Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 42, 1, pp. 83-93, Warsaw 2004 INFLUENCE OF THE MEAN LOADING ON FATIGUE CRACK GROWTH RATE AND LIFE UNDER BENDING Dariusz Rozumek Faculty of Mechanical and Engineering, Technical University of Opole e-mail: drozumek@po.opole.pl The paper contains the results of experimental tests of the fatigue crack growth rate of steels 10HNAP and 18G2A under bending for different load ratios. In the tests, plane specimens with external unilateral sharp notchesas stress concentratorswereused.Theresultsof the experimental tests are described by a nonlinear formula based on the ∆J-integral range. The proposed formula for description of the fatigue crack growth rate, including the ∆J-integral range, satisfactorily describes the results obtained experimentally. Key words: fatigue crack growth, ∆J-integral range, number of cycles, load ratio Notations a – length of the crack l – length of the specimen g – thickness of the specimen c – length of the actual specimen section before the crack front da/dN – fatigue crack growth rate N – actual (propagating) number of cycles ∆J – integral range R – load ratio Ma – amplitude of the moment B – coefficient in the equation n – exponent in the equation JIc – critical value of the integral Kt – stress concentration factor 84 D.Rozumek σu – ultimate tensile stress σy – yield stress εy – strain corresponding to the yield point α1 – material constant in the Ramberg-Osgood law n1 – exponent in the Ramberg-Osgood law E – elastic modulus ν – Poisson’s ratio K′ – coefficient of strain cyclic hardening n′ – exponent of strain cyclic hardening 1. Introduction Investigations of the fatigue crack growth rate and fatigue life constitu- te very important and complex problems in mechanics (Kocańda and Szala, 1997; Macha et al., 2000). Application of the stress intensity factor K or its range ∆K for description of the stress field before the crack front in elastic- plasticmaterials is burdenedwithmany errors causedbyoccuring large plastic strains. The coefficient K provides good results in the range of linear crack mechanics (Gasiak andRozumek, 2001). For description of elastic-plastic and plasticmaterials, the criterion based on the J-integral, definedbyRice (1968), is suggested. This criterion describes an energy state in the crack front area. TheDowling and Begley (1976) linear model is widely used for description of the ∆J-integral range. In the paper, authors compared the results obtained on the basis of their theoreticalmodelwith the experimental results.The agre- ement between these results was very good in the case of deflection control during the fatigue process. In the case of load control, significant differences were observed. The authors drew a conclusion that the proposed model did not give satisfactory results, and it was necessary to search for amore general criterion. Since there were many questions connected with the application of the J-integral for description of fatigue test results, Tanaka (1983) engaged in the research on that problem. He formulated an energy criterion based on the J-integral for the linearly-elastic and elastic-plastic ranges and its physi- cal interpretation. Tests of the fatigue crack growth were performed for five materials (aluminium, copper, nickel, titanium and steel) under tension. Pla- ne specimens with central slots were tested. It has been shown that for small strains and in the linear-elastic range, the J-integral can be determined inde- pendently of loading, as in the case of the monotonic J-integral. In the case of large strains in the elastic-plastic range, the cyclic J-integral depends on loading. However, in both cases, the cyclic J-integral remains constant while Influence of the mean loading... 85 loading (loading, unloading) if the first monotonic loading stage is elimina- ted. From the performed tests and considerations it results that the energy criterion based on the J-integral can be used under the fatigue crack growth. The aim of this paper is the experimental verification of the proposed relationship for description of the fatigue crack growth rate and fatigue life in the energetic approach with the ∆J-integral range. 2. Mathematical model The paper presents an empirical formula for description of the second and third ranges of the crack kinetics curve (Rozumek, 2002) including the ∆J-integral range da dN = B ( ∆J J0 )n (1−R)2JIc−∆J (2.1) where J0 =1MPa·m is a per unit value introduced to simplify the coefficient unit B. In Eq. (2.1), the ∆J integral range is calculated according to the finite element method ∆J = Jmax−Jmin Jmax(min) = Je+Jp = K2 E + 2Ap gc where the Je integral concerns the linear-elastic period, and the Jp integral concerns the elastic-plastic period, K = σ √ πa is the stress intensity factor, Ap – area under the force-displacement curve, and from relationship (2.2) ∆J =π ( M2k ∆σ2 E +1.13∆σ∆εp √ n′ ) a (2.2) where: Mk is the correction coefficient including finity of the specimen dimen- sions (Pickard, 1986), ∆σ – stress range at the root of the notch, ∆p – plastic strain range there. Formula (2.1) allows one to calculate the number of propagation cycles N for which the crack reaches the critical value or the required value. The equation is then integrated from an assumed or found initial length of the crack a0 up to the critical length af N = af ∫ a0 (1−R)2Jc−∆J B(∆J)n da (2.3) 86 D.Rozumek After integration of Eq. (2.3), we obtain N =B−1 [π∆σ2a E +α1∆σy∆εy (Pa P0 )n1+1 h1 ]−1 · (2.4) · {(1−R)2Jc n−1 [ 1 (∆J0)n−1 − 1 (∆Jf) n−1 ] − 1 n−2 [ 1 (∆J0)n−2 − 1 (∆Jf) n−2 ]} where ∆Jf is the range of Jf corresponding tomaterial failure, ∆J0 concerns the initial crack length a0, P0 is the limit loading (P0 =0.536σyc 2g/l), h1 – tabulated function specific for a givenmaterial and geometrical parameters of the specimen (Neimitz, 1998). The equation is valid for n 6= 1 and n 6= 2. If it is difficult to calculate the ∆J-integral range and correction coefficients, we can use numerical methods N = af ∫ a0 (1−R)2Jc−∆J B(∆J)n da≈ af ∑ a0 (1−R)2Jc−∆J B(∆J)n ∆a (2.5) 3. Materials and test procedure The subject of the investigations are construction steels 10HNAP and 18G2A characterised in Polish Standards, which concern low-alloyed and corrosion-resisting steels. Weldable steel 10HNAP is of general purpose and exhibits increased resistance to atmospheric corrosion. Steel 18G2A has im- proved mechanical properties. Their composition and properties are given in Table 1. Table 1.Characteristics of 10HNAP and 18G2A steels Steel Chemical compositions [%] Mechanical properties 10HNAP 0.14C 0.88Mn 0.31Si σy =418MPa, σu =566MPa, 0.066P 0.027S 0.73Cr E=2.15 ·105MPa, ν =0.29, 0.30Ni 0.345Cu JIc =0.178MPa·m (ASTME813, 1989) 18G2A 0.20C 1.49Mn 0.33Si σy =357MPa, σu =535MPa, 0.023P 0.024S 0.01Cr E=2.10 ·105MPa, ν =0.30, 0.01Ni 0.035Cu JIc =0.331MPa·m Influence of the mean loading... 87 The parameters of the cyclic strain curve are as follows: 10HNAP steel: K′ =832MPa, n′ =0.133 (cyclic softening); 18G2A steel: K′ =869MPa, n′ =0.287 (cyclic hardening). Fig. 1. Shape and dimensions of notched specimen The tested specimens were cut out from a of 10mm thick plate in the rolling direction. Figure 1 shows the dimensions and shape of the specimen. The specimens had an external, unilateral notch which was 5mm deep and ρ = 0.5mm in radius. The notches were made by milling, and the surface of the specimens was ground. The theoretical stress concentration factor in the specimen under bending, Kt = 3.27, was estimated with use of the model by Thum et al. (1960). The tests were done on the fatigue test stand MZGS- 100 (Achtelik and Jamroz, 1982), enabling realisation of cyclic loading with a mean value inducing the plane stress on thematerial surface. The propagating crack lengthwasmeasured by an optical device including a digitalmicrometer and amicroscopic telescopemagnifying 25 times.The fatigue crack lengthwas periodically measured after some thousands of cycles, with an accuracy not less than 0.01mm. Unilaterally restrained specimens were subjected to cyclic bendingmoment of the amplitude Ma =15.64N·m. 3.1. Numerical method The ∆J-integral range, i.e. values of Jmax and Jmin, was numerically de- termined by the finite elementmethod (FEM). For this purpose, the program FRANC2Dwas applied. It allows one to calculate the energy dissipatedduring the fatigue crack growth in an elastic-plastic material. The program includes nonlinear physical relationships obtained from the cyclic stress-strain curve of the materials tested. In this case, the cyclic strain curves for steels 10HNAP and18G2Adescribedby theRamberg-Osgood relationwereused.Calculations were done in an incremental way for flat notched specimens (Fig.1) loaded in a cyclic manner by a bending moment of the constant mean value Mm and amplitude Ma. 88 D.Rozumek Fig. 2. Finite element mesh in the notched area In the place of the notch, the fatigue crack (according to observations)was initiated, which propagated along the cross section of the specimen. Figure 2 shows a division of the area around the crack into finite elements. In the modelling, 6-nodal triangular finite elements were used; the triangles were of different dimensions. Calculations were done under the same loadings as those applied during the tests. 4. Experimental results The paper contains test results which allow one to show phenomena oc- curring in 10HNAP and 18G2A steels during the fatigue crack growth process under bending for different load ratios R. The tests were performed under controlled loading from the threshold value to the specimen failure. The expe- rimental results were presented as graphs of the crack growth rate da/dN versus ∆J-integral range. Fig.3 andFig.4 present graphs of the fatigue crack growth rates versus ∆J-integral range for two kinds of materials, three lo- ad ratios under constant loading, calculated with the finite element method and Eqs (2.1) and (2.2). It has been observed that in 10HNAP and 18G2A steels the stress ratio changes from −1 to 0 as the fatigue crack rate incre- ases, see Fig.3 andFig.4 (graphs 1,2,3). From a comparison of the test results for 10HNAP and 18G2A steels it appears that a slower rate of the increase Influence of the mean loading... 89 Fig. 3. Comparison of results found fromFEM and Eqs (2.1) and (2.2) for 10HNAP steel Fig. 4. Comparison of results found fromFEM and Eqs (2.1) and (2.2) for 18G2A steel 90 D.Rozumek of the fatigue crack growth is observed in the specimens of 10HNAP steel for R = −0.5 and 0. In formula (2.1) the coefficient B and exponent n for 10HNAP steel are B = 3.0 ·10−7MPa·m2/cycle and n = 0.63, respectively. For 18G2A steel they are B = 3.5 · 10−7MPa·m2/cycle and n = 0.62. The coefficients B and n, which should bematerial constants, are in practice de- pendent on other factors, for example the stress ratio R. In the presented tests, a relative error for the exponent n did not exceed 20%, and in the case of B it varied within ±20%. As for 10HNAP steel, the maximum relative error was 17% (Fig.3) under correlation at the significance level α = 0.05, r = 0.98 for R = −1, r = 0.96 for R = 0.5 and r = 0.91 for R = 0. For 18G2A steel the maximum relative error did not exceed 14% (Fig.4) under correlation r = 0.98 for R = −1, r = 0.99 for R = −0.5 and r = 0.98 for R=0. The correlation coefficients assume large values in all considered cases which means that there is significant correlation between the test results and the assumed relationship, see Eq. (2.1). The coefficient B and the exponent n were determinedwith the help of the least squaremethod. Froma comparison of the results obtained bymaking use of the finite element method and those found analytically from Eq. (2.2), it appears that the difference between the obtained results is less than 10%. From the analysis of experimental results (Fig.3 and Fig.4) for 10HNAP and 18G2A steels with different load ratios it appears that formula (2.1) de- scribes the fatigue crack growth rates in the consideredmaterials and loadings in a satisfactory way. Thus, it is possible to consider its applicability to the prediction of the fatigue life according to Eq. (2.4). In this equation, the pa- rameters from the Ramberg-Osgood equation are: α1 =10.74 and n1 =2.26 for 10HNAP steel and α1 = 15.98 and n1 = 2.60 for 18G2A (necessary for calculation of the Jp1-integral according to Neimitz (1998)). The strength properties of the material and the critical values of the JIc-integral given in Table 1 were used in calculations. Having these data, the calculation life was determined for the results from FEM. The results of analyses were shown in the from of graphs comparing the calculation life Ncal with the experimental life Nexp (Rozumek, 2002). InFig.5 andFig.6, the solid linemeansperfect agreement between the cal- culation and experimental lives, but thebroken lines express the intervalwhere the ratio of the calculation life to the experimental life is Ncal/Nexp =3.Con- sidering the calculation results obtained for load ratios R=−1,−0.5, 0 (Fig.5 andFig.6),wecan say thatbothconsidered steels showgoodagreementbetwe- en the calculation and experimental lives in the scatter band Ncal/Nexp =3. Thus, proposed nonlinear formula (2.1) can be applied to analysis of the consi- Influence of the mean loading... 91 Fig. 5. Comparison of calculated and experimental fatigue lives for 10HNAP steel Fig. 6. Comparison of calculated and experimental fatigue lives for 18G2A steel 92 D.Rozumek deredmaterials for determination of the fatigue crack growth rate and fatigue life in the propagation period. 5. Conclusion The tests of fatigue crack propagation in plane notched specimens of 10HNAP and 18G2A steel subjected to cyclic bending show that for load ratios varying from −1 to 0 the fatigue crack rate increases together with a change in R. The fatigue crack growth rate causes drop in the number of cyc- les. Presented empirical formula (2.1) describes the test results well for both the crack growth rate and life. Comparison of the results found from the finite elementmethod andEq. (2.2) indicates only small differences – less than 10%. Acknowledgements Thisworkwas supported by theCommission of theEuropeanCommunities under the FP5, GROWTHProgramme, contract No. G1MA-CT-2002-04058 (CESTI). References 1. 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PickardA.C., 1986,The application of 3-dimensional finite element methods to fracture mechanics and fatigue life prediction, London 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., 2002, Investigation of the influence of specimen geometry, con- centrator type and kind of the material on fatigue life under cyclic bending, Doctoral Thesis, Technical University of Opole (in Polish) 11. TanakaK., 1983,The cyclic J-integral as a criterion for fatigue crack growth, Int. J. Fracture, 22, 91-104 12. Thum A., Petersen C., Swenson O., 1960, Verformung, Spannung und Kerbwirkung, VDI, Duesseldorf Wpływ obciążenia średniego na prędkość wzrostu pękania zmęczeniowego i trwałość przy zginaniu Streszczenie W pracy przedstawiono wyniki badań doświadczalnych wzrostu prędkości pęk- nięć zmęczeniowych przy zginaniu w stalach 10HNAP i 18G2A dla różnych wartości współczynnika asymetrii cyklu R. Do badań użyto próbek płaskich z koncentratorem naprężeń w postaci zewnętrznego jednostronnego karbu ostrego. Wyniki badań do- świadczalnych opisano nieliniowym związkiem zawierającymzakres całki ∆J. Zapro- ponowany związek do opisu prędkości rozwoju pęknięcia zmęczeniowego, zawierający zakres całki ∆J, w sposób zadawalający opisuje wyniki uzyskane doświadczalnie. Manuscript received July 7, 2003; accepted for print September 10, 2003