Engineering Rock Mechanics

Questions and answers: stress 35

Note that the components of the first stress invariant,

11 = a,, + a,, + a,, = 01 + a2 + a3, are on the leading diagonal of the

stress matrix (the terms from top left to bottom right). Since I] is the sum

of the normal stresses, it is three times the mean normal stress. This first

invariance indicates that there is a constant mean normal stress as the

reference axes are changed.

43.9 By considering the rates of change of the stress components
in the answer to 43.1, establish force equilibrium in the x, y and
z directions and hence write down in differential form the three
equilibrium equations for an elemental cube.

A3.9 Consider the infinitesimal elemental cube shown in A3.1 and take
the rate of change6 of a,, in the x direction as aa,,/ax. Because of this
change, the stress on one side of the cube is larger than on the other, as
shown below.

The net stress in the x direction due to this change across the elemental
cube for a distance Sx is the stress applied to one side of the cube minus
the stress applied to the other side:

{ (axx + -6x - a,, or -6x.

ax ) I

Assuming that the cube has side lengths of 6x, hy and 6z, and mul-
tiplying this stress increment by the area of the face on which ax, acts, i.e.
6ySz, gives a force increment of (aaxx/ax)6x6y6z. Applying this principle
to the three stress components acting in the x-direction (see the left-hand
column of the stress matrix in A3.1), gives the three force increments:

Because the infinitesimal cube is in equilibrium, these forces must also
be in equilibrium (in the absence of any other forces) and hence have a
sum of zero. Equating the sum of the force increments to zero, cancelling
GxSySz, and assuming that no body forces such as gravity are acting,
provides one of the differential equations of equilibrium:

The mathematical operator, a/ax, represents differentiation with respect to x, dl other
variables being treated as constants.