Page 33 - Handbook of Properties of Textile and Technical Fibres

14 Handbook of Properties of Textile and Technical Fibres

h

Figure 1.5 A chain consisting of n links.

Now, if we consider the chain as a whole, without considering its structure made of
links we can write that the probability of the chain’s failure can be written as P n , so that
n
1 P n ¼ð1 P 0 Þ .
By taking the natural logarithm and then the exponential of the expression the prob-
ability of the chain’s failure, under an applied stress of s, becomes

P n ¼ 1 exp n lnð1 P 0 Þ (1.8)

Weibull defined n lnð1 P 0 Þ as the risk of failure “R”.
A material has a volume however, so if Weibull statistics are to be applied to real
materials, like fibers, we have to define what is analogous to a link. For a specimen of
volume V, consider it divided up into small volumes V 0 , which can contain a defect and
is an intrinsic characteristic of the material. The assumption here is that there is only
one type of defect population in the material. In this way we can write V/V 0 z n.
V
In this way Ra lnð1 P 0 Þ.
V 0
The risk of failure of an n elementary small volume dV is
1
dR ¼ lnð1 P 0 ÞdV
V 0
R
So that dR ¼ f s; 1 dV0R ¼ f s; 1 dV
V 0 V 0
From Eq. (1.8) we now write

1
Z
P V ¼ 1 exp f s; dV
V 0
m

Weibull put f s; 1 ¼ ðs s u Þ
V 0 = s 0