The fictitious radius as a tool for fatigue life estimation of notched elements

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1 The fictitious radius as a tool for fatigue life estimation of notched elements Grzegorz ROBAK 1, a, Marcel SZYMANIEC 1, b, Tadeusz ŁAGODA 1, c 1 Opole University of Technology, Faculty of Mechanical Engineering, ul. St. Mikołajczyka, Opole, Poland a g.robak@po.opole.pl, b m.szymaniec@wp.pl, c t.lagoda@po.opole.pl Keywords: fictitions radius, fatigue life, Abstract. In this paper, the fictitious radius - according to Neuber s method for determination of stresses at the notch root was used. Next, the fatigue lives of elements of the ring notches were calculated, and then compared with results of experimental tests of S235JR steel samples. However, the obtained fatigue lives did not bring satisfactory results. It has been demonstrated that the fictitious radius strongly depends on the expected fatigue life. Introdaction Estimation of fatigue life of machine elements has been the subject of intensive research for many years. Numerous failures that have been causing costly repairs of elements or even whole structures, have forced designers to perform this type of research in search for improvement. Moreover, in extreme cases, such failures led to life-threatening situations. At present, the problem of lifetime prediction of machine elements and structures has been given a proper recognition as an important and crucial issue addressed in almost every branch of a modern industry. In particular, the problem has been occuring in industrial activities, where the life, reliability and safety requirements are essential. One of the main reasons for conducting research on fatigue and creating complex calculation algorithms is optimization of machine parts [1]. The necessity of carrying out the optimization of machine elements has made the engineers to design structures with complicated geometry while maintaining lifetime and reliability properties. The aim of this paper is to compare the results of experimental tests performed on cylindrical elements with ring notches to the results of lifetime prediction determined by using the fictitious radius [2-4, 8]. Determination of the value and the fictitious radius for ring-notched elements In order to estimate the fatigue life, a concept of the fictional radius by Neuber was used [2]. The fictitious radius at the notch root allowed to calculate the maximum stresses. Determination of this radius results from the mean stresses at the notch root. Fictitious radius at the notch root was described in the following relation: f * s, (1) where: s multiaxial coefficient ρ* - equivalent microstructural length, ρ - real radius at the notch root. The multiaxial coefficient s for round elements under tension was calculated by referring to the Huber-Mises-Hencky s criterion, according to the equation:

2 s, (2) where ν is the Poisson's ratio. The equivalent microstructural length as proposed by Neuber ρ * for the steel is ca. 0.1 mm. Verification of the fictitious radius considerations based on experimental tests Verification of the fictitious radius assumptions was conducted by comparison of the calculated fatigue life with test results for the notched specimens (Fig. 1) made of stainless S235JR steel [5]. In Table 1, the mechanical properties of S235JR steel are presented. The theoretical stress concentration factor was calculated, according to Nody [6] K t = 3.288, and with the calculations obtained from FE analysis K t = However, it must be remembered that the notch factor is determined for the fatigue limit. In other cases, its value strongly depends on fatigue life [3, 4]. Table 1. Mechanical properties of steel S235JR R e, R m, MPa MPa E, GPa Experimental tests were performed on the fatigue stand type UFP ± 400 specimens were subjected to cyclic loading under tension-compression [7]. In view of the fact that the real radius ρ = 0.8 mm, and using equations (1) and (2), the value of fictitious radius ρ f = mm. Based on the obtained radius, a fictitious model of a specimen was created, then used for determination of stresses at the notch root by a finite element method. The specimen model was limited to only a quarter of the entire specimen, which enabled a greater grid density within the notch root. Software COMSOL was applied for calculations. The calculations were performed for five levels of nominal stresses in the notch cross-sections, at which the proper experimental tests were carried out: 125, 160, 200, 230, 300 and 330 MPa. The calculations were made for an elastic range. Figure 2 shows distribution of the applied finite elements at the notch root. Fig. 1. The geometry of specimen with a real ring notch.

3 Fig. 2. Distribution of finite elements at the notch root for the geometry of a specimen with the fictitious radius. Based on Neuber s assumptions, stress values obtained for a modified geometry of the specimen should be compared with fatigue characteristics of smooth specimens. The obtained values of the fatigue life should correspond to the results of experimental tests. Figure 3 shows two fatigue characteristics for notched specimens and smooth specimens made of S235JR steel. Relevant fatigue characteristic for smooth (3) and notched (4) specimens can be written in S-N as log( N log( N f f ) log( ), (3) a ) log( ), (4) an Fig. 3. Fatigue characteristics for smooth and notched specimens made of S235JR steel.

4 The obtained fatigue characteristics were then compared with the experimental ones - presented in figure 4. As can be seen, the values of calculated fatigue lives are much lower than the experimental ones. Observing the distribution of characteristics in Fig. 3, note that the higher the amplitude of the load, the difference between the calculated and experimental lives increases. In case of fatigue at about 106, the difference between the obtained results decreases. It should be seen, however, that in this case the model proposed by Nueber for the material analysed is not appropriate. This suggests that Neuber s assumptions are not correct for the steel for stress values at the fatigue limit, and even more to stresses above this level. In figure 3 comparison of experimental lives for nominal stresses equal to 300 and 330 MPa was not included, because the obtained values of calculated fatigue lives took values below one cycle. From figure 3, it is confirmed that distribution of fatigue characteristics for smooth and notched specimens, which coincide to the point below 3000 cycles, where S235JR material ceases to be sensitive to the notch effect. In such a case, the fictitious radius should aim at infinity, and the fatigue coefficient of the notch effect K f = 1. Fig. 4. Comparison of the experimental fatigue lives with the calculated ones for specimens made of S235JR steel. The fictitious radius as a fatigue life function According to the remarks discussed in the previous chapter, using fatigue characteristics for smooth and notched specimens, the geometry of the specimen was changed by modeling different fictitious radiuses, which resulted in obtaining various equivalent microstructural length, according to the equation (1) we obtain new formula for equivalent microstructural length : * f. (5) s

5 The geometry of a specimen was modified in such a way so that the stresses for notched specimens corresponded to stresses that occured for the same values as for smooth obtained from the fatigue characteristics. In the result, an ideal compliance of the estimated fatigue characteristics with experimental ones were therefore obtained. Based on the fatigue characteristics for calculations, two additional levels of nominal stresses of 80 and 100 MPa were determined. In figure 5 were estimated the obtained microstructures radiuses, according to the equation (5) as loading cycles function. From the analysis of this figure, it can be seen that the obtained values of the radius of microstructures are arranged according. to a linear function. This relation was described in the following equation: * log( ) log( N). (6) Fig. 5. The radius ρ * of the microstructure as a fatigue lifetime function. It should be noted that values of mircrostructure radius take quantities of over 1 millimeter. Only in the case of a nominal stress equal to 80 MPa, the obtained value was 0.85 mm. This result means that a ring notch causes much greater changes in microstructure of a material as compared to Neuber s assumptions. Summary From the obtained calculations, it can be observed that: 1. Application of the fictitious radius, according to Neuber s solution, for steel in elastic range under loadings above the fatigue limit does not allow to obtain results comparable with the test results. 2. The equivalent radius of microstructure strongly depends on the fatigue lifetime. 3. The microstructure radius are arranged in a linear function depending on the number of cycles.

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