Engineering Rock Mechanics

136 Discontinuities

between stress and displacement. However, for analysis it is convenient to
assume either one global linear stiffness value or a composite piecewise
linear approximation. Goodman has proposed a hyperbolic relation to
characterize the normal stress-displacement curve, viz.:

v=---C o

c +do,

where v and on are closure and normal stress, respectively, and c and d are

constants. This equation provides a good model for the discontinuity

closure curve illustrated in the upper diagram in Fig. 7.21. It is possible to

extend this basic concept to consider loading and unloading and many

other aspects of the joint behaviour. The comprehensive reference to the

mechanical behaviour of a single joint is the Proceedings of the Rock Joints

Symposium held in Norway (Barton and Stephansson, 1990).

To characterize the shear stress-shear displacement curve in the lower

diagram of Fig. 7.21, we can use the expression

6

a+bS

z=-

where z and S are the shear stress and shear displacement, respectively,

and a and b are constants. Again, there are many extensions to this basic

formula and the reader is referred to the Rock Joints text referenced

above.

It is interesting to note that the expressions for normal displacement and

shear displacement are mathematically similar, but have the stress and

displacement terms reversed. Whereas the normal displacement must

asymptote to a final closure value as the normal stress is increased, the shear

displacement can continue indefinitely, usually with a reduced shear stress.

Thus, the formula above refers to the deformation behaviour up to the peak

shear strength.

Despite the non-linearities of the two curves, as a first approximation we

can consider the linear stiffness representations as k,, for the normal case

and k,, for the shear case. We can also consider the possibility that a

normal stress will cause a shear displacement, using a constant k,,, and that

a shear stress will cause a normal displacement, using a constant ks,. These

stiffnesses have the dimensions of, for example, MPa/m, because they relate

stress to displacement. With these linear approximations for the

stiffnesses

or, in matrix notation,