Statistical analysis of the mechanical properties of composite materials
Introduction
Unlike most traditional structural materials, whose mechanical behaviour is assumed to be homogeneous and isotropic, mechanical properties of composite materials exhibit intrinsic statistical dependence. In particular, their strength properties are usually scattered due to their inhomogeneity and anisotropic characteristics and to the brittleness of the matrices and fibres. So, careful statistical analysis is indispensable for the understanding of the mechanical characterisation of these materials.
The Weibull statistic [1] has been widely used in the recent years to describe the statistical behaviour of the strength properties of many materials, such as advanced ceramics [2], metallic matrix composites [3], ceramic matrix composites [4], and polymeric matrix composites [5]. The Weibull distribution also describes the fatigue behaviour of materials and the scatter of the fracture toughness of steels in the ductile–brittle transition region, where failure occurs by cleavage [6], [7].
The two-parameter Weibull distribution function is given by:where F is the probability of rupture of the material under uniaxial tensile stress σ, m is the shape parameter or Weibull modulus, and σ0 is the scale parameter of the distribution. Weibull modulus, m, is related to the scatter of the data: the higher the m the lower the dispersion of fracture stress. It becomes the most important parameter of the distribution. The scale parameter is closely related to the mean fracture stress.
Since the evaluation of the parameters of Weibull distribution is made from a finite number of tests, the estimators of their true values have a statistical character, and thus the uncertainty of the estimation must be known. In this work a three-parameter Weibull distribution is also proposed for the estimator of the Weibull modulus. Formulae for the three parameters defining such a distribution, dependent only on the sample size, are given below.
To design structural and mechanical components, the determination of the σ-values, corresponding to a predefined failure probability, is of great interest to the engineer. These values coincide with their percentiles of the distribution. In particular, the values corresponding to the percentiles of 90 and 99%, estimated with a confidence level of 95%, are known [8] as the A-basis and B-basis material property, respectively. These values are obtained from the estimations of m, σ0, and in terms of two parameters, PA and PB, which depend only on the sample size and on the estimation method used, and are usually obtained from tables [8]. In this paper, to simplify the calculus of these parameters, fitted equations are developed for two estimation methods: maximum-likelihood and weighted regression.
Section snippets
Theoretical background
Several procedures are available for the determination of the Weibull distribution parameters. From the maximum likelihood method, the estimators of the Weibull parameters, and should satisfy the following equations:andAlthough Eq. (2) is non-linear, it has a unique positive solution [9], and may be solved by the Newton–Raphson iteration technique or by any other method.
Eq. (1) becomes a straight line if a double
Numerical simulation
To obtain the statistical distribution of the pivotal variables and a simulation procedure, based on the Monte Carlo method, has been used. In this procedure (see Fig. 1 for better understanding) a set of n values was generated asIn this work we select m=1 and σ0=1. Note that, as stated above, the analysis is independent of the true values [12].
R is a random variable with uniform distribution in the [0,1] interval. From each sample so obtained, {σ1,σ2,σ3,…,σn},
Parameter fitting
From the 20,000 values of the pivotal variable its average value, was calculated. In order to fit this value to the sample size, the following four-parameter function is proposed:where A, B, C, and D are the fit parameters given in Table 1, and n is the sample size. As shown in Fig. 2, the fit seems to be very good, with a maximum error of 0.23 %. If parameter D were taken as unity, the increment of the maximum error would become negligible and thus three
Practical examples
To illustrate the above methodology two examples are included in this paper, although it is worth noting that they can be applied in their integrity to any kind of composite material. In the first example, the estimation of the confidence intervals of the Weibull modulus for the static flexural strength of a CMC is considered, whereas in the second, the A-basis and B-basis values of the dynamic flexural strength of a CFRP are computed. In these examples, the loading rate and the sample size are
Summary
This work presents useful formulae to analyse the variability of the mechanical properties of composite materials. The study is focused on the two-parameter Weibull distribution, currently used to describe statistically the strength properties of many kinds of materials. In order to obtain the percentage points of the estimator of the Weibull modulus, published until now in tabular form, a three-parameter Weibull distribution is proposed. Empirical expressions for these three parameters,
Acknowledgements
The authors are indebted to the Fundación Ramón Areces (Área de Materiales, IX Concurso Nacional) for the financial support of this research.
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