Elasticity_ Barber

14 1 Introduction
representing the difference between the final and the initial position of P —
i.e. it is the distance that P moves during the deformation. In index notation,
the displacement is represented as u i .
The deformation of a body is completely defined if we know the displacement
of its every particle. Notice however that there is a class of displacements
which do not involve deformation — the so-called ‘rigid-body displacements’.
A typical case is where all the particles of the body have the same displacement.
The name arises, of course, because rigid-body displacement is the only
class of displacement that can be experienced by a rigid body.
1.2 Strains and their relation to displacements
Components of strain will be denoted by the symbol, e, with appropriate
suffices (e.g. e xx , e xy ). As in the case of stress, no special symbol is required
for shear strain, though we shall see below that the quantity defined in most
elementary texts (and usually denoted by γ) differs from that used in the
mathematical theory of Elasticity by a factor of 2. A major advantage of this
definition is that it makes the strain, e, a second order Cartesian Tensor (see
§1.1.4 above). We shall demonstrate this by establishing transformation rules
for strain similar to equations (1.15–1.17) in §1.2.4 below.
1.2.1 Tensile strain
Students usually first encounter the concept of strain in elementary Mechanics
of Materials as the ratio of extension to original length and are sometimes
confused by the apparently totally different definition used in more mathematical
treatments of solid mechanics. We shall discuss here the connection
between the two definitions — partly for completeness, and partly because the
physical insight that can be developed in the simple problems of Mechanics of
Materials is very useful if it can be carried over into more difficult problems.
Figure 1.6 shows a bar of original length L and density ρ hanging from the
ceiling. Suppose we are asked to find how much it increases in length under
the loading of its own weight.
It is easily shown that the tensile stress σ xx at the point P , distance x
from the ceiling, is
σ xx = ρg(L − x) , (1.33)
where g is the acceleration due to gravity, and hence from Hooke’s law,
e xx =
ρg(L − x)
E
. (1.34)