Page 138 - Plastics Engineering

Mechanical Behaviour of Plastics 121

2.16 The Concept of Stress Concentration
Any material which contains a geometrical discontinuity will experience an
increase in stress in the vicinity of the discontinuity. This stress concentration
effect is caused by the re-distribution of the lines of force transmission through
the material when they encounter the discontinuity. Causes of stress concen-
tration include holes, notches, keyways, comers, etc as illustrated in Fig. 2.62.

Fig. 2.62 Stress concentration

The classical equation for calculating the magnitude of the stress concentra-
tion at a defect of the type shown in Fig. 2.62(b) is
0, = a (1 + 243) (2.79)

where a, is the local stress, a is the nominal stress on the material, 2a is the
defect size and r is the radius of the defect at the area in question.
The parameter (1 + 2477) is commonly termed the stress concentration
factor (K,) and for a hole where a = r then K, = 3, i.e. the stresses around
the periphery of the hole are three times as great as the nominal stress in the
material.
It should be noted, however, that for a crack-like defect in which r + 0
then K, + 00. Obviously this does not occur in practice. It would mean that
a material containing a crack could not withstand any stress applied to it.
Therefore it is apparent that the stress concentration approach is not suitable
for allowing for the effects of cracks. This has given rise to the use of Fracture
Mechanics to deal with this type of situation.

2.17 Energy Approach to Fracture
When a force is applied to a material there is work done in the sense that a
force moves through a distance (the deformation of the material). This work is
converted to elastic (recoverable) energy absorbed in the material and surface
energy absorbed in the creation of new surfaces at cracks in the material. The
original work on Fracture Mechanics was done by Griffith and he proposed
that unstable crack growth (fracture) would occur if the incremental change in
the net energy (work done - elastic energy) exceeded the energy which could
be absorbed in the creation of the new surface. In mathematical terms this may