Engineering Rock Mechanics

2 1 0 Rock dynamics and time-dependent aspects

customary engineering constants E and v for isotropic materials through
A = Ev/[(l + v)(l - Zv)] and ,H = E/2(1 + v).
We can write the one-dimensional wave equation in terms of displace-
ment for wave propagation in each of the x-, y- and z-directions as

where C,, Cy and C, are the wave velocities for waves propagating in the

x-direction and with particle motion in the x-, y- and z-directions,

respectively. Two types of stress wave are propagated: one has particle

motion in the x-direction (longitudinal or P-waves), the other has particle

motion in the y- or z-directions (transverse or S-waves), with velocities given

by Cp” = (A + 2p)/p and C,” = p/p, respectively. A more complete analysis

of this topic is presented in, for example, Stress Waves in Solids by H. Kolsky.

Expressing these velocities as vp and VS, and using the engineering elastic

constants rather than Lame‘s constants, we find that

Also, with these relations, we find the ratio vdvP = [(l - 2v)/2(1- v)]”’.

We are also interested in the velocities of these waves when they occur

in thin bars. In this case, the longitudinal and shear wave velocities in a

bar, respectively, are

with the velocity ratio being VSBARIVPBAR = [1/2(1 + v)]”’.

The modes of transmission of the longitudinal and transverse waves are

shown in Figs 13.2(a) and (b). Two other types of stress wave which are

important are Rayleigh and Love waves. Both of these waves occur near

interfaces and free surfaces and have elliptical particle motion which is

polarized and perpendicular to the free surface: with Rayleigh waves the