Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 50, 4, pp. 1073-1086, Warsaw 2012 50th Anniversary of JTAM INTEGRAL FATIGUE CRITERIA EVALUATION FOR LIFE ESTIMATION UNDER UNIAXIAL COMBINED PROPORTIONAL AND NON-PROPORTIONAL LOADINGS Dariusz Skibicki, Łukasz Pejkowski University of Technology and Life Sciences in Bydgoszczy, Faculty of Mechanical Engineering, Bydgoszcz, Poland e-mail: dariusz.skibicki@utp.edu.pl; lukasz.pejkowski@utp.edu.pl The paper presents a review and verification of integral fatigue criteria. The review signals the key assumptions and criteria structure elements. The verification has been developed drawing on the experimental data reported in literature containing fatigue life for uniaxial, combined proportional and non-proportional loads. The verification involves a comparison of computational fatigue life with the experimental one. To determine the quality of the results generated, statistical parameters were used. As a result of the analysis the best and the worst criteria were pointed to. Key words: multiaxial fatigue, fatigue life, fatigue criteria, integral approach 1. Introduction A continuous attempt at cutting down machinery manufacturing and operation costs can be seen in the changes in the engineering design strategy from Infinite-Life Design through Safe- Life Design to Damage-Tolerant Design. The need to minimize the costs results in successive design strategies,withmoreandmoreprecise calculationmodels at their disposal, demonstrating lower and lower safety coefficient values. Bearing that inmind, a change in themachinery design strategy can trigger structure damage not found earlier. It surely concerns the effects of the non- proportional fatigue load. What is characteristic for that kind of load is rotation of the main axes of stresses and deformations throughout the fatigue process. The rotation of themain axes activates many slip systems and can have an essential effect on fatigue properties. Depending on the material type and the degree of load non-proportionality, this type of forced behaviour can result in even a 10-fold decrease in fatigue life (Ellyin et al., 1991; Socie, 1987) and a 25% decrease in fatigue limit (McDiarmid, 1987; Nishara and Kawamoto, 1945). It is assumed that the right approach to defining the fatigue criteria under non-proportional load conditions can be the integral approach (Weber et al., 2004). It is based on the assumption that for the right fatigue behaviour evaluation it is necessary to integrate the value of thedamage parameter in all the planes going through the material point considered. The aim of this paper is to evaluate the possibility to evaluate fatigue life with the use of fatigue criteria. The analysis was made applying the three most frequent integral criteria: the Zenner criterion (Zenner, 1983; Zenner et al., 2000) and the two Papadopoulos criteria (Papadopoulos, 1994, 2001). The results were compared with the McDiarmid fatigue criterion (McDiarmid, 1992), based on the competitive to the integral approach to critical plane and, commonly applied inmany fields of material fatigue, namely theHuber-Mises-Hencky criterion. Interestingly, there are many reports offering the analysis or computational verification of fatigue criteria. Themost essential reports of that type include e.g., the report byGarud (1981) with an extensive description of the computational models developed until 1981 and the paper by You and Lee (1996), with a presentation of the criteria developed 1980 through 1995. There 1074 D. Skibicki, Ł. Pejkowski are also papers available on specific groups of criteria, e.g. the reports by Macha and Sonsino (1999) on theenergycriteria andthe reportbyKarolczukandMacha (2005), beingadiscussionof criteriabasedon thecritical plane idea.Ahigh studyvalue isprovidedbythecomparative studies of multiaxial criteria including their computation verification, e.g. the paper by Papadopoulos et al. (1997), Wang and Yao (2004), Niesłony and Sonsino (2008), Walat et al. (2012) as well as by Łagoda and Ogonowski (2005). None of the above studies, however, focuses on integral criteria. The criteria covered by this analysis have been verified drawing on the results of the experi- mental tests of 7075-T651 aluminiumalloy (Mamiya et al., 2011), 1045 steel – for the data repor- ted inMcDiarmid (1992) aswell asVerremanandGuo (2007), and the tests ofX2CrNiMo17-12-2 steel (Skibicki et al., 2012). The types of materials have been selected in terms of their various sensitivity to non-proportional load; the lowest value for aluminium, average for carbon steels and the highest value for austenitic steels (Socie andMarquis, 2000). The experimental data derived from those papers provide fatigue life values for sinusoidally variable loads: uniaxial, namely tensile-compressive (markedwith R) aswell as torsion (S), pro- portional combined loads, namely compliant at the phase of tensile-compressive and torsion (P) and non-proportional combined loads obtained as a result of a simultaneous tensile-compressive and torsion with the phase shift equal 90◦ (N). For combined loads P and N, the ratio of the amplitudes of shear to normal stress is an important load-defining parameter λ= τxya σxa (1.1) Further in this paper, a description of the criteria analysed, the method of analysis of the calculation results, analysis of the load results and conclusions are to be found. 2. Description of the criteria analysed 2.1. McDiarmid criterion TheMcDiarmid criterion involved the use of the critical plane approach. In the case of that criterion, it is the plane determined by the tangent stress of the highest value τmax. To calculate the limit state, besides τmax, the effect of normal stress in the same plane σmax is considered (McDiarmid, 1992). Themathematical criterion can take the following form τmax τafA,B + σmax 2σu =1 (2.1) where τafA and τafB are torsion fatigue limits, for the case of an increase in cracking type A or B (Socie and Marquis, 2000), and σu is a monotonic tensile strength. By transforming formula (2.1), we obtain a relationship defining the equivalent stress σMD = τmax+kσmax ¬ τafA,B (2.2) where k= τafA,B 2σu (2.3) 2.2. Criterion according to Huber-Mises-Hencky The criterion according to the hypothesis byHuber-Mises-Hencky (abbreviated toHMH) for fatigue loads can be given as follows σHMH = √ 3J2 ¬σaf (2.4) Integral fatigue criteria evaluation for life estimation... 1075 where: σHMH is the value of equivalent stress, J2 – the second invariant of the deviator of stress state, and σaf – tensile-compressive fatigue limit. For axial load and torsion, J2 is expressed by the formula J2 = 1 3 σ2x+ τ 2 xy (2.5) where: σx and τxy are sinusoidally variable patterns of normal and shear stresses, respectively. In this paper, the criterion has been used to calculate the equivalent stress in two ways. The first approach assumes that the parameters representing the cycle of fatigue load are amplitudes of sinusoidal patterns, and then the equivalent stress can be calculated as follows σaHMH = √ σ2xa+3τ 2 xya (2.6) The secondapproach involves the occurrence of the in-phasedisplacement between load com- ponents, and so themathematical formula expresses the cycle-maximum value of the equivalent stress σmaxHMH =max t ( √ σ2x+3τ 2 xy ) (2.7) Among a few physical interpretations of the second invariant of deviator J2, there is an integral interpretation proposed by Novozhilow (in Zenner et al., 2000). It equates J2 with the root mean square of tangent stresses τγϕ calculated for all the possible planes passing through the neighbourhoodof the point considered (Fig. 1). Using that interpretation the idea of integral criteria presented further in this paper is given with the HMH criterion as an example. Fig. 1. Tangent stress in the plane Fig. 2. Coordinates of the normal line in the spherical coordinate system For the purpose of integration, it is convenient to define the position of plane ∆ as tangent to the sphere with unitary radius. In the contact point of the plane and the sphere, there is found a unitary normal vector n, the direction and the sense of which in the spherical system are described by angles ϕ and γ (Fig. 2). The square of the root mean square of all tangent stresses can be expressed as (Zenner et al., 2000) τ2rms = 1 Ω ∫ Ω τ2γϕ dΩ (2.8) where τγϕ is the tangent stress, Ω – unitary-radius sphere surface area Ω=4π (2.9) 1076 D. Skibicki, Ł. Pejkowski and dΩ is an elementary plane according to the following formula dΩ =sinγdϕdγ (2.10) Substituting (2.9) and (2.10) to (2.8), we receive τrms = √ √ √ √ √ 1 4π π ∫ γ=0 2π ∫ ϕ=0 τ2γϕ sinγ dϕdγ (2.11) In the case of the state of stress in two dimensions, the square of the tangent stress in the plane, the position of which is determined in the spherical coordinate system, is defined by the formula (Zenner and Richter, 1977) τ2γϕ =sin 2γ[(σ2x+ τ 2 xy)cos 2ϕ+ τ2xy sin 2ϕ+2σxτxy sinϕcosϕ] − sin4γ[σ2x cos4ϕ+4σxτxy sinϕcos3ϕ+4τ2xy sin2ϕcos2ϕ] (2.12) Having substituted τ2γϕ according to (2.12) to formula (2.11) and integrated, the following is obtained τrms = √ 2 15 (σ2x+3τ 2 xy)= √ 2 15 σHMH (2.13) It canbenoted that the term in roundbrackets is the squareof the equivalent stress according to theHMHhypothesis for the state of stress in twodimensions.Acomparisonof equations (2.11) and (2.13) provides √ 2 15 σHMH = √ √ √ √ √ 1 4π π ∫ γ=0 2π ∫ ϕ=0 τ2γϕ sinγ dϕdγ (2.14) After transformations, we obtain a formula for the integral form of the HMH criterion σHMH = √ √ √ √ √ 15 8π π ∫ γ=0 2π ∫ ϕ=0 τ2γϕ sinγ dϕdγ ¬σaf (2.15) 2.3. Zenner criterion The general form of the Zenner criterion is identical with notation (2.15). Zenner, however, considers the observation that besides the tangent stress, the fatigue life of the material is also affected by normal stress (Zenner, 1983). The author factors in that fact by generalising quantity τγϕ in a form of τγϕ = aτ 2 γϕa+bσ 2 γϕa (2.16) where the coefficients of the effect of the tangent stress τγϕ and normal stress σγϕ can be calculated as a= 1 5 [ 3 (σaf τaf )2 −4 ] b= 2 5 [ 3− (σaf τaf )2] (2.17) For the purpose of this paper, the effect of mean stress values, which are also considered in the Zenner criterion, has been disregarded. Thanks to coefficients a and b, the criterion can be applied for a greater group of materials. The HMH criterion is applied in the case of materials Integral fatigue criteria evaluation for life estimation... 1077 for which τaf/σaf = 1/ √ 3, whereas the Zenner criterion can be used for ductile materials for which the ratio of fatigue limits falls within the range 0.5<τaf/σaf < 0.8. Finally, the mathematical formula describing the equivalent stress according to Zenner as- sumes the form of (Karolczuk andMacha, 2005; Mamiya et al., 2011) σZ = √ √ √ √ √ 15 /8π π ∫ γ=0 2π ∫ ϕ=0 (aτ2γϕa+ bσ 2 γϕa)sinγ dγdϕ¬σaf (2.18) 2.4. Papadopoulos criterion 1 (1997) Papadopoulos based his criterion on the statement that plastic microdeformation along the slip direction in the plane of crystal slip is proportional to the tangent stress Ta acting in the slip direction (Papadopoulos, 1994). He notes, at the same time, that cracking of single plastically- flowing crystals is not themost critical event since, in the engineering approach, the initiation of cracking occursuponbreakingof a fewmaterial grains and successive coalescence of the emerging microcracks (Papadopoulos, 1994). The author states that the useful criterion in the engineering approach should consider an elementary volume V . That volume is defined by Papadopoulos as a cubic neighbourhood of the point investigated the size of which in the statistical sense ensures that grains of a various crystallographic orientation are equally represented. Besides the tangent stress, fatigue life is also affected by thenormal stress.To sumup, the criterion considers averaged values of the shear stress acting in the direction of slip Ta and themaximum values of normal stress N σP1 = √ 〈T2N〉+α(maxt 〈N〉)¬ τaf (2.19) (20) where √ 〈T2N〉 stands for the rootmean square of the amplitude of the tangent stress acting in the slip direction, maxt〈N〉 is themaximumvalue of themean for the normal stress, reported during the load cycle, while α is the quantity calculated based on material constants in the following way α= σaf−τaf√ 3 τaf 3 (2.20) The value of the amplitude of stress Ta depends not only on the position of plane ∆ but also on the direction of slip L, defined with angle χ (Fig. 3). To simplify the calculations, the author introduces auxiliary quantities a= τacosγcosϕcosθ b=−τacosγcosϕsinϑ c=σa sinγcosθ− τacos(2γ)sinϕcosθ d= τacos(2γ)sinϕsinθ Ca,b = √ √ √ √ a2+ b2+ c2+d2 2 √ (a2+ b2+ c2+d2 2 )2 − (ad− bc)2) (2.21) The symbol θ in the above notations stands for the phase shift angle. Using the above auxiliary quantities, the equation for root mean square √ 〈T2 N 〉 can assume the following form √ 〈T2 N 〉= √ 5 √ √ √ √ √ √ 1 4π π ∫ γ=0 2π ∫ ϕ=0 √ √ √ √ √ 1 2π 2π ∫ χ=0 (C2a cos 2χ+C2 b sin2χ) dχ)sinγ dγdϕ (2.22) 1078 D. Skibicki, Ł. Pejkowski Finally, the notation can be then simplified to √ 〈T2N〉= √ √ √ √ √ 5 8π2 π ∫ γ=0 2π ∫ ϕ=0 2π ∫ χ=0 (C2a cos 2χ+C2 b sin2χ)sinγ dγdϕdχ (2.23) The mean value of the normal stress has been defined as the mean of normal stresses in all possible positions of the plane ∆ passing through the elementary volume V , namely 〈N〉= 1 4π π ∫ γ=0 2π ∫ ϕ=0 N sinγ dγdϕ (2.24) 2.5. Papadopoulos criterion 2 (2001) In his second criterion, Papadopoulos (2001) gives up the considerations over microdamage in theelementary volume V . The criterion is further based on the integral approach and also relates the effect of shear and normal stresses to each other, but remains greatly simplified to the form of σp2 =maxTa+α∞σH,max ¬ γ∞ (2.25) where ,maxTa is denoted by the author as the value of generalised shear stress, while σH,max stands for the cycle-maximum hydrostatic stress. The quantity maxTa is a function of the position of plane ∆ in a spherical coordinate system, described with angles γ and ϕ (Fig. 2). The walue Ta is determined from the formula Ta = √ √ √ √ √ 1 π 2π ∫ χ=0 τ2a dχ (2.26) where τa is the amplitude of the tangent stress τ acting along the slip direction. The quantity τ is the projection of the vector of stress acting in the plane ∆ on the slip direction, represented by the vector m. The location of the vector m is describedwith the angle χwhich is formed by it together with the unitary vector l. In the plane ∆, the vectors l and r form an orthogonal frame of reference (Fig. 4). The coordinates of the vectors n and m, needed to determine τ, are as follows l=    −sinϕ cosγ 0    m= [ −sinϕcosχ−cosγ cosϕsinχ cosϕcosχ− cosγ sinϕsinχ sinγ sinχ ] (2.27) Fig. 3. Geometric interpretation of the amplitude Fig. 4. Description of the slip direction of tangent stress T a acting in the slip direction Integral fatigue criteria evaluation for life estimation... 1079 Stress τ can assume the following form τ =nσm (2.28) where σ stands for the stress state tensor. The value of amplitude τa is determined based on the maximum and minimum value reached by the vector τ in the time of cycle, which can be given as follows τa = 1 2 (maxτ−minτ) (2.29) The quantities α∞ and γ∞ are material parameters. The quantity γ∞ equals torsional fatigue limit τaf, and α∞ is defined from the following formula α∞ =3 (τaf σaf − 1 2 ) (2.30) Themethod of parameters determination method is described in Papadopoulos (2001). 3. Method of analysis of calculations results Theequivalent stresses calculatedwith thecriteria analysed, similarlyas inpapersbyMcDiarmid (1992) and Papadopoulos (2001), can become related with computational life by means of the Basquin equation (Stephens et al., 2001) σeq =AN B cal (3.1) where A and B are coefficients of the Basquin equation, and Ncal is the number of cycles calculated. The coefficients A and B are obtained from the approximation of the results of uniaxial sample tensile-compressive or torsion life testing. The choice which uniaxial samples should be used comes from nature of the equivalent stress. As for the HMHand Zenner criteria, the coefficients A and B have been calculated based on tensile-compressive fatigue life, and for the McDiarmid and Papadopoulos criteria, based on torsion life. By transforming equation (3.1), we obtain a relationship which allows determination of computational fatigue life Ncal = (σeq A ) 1 B (3.2) The criteria of analysis made in the present paper involve the comparison of experimental life Nexp with life Ncal calculated according to formula (3.2). The comparison was made using two statistical parameters described in paper byWalat and Łagoda (2011). The first of them is the mean statistical dispersion of life TN =10 E (3.3) where E is calculated from the formula E = 1 n n ∑ i=1 log Nexp,i Ncal,i (3.4) where n stands for the number of the results compared. The second parameter used is the life estimation mean-squared error TRMS =10 ERMS (3.5) 1080 D. Skibicki, Ł. Pejkowski where ERMS = √ √ √ √ 1 n n ∑ i=1 log2 Nexp,i Ncal,i (3.6) The measure TN assumes the following values: 1 in the case where the mean experimental and computational life are equal; more than 1, when the experimental life values are higher than the computational ones; lower than 1, when the experimental life values are lower than the computational ones. The measure TN is insensitive to the statistical dispersion of life. It can assume the same value for the results with a low and high statistical dispersion. The quantity TRMS is a measure of statistical dispersion. It assumes the value equal to 1 when themean and the statistical dispersion of experimental and computational life are identical as well as values higher than 1 in other cases. Unlike TN, based on TRMS, however, we have no information on whether the computational life values are higher or lower than the experimental ones. With the above properties ofmeasures inmind, it seems that tomake a complete evaluation of the results, both measures must be applied. 4. Analysis of the results The results of calculations have been presented in comparative computational and experimental life plots (Figs. 5, 6, 7 and 8). For each material, plots have been made for equivalent stress formulas: σMD, σ a HMH, σ max HMH, σZ, σP1, σP2. The points of the plot were marked compliant with the nature of the load, namely R,S,P and N. The number after the letter symbol stands for the value of coefficient λ. Solid lines mark the scatter band of factor 2, and dashed lines – the scatter band of factor 3. Fig. 5. Comparison of the experimental life values with the calculated ones for 7075-T651 (Mamiya et al., 2011) Integral fatigue criteria evaluation for life estimation... 1081 Fig. 6. Comparison of the experimental life values with those calculated for 1045 steel (McDiarmid, 1992) Fig. 7. Comparison of the experimental life values with the ones calculated for 1045 steel (Verreman andGuo, 2007) 1082 D. Skibicki, Ł. Pejkowski Fig. 8. Comparison of the experimental life values with the computational ones for X2CrNiMo17-12-2 For each material, criterion and the type of a sample, the measures TN and TRMS have been calculated. The results are broken down in Tables 1, 2, 3 and 4. For better understanding, the results for which the computational life falls within the scatter band of factor 2, namely for value TN falling in the range 0.5-2, and value TRMS in the range 1-2, aremarkedwith a double underscore. The results for which the computational life falls within the scatter band of factor between 2 and 3, namely for value TN falling in the range 0.3(3)-0.5 as well as 2-3 and values TRMS in the range 2-3, are marked with a single underscore. As for uniaxial loads R and S for aluminiumalloy 7075-T651, the resultsmost frequently fall within the scatter band of factor 2, for the Zenner criterion, σP1 andHMHaccording to σ a HMH. As for the proportional load, the evaluation of the criteria by McDiarmid and HMH according to σmaxHMH are relatively worst. For the non-proportional load, the best results were reported for the McDiarmid criterion and the first Papadopoulos criterion. For that material, the greatest errors are about 20-fold higher. Most often the Zenner and Papadopoulos criterion according to σP1 gives the results which fall within the scatter band of factor 2. Table 1.Values TN and TRMS for 7075-T651 aluminium alloy (Mamiya et al., 2011) σMD σ a HMH σ max HMH σZ σP1 σP2 TN TRMS TN TRMS TN TRMS TN TRMS TN TRMS TN TRMS R 1.00 1.36 0.56 1.70 1.00 1.36 1.00 1.36 0.60 2.36 0.33 3.67 S 4.45 5.95 1.00 2.13 4.45 5.95 0.92 2.66 1.00 2.13 1.00 2.13 P 2.01 2.29 1.21 1.70 2.01 2.29 0.83 1.46 1.11 1.67 0.80 1.80 N 1.79 2.59 0.05 19.06 0.16 7.50 0.09 12.56 0.52 2.03 0.11 9.20 As for uniaxial loads of 1045 steel (Verreman and Guo, 2007), only the life values predicted based on the McDiarmid and HMH criteria according to σmaxHMH do not fall within the scatter Integral fatigue criteria evaluation for life estimation... 1083 band of factor 3. For the proportional load, the results acceptable were only reported for the Papadopoulos criterion. As for the non-proportional load for which λ = 2, satisfactory results were recordedbasedon theHMHcriteria according to σmaxHMH andbothcriteriabyPapadopoulos. For the loads demonstrating the highest degree of non-proportionality, namelyN0.5, none of the criteria gives the results falling within the assumed scatter bands. For that group of data, the biggest errors reach about 6-thousand. For thatmaterial, the second criterion by Papadopoulos most frequently gives the results in the scatter band of factor 2. Table 2.Values TN and TRMS for 1045 steel (Verreman and Guo, 2007) σMD σ a HMH σ max HMH σZ σP1 σP2 TN TRMS TN TRMS TN TRMS TN TRMS TN TRMS TN TRMS R 1.00 1.84 0.98 1.91 1.00 1.84 1.00 1.84 0.95 1.92 0.95 1.92 S 4.62 5.06 1.00 1.64 4.62 5.06 1.12 1.73 1.00 1.64 1.00 1.64 P2 0.39 3.02 0.65 2.06 0.39 3.02 0.22 5.15 0.25 4.00 0.99 1.80 N2 5.85 6.00 0.39 2.56 1.28 1.47 0.31 3.34 1.90 1.93 0.91 1.19 N0.5 6.99 10.34 0.00 1439.35 0.00 6509.40 0.00 6509.40 3.24 4.78 0.01 130.31 For the experimental data for 1045 steel reported in paperbyMcDiarmid (1992), for uniaxial and proportional loads all the results fall within the scatter band of factor 3. For the non- proportional load, the satisfactory results in each case are reported by applying the second criterion byPapadopoulos. For that group of data, the greatest errors are about 20-folds higher. The best results were recorded for the HMH criteria according to σmaxHMH, the Zenner and the Papadopoulos criteria according to σP2. Table 3.Values TN and TRMS for 1045 steel (McDiarmid, 1992) σMD σ a HMH σ max HMH σZ σP1 σP2 TN TRMS TN TRMS TN TRMS TN TRMS TN TRMS TN TRMS R 1.00 1.60 1.00 1.93 1.00 1.60 1.00 1.60 1.18 1.97 1.00 1.93 S 1.06 1.92 1.00 1.71 1.06 1.92 0.98 1.92 1.00 1.71 1.00 1.71 P0.5 1.04 1.50 0.98 1.97 1.04 1.50 1.03 1.50 1.04 1.99 0.98 2.01 P1 1.11 1.47 1.22 2.05 1.11 1.47 1.09 1.47 1.08 2.03 1.48 2.19 P10 0.64 1.56 0.59 1.68 0.64 1.56 0.59 1.70 0.49 2.05 0.45 2.19 P2 1.33 1.57 1.91 2.67 1.33 1.57 1.29 1.56 1.40 2.27 1.66 2.58 P4 1.07 1.07 1.65 1.65 1.07 1.07 1.00 1.00 1.16 1.16 0.48 2.09 N2 9.63 9.88 0.28 3.73 0.38 2.63 0.35 2.86 13.94 15.18 0.60 2.14 N1 14.44 14.86 0.50 2.27 0.71 1.68 0.71 1.68 21.08 21.79 1.47 1.81 N0.5 10.03 10.10 0.42 2.64 0.50 2.07 0.49 2.07 17.72 18.36 1.24 1.65 The results for uniaxial loads for X2CrNiMo17-12-2 steel, for theMcDiarmid and the HMH criteria according to σmaxHMH are unacceptable. As for the proportional load with coefficient λ= 0.5, only the Zenner criterion gave acceptable results. The first criterion by Papadopoulos slightly exceeded the admissible values. Unfortunately, for the proportional loads demonstrating a greater share of tangible stresses, namely for λ = 0.8, both criteria give very bad results. Here, in turn, life values for HMH according to σmaxHMH fall within the acceptable limits. As for the non-proportional loads with λ=0.5, only the second criterion by Papadopoulos generated satisfactory results. For the non-proportional loadswith λ=0.8, none of the criteria gave results 1084 D. Skibicki, Ł. Pejkowski falling within the scatter bands of factors 2 and 3. The results obtained according to the Zenner criterion slightly exceeded that limit. The greatest errors for the non-proportional loads are in the range of 14-thousand times. For most of the results of that group of data, most frequently the Zenner criterion and the second criterion byPapadopoulos give the results fallingwithin the scatter band of factor 2. Table 4.Values TN and TRMS for X2CrNiMo17-12-2 steel (Skibicki et al., 2012) σMD σ a HMH σ max HMH σZ σP1 σP2 TN TRMS TN TRMS TN TRMS TN TRMS TN TRMS TN TRMS R 1.00 1.41 0.86 1.46 1.00 1.41 1.00 1.41 0.94 1.42 1.02 1.41 S 1903.4 1922.1 1.00 1.46 1903.4 1922.1 1.09 1.48 1.00 1.46 1.00 1.46 P0.5 13.00 13.21 3.31 3.44 13.00 13.21 0.99 1.33 2.38 2.50 4.74 4.88 P0.8 1.44 1.62 7.34 7.49 1.44 1.62 0.00 1214.8 0.00 1013.6 5.24 5.37 N0.5 146.64 148.04 0.39 2.68 0.44 2.38 0.44 2.38 24.89 25.27 1.48 1.65 N0.8 14075.7 14203.3 0.13 8.23 228.31 231.99 0.17 2.38 568.2 575.13 48.66 49.63 As for uniaxial loads, in most cases ,all the criteria analysed give satisfactory results. The worst results are reported by applying the HMH criterion for torsion, whichmust be due to the fact that this criterion is applied for a small group of materials resulting from a constant ratio of fatigue limits τaf/σaf =1/3. A similar situation is reported for proportional loads. The life values estimated according to the HMH criterion are most often encumbered with the greatest error, which is due to the failure in considering variable material properties expressed with the ratio τaf/σaf. As for the non-proportional loads formaterials sensitive to non-proportionality, namely 1045 and X2CrNiMo17-12-2 steels, the results can show a very high error. Most frequently, thebest resultswere reported for the second criterion byPapadopoulos, and theworst results, on the other hand, for theMcDiarmid and theHMHcriteria. Even though the Papadopoulos criterion is an integral criterion, however, it does not allowmaking the statement that the integral approach is the most adequate one to describe non-proportional loads; first of all, since the HMH criterion can be also considered as the integral criterion and, second of all, since it is the Papadopulos criterion, which for non-proportional loads often gave results demonstrating the statistical dispersion much greater than desired scatter bands of factors 2 or 3. 5. Conclusions For the experimental fatigue life data used, one can claim that: • Relatively the best resultswere reportedbyapplying the second criterion byPapadopoulos and the Zenner criterion, and the worst – according to the criterion byMcDairmid and by Huber-Mises-Hencky. • None of the criteria analysed can be applied to estimate fatigue life when exposed to non-proportional loads. • The integral approach can be effective under the non-proportional loads conditions, howe- ver, it does not always guarantee acceptable results. Acknowledgements The project has been financed by National Center for Science. Integral fatigue criteria evaluation for life estimation... 1085 References 1. Ellyin F., Gołoś K., Xia Z., 1991, In-phase and out-of-phase multiaxial fatigue, Transactions of the ASME, 113, 112-118 2. Garud Y.S., 1981, Multiaxial fatigue: a survey of the state of the art, Journal of Testing and Evaluation, 9, 165-178 3. 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Podejście całkowe bazuje na założeniu, że dla prawidłowej oceny zachowań zmęczeniowych konieczne jest zsumowanie (scałkowanie) wartości parametru zniszczenia na wszystkich płaszczyznach przechodzących przez rozpatrywany punkt materiału. Analizę przeprowadzono dla trzech najczęściej spotykanychkryteriówcałkowych: kryteriumZennera i dwóch kryteriówPapadopoulosa.Uzy- skanewyniki porównano z kryterium zmęczeniowymMcDiarmida, bazującymna konkurencyjnymw sto- sunku do całkowego podejściu płaszczyzny krytycznej, oraz powszechnie stosowanymw wielu obszarach wytrzymałości materiałów kryteriumHubera-Misesa-Hencky’ego. Manuscript received February 7, 2012; accepted for print March 5, 2012