Engineering Rock Mechanics

80 Intact rock: deformabilify, strength and failure

axial stress

yield point

(85,39.1)

confining pressure

(i) We are given information regarding the behaviour of the specimen
in a three-dimensional stress state: a relation between the axial stress,
the axial strain and the confining pressure is indicated. To use this
information, we start with Hooke's Law in three dimensions for an
isotropic material,
1
E1 = - [ai -
E (02 + a3)]
and substitute E] = E,, a1 = o, and 02 = a3 = p to obtain
1 1
E - - [a, - v (p + p)] = E [a. - 2vp].
a- E
We are told that in this test the axial strain is controlled to be zero. As
a result, the equation above reduces to 0 = (l/E) [a, - Zvp] or o, = 2vp.
We now have an equation linking a, and p which, on comparison with
the standard form of a straight line, y = mx + c, is found to pass through
the origin and has a gradient of 2u (when a, is plotted on the vertical
axis and p is plotted on the horizontal axis). The gradient of this part of
the a, - p plot can either be measured, or computed. Using co-ordinate
geometry to compute the gradient, we obtain

and hence v = 0.23.

Poisson's ratio.

Thus, from the first part of the curve we find that we can compute

(ii) If we assume that yield represents peak strength being reached,

then after yield the a, - p curve represents a plot of the principal stress

form of the Mohr-Coulomb criterion, with 0, a3 and p = 01. Thus we

have

2c cos # 1 +sin+ 1 +sin#

= ac + 0,

1 -sin#'

+ oa

'= l-sin# l-sin+

It is worth redrawing the curve with a, on the horizontal axis. Now, if

we extrapolate the failure criterion back to a, = 0, then the point where

it intersects the p axis is the compressive strength, a,. Using straight line

geometry, the compressive strength is found from

Y2 - YI^85 - a,

m=- + tm61" = - * 0, = 85 - 39.1 tm61,

x2 - x1 39.1 -^0

giving a, = 14.5 ma.