On Stochastic Evaluation of S N Models. Based on Lifetime Distribution

Transcription

1 Applied Mathematical Sciences, Vol. 8, 2014, no. 27, HIKARI Ltd, On Stochastic Evaluation of S N Models Based on Lifetime Distribution Abdellatif EL AMRAOUI ENSET Rabat University Mohamed V Souissi B.P Rabat institute Rabat, Morocco Abdellah EL GHARAD Moroccan laboratory of Innovation and Industrial Performance ENSET Rabat University Mohamed V Souissi B.P Rabat institute Rabat, Morocco Mohammed Ouadi BENSALAH Laboratory of Mechanic and Materials Department of Physic, Faculty of Science University Mohamed V Agdal, Rabat, Morocco Copyright 2014 Abdellatif EL AMRAOUI, Abdellah EL GHARAD and Mohammed Ouadi BENSALAH. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In general, fatigue test results are widely scattered even in the strictly controlled conditions. Therefore, the mean SN curve, representing the Stress in relation to the number of cycles, should be determined by a statistical method. The methodology by applying a mathematical model to S N curve is widely used. So many kinds of mathematical models are proposed in the literature and regression method for these models also has been discussed.

2 1324 Abdellatif EL AMRAOUI et al. The aim of this paper is the randomization of general deterministic models for predicting fatigue behavior for any stress level and range based on laboratory tests. More precisely, an existing Weibull model is generalized and adapted to this type of predictions via compatibility and functional equations techniques. A wide family of models is obtained, for which inference, testing hypotheses, model choice and model validation problems are dealt with. This provides a solution of the fatigue damage accumulation problem. Keywords: S N field modeling; damage model; lifetime; probabilistic evaluation; Weibull distribution. 1 Introduction To design structures or machines, a permissible stress is determined from the S N curve with the safety factor. Although S N curve is determined by experiment, the fatigue stress corresponding to a given fatigue life should be determined by statistical method considering the scatter of fatigue properties. For stochastic evaluation of fatigue test data, international standards are already established [4] and used in many research and industrial fields. On the other hand, the methodology by applying a mathematical model to S N relation is also widely used. There many kinds of mathematical models and regression method for these models also have been discussed. In the standard, S N models are classified into eight types and regression method for each S N model is also proposed. In this paper, a probabilization of models for S N curve based on evaluation of fatigue lifetime distribution has been proposed and comparisons between these models are done. 2 Probabilisation of S N models Types of models proposed in the literature. Many models for mathematical expression of S N curve have been proposed in literature. According to the standard evaluation method of fatigue reliability for metallic materials (see Hanaki and al. [3]), eight types of S N models g lnn, σ 0 were proposed in the literature. Fig.1 shows schematic illustrations of each model. The formulae of each S N model when the vertical axis of S N diagram is normal scale are also described in Fig.1. In these formulae, σ and N are stress amplitude and number of cycles to failure, respectively. A, B, C, D and E are constants. Four

3 On stochastic evaluation of S N models 1325 models in Fig.1 have normal and logarithmic scale in vertical axis, so that eight types of S N models are considered in total. ( σ E)( σ+a ln(n) B)=C σ = D (a) Hyperbola regression model. (b) Curve regression model σ + A ln( N ) = B si N < N c σ = E si N N c σ + A ln( N ) = B (c) Bilinear regression model (d) Linear regression model Fig. 1: Schematic illustrations of S N models Probabilization of models. The S N model depends on two variables: the fatigue lifetime, N, and the amplitude of stress σ The problem is to develop a non-deterministic regression model, to describe the SN curve and to estimate the model parameters. In this paper, we consider models, g( ln(n), σ ) = 0, proposed in Fig.1, which are deterministic and we add them a stochastic part: g( ln(n), σ ) = ε where ε is a random part.

4 1326 Abdellatif EL AMRAOUI et al. This choice is not arbitrary, but based on the following assumptions that lead to a functional equation (see Castillo and al. [1]): 1. Weakest link principle. If a longitudinal element is divided into n subelements, its fatigue life must be the fatigue life of the weakest element. 2. Independence. The fatigue stress of two non-overlapping subelements are independent random variables. 3. Stability. The cumulative distribution function (cdf) model must be valid for all lengths, but with different parameters. 4. Limit value. The cdf should encompass extreme lengths, i.e., the case of a length going to infinity. Thus, the cdf must belong to a family of asymptotic functions. 5. Limited range of the random variables involved. The variables Dr and N have a finite lower end, which must coincide with the theoretical lower end of the selected cdf. 6. Compatibility. In the S N field, the cumulative distribution function, G(ln(N); σ), of the lifetime given stress range should be compatible with the cumulative distribution function of the stress range given lifetime, F(ln(N); σ). Though in standard tests σ is fixed and the associated random lifetime N is determined, here σ is interpreted as the random stress that needs to be applied to produce failure at N. Only the Weibull distribution and the following S N field satisfies the above six conditions: g( ln(n), σ ) = δlog1p β (1) Where: N is the fatigue lifetime measured in cycles, σ is the stress range, P=F(ln(N); σ) is the probability of failure, and β and δ are the parameters to be estimated with the following meaning: β shape parameter of the Weibull distribution, δ scale parameter. The isoprobability curves, i.e., the curves joining points with the same probability of failure, are represented in Fig.2. The analytical expression of the S N field allows the probabilistic prediction of fatigue failure under constant amplitude loading. The cdf of the fatigue life N at the stress range σ becomes: ln; 1, 2

5 On stochastic evaluation of S N models 1327 ( σ E)( σ+a ln(n) B) = C + ε σ = D + ε (a) Hyperbola regression model. (b) Curve regression model σ + A ln( N ) = B+ε si N < N c σ = E+ε si N N c σ + A ln( N ) = B + ε (c) Bilinear regression model (d) Linear regression model Fig. 2: S N field with curves representing the same probability of failure Parameter estimation where ε = δlog1p β is the random part. All models allows, in a first step, the estimation of A, B, C, D and E. Next, in a second step, the other parameters β and δ can be estimated. Substituting the mean value of theweibull distribution K Γ 1 1, into expression (1) the following expressions are obtained:

6 1328 Abdellatif EL AMRAOUI et al. ( σ E)( σ+a µ B) C = K σ µ D = K σ+a µ B = K si N < N c σ E si N N c σ + A µ B = K for hyperbola regression model. for curve regression model. for bilinear regression model. for linear regression model. Where µ is the mean value of log(n) at stress amplitude σ. Expressions above suggests estimating first A, B, C, D and E by minimizing the following mean square errors: lnn 1 C σ B A σ E σ D lnn 1 A σb for hyperbola regression model for curve regression model. for bilinear regression model. 1 with respect to A, B, C, D, E and K, where n is the number of stress ranges and m the number of tests conducted at each stress range. Once the values of the parameters A, B, C, D and E are known, Eq. (2) shows that random variables V where V is: σ σ, for hyperbola regression model. σ, for curve regression model. σ, σ, for bilinear regression model. σ A B for linear regression model. have a two parameters Weibull distribution. Note that the hyperbola model can be transformed into bilinear model by taking a value of C = 0. Therefore, parameters of bilinear model can be estimated by abovementioned procedure with fixed value of C = 0. In regard to scatter of lifetime around S N curve, the following two points have been assumed. (i) Distribution of fatigue lifetime at different stress amplitude is approximately identical.

7 On stochastic evaluation of S N models 1329 (ii) The fatigue lifetime follows a Weibull distribution. These assumptions allows us, see Castillo and al. [2] and references therein, to pool all the fatigue data results for different stress ranges into a unique population (see Fig.3) and helps to overcome the limitation of the low number of results at each different test level, so that better estimates are thus achieved. In addition, the β and δ Weibull parameters can be estimated by using the maximum likelihood method (Lopez-Aenlle [5]). Since the probability density function (pdf) of a Weibull distribution is given by:,δ,β The logarithm of the pdf becomes:,δ,β 1 and the value of the parameters β and δ can be obtained by maximizing with respect to β and δ the expression: δ,β 1 Other methods for the estimation of the model parameters have been developed by Castillo and al. [1]. Fig. 3: S N field showing schematically the pdfs of lifetime N for different stress ranges and their conversion to the normalized distribution.

8 1330 Abdellatif EL AMRAOUI et al. 4 Practical example To check performance of each model and to illustrate its use in practical cases, the models are applied in this section to an example. We have considered the fatigue behavior using 15 specimens corresponding to some previously standard tests using prestressing wires. The data samples shown in table 1 are simulated by Castillo and al. [2]. The selected values of σ = 340, 476, 544 and 680 MPa correspond to combinations of stress values σ max = 1054 MPa and 1190 MPa, and stress values of σ min = 510 MPa and 714 MPa. Table 1 Simulated samples for each of the four values of σ. Level 1 σ min = 714 MPa σ max = 1054 MPa σ = 340 MPa Level 2 σ min = 714 MPa σ max = 1190 MPa σ = 476 MPa Level 3 σ min = 510 MPa σ max = 1054 MPa σ = 544 MPa Level 4 σ min = 510 MPa σ max = 1190 MPa σ = 680 MPa Table 2 shows coefficient of correlations, corresponding to failure probability of 0.5, on the plots of each model. It becomes maximum when the regression model is double logarithmic hyperbola model or bilinear model. In hyperbola model, parameter C takes 0 and there is no difference between hyperbola and bilinear model. It is confirmed that the distribution of equivalent fatigue stress can be expressed by double logarithmic hyperbola model or bilinear model. Table 2 Coefficient of correlations on the plots of each probability paper.n σ = 340 MPa σ = 476 MPa σ = 544 MPa σ = 680 MPa Semi logarithmic hyperbola Semi logarithmic bilinear Semi logarithmic curve Semi logarithmic linear Double logarithmic hyperbola Double logarithmic bilinear Double logarithmic curve Double logarithmic linear

9 On stochastic evaluation of S N models Conclusion This paper presents a general approach for constructing non linear probabilistic fatigue models. Models were studied and successfully applied, with different values of stress range, for prestressing wires and The following conclusions are drawn from this study: 1. The obtained S N curves agrees well the experimental data. 2. The procedure of the transformation of each fatigue data into equivalent fatigue stress enables to decide the distribution of fatigue strength using relatively small number of samples. 3. Any deterministic model of lifetime can be considered a special case of a probabilistic model of lifetime. 4. For prestressing wires, double logarithmic hyperbola model and bilinear model are better than all others models. References [1] Castillo E., Fernandez Canteli A., Hadi AS. On fitting a fatigue model to data. Int J Fatigue 1999(21): [2] Castillo E., Fernandez Canteli A., Ruiz-Ripoll M. A general model for fatigue damage due to any stress history. International Journal of Fatigue 30 (2008) [3] Hanaki S., Yamashita M., Uchida H., Zako M. On stochastic evaluation of S N data based on fatigue strength distribution. International Journal of Fatigue 32 (2010) [4] International Standard. Metallic materials fatigue testing statistical planning and analysis of data. ISO [5] Lopez Aenlle M., Ramos A., Fernandez Canteli A., Castillo E., Kieselbach R., Esslinger V. Specimen length effect on parameter estimation in modeling fatigue strength by Weibull distribution. International Journal of Fatigue 28, 2006: Received: January 3, 2014