JOURNAL OF THEORETICAL AND APPLIED MECHANICS 44, 1, pp. 127-137, Warsaw 2006 INFLUENCE OF NOTCH (TIP) RADIUS ON FATIGUE CRACK GROWTH RATE Dariusz Rozumek Ewald Macha Faculty of Mechanical Engineering, Technical University of Opole, Poland e-mail: drozumek@po.opole.pl; emac@po.opole.pl Paolo Lazzarin Department of Management and Engineering, University of Padova, Vicenza, Italy e-mail: plazzarin@gest.unipd.it Giovanni Meneghetti Department of Mechanical Engineering, University of Padova, Padova, Italy e-mail: meneghetti@unipd.it The paper deals with the problem of fatigue crack propagation from notches in plates made of FeP04 steel and AA356-T6 aluminium alloy. The tests were performed under different stress ranges by keeping the nominal load ratio (R=0.1) constant.The specimenswereweakenedby nearly-sharpandblunt two-sidednotches.The results of the fatigue tests were then reanalysed in terms of the J-integral range including influence of the notch. It was observed that the blunter notches are, the higher fatigue crack growth rate is. Key words: fatigue crack growth, J-integral range, load ratio, notch Notations E – Young’s modulus Kt – stress concentration factor K′ – cyclic strength coefficient Pa – amplitude of load R – load ratio 128 D.Rozumek et al. a – crack length b – fatigue strength exponent c – fatigue ductility exponent d – notch depth da/dN – fatigue crack growth rate n′ – cyclic strain hardening exponent t – specimen thickness w – specimen width ∆JI – integral range ∆KI – stress intensity factor range ε′f – fatigue ductility coefficient ν – Poisson’s ratio ρ – notch tip radius σnom – nominal stress σTS – ultimate tensile strength σY – yield stress σ′f – fatigue strength coefficient 1. Introduction Whenever fatigue loading occurs, stress concentrations or notches, nomat- ter macroscopic or microscopic, are preferred sites for crack initiation. Thus, the notch shape determines, as well known, the fatigue limit of the compo- nent (Macha and Rozumek, 2003). Recently, a diagram has been proposed in which the fatigue limit of the notched component was predicted on the basis of either the theoretical stress concentration factor Kt or the stress intensity factor range ∆Kth, with the notchmodelled as a crack of the equivalent depth (Atzori et al., 2003). In the low-medium fatigue range, the fatigue life is usually thought of as a sum of fatigue crack initiation and fatigue crack propagation, although the differentiation of the two stages is qualitatively distinguishable but quantita- tively ambiguous (Jiang andFeng, 2004). Residual fatigue life calculations are carried out bymeans of da/dN−∆K curves obtained from conventional, pre- cracked specimens, often considering the fatigue growth rate as independent of the notch shape. However, when a fatigue crack is initiated from a notch, the crack propagation rate is generally higher than that expected by using the stress intensity factor, mainly because tip there exists a prior accumulation of fatigue damage in initiating the fatigue crack ahead of the notch (Jiang and Feng, 2004). Influence of notch (TIP) radius... 129 In the presence of non-localised yielding, the parameter ∆K is no longer applicable and an energetic approach based on the J-integral (Rice, 1968) might be a more appropriate choice, mainly because local and global changes of the energyoccurduring fatigue loading. InpaperbyElHaddad et al. (1980), behaviour of small fatigue cracks was analysed and described with use of the stress intensity factor range and J-integral. The tests were done in elastic and plastic ranges on smooth specimens with blunt notches. The J-integral was applied for the description of plastic changes appearing near the notch. It was found that the J-integral is a good parameter for describing changes occurring in various notches. In the paper by Ogura et al. (1987), behaviour of the effective fatigue crack growth in notched elements was investigated. For the description of test results, ∆Keff and ∆Jeff parameters were used. It was concluded that ∆Keff is themost important parameter for the description of the fatigue crack growth for small cracks at the notch root. It was also found that ∆Jeff is a suitable parameter to be applied in the case of appearance of a plastic zone at the notch root and for small cracks. Themainaimof this paper is characterisation of fatigue crack growth rates in terms of the ∆J-integral, by using experimental data from plates of two different materials, weakened by rounded and nearly-sharp notches. 2. Materials and test procedure 2.1. Static properties and strain-based cyclic tests Present re-analyses are basedon sets of experimental data already reported in the literature (Lazzarin et al., 1997). Tests were carried out on plates made of FeP04-UNI 8092 deep-drawing steel and AA356-T6 cast aluminium alloy, weakenedby symmetric lateral notches of varyingacuity. For static and fatigue tests, a SchenckPSA100 servo-hydraulic devicewas used at theDepartment of Mechanical Engineering of thePadovaUniversity. Elastic and static properties of the two materials are summarised in Table 1. Table 1.Elastic and static strength properties Materials σY σTS E ν [MPa] [MPa] [GPa] FeP04 steel 185 310 191 0.30 AA356-T6 aluminium alloy 182 250 71 0.32 130 D.Rozumek et al. Strain-based fatigue curves are shown in Fig.1, where elastic and plastic components are given too, together with the best fitting values of thematerial properties. As usual, such curves are described by a linear law in a log-log diagram as suggested by theManson -Coffinmodel. In the same figures, some stabilised hystheresis loops are displayed too. Coefficients of the Ramberg- Osgood equation describing the cyclic strain curve under tension-compression conditions with Rε = −1 (a Schenck extensometer was used with a gauge length equal to 25mm) for FeP04 steel and AA356-T6 aluminium alloy were presented in Fig.1 (Lazzarin et al., 1997). Fig. 1. Fatigue curves under strain control and some stabilised histeresis loops; (a) FeP04, (b) AA356-T6 Influence of notch (TIP) radius... 131 2.2. Fatigue tests of notched specimens All fatigue tests were carried out under load control by imposing a con- stant value of the nominal load ratio, R=0.1, and a frequency ranging from 20 to 25Hz. The specimens were characterised by double symmetric lateral notches with a notch root radius ranging from 0.1–0.2mm to 10mm (Fig.2 and Table 2). Fig. 2. Geometry of specimens characterised by: (a) sharp and (b) blunt notches. All dimensions in mm The endurance limit for plain specimens was ∆σ0 =247MPa at 2 million cycles for FeP04 and 140MPa for AA356-T6. For the notched specimens the endurance limits ∆σA,g at NA =2 ·10 6 cycles are summarized in Table 2, to- gether with theoretical stress concentration factors evaluated by FE analyses. Kt,g and ∆σA,g in Table 2 is referred to the gross area of the specimens. Table 2.Geometrical parameters, stress concentration factors and fatigue strength range of notch specimens (Kt,g and ∆σA,g referred to the gross area, subscript g) Materials d w t ρ Kt,g ∆σA,g [MPa] [mm] [mm] [mm] [mm] at 2 ·106 cycles FeP04 steel 10 50 2 0.2 16.3 51.7 2.5 5.38 63.0 10 3.07 90.0 AA356-T6 8 40 4 0.1 20.2 29.2 aluminium alloy 2.5 4.92 39.9 132 D.Rozumek et al. In a number of fatigue tests, fatigue crack initiation and propagation pha- ses are controlled by means of a stereoscope (64×). The surfaces of the spe- cimens, 40-50mmwide, were accurately polished in order to make cracks ori- ginated from the notch tip easily distinguishable. In Lazzarin et al. (1997), a crack was considered ”significant” when its length was equal to a0, by using for a0 the definition given in (El Haddad et al., 1979). For a low- carbon steel, with ∆σA =247MPa and ∆Kth,0 ∼=10MPa·m 1/2 (both values referring to a nominal ratio R close to zero), the material length parame- ter a0 resulted to be 0.5mm. On the other hand, considering average values of thresholds among those reported in the literature concerning cast light al- loys (∆Kth,0 ∼= 5MPa·m 1/2) in combination with ∆σA = 140MPa, a0 was 0.41mm. In Lazzarin et al. (1997), the demarcation line between fatigue crack initiation and propagation phaseswas conventionally drawn just in correspon- dence of a0. Such a threshold length was found to be the limit value beyond which cracks became through-thickness and their propagation began to be suf- ficiently stable. This was not true in some light alloy specimens where, due to greater thickness andmaterial porosity, a number of different corner and sur- face microcrack initiation sites were detected, thus resulting in higher scatter of fatigue strength properties. 3. Experimental results Some plots of the crack length a versus the number of cycles N for FeP04 and AA356-T6 are shown in Fig.3. Different symbols are used depending on the notch root ρ. The specimens were subjected to cyclic tension at constant amplitudes of load: • FeP04 – Pa =3.9kN for ρ=0.2mm, Pa =4.2kN for ρ=2.5mm and Pa = 5.4kN for ρ = 10mm, corresponding to the nominal stress ran- ge amplitude to crack initiation ∆σnom = 145MPa for ρ = 0.2mm, ∆σnom = 156MPa for ρ = 2.5mm and ∆σnom = 200MPa for ρ=10mm • AA356-T6 – Pa = 4.8kN for ρ = 0.2mm and Pa = 6.72kN for ρ = 2.5mm, corresponding to the nominal stress range amplitude to crack initiation ∆σnom =50MPa for ρ=0.2mmand ∆σnom =70MPa for ρ=2.5mm. Influence of notch (TIP) radius... 133 Fig. 3. Crack length as a function of the number of cycles; (a) FeP04, (b) AA356-T6 Afterwards, the fatigue crack growth rate was evaluated according with the following equation (Dowling and Begley, 1976) da dN =B(∆JI) n (3.1) where ∆JI = Jmax − Jmin, and the parameters B and n need a best-fit analysis. ∆JI in Eq. (3.1) was calculated by using the following relationship (Rozumek, 2004) ∆JI = ∆K2I E +πY 2 ∆σ∆εp √ n′ a (3.2) where a is the crack length, Y =Y1Y2 a correction factor and ∆σ is the stress range corresponding to the plastic strain range ∆εp, both ranges evaluated ahead of the notch. In order to include the influence of crack length to gross area ratio as well as the non-uniform stress distribution due to notches, the stress intensity factor range was evaluated from the following expression ∆KI =Y1Y2∆σnom √ π(a+d) (3.3) where ∆σnom is evaluated on the gross area, and coefficients Y1 and Y2 are (other symbols in Fig.4) Y1 =1.12+0.203 2(a+d) w −1.197 (2(a+d) w )2 +1.930 (2(a+d) w )3 Y2 = √ 1− e−β with β= 6a(ρ+w−2d) ρ(w−2d) 134 D.Rozumek et al. Fig. 4. Description of symbols used for estimation of the range of stress intensity factors The coefficient Y2 was determined in Lazzarin et al. (1997) by means of a number of ad hoc FE analyses. Results from Eq. (3.3) are plotted in Fig.5. The effect of the non-uniform stress distribution due to different notch root radii on ∆KI is evident there. Fig. 5. Stress intensity factor range for different values of the notch root radius, see Eq. (3.3) Curves of the fatigue crack growth rate versus the J-integral range for steel and aluminium alloys are plotted in Fig.6. Plots 1a, 1b and 1c in Fig.6a make it clearly seen that the fatigue crack growth rate increases as the root radius ρ increases. It is especially visible at the final stage of cracking, for example for ∆J = 5 · 10−2MPa·m under a change of the notch root radius from 0.2mm to 10mm, the crack growth rate increases three times. This fact may be caused by a longer initiation period characterising specimens with blunt notches, accomplished by major local damage ahead of notches before of the appearance of a fatigue crack. Influence of notch (TIP) radius... 135 Moreover, comparing the two tested materials, a higher fatigue crack growth rate was noticed in the AA356-T6 aluminium alloy. In Fig.6, graph 1 is related to averaged values of the coefficients B and n found fromEq. (3.1). Fig. 6. A comparison between experimental results and values calculated according to Eq.(3.1); (a) FeP04, (b) AA356-T6 Table 3. Coefficients B and n, Eq. (3.1), and correlation index r for curves in Fig.6 Materials Figures B n r Graphs [m/(MPa·m)n cycle] FeP04 Fig.6a - 1 4.344 ·10−6 1.099 0.921 Fig.6a - 1a 2.274 ·10−6 1.041 0.969 Fig.6a - 1b 1.583 ·10−6 0.782 0.926 Fig.6a - 1c 4.541 ·10−6 1.062 0.975 AA356-T6 Fig.6b – 1 9.683 ·10−6 0.995 0.912 Fig.6b - 1a 7.523 ·10−6 0.899 0.959 Fig.6b - 1b 3.261 ·10−5 1.441 0.923 In the elastic-plastic range, stresses and strains were calculated by means of the finite element program FRANC2D. In the models, six-node triangular elements were used. In Fig.6, plots 1a, 1b for the AA356-T6 aluminium alloy show a trend similar to that of the FeP04 steel. The coefficients B and n, as well as the correlation index r, were determinedwith the least squaremethod for a confidence level α = 0.05. All coefficients are listed in Table 3, where 136 D.Rozumek et al. some variations of the coefficients B and n depending on the notch type are evident. High values of the correlation coefficient r clearly demonstrate that the correlation between test results and the assumedmodel is significant. 4. Conclusion The results of fatigue crack growth tests on notched specimens made of FeP04 steel andAA356-T6aluminiumalloy testedunder tension loading, allow the following conclusions to be drawn: • In elastic-plastic materials, the notch type strongly influences both the initial valueof J-integral rangeandthe shapeof crack growth rate curves. • In notched specimens made of the FeP04 steel and in the presence of a nominal load ratio R=0.1, the greater was the radius of a notch root, the greater the was fatigue crack growth rate. • Comparing the two materials, a higher crack growth rate was found in the AA356-T6 aluminium alloy with respect to the FeP04 steel. Acknowledgements Thisworkwas supported by theCommission of theEuropeanCommunities under the FP5, GROWTHProgramme, contract No. G1MA-CT-2002-04058 (CESTI). References 1. Atzori B., Lazzarin P., Meneghetti G., 2003, Fracture mechanics and notch sensitivity,Fatigue andFracture of EngineeringMaterials and Structures, 26, 257-267 2. Dowling N.E., Begley J.A., 1976, Fatigue crack growth during gross pla- sticity and the J-integral, In: Mechanics of Crack Growth, ASTM STP 590, American Society for Testing andMaterials, 82-103 3. El Haddad M.H., Dowling N.E., Topper T.H., Smith K.N., 1980, J- integral applications for short fatigue cracks at notches, International Journal of Fracture, 16, 15-30 4. El Haddad M.H., Topper T.H., Smith K.N., 1979, Prediction of non- propagating cracks,Engineering Fracture Mechanics, 11, 573-584 Influence of notch (TIP) radius... 137 5. Jiang Y., Feng M., 2004,Modelling of fatigue crack propagation, Journal of Engineering Materials and Technology, 126, 77-86 6. Lazzarin P., Tovo R., Meneghetti G., 1997, Fatigue crack initiation and propagation phases near notches in metals with low notch sensitivity, Interna- tional Journal of Fatigue, 19, 647-657 7. Macha E., Rozumek D., 2003, Fatigue crack path development in a one- sided restrained bar with a rectangular section and stress concentrator under bending, Proc. Int. Conference on Fatigue Crack Paths (FCP 2003), Parma, CD-ROM, 8 ps 8. Ogura K., Miyoshi Y., Nishikawa I., 1987, Fatigue crack growth and clo- sure of small cracks at the notch root,Current Research on Fatigue Cracks, Ed. Tanaka T. et al., Elsevier Applied Science, London and NewYork, 1, 67-91 9. Rice J.R., 1968, A path independent integral and the approximate analysis of strain concentration by notches and cracks, Journal of Applied Mechanics, 35, 379-386 10. Rozumek D., 2004, The ∆J-integral range applied for the description of fati- gue crack growth rate, Proc. 12th Int. Conference on Experimental Mechanics (ICEM12), Bari, 275-276 and CD-ROM, 8 ps Wpływ promienia (wierzchołka) karbu na prędkość wzrostu pęknięć zmęczeniowych Streszczenie W pracy przedstawiono propagację pęknięć zmęczeniowych, w próbkach płaskich z karbami, wykonanych ze stali FeP04 i stopu aluminium AA356-T6. Badania wy- konywano przy różnych zakresach naprężeń i stałym współczynniku asymetrii cyklu (R = 0.1). Próbki były osłabione przez dwustronne ostre i łagodne karby. Wyniki badań zmęczeniowych opisano za pomocą zakresu całki J z uwzględnieniem wpływu karbu. Stwierdzono, że im łagodniejszy karb, tym większa prędkość wzrostu pęknięć zmęczeniowych. Manuscript received July 13, 2005; accepted for print September 5, 2005