Linear elasticity

Linear elasticity models the macroscopic mechanical properties of solids assuming "small" deformations. It is a key part of elasticity theory, which described the initial part of deformation of solids under an external stress. Linear elasticity has a well-defined mathematical theory which describes well a wide range of solids. The current page summarizes some of the basic equations used to describe linear elasticity mathematically in tensor notation. For an alternative notation to describe linear elasticity, see the article on 3-D elasticity.

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
• 3-D elasticity
• Deformation

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)