Linear elasticity
| Table of contents |
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2 Basic equations 3 Wave equation 4 Plane waves 5 Isotropic media 6 References |
Linear elasticity
The linear theory of elasticity studies - with mathematical methods -
the macroscopic mechanical properties of solids, assuming "small" deformations.
Basic equations
Linear elastodynamics is based on three tensor equations:
- dynamic equation
- constitutive equation (anisotropic Hooke's law)
- kinematic equation
- is stress
- is body force
- is density
- is displacement
- is elasticity tensor
- is strain
Wave equation
From the basic equations one gets the wave equationPlane waves
A plane wave has the formand constitute an eigenvalue/eigenvector pair of theacoustic algebraic operator
denotes propagation direction
and is phase velocity.
with eigenvectors parallel and orthogonal to the propagation direction
, respectively.
In the seismological literature, the corresponding plane waves are called P-waves and S-waves (see Seismic wave).
Isotropic media
In isotropic media, the elasticity tensor has the form
where
is incompressibility, and
is rigidity.
Hence the acoustic algebraic operator becomes
where
denotes the tensor product,
is the identity matrix, and
are the eigenvalues of References
