Experimental determination of third- and higher-order elastic constants
Authier, A. and
Zarembowitch, A.,
International Tables for Crystallography
(2013).
Vol. D,
Section 1.3.7.6,
p.
[ doi:10.1107/97809553602060000902 ]
are deduced from the stress derivatives of . For instance, Table
1.3.7.1 gives the expressions for and for a cubic crystal. These quantities refer to the natural state free of stress. In this table, p denotes the hydrostatic pressure ...
Isotropic media
Authier, A. and
Zarembowitch, A.,
International Tables for Crystallography
(2013).
Vol. D,
Section 1.3.7.3.1,
p.
[ doi:10.1107/97809553602060000902 ]
properties
In this case, the strain-energy density becomes Differentiating (
1.3.7.6) with respect to the strains, we get All the other .
From (
1.3.7.5), we derive the stress components: Note ...
Equation of motion for elastic waves
Authier, A. and
Zarembowitch, A.,
International Tables for Crystallography
(2013).
Vol. D,
Section 1.3.7.2,
p.
[ doi:10.1107/97809553602060000902 ]
these equations are or where L is the Lagrangian per unit initial volume and are the elements of the Jacobian matrix.
For adiabatic motion where U is the internal energy per unit mass.
Combining (
1.3.7.2) and (
1.3.7.3 ...
Wave propagation in a nonlinear elastic medium
Authier, A. and
Zarembowitch, A.,
International Tables for Crystallography
(2013).
Vol. D,
Section 1.3.7.3,
p.
[ doi:10.1107/97809553602060000902 ]
density becomes Differentiating (
1.3.7.6) with respect to the strains, we get All the other .
From (
1.3.7.5), we derive the stress components: Note that this tensor is not symmetric.
For the particular problem discussed ...
Cubic media (most symmetrical groups)
Authier, A. and
Zarembowitch, A.,
International Tables for Crystallography
(2013).
Vol. D,
Section 1.3.7.3.2,
p.
[ doi:10.1107/97809553602060000902 ]
properties
In this case, the strain-energy density becomes Differentiating (
1.3.7.8) with respect to the strain, one obtains All other . From (
1.3.7.5), we derive the stress components: In this particular case ...
Harmonic generation
Authier, A. and
Zarembowitch, A.,
International Tables for Crystallography
(2013).
Vol. D,
Section 1.3.7.4,
p.
[ doi:10.1107/97809553602060000902 ]
and (
1.3.7.9), reduce to for an isotropic medium or for a cubic crystal (most symmetrical groups) when a pure longitudinal mode is propagated along [100].
For both cases, we have a one-dimensional problem; (
1.3.7.7) and (
1.3.7.9) can ...
Introduction
Authier, A. and
Zarembowitch, A.,
International Tables for Crystallography
(2013).
Vol. D,
Section 1.3.7.1,
p.
[ doi:10.1107/97809553602060000902 ]
...
Small-amplitude waves in a strained medium
Authier, A. and
Zarembowitch, A.,
International Tables for Crystallography
(2013).
Vol. D,
Section 1.3.7.5,
p.
[ doi:10.1107/97809553602060000902 ]
and are equal.
The simplest solutions of the equation of motion are plane waves. We now assume plane sinusoidal waves of the form where k is the wavevector.
Substitution of (
1.3.7.14) into (
1.3.7.13) results ...
Nonlinear dynamic elasticity
Authier, A. and
Zarembowitch, A.,
International Tables for Crystallography
(2013).
Vol. D,
Section 1.3.7,
p.
[ doi:10.1107/97809553602060000902 ]
U is the internal energy per unit mass.
Combining (
1.3.7.2) and (
1.3.7.3), it follows that which can be written since
Using now the equation of continuity or conservation of mass: and the identity of Euler ...
