74 Strain
especially in terms of the perhaps unexpected relation between the shear
modulus and Young’s modulus and Poisson’s ratio.
We have noted earlier that there may have been successive phases of
deformation of the rock mass during geological history. Thus, in decoding
such compound deformation into its constituent parts, as illustrated by
specific types of simple deformation in Fig. 5.4, we need to know whether
strain phases are commutative, i.e. if there are two deformational phases A
and B, and is the final result of A followed by B the same as B followed by
A? Similarly, in engineering, does the sequence of excavation have any
influence on the final strain state? Perhaps counterintuitively, the answer
is that the final strain state is dependent on the straining sequence in those
circumstances where shear strains are involved. An elegant example from
Ramsey and Huber (1983) is shown in Fig. 5.5, wherein the non-
commutative nature of shear strains is illustrated, both graphically and
mathematically. With reference to earlier emphasis on the significance of
the off-diagonal terms of the strain matrices, the reader should note that
it is these off-diagonal terms which give rise to the non-commutative
phenomemon. In Chapter 14, the concept of interactions in the off-diagonal
terms is introduced for a matrix with general state variables along the
leading diagonal, in the context of rock engineering systems.
It can be helpful to think about these strain operations in general and to
be able to identify the components of a general strain transformation matrix
for all circumstances. Such affine transformations are used in computer
graphics and we mention here the case of distorting any two-dimensional
shape. In order to introduce translation, i.e. movement of the entire
shape (without rotation) within the plane of the figure, ’homogeneous
co-ordinates’ are used. These are three co-ordinates, simply the two
Cartesian coordinates plus a third which allows translation to be intro-
duced. The transformation of co-ordinates is shown in Fig. 5.6.
[:I :[; :I[ :I
Simple shear........ I-‘------
I /ff ;@I
Pure sheart ................Simple shear followed by pure shear Pure shear followed by simple shear
Figure 5.5 Shear strain is not commutative (example from Ramsey and Huber, 1983).