Page 35 - Handbook of Properties of Textile and Technical Fibres

16 Handbook of Properties of Textile and Technical Fibres

m m m m
! ! ! !
So s 1 V 1 ¼ s 2 V 2 so that s 1 s 0 ¼ V 2
s 0 s 0 s 0 s 2 V 1
This gives

m
!
s 1 V 2
¼ (1.10)
s 2 V 1
Eq. (1.10) illustrates the dependence of strength on volume. Going back to the chain
analogy, it means that the bigger the volume, the longer the chain and the greater the
number of links. This increases the probability of there being an extra weak link in the
chain. In fibers it means that the longer the fiber, the greater the chance of there being a
major defect that weakens it.
Now, from Eq. (1.9), we obtain

m

ðs s u Þ
P V 1 ¼ P S ¼ exp V
s 0
Taking the natural logarithm

m

ðs s u Þ
ln P S ¼ V
s 0
Taking the natural logarithm again

1
ln ln ¼ ln V þ m ln s m ln s 0 (1.11)
P S
As m and s 0 are intrinsic material parameters, m ln s 0 is constant.
For a population of fibers of variable diameters D but all of the same length, Eq.
(1.11) becomes

1
ln ln ¼ m ln s þ 2ln D þ constant (1.12)
P S

If D can be considered constant, then Eq. (1.12) becomes
1
ln ln ¼ m ln s þ constant (1.13)
P S
Plotting ln ln 1 as a function of ln s allows the Weibull modulus, m,tobe
P S
determined.