32 Stress
-nine components of which six are independent;
-values which are point properties;
-values which depend on orientation relative to a set of reference axes;
-six of the nine components becoming zero at a particular orientation;
-three principal components; and finally
-complex data reduction requirements because two or more tensors
cannot, in general, be averaged by averaging the respective principal
stresses.
All this makes stress difficult to comprehend without a very clear grasp of
the fundamentals.
3.2 The difference between a scalar, a vector
and a tensor
As alluded to above, there is a fundamental difference between a tensor
and the more familiar quantities of scalars and vectors. We will explain this
first conceptually before the mathematical treatment.
A scalar is a quantity with magnitude only. Examples of scalars are
temperature, time, mass and pure colour-they are described completely
by one value, e.g. degrees, seconds, kilograms and frequency.
A vector is a quantity with magnitude and direction. Examples of vectors
are force, velocity, acceleration and the frequency of fractures encountered
along a line in a rock mass-they are described completely by three values,
for example, x, y, z components which together specify both direction and
magnitude.
A tensor is a quantity with magnitude, direction and ’the plane under
consideration’. Examples of tensors are stress, strain, permeability and
moment of inertia-they are described completely by six values, as
explained in Section 3.7.
It cannot be over-emphasized that a tensor quantity is not the same as
a scalar or vector quantity. This applies both in a conceptual sense and in
the mathematical sense. The reason why we emphasize this so much is that
both mathematical and engineering mistakes are easily made if this crucial
difference is not recognized and understood.3.3 Normal stress components and shear
stress components
On a real or imaginary plane through a material, there can be normal forces
and shear forces. These are illustrated directly in Fig. 3.l(a). The reader
should be absolutely clear about the existence of the shear force because it
is this force, in combination with the normal force, that creates the stress
tensor. Furthermore, it should be remembered that a solid can sustain such
a shear force, whereas a liquid or gas cannot. A liquid or gas contains a
pressure, i.e. a force per unit area, which acts equally in all directions and
hence is a scalar quantity.
The normal and shear stress components are the normal and shear forces
per unit area as shown in Fig. 3.l(b). We have used the notation F, and F,
for the forces, and cr and z for the corresponding stresses. However, many