyle \partial /\partial x_{i}}$  and ${\displaystyle \partial _{t}}$  is ${\displaystyle \partial /\partial t}$ .

The elastostatic equations are given by setting ${\displaystyle \partial _{t}}$  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

${\displaystyle (\delta _{kl}\partial _{tt}-A_{kl}[\nabla ])\,u_{l}={\frac {1}{\rho }}f_{k}}$ 

where

${\displaystyle A_{kl}[\nabla ]={\frac {1}{\rho }}\,\partial _{i}\,C_{iklj}\,\partial _{j}}$ 

is the acoustic differential operator, and ${\displaystyle \delta _{kl}}$  is Kronecker delta.

Plane waves

A plane wave has the form

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

with ${\displaystyle {\hat {\mathbf {u} }}}$  of unit length. It is a solution of the wave equation with zero forcing, if and only if ${\displaystyle \omega ^{2}}$  and ${\displaystyle {\hat {\mathbf {u} }}}$  constitute an eigenvalue/eigenvector pair of the acoustic algebraic operator

${\displaystyle A_{kl}[\mathbf {k} ]={\frac {1}{\rho }}\,k_{i}\,C_{iklj}\,k_{j}}$ 

This propagation condition may be written as

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

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

Isotropic homogeneous media

In isotropic media, the elasticity tensor has the form

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

where ${\displaystyle \kappa }$  is incompressibility, and ${\displaystyle \mu }$  is rigidity, two modulus of elasticity. If the material is homogeneous (i.e. the elasticity tensor is constant throughout the material), the acoustic operator becomes:

${\displaystyle A_{ij}[\nabla ]=\alpha ^{2}\partial _{i}\partial _{j}+\beta ^{2}(\partial _{m}\partial _{m}\delta _{ij}-\partial _{i}\partial _{j})\,}$ 

and the acoustic algebraic operator becomes

${\displaystyle A_{ij}[\mathbf {k} ]=\alpha ^{2}k_{i}k_{j}+\beta ^{2}(k_{m}k_{m}\delta _{ij}-k_{i}k_{j})\,}$ 

where

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

are the eigenvalues of ${\displaystyle A[{\hat {\mathbf {k} }}]}$  with eigenvectors ${\displaystyle {\hat {\mathbf {u} }}}$  parallel and orthogonal to the propagation direction ${\displaystyle {\hat {\mathbf {k} }}}$ , respectively. In the seismological literature, the corresponding plane waves are called P-waves and S-waves (see Seismic wave).

The biharmonic equation

For a static situation (${\displaystyle \partial _{t}=0}$ ) in isotropic materials, the wave equation becomes the elastostatic equation :

${\displaystyle A_{ij}u_{j}=(\alpha ^{2}-\beta ^{2})\partial _{i}\partial _{j}u_{j}+\beta ^{2}\partial _{m}\partial _{m}u_{i}=-f_{i}/\rho }$ 

Taking the divergence of both sides of the elastostatic equation and assuming a conservative force, (${\displaystyle \partial _{i}f_{i}=0}$ ) we have

${\displaystyle \partial _{i}A_{ij}u_{j}=(\alpha ^{2}-\beta ^{2})\partial _{i}\partial _{i}\partial _{j}u_{j}+\beta ^{2}\partial _{i}\partial _{m}\partial _{m}u_{i}=0}$ 

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:

${\displaystyle \partial _{i}A_{ij}u_{j}=\alpha ^{2}\partial _{i}\partial _{i}\partial _{j}u_{j}=0}$ 

from which we conclude that:

${\displaystyle \partial _{i}\partial _{i}\partial _{j}u_{j}=0}$ 

Taking the Laplacian of both sides of the elastostatic equation, a conservative force will give ${\displaystyle \partial _{k}\partial _{k}f_{i}=0}$  and we have

${\displaystyle \partial _{k}\partial _{k}A_{ij}u_{j}=(\alpha ^{2}-\beta ^{2})\partial _{k}\partial _{k}\partial _{i}\partial _{j}u_{j}+\beta ^{2}\partial _{k}\partial _{k}\partial _{m}\partial _{m}u_{i}=0}$ 

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

${\displaystyle \partial _{k}\partial _{k}A_{ij}u_{j}=\beta ^{2}\partial _{k}\partial _{k}\partial _{m}\partial _{m}u_{i}=0}$ 

from which we conclude that:

${\displaystyle \partial _{k}\partial _{k}\partial _{m}\partial _{m}u_{i}=0}$ 

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

See also

• Castigliano's method

References

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