Page 83 - Numerical Analysis and Modelling in Geomechanics

64 D.S.JENG

(3.5c)

In (3.5), and are the effective normal stresses in the x- and z-directions,
respectively, and τ xz is the shear stress in the x-z plane. It is noted that
compressive stresses are taken as positive here. Therefore, the effective stress is
defined by

(3.6)

in which σ and σ are the total stresses in the x- and z-directions, respectively.
z
x
In (3.5), the C coefficients are given by
ij
(3.7a)

(3.7b)

(3.7c)

(3.7d)

Based on the concept of effective stresses, the equations of overall equilibrium
for a poro-elastic medium with the absence of body forces can be expressed as

(3.8)

It is noted that the governing equations (1) and (8) do not include inertial effects.
In general, such effects can be ignored for most cases with small amplitude
waves (Jeng et al., 1999). It is also noted that the body force (the soil self-
weight) is ignored in this study.
Substituting (3.5) into (3.8), the equations for the force equilibrium in a porous
media can be re-written as

(3.9a)