Page 34 - Handbook of Properties of Textile and Technical Fibres

Introduction to the science of fibers 15

In which s is the applied stress, s u is a stress threshold below which there is no pos-
sibility of failure, and s 0 and m are material parameters. The scatter of the strengths is
quantified by m, which is known as the Weibull modulus.
We can now write

Z m
P V ¼ 1 exp ðs s u Þ dV
= s 0
V
For an evenly distributed stress throughout the body

m
P V ¼ 1 exp ðs s u Þ V (1.9)
= s 0
The Weibull modulus, m, allows the scatter in fiber strengths to be quantified. For
example, the average strength of two materials could be the same but the two materials
could have very different scatter in their strengths and that could be important in
assessing the risk of failure of a structure (Fig. 1.6).
Now let us consider two populations of the same material, for example, a type of
fiber, but with different volumes because they are of different lengths. If the volumes
are V1 and V2 we could test a number of the specimens and determine at which stresses
half of each group was broken. That is to say, when the probabilities of each group are
both equal to a half. These are known as the median strengths of each population, s 1
and s 2 .
If we consider that s u ¼ 0 we can now write, from Eq. (1.9)

m
!
m
s 1 s 2
1=2 ¼ 1 exp V 1 ¼ 1 exp V 2
s 0 s 0
Figure 1.6 Two materials could have
the same mean strength but different
scatter in strength. The greater the
Weibull modulus, m, the smaller the
scatter.
Probability of failure
M = 8

M = 20

Applied stress