Engineering Rock Mechanics

78 Strain

The transformation properties of the strain matrix allow us to determine
the complete in situ or laboratory strain tensor from measurements which
are made with strain gauges and which are normal strain measurements
only. In the same way that shear stresses cannot be measured directly,
neither can shear strains, and hence the complete strain matrix must be
established from normal strain measurements.

5.5 The elastic compliance matrix

One may be tempted to ask, given the mathematical equivalence of the

strain matrix developed in this chapter and the stress matrix developed in

Chapter 3, whether there is any means of linking the two matrices together.

Clearly, this would be of great benefit for engineering, because we would

then be able to predict either the strains (and associated displacements)

from a knowledge of the applied stresses or vice versa. As we will be

discussing in Chapter 6, it is often critical to be able to consider whether it

is stress, or strain, which is being applied and hence whether it is strain, or

stress, which is the result.

A simple way to begin would be to assume that each component of the

strain tensor is a linear combination of all the components of the stress

tensor, i.e. each stress component contributes to the magnitude of each

strain component. For example, in the case of the E,, component, we can

express this relation as

Because there are six independent components of the strain matrix, there

will be six equations of this type. If we considered that the strain in the x-

direction were only due to the stress in the x-direction, the previous

equation would reduce to

or

oxx = E/S = E&,,, where E = USll.

This form of the relation, where the longitudinal strain is linearly

proportional to the longitudinal stress, as is the case for a wire under

tension, was first stated by Robert Hooke (the first President of the Royal

Society) in 1676. He published the relation as an anagram in The Times of

London as CEIIINOSSSTTUU and three years later revealed this to mean

UT TENS10 SIC UIS, i.e. as the extension so the force. For this reason, the

more complete expression where E,, is related to all six components of stress

is known as the generalized Hooke’s law.

Hence, the complete set of relations between the strain and stress

components is: