Page 139 - Plastics Engineering

122 Mechanical Behaviour of Plastics
be expressed as

(2.80)

where y is the surface energy per unit area.
Note that for a situation where the applied force does no work (Le. there is
no overall change in length of the material) then W = 0 and equation (2.80)
becomes au
-'Yz aA (2.81)
aa
Now, for a through crack propagating in a sheet of material of thickness, B,
we may write
aA = 2Baa

So equation (2.80) becomes
a
-(W - U) > 2yB (2.83)
aa
In the context of fracture mechanics the term 2y is replaced by the G, so
that the condition for fracture is written as
ia
--(W - U) > G, (2.84)
Baa
G, is a material property which is referred to as the toughness, critical
strain energy release rate or crack extension force. It is effectively the energy
required to increase the crack length by unit length in a piece of material of
unit width. It has units of J/m2.
Equation (2.84) may be converted into a more practical form as follows.
Consider a piece of material of thickness, B, subjected to a force, F, as shown
in Fig. 2.63(a). The load-deflection graph is shown as line (i) in Fig. 2.63(b).
From this the elastic stored energy, U 1, may be expressed as

U1 = ZFS (2.85)
1
If the crack extends by a small amount aa then the stiffness of the material
changes and there will be a small change in both load, aF, and deflection,
86. This is shown as line (ii) in Fig. 2.63(b). The elastic stored energy would
then be
U2 = i(F + aF)(6 + 36) (2.86)
From equations (2.85) and (2.86) the change in stored energy as a result of
the change in crack length 8~ would be given by
au = u2 - u1 = $(Fad + 6aF + aFas) (2.87)