Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 51, 3, pp. 581-592, Warsaw 2013 FATIGUE LIFE OF METALLIC MATERIAL ESTIMATED ACCORDING TO SELECTED MODELS AND LOAD CONDITIONS Krzysztof Kluger, Tadeusz Łagoda Opole University of Technology, Opole, Poland e-mail: k.kluger@po.opole.pl; t.lagoda@po.opole.pl The authors present results of a fatigue test for specimens made of the aluminium alloy 2017A-T4 and alloy steels S355J2WP and S355J2G3 subjected to constant-amplitude pro- portional combinedbendingwith torsion includingmean stressvalues and for theS355J2WP alloy steel under uniaxial constant-amplitude and random loading with both zero and non- zero mean stress values. The test results were compared with the results of calculations according to the models proposed by Goodman, Gerber and Morrow as well as the stress- strain parameter. In the case of calculations based on the stresses, themultiaxial stress state was reduced to a uniaxial one using the Huber-Mises relationship. As for the method based on strain energy density, the multiaxial stress state was reduced to the uniaxial one with use of the stress-strain parameter. The plane in which the stress-strain parameter of shear loadings reaches its maximum value is assumed to be the critical plane. Key words: uniaxial fatigue, multiaxial fatigue, mean stress 1. Introduction Currently, according to the parameters used in the fatigue criterion, multiaxial fatigue damage models which take into consideration the influence of the mean stress can be mainly classified into three categories, namely the stress or strain-based approaches (Gerber, 1874; Goodman, 1899; Morrow, 1968; Lazzarin and Susmel, 2003; Kluger and Łagoda, 2004) and the energy critical plane criteria approach (Smith et al., 1970; Glinka et al., 1995; Palin-Luc and Lasserre, 1998; Fatemi andSocie, 1988; Papadopoulos, 1998; Łagoda, 2001a,b;Kardas et al., 2008; Sonsino et al., 2004; Macha et al., 2006). The known stress, strain or energy models refer to a specified loading. The stress models are applied under high numbers of cycles, the strain models can be used in the case of low numbers of cycles, and the energy models refer to both high and low numbers of cycles (Łagoda, 2001a,b; Kardas et al., 2008). Moreover, it is not easy to choose an adequate method for calculations. The methods based on stresses or strains are simple and their application does not requ- ire much time, but they do not guarantee adequate accuracy of the results. In such models, for computing fatigue life only, the stress or strain amplitude and its mean value are directly taken into consideration (Gerber, 1874; Goodman, 1899; Morrow, 1968; Lazzarin and Susmel, 2003). On the other hand, the methods based on strain energy density in the critical plane need more time and work, but they give much better results in comparison to the methods based on the stresses (Sonsino et al., 2004). The approaches to fatigue life prediction using the concept of the critical plane have been found very effective because the critical plane concept is based on physical observations that cracks initiate and grow on favourable planes. The value of energy dissipated in the material during one cycle of loading or during all the cycles up to the failure are usually calculated from a history of the changes in cyclic strain and stress together with the 582 K. Kluger, T. Łagoda number of cycles. This energy is connected with the area of the hysteresis loop (σ-ε) in the case plastic strains occur in the material. It is assumed that this area is proportional to the strain energy dissipated in the material during one cycle of loading. The total energy is the sum of areas of the hysteresis loop. In this paper, the authors present the stress-strain parameter (Wσε) based on Lagoda- -Macha’s strain energy density parameter (Kluger and Łagoda, 2007; Łagoda and Ogonow- ski, 2005) and themodels proposed byGoodman (Goodman, 1899; Lazzarin and Susmel, 2003), Gerber (Gerber, 1874; Lazzarin and Susmel, 2003) andMorrow (1968) for estimation of fatigue life of structure elements and machine sub-assemblies under combined bending with torsion, including the mean stress and strain values. The authors present results of calculation and experimentation for the S355J2WPalloy steel under uniaxial constant-amplitude and random loadingwith both zero andnon-zeromean stress values as well. For the registered stress histories in theuniaxial loading for theS355J2WPalloy steel, elastic- -plastic strains were calculated with the incremental kinematic model of material hardening formulated byMroz (1967) and Garud (1982). The paper also contains experimental verification of the considered model based on the obtained fatigue test results and show which model is the best for all kinds of loading. 2. Experimental data Specimens made of the aluminium alloy 2017A-T4 (Kardas et al., 2008) and the alloy steels S355J2WP (Łagoda et al., 2001) and S355J2G3 (Gasiak and Pawliczek, 2001) were tested. Static and fatigue properties of the tested materials are given in Table 1. Table 1. Static and fatigue properties of the analyzed materials Material ε′f c σ′f B K′ n′ E σ0.2 σUTS ν (EN) [MPa] [MPa] [GPa] [MPa] [MPa] 2017A-T4 1.879 −0.988 643 −0.065 617 0.066 72 395 545 0.32 S355J2WP 0.114 −0.420 1012 −0.105 853 0.156 215 414 556 0.29 S355J2G3 2.822 −0.491 1190 −0.143 869 0.287 213 394 611 0.31 For the uniaxial cyclic loading of the S355J2WP alloy steel (Sonsino et al., 2004), the tests were performed for five different stress amplitudes and three levels of the mean loading, σm = 75MPa, 150MPa and 225MPa. The tests were realized under force controlled. Under random loading, the tests were performed for 14 different values of the root mean square of stress σRMS and mean values σm (zero, compressive and tensile). The observation time for random loading was T0 =649s. As for the aluminium alloy 2017A-T4, two combinations of proportional constant-amplitude bending with torsion were considered (with constant mean value), where τ(t) = σ(t) and τ(t) = 0.5σ(t). In the case of steel alloy S355J2WP (with constant mean value) and S355J2G3 (with constant R ratio), only combined proportional constant-amplitude bending with torsion where τ(t)= σ(t) was taken into account. All tests were carried out under bending and torque controlled. In all multiaxial analyses, the authors used the elastic model (high cycle fatigue) in which the mean stress and amplitudes were counted as the nominal stress. Fatigue life of metallic material estimated ... 583 3. Models of fatigue life calculation 3.1. Energy model (Wσε) 3.1.1. Uniaxial constant-amplitude loading The stress-strain parameter (Wσε) under the uniaxial stress state is the base of formulation of the energy model under complex loading states including the mean value. This parameter is defined as Wσε(t)= 1 4 {|σ(t)|[ε(t)−εm]+σ(t)|ε(t)−εm|} (3.1) Theabsolutevalueof σ(t) and ε(t)−εm defines the tensile andcompressivephases of loading. The application of the absolute value of σ(t) and ε(t)−εm in calculations brings about a change of the history of the stress-strain parameter in time in a symmetric waywhile cyclic stresses and strains change in relation to the mean values. Figure 1 shows the histories of the stress-strain parameter Wσε(t) with and without the absolute value of σ(t) and ε(t)−εm W∗(t). From the graphs, it appears that application of the absolute value of σ(t) and ε(t)−εm reduces themean value of Wm. Fig. 1. Exemplary histories of the stress-strain parameter W∗(t), W(t) for aconstant-amplitude loading: (a) W∗(t)= 0.5σ(t)[ε(t)−εm], (b) Wσε(t)=0.5{|σ(t)|[ε(t)−εm]+σ(t)|ε(t)−εm|} The stress-strain parameter was based on themultiaxial Łagoda-Macha model. Only in one case of loading (uniaxial tension-compression for small elastic-plastic strain and σm > 0), stress- -strainparameter (3.1) is in away similar to theSmith-Watson-Topperparameter (SWT)(Smith et al., 1979) according to the following equation Weq = 1 2 PSWT = 1 2 σmaxεa = 1 2 (σa+σm)εa (3.2) In other case of loadings (bending, torsion, combination bending and torsion) the stress- -strain parameter works differently than SWT. The absolute value of |σ(t)| and |ε(t)− εm| changes history of the stress-strain parameter (see Fig. 1). If the elastic-plastic strain is higher, than difference between SWT and Wσε is higher, too. The transformed amplitudes Weq,aT of the strain energy density parameter for tensile and compressive states were calculated from the following formula Weq,aT = (σa+ψσm)εa 2 =        (σa+σm)εa 2 = Wa+Wm 2 for σm ­ 0 ∧ ψ =1 σaεa 2 = Wa 2 for σm < 0 ∧ ψ =0 (3.3) where Wa = σaεa and Wm = σmεa. 584 K. Kluger, T. Łagoda The number of cycles to failure was calculated according to Eqs. (3.1), (3.6), (Nf = Ncal) Weq,aT = 1 4 [|σa+ψσm|εa+(σa+ψσm)|εa|] = σ′f 2 2E (2Nf) 2b+ 1 2 ε′fσ ′ f(2Nf) b+c (3.4) Fig. 2. Algorithm of fatigue life determination for a uniaxial constant-amplitude loading 3.1.2. Uniaxial random loading Figure 3 shows the algorithm for the determination of the fatigue life of S355J2WP steel according to the stress-strain parameter. Fig. 3. Algorithm of fatigue life determination for a uniaxial random loading The rain flow algorithm (Downing and Socie, 1982, [1]) was used for determination of am- plitudes W (i) a andmean values W (i) m of cycles and half-cycles. The transformed amplitudes of the strain-stress parameter including the mean load values could be calculated from the distinguished amplitudes of cycles of the parameter Wa. The correspondingmean values Wm are determined from the history according to WaT = { Wa+Wm for Wm ­ 0 Wa for Wm < 0 (3.5) where Wm is determined from the rain flow algorithm. Fatigue life of metallic material estimated ... 585 For cyclic loadings, (3.5) canbeexpressedbymeans of the stress criteria. For elasticmaterials under a high number of cycles to failure, the transformed amplitude of the strain energy density parameter in the stress approach can be written as σaT = √ (σm+σa)σa (3.6) This notation conforms with the models based on the Smith-Watson-Topper parameter (PSWT) (Smith et al., 1970). The amplitudes of elastic-plastic strains are obtained from the Ramberg-Osgood relationship, and in the stress approach can be expressed as σaT 2 [ σaT E + (σaT K′ ) 1 n′ ] = (σa+σm)σa 2E + σa+σm 2 (σa K′ ) 1 n′ (3.7) As a result, it is possible to determine the requested value σaT numerically. Differences between the elastic-plastic strain and the elastic strain depend on the loading level. For determination of the damage degree, the Palmgren (1924) andMiner (1945) hypothesis was used S(T0)= n ∑ i=1 1 N (i) f (3.8) where N (i) f is determined from Eq. (3.9) for the transformed amplitudes W (i) aT W (i) aT = σ′f 2 2E ( 2N (i) f )2b + 1 2 ε′fσ ′ f ( 2N (i) f )b+c (3.9) and fatigue life determination was realized according to the following relationship Tcal = T0 S(T0) (3.10) where T0 is the observation time. 3.1.3. Multiaxial constant-amplitude loading In the presented stress-strain parameter approach (3.1), the mean stress value strongly in- fluences the fatigue life and themean strain value does not influence the fatigue process. Under multiaxial loading, the material effort is determined by the maximum value of the linear combination of the stress-strain parameters Wη(t) (normal loadings) and Wηs(t) (shear loadings). It leads to the equation for the equivalent value Wσε,eq in the form Wσε,eq(t)= βWηs(t)+κWη(t) (3.11) where β is the constant value for selection of a particular form (Łagoda andOgonowski, 2005), see (3.14)2, κ –material constant obtained from uniaxial fatigue tests (Łagoda andOgonowski, 2005), see (3.14)1 Wη(t)= 1 4 {|ση(t)+σηm|[εη(t)−εηm]+ [ση(t)+σηm]|εη(t)−εηm|} Wηs(t)= 1 4 {|τηs(t)|[εηs(t)−εηsm]+ τηs(t)|εηs(t)−εηsm|} (3.12) and ση –normal stress to the critical plane (Fig. 4), σηm –meanvalue of thenormal stress in the critical plane (Fig. 4), εη – normal strain in the critical plane (Fig. 4), τηs,εηs =0.5γηs – shear 586 K. Kluger, T. Łagoda Fig. 4. Orientation of particular stress and strain on the critical plane (Łagoda and Ogonowski, 2005; Macha et al., 2006) stress andahalf of the shear strain in thecritical plane (Fig. 4), respectively, εηm,εηsm =0.5γηsm – mean normal strain and a half of the shear strain in the critical plane (Fig. 4), respectively. In this case, the critical plane is determined by themaximumvalue of the shear stress-strain parameter max{Wηs(t)}. For high cycle fatigue (HCF), stress amplitudes are calculated directly from loading history (for example for bending as nominal stress amplitude). Themean value of stress is determined as a global expected non-zero value from the load history (Łagoda, 2001a,b). To determine the strain amplitudeand itsmeanvalue inHCFwereplastic strains are almost zero,we can calculate it from the stress history. In the case the strain history was registered, the stress history could be derived from calculation in the reverse direction. The equivalent value of the shear loading according toEq. (3.11) for the critical plane defined by the maximum stress-strain parameter takes the form Wσε,eq(t)= 1 4 β{|τηs(t)|[εηs(t)−εηsm]+ τηs(t)|εηs(t)−εηsm|} + 1 4 κ{|ση(t)+σηm|[εη(t)−εηm]+ [ση(t)+σηm]|εη(t)−εηm|} (3.13) The analysis of the stress and strain states for pure torsion and alternating bending under constant-amplitude loadings offers grounds for determination ofweighted coefficients for the par- ticular components in the combination. Theweights can bewritten as (Łagoda andOgonowski, 2005) κ(Nf)= 4−k(Nf) 1−ν β(Nf)= k(Nf) 1+ν k(Nf)= (σa(Nf) τa(Nf) )2 (3.14) Depending on the constant k and the assumed lifeNf determined fromEq. (3.14)3, it is possi- ble to derive special forms of the criteria presented and verified in Łagoda (2001a,b), Karolczuk et al. (2002). Inageneral case, the coefficient k(Nf) (ŁagodaandOgonowski, 2005) is givenby the relation between the amplitude of the normal stress σa and amplitude of the shear stress τa for a given number of cycles. The values σa(Nf) and τa(Nf) are calculated from the fatigue curves S-N for simple loadings: tension (bending), shearing (torsion). If there are nodistinct irregularities in the S-N curves (σa-Nf, τa-Nf), for the purpos of simplificationwe can assume that k(Nf)= const, e.g. 105 or 106 cycles, or usually for the fatigue limit level. The plane where the stress-strain parameter of shear loading Wηs assumes its maximum value as the critical plane for the aluminum alloy 2017A-T4 and both steels S355J2WP and S355J2G3. The proposed algorithm includes the effect of the mean values of stress and strain during fatigue life calculations. Fatigue life of metallic material estimated ... 587 Fig. 5. Algorithm of fatigue life determination for multiaxial constant-amplitude loading Table 2.Computational data for considered materials under multiaxial loading Material k β κ S355J2WP 3.63 2.77 0.53 S355J2G3 2.14 1.63 2.70 2017A-T4 2.79 2.11 1.78 Table 2 contains values of the coefficients k, β and κ fromEqs. (3.14). In this case, parallel fatigue characteristics were assumed for bending and torsion. 3.2. The Goodamn, Gerber and Morrow models The stressmodels are often applied by engineers. In this paper, they are used for the purpos of comparison of the existing models. These models describe the boundary between the mean value of cycle, its amplitude andmaterial constants, the calculations are quite simple and they do not require much time. The applied models were formulated by: —Goodman σa σaT + σm Rm =1 (3.15) —Gerber σa σaT + (σm Rm )2 =1 (3.16) —Morrow σa σaT + σm σ′f =1 (3.17) where σaT is the normalized stress amplitude. The familiar andwidely usedHuber-Mises criterionwas applied for reduction of the complex stress state, namely bending combined with torsion. A number of cycles to failure was calculated from Nf =10 A−m logσaTH−M (3.18) 588 K. Kluger, T. Łagoda where m is the coefficient of the S-N curve slope (for bending), A – constant of the fatigue curve regression (for bending), σaTH−M – transformed amplitudes related to the stress mean value under the uniaxial stress state. The relationship abovewas obtained fromconversion of the knownS-N fatigue characteristic S-N [2] logNf = A−m logσa (3.19) on the assumption that σa = σaTH−M. In the stressmodels, just as in the case of the energymodel, after reduction fromthe complex stress state to theuniaxial one, theobtained resultswere comparedwith the simple state, namely bending in the considered case. In the case of random loading, for fatigue life determination, the cycle countingmethodwas adopted. In this method, schematization of the random stress historywas undertaken bymeans of the rain flow algorithm, damages were accumulated according to Palmgren (1924) andMiner (1945) hypothesis including the influence of amplitudes less than the fatigue limit σaf SPM(To)= k ∑ i=1 ni N0 ( σaf σai )m (3.20) The damage degree S(To) at observation time To of stress historywas calculated by summation of damages from successive amplitudes of cycles and half-cycles σai according to relationship (3.10). 4. Verification of the models A statistical analysis understood as ameasure of usability was performed for all the considered models and materials. The analysis included determination of the mean scatter expressed by (Macha et al., 2006) TN =10 E (4.1) where Ei = log Nexpi Ncali E = 1 n n ∑ i=1 Ei (4.2) and the scatter coefficient expressed as TN =10 tn−1α 2s (4.3) where s is the standard deviation defined according to s = √ s2 s = 1 n−1 n ∑ i=1 (Ei−E)2 (4.4) The required confidence level for each desired quantity, for most applications, at the level of 95% is retained as the default value, the significance level is usually assumed as α = 5% ([20], Sutherland and Veers, 2000). Thus, the mean value should be included in the interval −t(n−1),α/2(s) ¬ E ¬ t(n−1),α/2(s), where t(n−1),α/2 is the constant from t-Student’s distribu- tion. Fatigue life of metallic material estimated ... 589 The constant t(n−1),α/2 from t-Student’s distribution is determined for a half of the signifi- cance level α/2 because of the boundary section of the normal distribution. Figures 6-8 show a comparison between the calculated and experimental fatigue lives for all the considered materials. The solid line means ideal conformity of the results. The dashed lines represents a scatter bandwith a coefficient of 3, i.e. Nf exp/Nf cal =3 (1/3). In the case of stress methods, the large scatter of the results is attrributable to small material constants which were used in the calculations (σTS,σ ′ f). In the case of the energymodel,morematerial constants and strain history were applied in the calculations. This data offered better results of calculation as we can see in Table 3, however, it has a big influence on the calculation time (longer calculation time as in the stress models). Fig. 6. Comparison of the calculated and experimental fatigue lives for specimens made of alloy steel S355J2WPunder tension-compression cyclic loading (a) and random loading (b) Fig. 7. Comparison of the calculated and experimental fatigue lives for specimens made of aluminium alloy 2017A-T4 under combined bending with torsion; (a) τ(t)= 0.5σ(t), (b) τ(t)= σ(t) FromTable 3 it appears that themodels for fatigue life estimation basedonly on stresses have a highermean scatter. The perfect conformity of the results is the case for T =1. Themajority of the results are found in the safe region as calculation results are greater than the experimental, but when T is too big, the mechanical values are no longer familiar with them. TN coefficient 590 K. Kluger, T. Łagoda Fig. 8. Comparison of the calculated and experimental fatigue lives for specimens made of alloy steel S355J2WPunder combined bending with torsion; (a) τ(t) = σ(t), (b) τ(t) = σ(t) Table 3. Statistic analysis of the considered models Uniaxial loading Multiaxial loading S355J2WP S355J2G3 S355J2WP 2017A-T4 cylic random τ = σ τ =0.5σ T TN T TN T TN T TN T TN T TN T TN Goodman 3.31 6.44 1.51 47.59 2.63 4.43 4.65 10.69 8.96 3.28 20.37 5.85 9.15 6.06 Gerber 3.22 2.93 2.53 6.11 7.20 3.90 24.36 8.71 21.75 4.11 50.31 9.99 25.91 6.68 Morrow 1.25 2.65 2.12 5.31 3.74 3.80 15.22 8.91 15.31 3.84 38.70 9.14 18.24 6.42 Wσε 1.02 2.94 1.04 3.75 1.04 4.79 1.02 2.65 1.15 3.52 1.04 3.28 1.06 3.56 describes the range of the scatter band. If TN assumes a higher value, the calculation method is worse. Table 4 show comparisons the advantages and faults of the considered models. Table 4.Usability analysis of the considered models Goodman Gerber Morrow Wσε Usable in multiaxial loading in base form no no no yes Usable in uniaxial loading in base form yes yes yes yes Easy to use yes yes yes no Time counting in cyclic loading short short short short Time counting in random loading short short short long Scatter band in multiaxial cyclic loading medium poor poor good Scatter band in uniaxial cyclic loading medium good good good Scatter band in uniaxial random loading poor medium medium good 5. Conclusions The undertaken verification of the energy model offered satisfactory results regarding the com- parison of the calculated and experimental data for aluminum alloy 2017A-T4 for combined bending with torsion including mean values. The calculated results are included in the scatter band for tests of cyclic bending with the zero mean value. Fatigue life of metallic material estimated ... 591 The application of Wσε including the influence of the stress mean value for fatigue life determination of steels S355J2WP and S355J2G3 makes it possible to obtain results close to those obtained from tests. The energy model gives satisfactory results because, in this research, a low mean scatter T and a small scatter band TN has been noted. The Goodman model seems to be the most appropriate for the considered materials, and the Gerber model seems to be the worst one. Models based only on stress history should only be used in engineering calculations, in the design process of machines which do not bear heavy loads. Concurrently, when we have to do with structures or parts of machines that are crucial for human life, designers should use new methods based on the stress and strain history together. 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