58
Y. Qin et al.
role, because of a higher S R. This conclusion can also be supported by the lower
tenacity of rPET-B fibres produced at a higher processing temperature of 280 °C,
drawn at 3.0 bar, in Fig. 18, where more oriented molecular chains are relaxed during
the longer cooling time. While regarding vPET-1 fibres, its mechanical properties
mainly depend on its higher crystallinity X c , together with the secondary influence
of increased inner molecular chain orientation.
5 Performance-Reliability Plot
Weibull distribution, normally applied to analyze the durability, can provide a statistical method to present the performance and reliability of specimens. This has been
successfully applied to mechanical performance in the field of polymeric materials
[59, 60]. With regard to the stress at break, the Weibull distribution can be written as
P = 1 − exp[−
σ b /σ b,0
m ]
(12)
where P means the cumulative probability of failure, σ b refers to the stress at break,
σ b,0 refers to the characteristic stress at break where 63.2% of the specimens break,
and the Weibull modulus m can be obtained to evaluate the width of the distribution. When the σ b,0 keeps constant, an increased Weibull modulus m corresponds
to an increased slope of the cumulative probability curve, mirroring a remarkable
reproducibility, i.e., better reliability.
In order to assess the test group by the Weibull distribution, all the performance
values of specimens need to be ordered, such as σ 1 ≤ σ 2 ≤ · · · σ i · · · ≤ σ n . Afterwards, a probability P i is successively distributed to each value of stress at break,
where P i is obtained using Eq. (13), based on the Bergman method [61]. This would
offer the lowest coefficient of variation of m and any Weibull parameter x 0 for a
total number of n specimens. The Weibull modulus m and the characteristic Weibull
parameter x 0 can be revealed by Eq. (12).
P i = (i − 0.5)/n
(13)
As an important industrial parameter showing the distribution of the material
characteristics, the Weibull moduli m of different measured properties from rPET-B
and vPET-1 fibres were exemplified to explore their performance and reliability. The
highest values of Weibull moduli m represent the narrowest distribution of all the
measured results, i.e., the best reliability. As mentioned above, at least 30 specimens
of PET melt-spun fibres for each processing condition were conducted ‘single fibre
tensile test’, and then the characteristic Weibull parameter X 0 and Weibull moduli m
were obtained by Eqs. (12) and (13). In order to better assess the performance and
reliability of the two selected fibres, the average values of the characteristic Weibull
parameter X 0 and Weibull modulus m from three virgin PET fibres (fabricated at
270 °C, 2.0 bar and 280 °C, 2.0 bar) were defined as the reference points respectively
Y. Qin et al.
role, because of a higher S R. This conclusion can also be supported by the lower
tenacity of rPET-B fibres produced at a higher processing temperature of 280 °C,
drawn at 3.0 bar, in Fig. 18, where more oriented molecular chains are relaxed during
the longer cooling time. While regarding vPET-1 fibres, its mechanical properties
mainly depend on its higher crystallinity X c , together with the secondary influence
of increased inner molecular chain orientation.
5 Performance-Reliability Plot
Weibull distribution, normally applied to analyze the durability, can provide a statistical method to present the performance and reliability of specimens. This has been
successfully applied to mechanical performance in the field of polymeric materials
[59, 60]. With regard to the stress at break, the Weibull distribution can be written as
P = 1 − exp[−
σ b /σ b,0
m ]
(12)
where P means the cumulative probability of failure, σ b refers to the stress at break,
σ b,0 refers to the characteristic stress at break where 63.2% of the specimens break,
and the Weibull modulus m can be obtained to evaluate the width of the distribution. When the σ b,0 keeps constant, an increased Weibull modulus m corresponds
to an increased slope of the cumulative probability curve, mirroring a remarkable
reproducibility, i.e., better reliability.
In order to assess the test group by the Weibull distribution, all the performance
values of specimens need to be ordered, such as σ 1 ≤ σ 2 ≤ · · · σ i · · · ≤ σ n . Afterwards, a probability P i is successively distributed to each value of stress at break,
where P i is obtained using Eq. (13), based on the Bergman method [61]. This would
offer the lowest coefficient of variation of m and any Weibull parameter x 0 for a
total number of n specimens. The Weibull modulus m and the characteristic Weibull
parameter x 0 can be revealed by Eq. (12).
P i = (i − 0.5)/n
(13)
As an important industrial parameter showing the distribution of the material
characteristics, the Weibull moduli m of different measured properties from rPET-B
and vPET-1 fibres were exemplified to explore their performance and reliability. The
highest values of Weibull moduli m represent the narrowest distribution of all the
measured results, i.e., the best reliability. As mentioned above, at least 30 specimens
of PET melt-spun fibres for each processing condition were conducted ‘single fibre
tensile test’, and then the characteristic Weibull parameter X 0 and Weibull moduli m
were obtained by Eqs. (12) and (13). In order to better assess the performance and
reliability of the two selected fibres, the average values of the characteristic Weibull
parameter X 0 and Weibull modulus m from three virgin PET fibres (fabricated at
270 °C, 2.0 bar and 280 °C, 2.0 bar) were defined as the reference points respectively