Engineering Rock Mechanics

Interaction matrices 225

these concepts or subjects. The matrix presentation is not merely a peda-

gogic device: it serves to identify and highlight the interactions between

subjects, and forms the structure for coupled modelling.

14.2 Interaction matrices
The idea that there may be a relation between all things has been expressed
by Francis Thompson, the English Victorian poet, who wrote the lines

All things by immortal power,
Near or far,
Hiddenly
To each other link6d are,
That thou canst not stir a flower
Without troubling of a star.
In fact, the concept of considering the relations between quantities which
have the property of perpendicularity is extremely old. The margin sketch
is from Ch'ou-pei Suan-king (an ancient Chinese treatise dating from circa

BIZ

m

R

t

1100 BC, which is housed in the British Museum), and is an early illus- W

tration of the proof of what is now known as Pythagoras' theorem. In the

history of mathematics there has been considerable development of the %

mathematics associated with orthogonality (n-dimensional perpendicular- d

ity) via matrix and tensor analysis, so that the foundation for many subjects

has already been laid. For example, when considering the variability of

many different parameters, the subject of multivariate analysis is used,

where n individual parameters are considered along n orthogonal axes in

n-dimensional space, and there are other examples of subjects built on this

mathematical foundation, e.g. Fourier analysis.

The basic concept here is to study the combination, interaction or

influence of one subject on another. We begin with 2 x 2 matrices, but it

should be remembered that all the ideas can be extended to an n x n matrix.

In Fig. 14.2, the main subjects, here denoted by A and B, are placed in the

leading diagonal positions, i.e. from the top left to the bottom right of the

matrix. A matrix is a list, and we are considering subjects, rather than the

A.

3

D b

b

a

more usual numerical quantities. We are also considering the interactions-

shown in the off-diagonal boxes-that are studied by clockwise rotation,

(a + b? = c2 + 4 (tab)

a2 + Zab + b2 = c2 + 2ab

a2 + b2 = c2

as indicated by the arrows in the figure.

In the construction of such matrices, the primary parameters are always

listed along the leading diagonal, as in Fig. 14.2. The off-diagonal terms could

represent the combination, influence or interaction of the primary para-

meters, as shown in Fig. 14.3. Combination can be demonstrated simply by

inserting numbers in the leading diagonal with the off-diagonal terms being,

for example, their sums. Similarly, influence is demonstrated by considering

discontinuity aperture and water flow and, finally, interaction by consider-

ing how, for a given stress state, normal stresses give rise to shear stresses.

In the first matrix of Fig. 14.3, the off-diagonal terms in the matrix repre-

sent the addition of the leading diagonal numbers. Because 3 + 2 = 2 + 3

= 5, the two off-diagonal terms are the same and the matrix is therefore

symmetrical about the leading diagonal.