Statistical treatment of results of measurement of strength of complex fibers and individual filaments of fibers in the practice of technological investigations almost always relies on three parameters, namely, average value, dispersion (or rootmean-square deviation), and coefficient of variation. These parameters do not depend on a specific distribution that is adequate for experimental data, but often confidence interval is given, as a complement to it, for average value, which is determined by using Student’s distribution, and its use is proper only for quantities distributed in keeping with normal distribution.

The proof that normal distribution is adequate for experimental data, however, is almost never given. In review paper [1] specific data are cited, although commonly experimental data are not described by normal distribution and, consequently, numerical values of confidence intervals calculated by using Student’s distribution will be erroneous in this case. Such a situation in fiber strength measurement data processing cannot be considered acceptable.

The technoeconomic costs for obtaining samples of high-strength fibers for tests and subsequent rectification of the technological process are high at present. Additional information may provide triparametric Weibull”Gnedenko distributions or gamma-distribution (provided it is proved that they are adequate) because they contain shear parameter, the physical meaning of which is minimum value of mechanical stress, at which the probability of fiber breaking is different from zero. It is an important parameter, but it shows only the lower theoretical strength limit.

In [2], a distribution was proposed for determination also of the upper theoretic strength limit, which is the mechanical stress upon attainment of which the probability of breaking is equal to unity. The distribution function in this case has the form

$$ {\displaystyle \begin{array}{c}F(x)=\exp \left[-k\frac{{\left({x}_{\mathrm{max}}-x\right)}^{n1}}{{\left(x-{x}_{\mathrm{min}}\right)}^{n2}}\right]\\ {}F(x)=0\dots \kern0.5em if\dots x\le {x}_{\mathrm{min}}\\ {}F(x)=1\dots \kern0.5em if\dots x\ge {x}_{\mathrm{min}}\end{array}} $$
(1)

where xmin and xmax – theoretically possible minimum and maximum strength values; x – measurement result (a series of parallel measurement results forms what is called sample); k, n1, n2 – adjustable parameters. Physical meaning of distribution function F(x) – fraction of measurement results not exceeding a specific x value.

From the point of mathematical statistics, the sample is fully characterized by a general set (in the studied case, by a very large, almost infinite number of strength measurements). In practice, it is impossible to get such a number of strength measurements, and there is a specific methodology of getting distribution parameters that actually are characteristics of the general set, based upon a limited set consisting of a real number of parallel measurements, i.e., of a sample. The procedure for determining distribution constants (equation (1); appropriate term of mathematical statistics is estimation) is described in [2]. Estimation of parameters was made in this work in accordance with the commonly accepted laws of nonparametric statistics [3] with due regard for the recommendations in [4].

The distribution parameters for the strength of filaments of basalt fiber (measurement results in Table 1) were xmin = 0.951 GPa, xmax = 2.823 GPa, k = 0.8928, n1 = 0.5262, n2 = 1.7558. The root-mean-square deviation was 0.0199. The results of analysis of strength of filaments of basalt fiber are plotted in Fig. 1.

Table 1. Values* o f Strength Limit of Individual Filaments of Basalt Fiber (in GPa)
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Fig. 1.
figure 1

Stepped curve 1 – empirical distribution function; 2 – curve calculated by equation (1) for filaments of basalt fiber.

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For filaments of Taparan aramid fiber (strength measurement results are presented in Table 2 and Fig. 2), the following values of the constants of equation (1) were obtained: xmin = 3.574 GPa, xmax = 4.485 GPa, k = 4.0697, n1 = 2.0106, n2 = 0.0014. The root-mean-square deviation was 0.0397.

Table 2. Values* of Strength Limit of Individual Filaments of Taparan Aramid Fiber (in GPa)
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Fig. 2.
figure 2

Multistep curve 1 – empirical distribution function; 2 – curve calculated by equation (1) for filaments of Taparan aramid fiber

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