blunted[i.e., a larger value ofRin Eq. (17.7)], which means that the stress
concentration becomes smaller and a higher overall stress is needed to
achieve fracture. (c)The work of fracture increases, since it now includes the
energy dissipated owing to the local yielding. It may be clear that ‘‘soft
solids’’ by their nature cannot exhibit linear-elastic fracture, but some do
show plastic-elastic fracture.
Grinding.An important aspect is often whether such a material can be
ground, and how small then are the particles obtained. It can be derived
from the theory that the thickness of the zone near a crack in which plastic
deformation (yielding) occurs in a homogeneous isotropic material is given
by
z¼32
3
WfrAE
s^2 yð 17 : 10 Þwheresyis the yield stress. If the diameter of a particle of the material is
smaller thanz, it cannot be broken up and hence it cannot be ground any
finer. Expressed in another way, not enough elastic energy can be stored in
the particle to provide the total work needed for fracture propagation.
Values forzmostly range between 1 and 100mm. For materials that exhibit
plastic-elastic fracture, decreasing the temperature leads to a smallerzvalue,
since the yield stress tends to markedly decrease while the modulus is not
strongly affected (the work of fracture will also decrease). If the temperature
decrease causes a glass transition, the regime even changes to linear-elastic
fracture, greatly facilitating the grinding process.
Thespecific work of fractureis in the simplest case given by 2g, whereg
is the surface tension of the material; the factor 2 is because fracture leads to
the formation of two new surfaces. For foods,gwill rarely be larger than the
value for water (70 mN?m^1 ), which would imply thatWfrA&0.1 J?m^2.
However, values observed are generally between 1 and 10 J?m^2 , one to two
orders of magnitude higher. Possible reasons are (a) that the fracture surface
is often uneven, causing the surface area of the crack to be larger than it
appears on a macroscopic scale; (b) that a crack is often accompanied by
small side cracks or a few small cracks about parallel to the main one, which
also increases the effective surface area; and (c) that local yielding causes
energy to be dissipated during fracture, and the increase of the work of
fracture will be about proportional to the value of z in Eq. (17.10).
Prediction of the magnitude of the excess is very difficult, because the
rheological behavior of the yielding material is highly nonlinear.
Values ofWfrAthus have to be determined experimentally. This can be
done by means of various cutting tests, but it is often difficult to obtain
reliable results.