Physical Chemistry of Foods

must be taken into account. Fracture means an increase in surface area and
hence an increase in surface free energy, and this energy must somehow be
provided. When an elastic material is put under stress, the energy applied to
do so (Wel) is stored in the test piece and can be available for realizing
fracture. In the situation as given in Figure 17.7a (assuming now that the
crack depicted was formed by putting the test piece under tension), the
energy that had been stored in the shaded region was available to cause the
fracture by being transported to the tip of the crack. Its amount is roughly
proportional toL^2. The total fracture energy or work of fractureWfrwould
be proportional toL. The crack will now be propagated, i.e., fracture will
spontaneously proceed, if the differential energy released surpasses the
differential energy required. The condition thus is

1

2

pb½ðLþdLÞ^2 
L^2 ŠWVel>bWfrAdL ð 17 : 8 Þ

The symbolWis used for the total amount of work done or energy released,
and the specific values are denoted by the superscripts A or V for amount
per unit area and volume, respectively.
The energies involved are illustrated in Figure 17.7b, and it is seen that
a maximum in free energy occurs at a crack lengthLcr, beyond which the
crack will grow spontaneously if the test piece remains under stress. From
Formula (17.8) it follows that thecritical crack lengthis given by

Lcr¼

WfrA

pWelV

¼

2 WfrAE

ps^2 fr

ð 17 : 9 Þ

where the part after the second equals sign only holds for an ideally elastic
solid, whereWelV¼^12 seandE¼s=e. Since the modulus (E) and the specific
work of fracture are material constants, this means that in this casesfris
proportional toL
0.5. It should be noted thatsfris defined as theoverall
stress—i.e., the force divided byw 6 b—acting on the specimen at fracture.
In linear-elastic fracture, crack propagation (after Lcr has been
reached) proceeds roughly at the speed of sound in the material (of order
1km?s
^1 ). This gives rise to a snapping sound, typical for fracture of brittle
materials.

Elastic-Plastic Fracture. Many elastic solids can show yielding at
high stress. During fracturing, the local stress near the crack tip can be very
high, as mentioned. This then may lead to local yielding, which has several
consequences. (a)Local plastic deformationoccurs, which means that the
pieces remaining after fracture do not precisely fit to each other, as they
would do after linear-elastic fracture. (b)The tip of the crack becomes