Thespecific work of fractureof the materialWfr(in J?m^3 ) can be
called ‘‘toughness.’’ It is given byWfr¼Zefr0sde ð 17 : 5 ÞA curve like that in Figure 17.2 can be obtained in various ways,
generally not leading to the same result. To name the most important points:
(a) either the stress is increased (in a controlled manner) or the strain; (b) the
rate of doing so, generally to be expressed as a strain rateC, can be varied;
and (c) the deformation mode can be like any depicted in Figure 17.1 or yet
otherwise (bending, penetration, etc.). One should always choose these
variables in accordance with the situation of interest. For instance, if the
quantity to be determined is the resistance of a bread dough to the growth of
aCO 2 bubble in it, it makes little sense to study deformation in shear, since
it is biaxial extension that occurs around the bubble; moreover, it makes
little sense to do experiments at a strain rate of, say, 10 min^1 , since increase
of bubble radius by a factor of two takes a far longer time than 0.1 min
during a typical baking process.
Many rheological tests yield results that depend on the size and shape
of the test piece. To obtain truematerial properties, some conditions must be
fulfilled. First, the deformation should be homogeneous, i.e., the same
everywhere in the test piece; this generally implies that the material has to be
homogeneous above the scale of the smallest structural elements. Second,
true stress and strain should be calculated. Consider the compression of a
test piece, as depicted in Figure 17.1d. The cross-sectional areaA will
increase with increasing compression, which means that the stress (s¼F/A)
will not be proportional to the force F, and a correction is needed.
Moreover, the strain is often given as the linear or CauchystraineC¼
(LL 0 )/L 0 , where Lis length andL 0 original length. As discussed in
Section 5.1.1, this is not a good representation of the strain, and the natural
orHenckystrain should be used. It is defined as
eH¼ZL
L 01
L
dL¼lnL
L 0
ð 17 : 6 Þ*The strains often are given as the absolute values, and then for compression
eH>eC, and for extension the other way around.
An example is given in Figure 17.3, and it is seen that the effect is very
large for large deformation. Even the shape of the curve can be strongly