which can be rewritten as
dlns
deu> 1 ð 17 : 13 ÞThe condition for stability is thus that the stress in the material increases with
the strain, a phenomenon calledstrain hardening, and that the relative
increase is more than proportional to the increase in the uniaxial Hencky
strain. Of course, when elongation goes on, the thread will finally break, but at
a far larger strain value than would occur in the absence of strain hardening.
A comparable situation occurs forbiaxial extensionof a thin film. This
is, for instance, important in a bread dough. During fermentation and in the
beginning of baking, many large gas cells are formed, which soon deform each
other. The thin film between them is being extended and may readily break,
leading to coalescence. Again, the formation of thin spots can be prevented by
sufficient strain hardening. The condition for stability now becomes
dlns
deb> 2 ð 17 : 14 ÞThe magnitude of the derivative is called thestrain hardening index. A good
correlation is observed between its value (between 0.5 and 2.5 in most doughs)
and the strain at rupture. There is also a good correlation between the index
and the gas holding capacity of the bread, since early coalescence of the cells
leads to loss of gas to the atmosphere.
Strain hardening is generally ascribed to materials containing strongly
entangled polymers, which become stretched upon extension because they
resist disentanglement. This can especially happen with polymers of which
the backbone contains large (groups of) side chains. This is the case, for
instance, for wheat gluten and xanthan gum, materials that exhibit strong
strain hardening. There are some additional factors of importance. First, the
strain hardening also depends on strain rate, and bread dough tends to be
strain rate thinning; consequently, the strain hardening index will be smaller,
to an extent that depends on the rate of increase of the strain rate during
deformation. Second, the apparent viscosity of the material must be between
certain limits at the conditions applied: at lowZavalues, the extension will
proceed too fast, while at high values the stress needed for extension is too
high. Third, for a very thin film, say< 10 mm, the Gibbs mechanism of film
stability (see Figure 10.29c) can become significant, provided that the
material is homogeneous at the scale of the film thickness (which is not the
case in bread dough).