JOURNAL OF THEORETICAL AND APPLIED MECHANICS 42, 2, pp. 285-294, Warsaw 2004 INFLUENCE OF THE MATERIAL SENSITIVITY FACTOR ON THE STRESS RATIO FOR DIFFERENT SPECIMENS GEOMETRIES AND MATERIALS UNDER BENDING Roland Pawliczek Dariusz Rozumek Faculty of Mechanical Engineering, Technical University of Opole e-mail: rolandp@po.opole.pl; drozumek@po.opole.pl In the paper, the influence ofmean loads on the lowcycle fatigue for not- ched specimens of different shapes of the cross section is analysed. The specimens are characterizedbya similar value of the stress concentration factor (Kt =2.46 for round specimens and Kt =3.27 for specimens of the rectangular cross-section). A mathematical model describing the li- miting stress plane under fatigue bending is applied, taking into account the influence of mean stresses. The model has been verified by 18G2A steel and PA6 aluminium alloy specimens via experimental test results under cyclic bending for stress ratios R =−1,−0.5, 0. The relationship between the stress amplitude and the stressmeanvalue taking into acco- unt the number of cycles N before failure is described by a linearmodel, for which good agreement of the calculationswith the experimental test results is achieved. Key words: stress ratio, numbers of cycles, bending Notations R – stress ratio σa – amplitude of stress σm – mean stress N – number of cycles to failure σy – yield stress σu – ultimate stress E – Young’s modulus ν – Poisson’s ratio 286 R.Pawliczek, D.Rozumek 1. Introduction Additional static loadings occurring in structure elements strongly influen- ce fatigue behaviour of these elements (Glinka et al., 1995). From the known literature it appears that material reactions to the occurring additional me- an loading can be different (Stephens et al., 2001). Morrow (1968) proposed a widely used relation including the influence of the mean stress value and proved strong influence of this stress on the fatigue life. Application of the Morrow equation to some materials was discussed by Manson (1975). The author proved that the formula was useful for fatigue tests with mean stress values. Description of the influence of themean loading often uses a coefficient expressing the material sensitivity to the cycle asymmetry. That coefficient is applied for formulation of a simplified Haigh graph, showing the dependen- ce between the stress amplitude and the cycle mean value (Stephens et al., 2001). Its value can be determined from tests. Serensen (1975) joined the safety coefficient related to the mean stresses with the coefficient of sensiti- vity to the cycle asymmetry. Bergman and Seeger (1979) introduced an ad- ditional coefficient including material sensitivity to the influence of the mean stress into the Smith-Watson-Topper energy equation. Ellyin and Kujawski (1993) modified the criterion presented by Ellyin for the critical energy of non-dilatational strain. In the paper, the coefficient of sensitivity to the cycle asymmetry was introduced. The criterion was verified for A516 steel in the form of cylindrical thin-walled specimens subjected to a combination of ten- sion with internal and external pressure applied to the specimen walls. The coefficient of sensitivity to the cycle asymmetrywas also includedbyGołoś and Esthevi (1997). In this paper, the presented criterion was verified by fatigue test results for St5 steel. Cylindrical thin-walled specimenswere tested under tensionwith internal and external pressure acting on the specimenwalls. The tests perfor- med according to this criterion gave satisfactory results as comparedwith the experimental results.The coefficient ofmaterial sensitivity presentedbyHaigh is right for the high-cycle fatigue limit and it takes a constant value. From the results obtained by Pawliczek and Rozumek (2003) we can draw a conclusion that the material sensitivity to the cycle asymmetry depends not only on the material. It is also dependent on the number of cycles to failure for both low and high cycle ranges. Theaimof this paper is to determine thepermissible fatigue loading of spe- cimensmade of 18G2A steel andPA6 aluminiumalloy for different geometries of cross-sections and stress ratios. Influence of the material sensitivity factor... 287 2. A mathematical model of the limiting stress surface In the case of a cyclic loading with participation of a mean stress, the limiting stress surface is presented in the coordinate system σa − σm − N. Pawliczek (2003) proposed to describe the mean stress value with the use of a function of the change in the factor of the material sensitivity to the stress ratio depending on the number of cycles N to failure. This function can be written as ψ(N)= γNλ (2.1) where γ, λ are parameters which are determined from fatigue tests under alternating loading (R =−1) and pulsating loading (R =0), N denotes the number of cycles to failure. Values of the factor of thematerial sensitivity to the stress ratio ψ(N) for a given N can be determined from experiments carried out according to the following formula ψ(N)= 2σa −1 (N)−σa0(N) σa0(N) (2.2) where σa −1 (N) is the stress amplitude for the fatigue life level corresponding to N cycles for loadings with the stress ratio R = −1 (symmetric cycles), σa0 is the maximum stress for this level and loadings with the stress ratio R =0 (pulsating cycles). Assuming a linear relationship between the stress amplitude and themean stress σa = σa −1 (N)−ψ(N)σm andusingEq. (2.1) for description of the factor of thematerial sensitivity to the stress ratio togetherwith thenumberof cycles to failure, we obtain a relationship for calculation of the stress amplitude σa = σa −1 (N)−γNλσm (2.3) TheWõhler equation for symmetric loadings (R =−1) takes the following form ( σa −1 (N) )k N = σkafN0 (2.4) where σaf is the fatigue limit, k – coefficient of the regression equation, N0 – limiting number of cycles. Substituting expression (2.4) to relationship (2.3), we obtain σa(σm,N)= (σkafN0 N ) 1 k −γNλσm (2.5) Equation (2.5) expresses a mathematical model of the limiting stress surface determined from the fatigue test results obtained for symmetric and pulsating 288 R.Pawliczek, D.Rozumek courses on the assumption that the influence of the mean value on the stress amplitude takes the form of a linear function. 3. Materials and the test procedure The subject of the investigations are construction steel 18G2A and PA6 aluminium alloy. Round specimens were cut off from a drawn bar, flat spe- cimens were cut off from a sheet, according to the rolling direction. Their chemical compositions andmechanical properties are given in Table 1. Table 1.Characteristics of 18G2A steel and PA6 aluminium alloy Material Chemical compositions [%] Mechanical properties 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 PA6 4.15Cu 0.69Mg 0.45Si σy =395MPa, σu =545MPa, 0.65Mn 0.7Fe 0.2Ti E =7.2 ·104MPa, ν =0.32 0.5Zn 0.1Cr The fatigue testswere performedat the fatigue test standMZGS-100 (Ach- telik andJamroz, 1982) enabling realization of cyclically variable stress courses with participation of a mean stress. The tests included sinusoidally variable bending with participation of a loading mean value. The tests were done for the stress ratios R =−1,−0.5, 0, changing the amplitude of themoment Ma and the mean values of the moment Mm for bending. The tests were perfor- med for round and flat notched specimens (Fig.1). The notches weremade by machining (turning andmilling), and next the specimen surfaceswere ground. The theoretical stress concentration factor in the round specimens Kt =2.46 and flat specimens Kt =3.27, were estimated with use of themodel byThum et al. (1960). The regression equations for specimens made of 18G2A steel and PA6 aluminiumalloy subjected to cyclic bending for various stress ratios are shown in Table 2. From the performed tests for R = −1 and R = 0, Eq. (2.1) was determined under bending for 18G2A steel and round smooth specimens ψ(N) = 3.124N−0.162, for round notched specimens ψ(N) = 1.363N−0.105, for plane notched specimens ψ(N) = 0.046N−0.197 and for PA6 aluminium alloy and plane notched specimens ψ(N) = 0.238N−0.076. Different values of Influence of the material sensitivity factor... 289 Fig. 1. Specimens subjected to tests, (a) round, (b) round notched, (c) flat notched the coefficients of the equation ψ(N) prove that the influence of the cross section geometry and the material on the fatigue process intensity is signifi- cant. The correlation coefficients r for the equations presented inTable 2were calculated with the least square method. In all considered cases, the correla- tion coefficients take great values ranging from 0.97 to 0.99, which bespeaks a significant correlation of the test results. Table 2. Parameters of the regression equation for bending in round and flat specimens R Round specimens – 18G2A Round notched specimens – 18G2A −1 logN =23.93−7.19logσa logN =19.94−6.02logσa −0.5 logN =23.71−7.40logσa logN =22.46−7.34logσa 0 logN =31.40−10.73logσa logN =21.86−7.23logσa R Flat notched specimens – 18G2A Flat notched specimens – PA6 −1 logN =19.96−6.70logσa logN =17.33−6.05logσa −0.5 logN =14.19−4.27logσa logN =14.37−4.79logσa 0 logN =14.35−4.52logσa logN =14.44−5.13logσa 4. Experimental results and discussion Figures 2 and 3 show the test results in the form of a surface of boundary stresses for the consideredmaterials anddifferent cross section geometries, and 290 R.Pawliczek, D.Rozumek for smooth and notched specimens, obtained from the regression equations presented in Table 2. Moreover, Fig.2 and Fig.3 show theoretical contours of the stress amplitudes σa in the plane of the material life N and the stress ratio R, calculated fromEq. (2.5). Figures 2 and 3 also show the relative error distribution determined in an analytical way in relation to the experimental amplitude. In the case of the round smooth specimens (Fig.2a) for N = 105 cycles, a change in the stress ratio from R = −1 to R = 0 causes a large decrease in the permissible stress amplitude (from440 to 280MPa). The stress amplitude decreases as R increases and is almost uniform in the whole tested range. However, for N > 1.5 · 106 cycles, a change in the stress ratio from −1 to −0.5 causes large drops of the permissible stress amplitudes, and any further increase of themean stress value (R =−0.5 to 0) hardly influences the stress amplitudes. Analysing the round notched specimens (Fig.2b), we can observe that for N =105 cycles, the change of the stress ratio from R =−1 to R =0 takes place similarly as in the case of the specimens shown inFig.2a. Fig. 2. The limiting stress surface under bending determined from tests, the theoretical contour and error distribution, (a) round specimens – 18G2A, (b) round notched specimens – 18G2A Influence of the material sensitivity factor... 291 Lower stress amplitudes are caused by the notch (decrease of the stress amplitude from 280 down to 215MPa). For N > 1.5 ·106 cycles, a change in the stress ratio from −1 to 0 causes uniform linear drops of the stress am- plitude. Further results obtained for flat notched specimens made of 18G2A steel (Fig.3a) had the maximum stress amplitudes of 170MPa. It can be ob- served that for N = 105 cycles a change in the stress ratio from R = −1 to R = −0.5 causes a small reduction of the stress amplitude, and for R in the range of −0.5 to 0, we observe large changes of the stress amplitude from about 167 down to 115MPa. For N > 1.5 ·106 cycles, a change in the stress ratio from −1 to −0.5 and from −0.5 to 0 causes uniform linear drops of the stress amplitude. Fig. 3. The limiting stress surface under bending determined from tests, the theoretical contour and error distribution, (a) flat notched specimens – 18G2A, (b) flat notched specimens – PA6 In the case presented in Fig.3b, we can find that for N = 105 cycles a change in the stress ratio from R = −1 to R = 0 causes linear reduction of the stress amplitude from 100 down to 67MPa. Similar behaviour is observed for N > 1.5 ·106 cycles. From the analysis of the experimental results it also appears that for flat notched specimens (Fig.3) comparedwith round smooth 292 R.Pawliczek, D.Rozumek andnotched specimens (Fig.2) we can observe smaller differences of the stress amplitudes passing from N = 105 to N = 2.5 · 106 cycles. From the fatigue test results for 18G2A steel and PA6 aluminium alloy, if the boundary stress surface is assumed according to Eq. (2.5), the equations of that surface have been determined for the considered cases: — for round specimens (18G2A) σa = ( N 1023.93 ) −1 7.19 −3.124N−0.162σm (4.1) — for round notched specimens (18G2A) σa = ( N 1019.93 ) −1 6.02 −1.363N−0.105σm (4.2) — for flat notched specimens (18G2A) σa = ( N 1019.96 ) −1 6.70 −0.046N0.197σm (4.3) — for flat notched specimens (PA6) σa = ( N 1017.33 ) −1 6.05 −0.238N−0.076σm (4.4) From the above equations, it results that the specimen geometry and the kind of the material strongly influence the life of the tested element. The greatest changes of the regression equation coefficient mare observed for roundsmooth specimens. In the other cases, the coefficients m have similar values (surface equation – Eq. (2.5)). Also, the coefficient of the material sensitivity to the cycle asymmetry changed – the highest fatigue parameters γ, λ (Eq. 2.5) are observed for round smooth specimens and the least ones – for flat notched specimens (18G2A steel). Themaximum relative error (Fig.2a and Fig3) was 10% under N > 1.5 · 106 cycles, and for round notched specimens (Fig.2b) that error was 5% for N ≈ 105 cycles. 5. Conclusion From the analysis of the obtained test results we can draw the following conclusions: Influence of the material sensitivity factor... 293 • The proposed mathematical relation, (2.5), well describes boundary stress surfaces under bendingwhere themaximumerror does not exceed 10% for both kinds of specimen geometries andmaterials. • A change in the stress ratio from R = −1 to R = −0.5 for 105 cycles for round specimens of 18G2A steel and flat specimens of PA6 alumi- nium alloy causes a significant drop of the stress amplitude, but for flat specimens of 18G2A steel a certain insensitivity can be observed. • In the range from R =−0.5 to 0 and N = 105 cycles for round speci- mens, we observe a small drop of the stress amplitude, but in the case of flat specimens greater drops of the permissible stress amplitudes are observed. • For the fatigue life 2.5 · 106 cycles and both round and flat specimens similar behaviour of the material can be observed. • Passing from the fatigue life 105 cycles to 2.5·106, we observe significan- tly smaller differences in the stress amplitude reduction in flat specimens as compared with the round ones. Acknowledgements This workwas supported by the Commission of the EuropeanCommunity under the FP5, GROWTHProgramme, contract No. G1MA-CT-2002-04058 (CESTI). References 1. Achtelik H., Jamroz L., 1982, Patent PRL nr 112497, CSR nr 200236 i HDR nr 136544, (in Polish) 2. Bergmann J.W., Seeger T., 1979, On the influence of cyclic stress-strain curves, damage parameters and various evaluation concepts on the prediction by the local approach, Proc. of 2nd European Coll. on Fracture, Darmstadt, FRG, 18 3. Ellyin F.,Kujawski D., 1993,Amultiaxial fatigue criterion includingmean- stress effect,Advances in Multiaxial Fatigue, D.L.McDowell andR.Ellis, Eds., American Society for Testing and Materials, Philadelphia, ASTM STP 1191, 55-66 4. Glinka G., Shen G., Plumtree A., 1995, Mean stress effect in multiaxial fatigue,Fatigue andFractureEngineeringMaterials andStructures,18, 755-764 294 R.Pawliczek, D.Rozumek 5. GołośK., Esthewi S., 1997,Multiaxial fatigue andmean stress effect of St5 medium carbon steel, Proc. of 5h Int.Conf.on Biaxial/Multiaxial Fatigue and Fracture, Eds. E.Macha and Z.Mróz, Cracow, Poland, 1, 25-34 6. Manson S., 1979, Inversion of the strain-life and strain-stress relationships for use inmetal fatigue analysis,Fatigue of Eng.Materials and Structures,1, 37-57 7. Morrow J., 1968, Fatigue properties in metal, Fatigue Design Handbook, Advances in Engineering, Society of Automotive Engineers, 4, 21-29 8. PawliczekR.,RozumekD., 2003, Influence of stress ratioon life under cyclic bending for different specimens geometry,Proc. of 5th International Conference on Low Cycle Fatigue, Abstracts, DVM, Berlin, Germany, P82, 2ps 9. Pawliczek R., 2003, Application of the modified linear function for mean stress effect description in fatigue of materials,Proc. of 2nd Youth Symposium on Experimental Solid Mechanics, Milano-Marittima, University of Bologna, Bologna, Italy, 133-134 10. Sersnsen S.V., 1975, Soprotivlenie materialov ustalostnomu i khrupkomu rozrusheniyu, Atomizdat, Moskva 11. Stephens R.I., Fatemi A., Stephens R.R., Fuchs H.O., 2001, Metal Fatigue in Engineering, Wiley-Interscience Publication JohnWiley & Sons 12. Thum A., Petersen C., Swenson O., 1960, Verformung, Spannung und Kerbwirkung, VDI, Düsseldorf Wpływ współczynnika wrażliwości materiału na asymetrię cyklu dla różnych geometrii i materiałów próbek przy zginaniu Streszczenie W pracy przeanalizowanowpływ obciążeń średnich przy niskocyklowym zmęcze- niu dla próbek z karbem o różnych kształtach przekroju poprzecznego. Próbki cecho- wały się zbliżonąwartością współczynnika kształtu (Kt =2.46 dla próbek okrągłych i Kt =3.27 dla próbek o przekroju prostokątnym). Zastosowanomodel matematycz- ny opisujący płaszczyznę naprężeń granicznych przy cyklicznie zmiennym zginaniu z uwzględnieniemwpływu naprężeń średnich. Model zweryfikowanowynikami badań eksperymentalnych próbek ze stali 18G2A i stopu aluminium PA6 w warunkach cy- klicznego zginania dla przebiegówowspółczynniku asymetrii cyklu R =−1;−0,5; 0. Zależność amplitudy naprężeń od wartości średniej naprężeń z uwzględnieniem licz- by cykli N do zniszczenia opisano linowym modelem, dla którego uzyskano dobrą zgodność obliczeń z wynikami badań eksperymentalnych. Manuscript received November 13, 2003; accepted for print January 13, 2004