Engineering Rock Mechanics

422 Appendix A: Stress and strain analysis

The directions x’ and y’ corresponding to this value of p are known
as principal directions of strain. These directions are orthogonal. The
longitudinal strains e:, and e’yy in the principal directions are known as
principal strains.
In the example,

= 1.5

2e, - - 2 x 4500

( erx - err) (8000 - 2000)

:. p = 1/2 tanw1 1.5 = 28.2”.

Negative

shear

strain

Positive

shear

strain

Mohr’s circle of strain

If the global axes x and y are chosen to coincide with the principal

directions, the strain transformation equations become

compare to the stress transformation

equations.

e:, = e, cos’ e + ew sin’ e

e; = e, sin’ e + eyy cos’ e

e& = -(exx - ew ) cos e sin2 e

4

B

T

Now let r$ = 28. By analogy with the stress tensor,

e:, = Y2 (exx + ew) + Y2 (exx - ew) cos @

e& = - 1/z (exx - ew) sin Cp.

By analogy with Mohr‘s circle of stress, each point on the circle represents

a direction in the material in which the longitudinal strain is eiX: