Linear elasticity

Linear elasticity

The linear theory of elasticity models the macroscopic mechanical properties of solids assuming "small" deformations.

Basic equations

Linear elastodynamics is based on three tensor equations:

• dynamic equation

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

• constitutive equation (anisotropic Hooke's law)

${\displaystyle T_{ij}=C_{ijkl}\,E_{kl}}$ 

• kinematic equation

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

where:

• ${\displaystyle T_{ij}=T_{ji}}$  is stress
• ${\displaystyle f_{i}}$  is body force
• ${\displaystyle \rho }$  is density
• ${\displaystyle u_{i}}$  is displacement
• ${\displaystyle C_{ijkl}=C_{klij}=C_{jikl}=C_{ijlk}}$  is the elasticity tensor
• ${\displaystyle E_{ij}=E_{ji}}$  is strain

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 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. Hence the acoustic algebraic operator becomes

${\displaystyle A[{\hat {\mathbf {k} }}]=\alpha ^{2}\,{\hat {\mathbf {k} }}\otimes {\hat {\mathbf {k} }}+\beta ^{2}\,(\mathbf {I} -{\hat {\mathbf {k} }}\otimes {\hat {\mathbf {k} }})}$ 

where ${\displaystyle \otimes }$  denotes the tensor product, ${\displaystyle \mathbf {I} }$  is the identity matrix, and

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

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).

References

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