Engineering Rock Mechanics

62 Strain and the theory of elasticity

material - why? Hence explain why five elastic constants are re-

quired for a transversely isotropic material rather than six.

A5.4 We explain this using the diagram below for the case of pure
shear, in which a square element of rock is subjected to compression
in the xdirection and tension in the y-direction of equal amounts.
However, the analysis is general in that it is applicable to all coupled
stress and strain states.
J/

Pure shear on .,o+

E

”+z

The two Mohr’s circle diagrams show the stress and strain states for

the square element. Because the circles are both centred at the origin, we

can see that t = u1 and y/2 = €1. Now, the shear modulus, G, is defined

as G = r/y which, on substituting for the two terms, gives G = ~((T~/E~).

For a 2-D stress state such as the one illustrated, we can write Hooke’s

Law as E, = (l/E)(ux - vu,). In this case, we have E, = EI, ox = ul, and

a, = u2 = -81. Substituting these values gives c1 = ( l/E)(al + uq), from

which we find that ~/EI = E/(l+ u).

Substituting these into the earlier expression for the shear modulus

gives

1 UI E

2El 2(1 +u)

G=--=-

as required.

This approach is only valid if the Young’s moduli and Poisson’s ratios

in the u1 and 02 directions are the same, i.e. for an isotropic material

having the same properties in all directions. If this were not the case, the

strain EI could not be expressed as simply as

81 = - (01 + vu]).

In the case of a transversely isotropic rock, say an unjointed rock

with a set of bedding planes perpendicular to the z-axis, as shown in

the adjacent sketch, the relation G = E/2(1 + u) will apply in the x-y

plane but not in the x-z and y-z planes, because the rock has different

properties in different directions in these planes. Thus, the six elastic

constants required to characterize a transversely isotropic rock, E, = E,,

E, , u,,, uxz = uyz , G,, , and G ,, = G,, , are reduced to five, E, , E,, u,, , v,, ,

and G,, because in the x-y plane of isotropy G,, = E,/2(1 + uxy) (see

diagram below).

1

E