in tension. The two samples of cheese discussed give good examples,
illustrated in Figure 17.8c. The ‘‘short’’ sample (1) fractures in shear. Before
fracture is seen at the outside, cracks have formed inside, in planes that are
about 45 8 from the direction of the compression. The ‘‘long’’ sample (2)
shows cracks at the bulging surface after the test piece has been considerably
compressed. Fracture occurs obviously in tension, and it starts at the
outside, where the elongational strain is largest.
It will be clear now that the relations governing fracture of viscoelastic
materials are far more complex than those for elastic solids. The discussion
above gives some qualitative relations that generally hold. Quantitative
prediction of the behavior from first principles is mostly not possible. It all
becomes even more complex for inhomogeneous materials.
Notch Sensitivity. Consider testing a material in the geometry as
depicted in Figure 17.7a. Test pieces of the same shape, but containing
horizontal notches of various lengthsL, are used. The notches have been
made before applying the stress. Assuming for the moment that there is no
stress concentration at the tip of the notch, the stress in the plane of the
notch would be given bys 0 w=ðwLÞ, wheres 0 gives the overall stress. This
implies that fracture will occur at an overall stress proportional to (1L/w),
as shown by curve 1 in Figure 17.9. Most soft solids do not show such a
relation, since stress concentration will occur, leading to notch sensitivity.
To explain this we consider the overall stress needed topropagatea
crackspr(not the fracture stresssfr) in a homogeneous isotropic elastic
material. Equation (17.9) gives the value ofLcr, further illustrated in Figure
17.7b. The equation can also be ‘‘inverted’’: assuming that a notch of
sufficient length has been applied, we then obtain for the crack propagation
stress
spr¼2 WfrAE
pL0 : 5
L>L^0 cr ð 17 :12aÞwhereL^0 cris the critical crack length for fracture propagation in the absence
of a notch. A relation like that of curve 2 in Figure 17.9 is obtained: the
curve starts atL^0 crand then steeply decreases. The equation will hold as long
asLis significantly smaller thanw. Since atL¼L^0 cr, we havespr¼s^0 pr(the
stress needed for fracture propagation in the absence of a notch), Eq.
(17.12a) can be rewritten as
spr¼s^0 pr
L^0 cr
L0 : 5
ð 17 :12bÞ