Engineering Rock Mechanics

Questions and answers: stress 31

43.2 When is a matrix symmetrical? Why is the stress matrix sym-

metrical?

A3.2 A term in the ith row and jth column of a matrix can be written as

xij. A matrix is symmetrical when all the complementary terms xjj and

xji are equal.

The elemental cube shown in answer A3.1 is in equilibrium, and

taking moments about the axes shows that the corresponding shear

stresses must be equal.

In the stress matrix

?ry = ryx

=yz ==zy

272 ==xz

Leading diagonal

So, for a cube of side AZ, taking moments about the z-axis gives

tx,AZ/2 - t,,AZ/2 = 0, or txy = tyx. Because the complementary shear

stresses are equal, the stress matrix has six independent components, e.g.

the normal stresses on the leading diagonal and the three shear stresses

above the leading diagonal: a,,, a,,, azz, txy, txz, ty2.

43.3 Explain the differences between scalar, vector and tensor

quantities. Why is stress a tensor quantity?

A3.3 A scalar is a quantity with magnitude only, e.g. temperature or
pressure. A vector is a quantity with magnitude and direction, e.g.
velocity or force. A tensor is a multi-component quantity, each of the
components having magnitude and direction, e.g. stress.
Stress is a tensor because the rules which govern the changes in the
stress components as the reference axes are changed are those of a tensor.
More mathematically, a tensor is a "multilinear differential form invari-
ant with respect to a group of permissible co-ordinate transformations in
n-space" (Borowski and Borwein, 1989 4).

43.4 How are normal and shear stress components plotted on

Mohr's circle?

A3.4 The components are plotted as shown in the diagram below.
Remember that compressive stresses are positive in rock mechanics, and
notice the signs of the shear stresses. (A detailed explanation of Mohr's
circle and its use may be found in ERM 1.)

4Borowski E. J. and Borwein J. M. (1989) Dictionay of Mathematics. Harper Collins,

London, 659pp.