International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. D. ch. 1.3, pp. 95-96
Section 1.3.7.3. Wave propagation in a nonlinear elastic medium
a
Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France, and bLaboratoire de Physique des Milieux Condensés, Université P. et M. Curie, 75252 Paris CEDEX 05, France |
As an example, let us consider the case of a plane finite amplitude wave propagating along the axis. The displacement components in this case become Thus, the Jacobian matrix reduces to
The Lagrangian strain matrix is [equation (1.3.6.8)] The only nonvanishing strain components are, therefore, and the strain invariants reduce to
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 that this tensor is not symmetric.
For the particular problem discussed here, the three components of the equation of motion are
If we retain only terms up to the quadratic order in the displacement gradients, we obtain the following equations of motion:
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, the three components of the equation of motion are
If we retain only terms up to the quadratic order in the displacement gradients, we obtain the following equations of motion: which are identical to (1.3.7.7) if we put