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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:
- constitutive equation (anisotropic Hooke's law)
where:
-
 is stress
-
 is body force
-
 is density
-
 is displacement
-
 is elasticity tensor
-
 is strain
Wave equation
From the basic equations one gets the wave equation
where
is the acoustic differential operator, and  is Kronecker delta.
Plane waves
A plane wave has the form
with  of unit length.
It is a solution of the wave equation with zero forcing, if and only if
 and constitute an eigenvalue/eigenvector pair of the
acoustic algebraic operator
This propagation condition may be written as
where
denotes propagation direction
and  is phase velocity.
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
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).
References
- Gurtin M. E., Introduction to Continuum Mechanics, Academic Press 1981
- L. D. Landau & E. M. Lifschitz, Theory of Elasticity, Butterworth 1986
Referenced By
List of mathematical topics (J-L) | Solid mechanics | Theory of elasticity
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