Strain and the theory of
5 elasticity
5.1 Stress and strain are both tensor quantities
The concept of strain is required to understand and quantify how a rock
mass has been deformed. There are many applications of strain analysis.
For example, some in situ stress determination techniques measure strain
and hence compute stress, so it is necessary to be able to manipulate
the strain values, and understanding the anticipated deformation of
structures built on and in rock masses is critical for rock engineering
design and back analysis.
Strain is a measure of the deformation of a body and is the same
type of mathematical quantity as stress: a second-order tensor. There are
normal strains and there are shear strains, as shown below*, which are
directly analogous to normal stresses and shear stresses.
The nature of stress as a tensor is explained in Chapter 3, where we
see that the stress state can be specified either as the six independent
components of the stress matrix or as the magnitudes and orientations
of the three principal stresses. The strain state is expressed using the
symbol E for normal strains and either the symbol E or y for shear
strains. Note that the engineering shear strains, e.g. yyx, as illustrated
in Fig. 5.1 are divided by two to give the components of the strain
matrix, e.g. syx. Intuitively, this is because engineering shear strain must
be attributed equally to both ryx and its equal-value complement rxy;
unstrained state +
Lstrained state
Figure 5.1 Normal strains and shear strains.'In Fig. 5.1, the normal strain shown is positive, but the shear strain is negative.
ERM 1 contains a detailed discussion of the sign of the shear strain.