PSI - Issue 28

L.R. Botvina et al. / Procedia Structural Integrity 28 (2020) 1686–1693 L.R. Botvina et al. / Structural Integrity Procedia 00 (2019) 000–000

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In connection with this, the purpose of the work consisted in the search for a criterion, which characterizes statistical properties of materials and is connected to their mechanical properties, fracture mechanisms and loading conditions. In most studies, for statistical analysis of experimental data, the two- or three-parameter Weibull equation is used, which, in turn, used relation proposed earlier by P. Rosin and E. Rammler (1933). Moreover, Weibull noted that his equation has no physical meaning, and it is based on a linear approximation of the data, which makes the processing of experimental results easier. Since then, many researchers have set themselves the task of finding the relationship between the parameters of the Weibull equations with stress (Sakai et al., 1992), fracture toughness (Gao et al., 1998), and fracture mechanisms (Harlow, Sakai et al., 2001, 2006). However, in most cases these efforts lead to equations requiring the estimation of many parameters that complicate the processing of the results, which is not surprising, since Weibull’s distribution was proposed only for a linear approximation of experimental data. To solve the above problem, the following approaches are used, in particular: • Separation into at least two modes of damage, characterized by a specific fracture mechanism, followed by a description of each of them and using the mixture rule. Weibull was the first to use such an approach (Weibull, 1951) by isolating from the data set two sub-distributions corresponding to close values of the initial material strength. T.P. Zakharova in her work (1974) also proposes dividing the normal distribution of fatigue lifetime into two parts based on a change in the deformation mechanism, D.G. Harlow uses Weibull distributions to describe fatigue fracture due to internal and surface damage accumulation (2006, 2011). In the field of dynamic fragmentation V.A. Odintsov proposed to use the hyperweibull bimodal distribution, which makes it possible to describe the full spectrum of fragments corresponding to the mixed fracture mechanism (1991). • Using other statistical distributions instead of the Weibull equations: for this purpose the normal distribution is widely used: the work (Sakai et al., 1979) is dedicated to the creation of a normal distribution based on Paris's law. In many cases the Weibull distribution cannot always be applied in principle: Z.P. Bažant and S.D. Pang (2007) showed that two types of distributions (Weibull and Gauss) would be more accurately used to describe quasibrittle materials; however, in later work (2019) Bažant noted that it is necessary to apply a fundamentally different approach – fishnet statistics - instead of Weibull's theory of the weakest link and Gauss bundles. • Application of a methodology that combines the Weibull distribution with the analysis of experimental data was studied by D.G. Harlow (2007) by integration of existing, but limited experimental data with scientific modeling; Alternatively, the other possible statistical approach to material state evaluation can be used. This method is based on using inverse coordinates for cumulative distribution curves and exponential function for its description. The alternative method was proposed in (Botvina and Demina, 2010). The resulting distribution functions are described by exponential functions for both fatigue process (eq. 1) and fracture process in ductile-brittle transition (eq.2): P N Ae   (1),

P 

c J Be 

(2),

where N – number of cycles until fatigue failure, J c – fracture toughness, P – cumulative probability of failure, γ – slope coefficient of cumulative curves, and A , B – constants; In their work dedicated to fatigue failure analysis it was shown, that use of the proposed approach is more simple and slope coefficient γ has strong physical meaning – this coefficient depends on stress amplitude during fatigue loading. For development of this model authors used data obtained by Zakharova (1974) and Okada et al. (1999). The aim of the study is application of a probabilistic assessment of the material state by the method of inverse coordinates to analysis of microhardness and fatigue properties of different materials.