Engineering Rock Mechanics

The stress components on a small cube within the rock 35

1

<

/

UYY

Direct stress Acting on a

plane normal

uxx -- to the x-axis

\

Shear stress Acting on a

/' plane normal

to the x-axis

\

k7

Figure 3.4 The normal and shear stress components on an infinitesimal cube in the

rock aligned with the Cartesian axes.

shown in Fig. 3.4; however, the case of the shear stresses is not so direct,
because the resulting shear stress on any face will not generally be aligned
with these axes. To overcome this problem on any face, the shear force in
Fig. 3.3(b) is resolved into two perpendicular components which are
aligned with the two axes parallel to the edges of the face. Two components
of shear stress are then defined on each of the planes in Fig. 3.4, as the
diagrams shown in Fig. 3.5 demonstrate. Thus, we arrive at nine stress
components comprised of three normal components and six shear
components.
It should be noted that this discussion has been related only to the
development and definition of the nine stress components. So far, we have
not discussed how these components are affected by rotation of the cube
relative to the reference axes: we are only defining them. The next step,
therefore, is to list the components in a logical way. We have adopted the
convention that the first subscript refers to the plane on which the com-
ponent acts, and the plane is defined by the reference axis perpendicular
to it, e.g. zzy acts on a plane perpendicular to the z-axis. The second subscript
denotes the direction in which the stress component acts, e.g. zzy acts in the
y-direction.
Hence, it is convenient to collate the stress components in a matrix with
the rows representing the components on any plane, and the columns
representing the components acting in any given direction. This is
illustrated as:

There are many conventions in use for designation of the matrix

components.

As an example, the component zXy in the middle of the top row could be

designated as oxy, 072, S,, Pxy( or indeed any expression, say Rob. The most

important aspect of the notation is that the reader should recognize which

notation is being used and not be over-concerned about differences of

nomenclature.