Equation of motion for elastic waves

Authier, A.; Zarembowitch, A.

doi:10.1107/97809553602060000630

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. D. ch. 1.3, pp. 94-95

Section 1.3.7.2. Equation of motion for elastic waves

A. Authier^a ^* and A. Zarembowitch^b

^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
Correspondence e-mail: aauthier@wanadoo.fr

1.3.7.2. Equation of motion for elastic waves

| top | pdf |

For generality, these equations will be derived in the X configuration (initial state). It is convenient to obtain the equations of motion with the aid of Lagrange's equations. In the absence of body forces, these equations are $[{{\rm d} \over {\rm d}t}{\partial L \over \partial x^{\prime}_i} + {\partial \over \partial X_i}{\partial L \over \partial \left( \partial x_i/\partial X_j\right)} = 0 \eqno(1.3.7.1) ]$ or $[{{\rm d} \over {\rm d}t}{\partial L \over \partial x^{\prime}_i} + {\partial \over \partial X_i}{\partial L \over \partial \alpha_{ij}} = 0, \eqno(1.3.7.2) ]$ where L is the Lagrangian per unit initial volume and $[\alpha_{ij} = \partial x_i/\partial X_j]$ are the elements of the Jacobian matrix.

For adiabatic motion $[L = \textstyle{1\over 2}\displaystyle \rho_0 x_i^{'2} - \rho_0 U, \eqno(1.3.7.3) ]$ where U is the internal energy per unit mass.

Combining (1.3.7.2) and (1.3.7.3), it follows that $[\rho_0 x{''}_i^{2} = {\partial \over \partial X_j} \left( \rho_0 {\partial U \over \partial S_{l m}} {\partial S_{l m}\over \partial \alpha_{ij}}\right), ]$ which can be written $[\rho_0 x{''}_i^{2} = {\partial \over \partial X_j} \left( \alpha_{il}\alpha_{jm}\rho_0 {\partial U \over \partial S_{l m}}\right) ]$ since $[{\partial S_{l m}\over \partial \alpha_{ij}} = \textstyle{1\over 2}\displaystyle \left( \alpha_{im}\delta_{jl} + \alpha_{il} \delta_{jm}\right). ]$

Using now the equation of continuity or conservation of mass: $[{\rho_0\over \rho}= J = {\rm det} (a_{ij}),]$ and the identity of Euler, Piola and Jacobi: $[{\partial \over \partial x_j} \left( {1\over J} {\partial x_j \over \partial X_i}\right) = 0, ]$ we get an expression of Newton's law of motion: $[\rho x_i'' = {{\rm d}T_{ij}\over {\rm d} X_j}\ {\rm or} \ \rho u_i'' = {{\rm d}T_{ij}\over {\rm d} X_j} \eqno(1.3.7.4) ]$ with $[T_{ij} = {\rho_0 \over J} \alpha_{ik}\alpha_{jl} {\partial U \over \partial S_{kl}} = \rho \alpha_{ik}\alpha_{jl} {\partial U \over \partial S_{kl}}. ]$ $[T_{ij}]$ becomes $[T_{ij} = {1\over J} \alpha_{ik} \alpha_{jl} t_{kl} ]$ since $[t_{kl} = \rho_0 {\partial U \over \partial S_{kl}}. ]$ $[t_{kl}]$ , the thermodynamic tensor conjugate to the variable $[S_{kl}/\rho_ 0]$ , is generally denoted as the `second Piola–Kirchoff stress tensor '.

Using Φ, the strain energy per unit volume, Newton's law (1.3.7.4) takes the form $[\rho x_i'' = {\partial \over \partial X_j} \left(\alpha_{jk} {\partial \Phi \over \partial S_{ik}}\right) \ \ {\rm or} \quad \rho u_i'' = {\partial \over \partial X_j} \left(\alpha_{jk} {\partial \Phi \over \partial S_{ik}}\right) ]$ and $[T_{ij} = \alpha_{jk}{\partial \Phi \over \partial S_{ik}}. \eqno(1.3.7.5) ]$

References

International Tables for Crystallography (2006). Vol. D. ch. 1.3, pp. 94-95