58 Strain and the theory of elasticity
stressStrainComponents of the stress and strain
matnces - relatlve to
x, y and z reference axesComponents of the stress and strain
stress and strain directions.matrices -relative to the three principalcxc ?ty 7,: 00[vmm 0, [s:mc2 :,]
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where = (r&!mathematically, the division by two is required to give similarity to both
the stress and the strain transformation equations.
In Table 5.1, the correspondence between the stress and strain tensors
is evident; the normal and shear strains correspond directly with the
normal and shear stresses, and the principal strains correspond directly
with the principal stresses. In fact, stress and strain are mathematically
identical; only the physical interpretations are different. As a result, there
are strain analogies to all the stress aspects described in Chapter 3.
The next step is to consider how stress and strain might be related.
Historically this has been considered using two approaches: one for
finite strain and one for infinitesimal2 strain. Both are useful in rock
mechanics, but here we will highlight infinitesimal strain and the theory
of elasticity because of their widespread use.
The theory of elasticity relates the stress and strain states for infin-
itesimal strains. Hooke’s Law in its original form states that when, say,
a wire is stretched, the strain is proportional to the stress or, as Hooke
himself stated3, ”ut tensio sic vis”. Given the tensors in Table 5.1, we
can extend this concept to the generalized Hooke’s Law relating all
the components of the strain matrix to all the components of the stress
matrix. The assumption is that each component of the strain matrix is
linearly proportional to each component of the stress matrix by a factor
Sij. This means that the value of a specific strain component can be
found from the six contributions that the stress components make to
it. For example, the contribution to the strain component E, (or ql)
made by the stress component a,, (or 4 is S13az,. The relation between
the stress and the strain components may not be a simple linear re-
lation, but this is the first-order approximation made in the theory of
elasticity.* The word ’infinitesimal’ means infinitely small, or approaching zero as a limit.
This expression was first published by R. Hooke in The Ties of London newspaper
in anagram form in 1676.