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)

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

• 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

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

where ${\displaystyle K}$  is the bulk modulus (or incompressibility), and ${\displaystyle \mu }$  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:

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

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

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

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
• Elastic limit
• Proportional limit

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)