The Modified Manson-Coffin Curve Method to estimate fatigue lifetime under complex constant and variable amplitude multiaxial fatigue loading

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1 Please, cite this paper as: Wang, Y., Susmel, L. he Modified Manson Coffin Curve Method to estimate Journal of Fatigue 83 (), pp , 6. he Modified Manson-Coffin Curve Method to estimate fatigue lifetime under complex constant and variable amplitude multiaxial fatigue loading Yingu Wang, and Luca Susmel Ke Laborator of Fundamental Science for National Defense-Advanced Design echnolog of Flight Vehicle, Nanjing Universit of Aeronautics and Astronautics, Nanjing, 6, China Department of Civil and Structural Engineering, the Universit of Sheffield, Sheffield S 3JD, UK Corresponding Author: Prof. Luca Susmel Department of Civil and Structural Engineering he Universit of Sheffield, Mappin Street, Sheffield, S 3JD, UK elephone: +44 () Fax: +44 () l.susmel@sheffield.ac.uk ABSRAC his paper investigates the accurac of the so-called Modified Manson-Coffin Curve Method (MMCCM) in estimating fatigue lifetime of metallic materials subjected to complex constant and variable amplitude multiaxial load histories. he MMCCM postulates that fatigue damage is maximised on that material plane experiencing the maximum shear strain amplitude. In the present investigation, the orientation of the critical plane was determined through that direction along which the variance of the resolved shear strain reaches it maximum value. Under variable amplitude complex load histories, this direction was also used to count the resolved shear strain ccles via the classic Rain-Flow method. Further, the degree of multiaxialit and nonproportionalit of the time-variable stress states at the assumed critical locations was directl quantified through a suitable stress ratio which accounts for (i) the mean value and the variance of the stress perpendicular to the critical plane as well as for (ii) the variance of the shear stress resolved along the direction experiencing the maximum variance of the resolved shear strain. he accurac and reliabilit of the proposed approach was checked against approximatel 65 experimental data taken from the literature and generated b testing un-notched metallic materials under complex constant and variable amplitude multiaxial load histories. he sound agreement between estimates and experimental results which was obtained strongl supports the idea that the proposed design technique is a powerful engineering tool allowing metallic materials to be designed against constant and variable amplitude multiaxial fatigue b alwas reaching a remarkable level of accurac. his approach offers a complete solution to the strain based multiaxial fatigue problem. Kewords: Multiaxial fatigue, Variable amplitude loading, Critical plane

2 Please, cite this paper as: Wang, Y., Susmel, L. he Modified Manson Coffin Curve Method to estimate Journal of Fatigue 83 (), pp , 6.. Introduction In situations of practical interest, engineering components and structures are subjected to complex time-variable load histories, the applied time-dependent sstems of forces/moments resulting in local variable amplitude (VA) multiaxial stress/strain states. Estimating fatigue strength of metallic materials subjected to VA multiaxial load histories is a complex design problem which must be addressed properl in order to avoid unwanted breakages during in-service operations. Owing to the high costs associated with fatigue failures, since the beginning of the last centur a tremendous effort has been made b the international scientific communit to devise appropriate engineering tools suitable for estimating fatigue damage under complex loading paths. If attention is focused on the low/medium-ccle fatigue regime, examination of the state of the art [-7] suggests that, so far, this intractable design problem has being addressed mainl b tring to extend the use of wellknown constant amplitude (CA) multiaxial fatigue criteria to those situations involving multiaxial VA load histories. In this context, among the methods which have been emploed so far, certainl the SW parameter [8, 9], Brown & Miller s criterion [, ], and Fatemi & Socie s critical plane approach [, 3] deserve to be mentioned explicitl. As far as VA multiaxial load histories are concerned, accuratel performing the ccle counting certainl represents one of the trickiest aspects, the scientific communit being still debating to agree a commonl accepted strateg. As to the ccle counting issue, examination of the state of the art suggests that the most successful methodologies [, 3-5] which have been formalised and validated so far are all based on the use of the classic Rain-Flow Method (this method being originall developed to address simple uniaxial situations [6]). When it comes to designing components and structures against VA multiaxial fatigue, another trick problem that must be addressed properl is the definition of an appropriate rule suitable for estimating cumulative damage. Even though a variet of methods have been proposed so far [7], certainl, in situations of practical interest, the most used rule is still the linear one devised b Palmgren [8] and Miner [9]. According to this classic approach, fatigue failure takes place as soon as the damage sum becomes equal to unit. However, accurate experimental investigations have proven that the critical value of the damage sum, D cr, var in the range. 5, its average value being equal to.7 for steel and to.37 for aluminium []. Further, given the material, D cr is seen to var as the geometr of the component, the degree of multiaxialit of the assessed VA load histor, and the profile of the considered load spectrum change [-]. hus, sstematicall taking the critical value of the damage sum equal to unit ma lead, under particularl unfavourable circumstances, to non-conservative estimates. his suggests that D cr can be evaluated accuratel for the specific material/component/load histor being assessed solel via expensive and time-consuming experimental trials. In this complex scenario, this paper reports on an attempt of extending the use of a multiaxial fatigue criterion we have recentl proposed [3-5] - here called the Modified Manson-Coffin

3 Please, cite this paper as: Wang, Y., Susmel, L. he Modified Manson Coffin Curve Method to estimate Journal of Fatigue 83 (), pp , 6. Curve Method (MMCCM) - to those situations involving complex CA and VA loading paths. In more detail, such a strain based critical plane approach is attempted here to be applied along with the Maximum Variance concept [6-8] in order to formalise a robust fatigue assessment technique suitable for estimating fatigue lifetime of metallic materials subjected to complex CA and VA multiaxial load histories.. Fundamentals of the MMCCM As far as CA loading paths are concerned, the MMCCM [3-5] postulates that fatigue damage in the low/medium-ccle fatigue regime can accuratel be estimated via the stress and strain components acting on that material plane (i.e., the so-called critical plane) experiencing the maximum shear strain amplitude, a. he degree of multiaxialit and non-proportionalit of the applied load histor as well as the presence of non-zero mean stresses are quantified b the MMCCM via the shear stress amplitude, a, relative to the plane of maximum shear strain amplitude and the amplitude, n,a, and the mean value, n,m, of the stress normal to the critical plane. he definitions which are proposed here as being adopted to calculate the stress/strain quantities of interest not onl under CA, but also under VA multiaxial fatigue loading will be discussed in the next section in great detail. he fatigue damage model on which the MMCCM is based is shown in Figure a. According to this schematisation, Stage I cracks are assumed to initiate on those crstallographic planes most closel aligned with the maximum shear strain direction [9]. he subsequent propagation phenomenon is strongl influenced b the stress perpendicular to the critical plane [9, 3]. In particular, the amplitude of the stress normal to the critical plane, n,a, favours the growth process b cclicall opening and closing the micro/meso fatigue cracks [3]. he propagation phase is also influenced b the mean stress, n,m, normal to the plane of maximum shear strain amplitude. In fact, a tensile superimposed static normal stress tends to keep the micro/meso fatigue cracks open b minimising the interactions amongst the crack surfaces asperities [9, 3]. his favours the effect of the cclic shear stress which pushes the tips of the cracks themselves [3]. On the contrar, under compressive mean normal stresses, the resulting additional frictional phenomena between the crack surfaces [9, 3] mitigate the action of the cclic shear stress [3], this resulting in a reduction of the crack growth rate. According to the fatigue damage model depicted in Figure a, the degree of multiaxialit and nonproportionalit of the stress state damaging the assumed crack initiation locations is quantified b the MMCCM via the following critical plane stress ratio [3]: n,m n,a n,max () a a

4 Please, cite this paper as: Wang, Y., Susmel, L. he Modified Manson Coffin Curve Method to estimate Journal of Fatigue 83 (), pp , 6. In definition () a is the shear stress amplitude relative to the critical plane, whilst n,m, n,a and n,max are the mean value, the amplitude and the maximum value of the stress perpendicular to the plane of maximum shear strain amplitude, respectivel. Ratio is seen to be capable of modelling not onl the presence of superimposed static stresses, but also the degree of multiaxialit and nonproportionalit of the applied load histor [3, 3]. In particular, as suggested b Socie [9, 3], the effect of the stress components perpendicular to the critical plane can efficientl be modelled b simpl using the maximum normal stress, since n,max = n,m + n,a. his simple strateg was followed b Socie himself to reformulate the SW parameter to make it suitable for performing the multiaxial fatigue assessment of those metals whose mesoscopic cracking behaviour is mainl Mode I governed [9]. Similarl, the normal maximum stress, n,max, was emploed b Fatemi and Socie to devise their shear strain based critical plane approach []. he well-known accurac and reliabilit of these two criteria should full support the idea that n,max is a stress quantit capable of accuratel taking into account the mean stress effect in multiaxial fatigue. his holds true provided that n,max is consistentl used with an appropriate fatigue damage model, the damage model on which the MMCCM is based being shown in Figure a. urning back to definition (), it is evident that under VA fatigue loading the value of ratio ma var ccle b ccle. Accordingl, appropriate definitions for the stress quantities of interest are required in order to consistentl calculate the ratio also in the presence of complex VA multiaxial load histor. he strateg we propose to address the VA problem will be discussed in Section 3 in detail. Under fatigue loading, there are alwas at least two planes experiencing the maximum shear strain amplitude, this holding true independentl from the complexit of the assessed load histor. herefore, amongst all the potential critical planes, the one which must be considered to estimate fatigue lifetime is the one characterised b the largest value of ratio ρ [3]. o quantif the fatigue damage extent, the MMCCM makes use of non-conventional Manson- Coffin curves, where the values of the required calibration constants var as the critical plane stress ratio,, changes [5]. he wa this method works can be depicted in a log-log diagram where the shear strain amplitude, a, relative to the critical plane is plotted against the number of reversals to failure, N f (Fig. b). As shown in Figure b, different modified Manson-Coffin curves are obtained as ratio varies, each of these curves being described mathematicall as follows: a ' f ( ) G b c N ' ( ) N f f f ()

5 Please, cite this paper as: Wang, Y., Susmel, L. he Modified Manson Coffin Curve Method to estimate Journal of Fatigue 83 (), pp , 6. Given the material, constants f (), f (), b() and c() can directl be derived from the fullreversed uniaxial and torsional Manson-Coffin fatigue curves re-written according to resca s hpotheses, i.e. [3]: ' f b c a e Nf p ' f Nf (Uniaxial case, =) (3) E ' G b c N ' N f a f f f (orsional case, =) (4) where e and p are Poisson s ratio for elastic and plastic strain, respectivel. B manipulating Eqs (3) and (4) under some simplifing hpotheses [3], the material constants in Eq. () can be expressed explicitl as follows [3, 3]: ' f ( ) G f ' f ' G e E p ' f ' f f ' (6) bb b (7) (b b) b c c c (8) (c c) c (5) o conclude, it is worth observing that, according to the above definitions, the modified Manson- Coffin curves move upwards as ratio decreases (see Figure b). In other words, the MMCCM estimates fatigue lifetime b assuming that, for a given value of the shear strain amplitude acting on the critical plane, the fatigue damage extent increases with increasing of ratio : this explains wh when selecting the critical plane amongst those experiencing the maximum shear strain amplitude, the one to be used is that characterised b the largest value of ratio. 3. he Maximum Variance concept to determine the stress/strain quantities relative to the critical plane In order to appl the MMCCM to estimate fatigue lifetime of metallic materials subjected to uniaxial/multiaxial fatigue loading, the first problem to be addressed is the determination of those stress/strain quantities relative to the critical plane which are required to quantif the fatigue damage extent. As discussed in what follows, in the present investigation such quantities are suggested as being estimated b taking full advantage of the maximum variance concept [6].

6 Please, cite this paper as: Wang, Y., Susmel, L. he Modified Manson Coffin Curve Method to estimate Journal of Fatigue 83 (), pp , 6. From a statistical point of view, the variance of a time-variable signal is equal to the expected value of the squared deviation from the mean. In other words, b definition, the variance quantifies the amount of variation (within the two extremes delimiting the maximum range) associated, over the time interval,, of interest, with the signal being investigated. According to this definition, the variance is a statistical quantit which is independent from the mean value of the considered signal. If attention is focussed specificall on time-variable load histories, the variance of a stress/strain signal is seen be related to the amount of damage caused b the signal itself [33, 34]. he variance approach assumes that the damage in the candidate plane can be estimated from a root mean square calculation. It is well-known that the dependence between shear amplitudes and damage ma involve exponents having value different from two. Accordingl, under ver specific circumstances, more accurate estimates ma be obtained b adopting higher central moments of the resolved shear strain histor, instead of the second central moment. However, much experimental evidence (please, see Refs [6, 33, 34] and references reported therein) suggests that, in situation of practical interest, the use of the variance concept allows VA load histories to be postprocessed b alwas reaching an adequate level of accurac. Recentl, this idea [7, 8] has successfull been applied along with the stress based critical plane concept to estimate fatigue lifetime under uniaxial/multiaxial VA multiaxial fatigue loading [35-4]. In light of the high level of accurac obtained in the long-life fatigue regime, the next logical step is then reformulating the maximum variance idea to make it suitable for being applied in terms of cclic strains. he fundamental concepts on which the Shear Strain Maximum Variance Method (-MVM) is based are summarised in what follows, whereas its mathematical formalisation is discussed in detail in Appendix A. he -MVM takes as a starting point the assumption that the critical plane is that material plane containing the direction, MV, that experiences the maximum variance of the resolved shear strain, MV (t) see Figures a and b. If this direction as well as the orientation of the associated material plane are known, direction MV together with the unit vector normal to the critical plane can directl be used to calculate the stress/strain quantities of interest. In more detail, under CA fatigue loading, the amplitudes and the mean values of the shear strain and shear stress component relative to the critical plane can directl be calculated as follows (see also Figure c): a MV,max MV,min (9) m MV,max MV,min () a MV,max MV,min ()

7 Please, cite this paper as: Wang, Y., Susmel, L. he Modified Manson Coffin Curve Method to estimate Journal of Fatigue 83 (), pp , 6. m MV,max MV,min () In definitions (9) and () MV,max and MV,min are the maximum and minimum value of MV (t), respectivel. Similarl, in definitions () and () MV,max and MV,min denote the maximum and minimum value of the shear stress, MV (t), resolved along direction MV, respectivel (Fig. c). B following the same strateg as above, under CA fatigue loading the amplitude, n,a, and the mean value, n,m, of the stress normal to the critical plane, n (t), can be calculated as follows: n,a n,max n,min (3) n,m n,max n,min, (4) where n,max and n,min are used to denote the maximum and minimum value of normal stress n (t), respectivel (Fig. c). urning to VA situations, assume now that the critical point O in the component being assessed (Fig. a) is damaged b a stress/strain state whose components var randoml in the time interval of interest, i.e., time interval [, ]. As soon as the orientation of the direction, MV, experiencing the maximum variance of the resolved shear strain is known (Fig. b), the mean value and the variance of the shear strain and shear stress component relative to the critical plane can directl be determined as follows (see also Figure d): m MV (t) dt (5) Var MV (t) MV (t) m dt (6) m MV (t) dt (7) Var MV(t) MV(t) m dt (8) he variance terms determined as above allow the equivalent amplitude of the shear stress and shear strain resolved along direction MV to be determined according to the following trivial relationships:

8 Please, cite this paper as: Wang, Y., Susmel, L. he Modified Manson Coffin Curve Method to estimate Journal of Fatigue 83 (), pp , 6. Var (t) (9) a MV Var (t) () a MV In a similar wa, the equivalent amplitude and the mean value of the stress perpendicular to the critical plane, n (t), can be determined as follows: Var (t) () n,a n n,m n (t) dt () where: (t) (t) Var n n n,m dt (3) o conclude, it is worth observing that the -MVM has two ke advantages over the other existing methods. First, it is ver efficient from a computational point of view. In fact, as soon as the variance and covariance terms of the components of the strain tensor being post-processed are known, the time required to determine the orientation of the critical plane is almost independent from the length of the assessed load histor. Second, since MV (t) and MV (t) are monodimensional quantities, the ccle counting under multiaxial fatigue loading can be performed rigorousl b using one of those techniques specificall devised b considering uniaxial fatigue situations (and, in particular, via the Rain-Flow counting method [6]). 4. he MMCCM to estimate fatigue lifetime under CA and VA multiaxial fatigue loading he last step in the formalisation of the proposed design technique is the definition of standard procedures suitable for using this approach in situations of practical interest to estimate lifetime of metallic materials subjected to in-service CA and VA load histories. Consider then the component sketched in Figure 3a which is assumed to be subjected to a CA load histor. As soon as the direction experiencing the maximum variance of the resolved shear strain is known (Fig. 3b), the amplitude of the shear strain relative to the critical plane, a, can easil be determined according to definition (9) see Figure 3c. Similarl, the shear stress amplitude, a, and the maximum normal stress, n,max, relative to the critical plane (Figs 3d and 3e) can directl be calculated through Eqs (), (3) and (4). hese stress quantities allow the critical plane stress

9 Please, cite this paper as: Wang, Y., Susmel, L. he Modified Manson Coffin Curve Method to estimate Journal of Fatigue 83 (), pp , 6. ratio, Eq. (), characterising the load histor being assessed to be determined unambiguousl (Fig. 3f). hrough the calculated value for, the constants in the MMCCM can now be estimated (Fig. 3g) from the full-reversed uniaxial and torsional fatigue properties via relationships (5) to (8). hese constants can then be used to derive, for the calculated value of, the required Modified Manson-Coffin curve (Fig. 3h). Finall, the shear strain amplitude relative to the critical plane, a, together with Eq. () allows the number of ccles to failure to be estimated directl (Fig. 3h). urning to VA multiaxial fatigue situations, the first problem which has to be addressed is the formalisation of a strateg allowing the ccle counting to be performed so that the fatigue damage extent can be quantified accuratel. As briefl discussed in Section, the MMCCM assumes that the critical plane is the one containing the direction experiencing the maximum variance of the resolved shear strain (Fig. a). According to Kanazawa, Miller and Brown [9], Stage I cracks form on those slip planes most closel aligned to the macroscopic planes of maximum shear. Accordingl, the hpothesis can be formed that, under VA fatigue loading, the resolved shear strain, MV (t), is the strain quantit to be used to perform the ccle counting. herefore, owing to the fact that MV (t) is a monodimensional strain quantit, given a load histor, the corresponding cumulative shear strain spectrum can directl be determined b using the Rain-Flow method [6]. Having formed this initial hpothesis, consider now the component of Figure 4a which is assumed to be subjected to a complex sstem of time variable forces and moments which result in a local VA state of stress/strain at critical point O. As soon as the direction of maximum variance is known (Fig. 4b), the equivalent amplitude of the shear stress relative to the critical plane, a, can be determined according to definition (), whereas the equivalent value of the maximum stress perpendicular to the critical plane can directl be estimated via definitions () and (), where n,max = n,m + n,a (see Figures 4d and 4e). Stress quantities a and n,max allow then the critical plane stress ratio,, to be determined under VA fatigue loading via definition () see Figure 4f. his ratio can now be used to estimate the constants in the MMCCM according to definitions (5) to (8) see Figures 4g and 4h. In parallel, b taking full advantage of the Rain-Flow Method (Fig. 4i), signal MV (t) has to be post-processed in order to build the corresponding shear strain spectrum (Fig. 4j). his spectrum along with the estimated modified Manson-Coffin curve, Eq. (), allow the damage content associated with an counted shear strain ccles to be quantified (see Figures 4h and 4k). Finall, the number of ccles to failure can directl be estimated as follows: n j j i cr Dtot f,e i N i f,i D tot D N n. (4) i where D tot is the total value of the damage sum. D cr is instead the critical value of the damage sum, i.e., the value of D tot resulting in the initiation of a fatigue crack in the metallic material being

10 Please, cite this paper as: Wang, Y., Susmel, L. he Modified Manson Coffin Curve Method to estimate Journal of Fatigue 83 (), pp , 6. assessed. It is worth observing here that according to the classic theor due to Palmgren [8] and Miner [9], D cr should be invariabl equal to unit. On the contrar, as observed b Sonsino [5], in situations of practical interest its value is seen to range in the interval.-5. Under VA multiaxial fatigue loading, ratio is determined b post-processing the entire local load histor projected along the direction of maximum variance of the resolved shear strain as well as along the direction perpendicular to the critical plane. he value for ratio calculated according to this strateg is then used to estimate the specific modified Manson-Coffin curve that has to be used to quantif the damage content associated with the counted ccles. his means that, given the material, different VA multiaxial load histories are assessed b using the same modified Manson- Coffin curve as long as the are characterised b the same value of ratio. However, even if is the same, in the most general case, different VA load histories are expected to result in different values of the estimated number of ccles to failure, this depending on the specific profile of the corresponding shear strain spectrum (Fig. 4j). Another important aspect is that the shear strain spectrum is built b post processing the actual strain histor resolved along the direction experiencing the maximum variance of the resolved shear strain. his allows the sequence effect to be taken into account effectivel. he above ideas represent the ke concepts on which the proposed approach is based, the validit of this modus operandi being checked in the next section b post-processing a large number of experimental results taken from the literature. o conclude, it is worth emphasising the fact that both under CA and VA fatigue loading the MMCCM must be applied b post-processing the actual elasto-plastic time-variable stresses and strains damaging the assumed critical location. his implies that appropriate techniques must be emploed in order to perform the stress/strain analsis b accuratel quantifing/modelling important phenomena such as strain hardening/softening, non-proportional hardening, ratcheting, memor effect, mean stress relaxation, etc. 5. Validation b experimental data In order to check the accurac and reliabilit of the proposed design methodolog, about 65 experimental results were selected from the technical literature. able summarises the static and fatigue properties of the investigated metallic materials. For the majorit of the considered data sets, the required material properties were directl available in the original sources. In some cases, even though the required fatigue constants were not listed explicitl, K, n, f, f, b, c, f, f, b, and c were directl calculated b post-processing the full-reversed uniaxial and torsional experimental results being provided. In particular, given the population of data, the required parameters were determined, for a probabilit of survival equal to 5%, via a linear regression model (in a log-log representation) optimised b using the least-squares method [58]. he material parameters quantified according to this procedure are clearl indicated in able. When solel the constants of the full-reversed torsional Manson-Coffin curves were not available, the were

11 Please, cite this paper as: Wang, Y., Susmel, L. he Modified Manson Coffin Curve Method to estimate Journal of Fatigue 83 (), pp , 6. estimated from the corresponding uniaxial fatigue constants b using von Mises criterion [7, 3], i.e.: ' f ' f ; ' f 3' f ; b b ; c c (5) 3 All the considered data were generated b controlling the local deformations during testing, so that in the majorit of the sources the corresponding stress histories were not given. In order to calculate the stress components relative to the critical plane when the relevant time-variable stress histories were not directl available, the required elasto-plastic stress states were estimated b using Jiang and Sehitoglu s method [59, 6]. Finall, in those cases in which the values of the nonproportional cclic strength coefficient, K NP, and the non-proportional cclic strain hardening exponent, n NP, were not determined experimentall, non-proportional hardening was modelled b taking the constants in the corresponding Ramberg-Osgood tpe equation as follows [7]: K'.5 ; n' NP n' (6) NP K NP Since a number of assumptions were made to make the selected experimental data suitable for performing a sstematic validation exercise, the first problem to address was the definition of a reference error band allowing the accurac of the proposed fatigue design approach to be assessed quantitativel. he estimated, N f,e, vs. experimental, N f, number of ccles to failure diagram reported in Figure 6a shows the accurac of the conventional approach due to Manson and Coffin in estimating the fatigue lifetime of the considered materials under pure axial and pure torsional full-reversed loading, the corresponding loading paths being sketched in Figure 5. his chart shows that the experimental points fall within an error band of 3. Accordingl, such an error band will be used in what follows to quantif the accurac of the proposed approach. his choice can be justified b observing that, from a statistical viewpoint, the sstematic usage of a predictive method cannot obviousl result in an accurac level which is higher than the intrinsic scatter characterising the information used to calibrate the method itself. Another important aspect which deserves to be recalled here is that under non-proportional/complex loading paths the principal directions rotate during the loading ccle. his results in the simultaneous activation of several slip sstems so that Stage I cracks tend to initiate on several material planes b subsequentl propagating along certain paths whose orientation depends not onl on the characteristics of the applied stress/strain histor, but also on the local material morpholog [9]. On the contrar, under proportional loading microcracks are seen to initiate on preferential material planes, resulting in smaller deviations of the propagation directions with respect to the one of maximum shear strain [9]. According to the

12 Please, cite this paper as: Wang, Y., Susmel, L. he Modified Manson Coffin Curve Method to estimate Journal of Fatigue 83 (), pp , 6. above considerations, complex/non-proportional load histories are expected to be characterised b a larger degree of scattering compared with the corresponding proportional/simple cases. In order to checked the accurac of the MMCCM applied along with the -MVM, initiall our fatigue assessment methodolog was used to estimate fatigue lifetime under CA fatigue loading, the considered loading paths being shown in Figure 5. he error diagram reported in Figure 6b makes it evident that our approach was highl accurate, its sstematic usage resulting in predictions failing mainl within the target error interval. It is worth observing that such a high level of accurac was reached not onl in the presence of proportional and non-proportional sinusoidal/triangular strain paths, but also under complex CA load histories. Further, the MMCCM used in conjunction with the -MVM was seen to be capable of accuratel taking into account the effect of superimposed static strains as well (see Figure 6b). Subsequentl, our fatigue assessment method was used to predict the fatigue lifetime of samples of Al [4], S46N [48] and 34SS [55] tested under combined CA axial and torsional sinusoidal/triangular strain signals of different frequencies. B focussing attention on the considered sinusoidal load histories, it is possible to observe that under a ratio between the frequenc of the axial channel, F x, and the frequenc of the torsional channel, F x, equal to, the resulting stress/strain histor relative to the critical plane was composed of one shear stress/strain ccle and two normal stress ccles (see Path F in Figure 7). On the contrar, the use of the proposed method resulted in two shear stress/strain ccles under a F x to F x ratio equal to.5 (Path G in Figure 7) and in four shear stress/strain ccles under F x /F x =.5 (Path H in Figure 7). In other words, for these two loading paths one nominal ccle was composed of two and four shear ccles under a F x to F x ratio equal to.5 and.5, respectivel. Since a similar reasoning applies also to the triangular strain paths sketched in Figure 7 (see Paths I to M), this explains the reason wh the error diagram in Figure 7 was plotted in terms of number of blocks to failure, one block corresponding to one nominal ccle. he N f,e vs. N f chart of Figure 7 shows that the proposed approach was highl accurate in modelling the damaging effect of combined CA axial and torsional strain signals of different frequencies. As to the obtained level of accurac, it is important to point out that it was reached b taking the critical value of the damage sum, D cr, equal to unit for Al [4] and S46N [48], whereas, as suggested b Sonsino [], it was set equal to.7 for 34SS [55]. Subsequentl, attention was focussed on the accurac of the proposed design approach in modelling the sequence effect. In particular, our method was attempted to be used to estimate the fatigue lifetime of specimens of 34SS [54], pure titanium [4], titanium B9 [4], SNCM439 [55], SNCM63 (A) [56] and SNCM63 (B) [55] tested under different combinations of axial (A), torsion (), in-phase (I) and 9 out-of-phase (O) axial/torsion ccles. Such load histories were built as a sequence of full-reversed CA loading blocks containing a predefined number of ccles. Sequences AA and were formed of two axial and two torsional blocks characterised b different

13 Please, cite this paper as: Wang, Y., Susmel, L. he Modified Manson Coffin Curve Method to estimate Journal of Fatigue 83 (), pp , 6. amplitudes. he error diagram reported in Figure 8 confirms that the MMCCM applied along with the -MVM was capable of accuratel predicting the sequence effect in fatigue, the critical value of the damage sum, D cr, being taken equal to unit for all the investigated materials. he next step in the performed validation exercise was considering short VA load histories. he obtained results are summarised in the error diagram of Figure 9, whereas the profile of the considered nominal loading paths are sketched in Figure 5. In order to show how our design technique works in the presence of short VA loading blocks, the load histories re-calculated in terms of stress/strain quantities relative to the critical plane are reported in Figure 9 for Paths R, R and R3. his figure makes it evident that, although Path R and R ma appear ver similar, in the case of Path R the use of our approach resulted in four shear ccles, whereas for Path R a nominal loading block contained one shear stress/strain ccle. he estimated vs. experimental number of blocks to failure diagram of Figure 9 confirms that the MWCM applied along with the -MVM was highl accurate also in estimating lifetime under short VA load histories, with predictions falling within the target error band. Subsequentl, we focussed our attention on the results generated b Shamsaei, Fatemi and Socie [5] b testing thin-walled tubular specimens of 5 Q and 34L stainless steel under the discriminating strain paths shown both in Figure 5 and in Figure. In more detail, Paths FR contained a series of full-reversed in-phase axial/torsion ccles applied b making angle var (see Figure ). Both Path FRI and Path FRR were characterised b a step angle equal to, with ranging in the interval -36. Paths FRI were generated b graduall increasing angle from to 36, whereas Paths FRR contained ccles applied in random order. Path FRI5 had =5 with 36, whereas for Path FRI9 was equal to 9, with graduall increasing from to 7. Paths PI contained pulsating axial/torsion in-phase ccles with = and 36. Similarl, Paths PI9 was formed of four (=9 ) pulsating in-phase axial/torsion ccles with 7. Some examples showing the resolved shear strain spectra determined b appling the -MVM are reported in Figure. he error diagram of Figure confirms that the MMCCM was capable of estimates falling within the target error band. As to the made predictions, it is worth observing that a higher degree of conservatism could have been obtained b simpl setting, as suggested b Sonsino [], the critical value of the damage sum, D cr, equal to.7. Finall, the accurac of the MMCCM applied along with the -MVM was checked against the results generated b Vormwald and co-workers [49] b testing tubular samples of Al583 and S46N under the in-phase and 9 out-of-phase strain spectrum reported in Figure. his linear spectrum with an omission level of % contained 456 full-reversed ccles, the ratio between the amplitude of the axial strain, x,a, and the amplitude of the shear strain, x,a, being constant and equal to.577. In the strain spectrum reported in Figure x,max, x,max and MV,MAX are used to denote, for an tests, the corresponding maximum value in the applied loading blocks. he error

14 Please, cite this paper as: Wang, Y., Susmel, L. he Modified Manson Coffin Curve Method to estimate Journal of Fatigue 83 (), pp , 6. diagram of Figure proves that the use of our multiaxial fatigue lifetime estimation technique resulted in estimates falling within the target error band. It is possible to conclude b observing that, as recommended b Sonsino [], the above predictions were made b taking D cr equal to.7 for S46N and equal to.37 for Al Conclusions he MMCCM applied along with the -MVM is seen to be highl successful in estimating lifetime of metallic materials subjected to CA and VA multiaxial load histories. he use of the -MVM allows the computational time required to calculate the stress/strain quantities relative to the critical plane to be reduced remarkabl: this can help to minimise the costs associated with the design process, this being done b alwas reaching a remarkable level of accurac. Under VA load histories, if the critical value of the damage sum cannot be determined experimentall, the proposed approach should be applied b taking D cr equal to (or lower than).7 for steel and.37 for aluminium. As far as un-notched metallic materials are concerned, the MMCCM applied along with the -MVM offers a complete solution to the strain based multiaxial fatigue problem. More works need to be done in this area to extend the use of the proposed multiaxial fatigue life estimation technique to metallic components containing notches. Appendix A. Mathematical formalisation of the -MVM he bod of Figure a is subjected to a complex sstem of forces and moments resulting in tri-axial time-variable states of stress and strain damaging internal reference point O. his point is used to define also a suitable local sstem of coordinates, Oxz. he following tensors are used to summarise the states of stress and strain at point O (where t ): (t) x (t) x (t) xz (t) x z (t) (t) (t) xz (t) z (t) (t) z (A) (t) x (t) x (t) xz (t) x z (t) (t) (t) xz (t) z(t) z (t) (A)

15 Please, cite this paper as: Wang, Y., Susmel, L. he Modified Manson Coffin Curve Method to estimate Journal of Fatigue 83 (), pp , 6. In the above tensors i (t) and i (t) (i=x,, x) are the normal stress and normal strain components, whereas ij (t) and ij (t) (i, j=x,, x) are the shear stress and shear strain components. Angles and as shown in Figure A can be used to define the orientation of a generic material plane, via its normal unit vector, n. According to the above schematisation, is the angle between the projection of unit vector n on the x- plane and the x-axis, whereas is the angle between unit vector n and the z-axis. A second frame of reference, Oanb, can also be defined as shown in Figure A, where the unit vectors giving the orientation of the three axes are as follows: n n n n x z sin sin cos sin cos ; a a a a x z sin cos ; b b b b x z cos cos cos sin sin (A3) Given a generic direction on the plane which passes through point O (Fig. A), the associated unit vector, q, is as follows: q q q q x z cos cos sin sincoscos( ) cos sincossin( ) sin sin (A4) In definition (A4) is the angle between direction q and the a-axis (Fig. A). he instantaneous values of the stress, n (t), and strain, n (t), normal to the plane can directl be determined as: x (t) x (t) xz (t) n x n ( t) n x n n z x (t) (t) z(t) n (A5) xz (t) z(t) z (t) n z x (t) xz (t) x (t) n x x (t) z(t) n ( t) n x n n z (t) n (A6) n z(t) z xz (t) z (t) he shear stress, q (t), and shear strain, q (t), resolved along direction q can instead be determined via the components of unit vector q, i.e.:

16 Please, cite this paper as: Wang, Y., Susmel, L. he Modified Manson Coffin Curve Method to estimate Journal of Fatigue 83 (), pp , 6. x (t) x (t) xz (t) n x q ( t) q x q q z x (t) (t) z(t) n (A7) xz (t) z(t) z (t) n z q (t) q x q q z x (t) x (t) xz (t) x z (t) (t) (t) xz (t) n z(t) n n z (t) x z (A8) Alternativel, shear strain q (t) can also be expressed via the following scalar product: q t d e t, (A9) where d is the vector of direction cosines: d n q n q n q n q n q n q n q n q n q, (A) x x z z x x x z z x z z and e(t) a six-dimensional vector process depending on the components of strain tensor [(t)], i.e.: t t x xz t z e t x t t z t. (A) Vector of direction cosines d can also be expressed through angles, and, as follows [7]: sin( )sin()cos( ) sin( )sin()cos( ) d d sin( )sin( )cos( ) sin( )sin( )sin( ) d 3 d sin( )sin(). (A) d 4 d5 sin( )sin()sin() cos( )cos()sin( ) d6 sin( )cos( )cos() cos( )sin( )cos( ) sin( )sin( )cos() cos( )cos( )cos( ) herefore, the variance of the shear strain resolved along generic direction q can be expressed in the following simplified form:

17 Please, cite this paper as: Wang, Y., Susmel, L. he Modified Manson Coffin Curve Method to estimate Journal of Fatigue 83 (), pp , 6. t q Var Var d kek t didjcov ei t,e j t (A3) k i j B defining smmetric matrix [C] as: Vx Cx, Cx,z Cx,x Cx,xz Cx,z Cx, V C,z C,x C,xz C,z C x,z C,z Vz Cz,x Cz,xz Cz,z [ C] (A4) Cx,x C,x Cz,x Vx Cx,xz Cx,z C x,xz C,xz Cz,xz Cx,xz Vxz Cxz,z Cx,z C,z Cz,z Cx,z Cx,z Vz where (for i, j=x,, z): t V Var (A5) i i t ij Vi Var (A6) C i, j t, t CoVar (A7) i t j ij Cij,i CoVar, i t (A8) t ij Ci,ij CoVar i t, (A9) t t ij ij Cij,ij CoVar, (A) Eq. (A3) can easil be rewritten as follows: q Var t d [C]d. (A) he global maxima of Eq. (A) allow the directions experiencing the maximum variance of the resolved shear stress to be determined directl. In particular, the problem to be solved is nothing

18 Please, cite this paper as: Wang, Y., Susmel, L. he Modified Manson Coffin Curve Method to estimate Journal of Fatigue 83 (), pp , 6. but a conventional optimisation problem which can easil be addressed b using standard methodologies such as the gradient ascent technique [7]. Amongst all the selected planes containing a direction experiencing the maximum variance of the resolved shear strain, according to the fatigue damage model depicted in Figure a, the critical plane is then the one associated with the largest value of ratio, Eq. (). Finall, if *, * and * are used to denote the angles defining the orientation of this plane together with the associated critical direction MV, the stress and strain components relative to the critical plane can directl be determined via Eqs (A5) to (A8), where =*, =* and =*. Acknowledgements he Jiangsu Oversea Research & raining Program for Universit Prominent Young & Middle-aged eachers and the National Natural Science Foundation of China are acknowledged for supporting the present research project (Project No.: 77). References [] Langlais E, Vogel JH, Chase R. Multiaxial ccle counting for critical plane methods. Int J Fatigue 3;5: [] Wang CH, Brown MW. Life prediction techniques for variable amplitude multiaxial fatigue Part : heories. rans. ASME, J. Eng. Mat. echn. 996;8: [3] Wang CH, Brown MW. Life prediction techniques for variable amplitude multiaxial fatigue Part : Comparison with experimental results. rans. ASME, J. Eng. Mat. echn. 996;8: [4] Kim KS, Park JC, Lee JW. Multiaxial fatigue under variable amplitude loads. rans. ASME, J. Eng. Mat. echn. 999;: [5] Kim KS, Park JC. Shear strain based multiaxial fatigue parameters applied to variable amplitude loading. Int J Fatigue 999;: [6] Chen X, Jin D, Kim S. A weight function-critical plane approach for low-ccle fatigue under variable amplitude multiaxial loading. Fatigue Fract Engng Mater Struct 6;9: [7] Socie DF, Marquis GB. Multiaxial Fatigue, SAE, Warrendale, PA,. [8] Smith KN, Watson P, opper H. A stress-strain function for the fatigue of metals. J Mater. 97;5: [9] Socie DF. Fatigue damage models. rans. ASME, J. Eng. Mat. echn. 987;9: [] Kandil FA, Brown MW, Miller KJ. Biaxial low-ccle fatigue fracture of 36 stainless steel at elevated temperature. Met. Soc. London 98;8:3-. [] Wang CH, Brown MW. A path-independent parameter for fatigue under proportional and nonproportional loading. Fatigue Fract Engng Mater Struct 993;6: [] Fatemi A, Socie DF. A critical plane approach to multiaxial fatigue damage including out-ofphase loading. Fatigue Fract Engng Mater Struct 988;: [3] Bannantine JA, Socie DF. A variable amplitude multiaxial life prediction method. In: Fatigue under Biaxial and Multiaxial Loading, Edited b K. Kussmaul, D. McDiarmid and D. Socie, ESIS, Mechanical Engineering Publications, London, pp. 35-5, 99.

19 Please, cite this paper as: Wang, Y., Susmel, L. he Modified Manson Coffin Curve Method to estimate Journal of Fatigue 83 (), pp , 6. [4] Shamsaei N, Gladski M, Panasovski K, Shukaev S, Fatemi A. Multiaxial fatigue of titanium including step loading and load path alteration and sequence effects. Int J Fatigue ;3: [5] Shamsaei N, Fatemi A, Socie DF. Multiaxial fatigue evaluation using discriminating strain paths. Int J Fatigue ;33: [6] Matsuishi M, Endo. Fatigue of metals subjected to varing stress. Presented to the Japan Societ of Mechanical Engineers, Fukuoka, Japan; 968. [7] Fatemi A, Yang L. Cumulative fatigue damage and life prediction theories: a surve of the state of the art for homogeneous materials. Int J Fatigue 998;():9 34. [8] Palmgren A. Die Lebensdauer von Kugellagern. vol. 68. Verfahrenstechnik, Berlin, 94. p [9] Miner MA. Cumulative damage in fatigue. J Appl Mech 945;67:AI [] Sonsino CM. Fatigue testing under variable amplitude loading. Int J Fatigue 7;9 6:8 9. [] Sonsino CM, Kueppers M. Multiaxial fatigue of welded joints under constant and variable amplitude loadings. Fatigue Fract Eng Mater Struct ;4:39 7. [] Chen X, Jin D, Kim KS. Fatigue life prediction of tpe 34 stainless steel under sequential biaxial loading. Int J Fatigue 6;8: [3] Susmel L, Meneghetti G, Atzori B. A simple and efficient reformulation of the classical Manson-Coffin curve to predict lifetime under multiaxial fatigue loading. Part I: plain materials. rans ASME, J Eng Mat echn 9;3():9-/9. [4] Susmel L, Meneghetti G, Atzori B. A simple and efficient reformulation of the classical Manson-Coffin curve to predict lifetime under multiaxial fatigue loading. Part II: notches. rans ASME, J Eng Mat echn 9;3():-/8. [5] Susmel L, Atzori B, Meneghetti G, alor D. Notch and Mean Stress Effect in Fatigue as Phenomena of Elasto-Plastic Inherent Multiaxialit. Engineering Fracture Mechanics ;78: [6] Macha E. Simulation investigations of the position of fatigue fracture plane in materials with biaxial loads. Materialwiss Werkstofftech 989;(4):3 6. [7] Susmel L. A simple and efficient numerical algorithm to determine the orientation of the critical plane in multiaxial fatigue problems. Int J Fatigue ;3: [8] Susmel L, ovo R, Socie DF. Estimating the orientation of Stage I crack paths through the direction of maximum variance of the resolved shear stress. Int J Fatigue 4;58:94. [9] Kanazawa K, Miller KJ, Brown MW. Low-ccle fatigue under out-of phase loading conditions. rans ASME, J Eng Mat echn 977: 8. [3] Socie D, Bannantine J. Bulk Deformation Damage Models. Materials Science and Engineering 988;A3:3-3. [3] Susmel L. Multiaxial Notch Fatigue: from nominal to local stress-strain quantities. Woodhead & CRC, Cambridge, UK, 9. [3] Kaufman RP, opper. he influence of static mean stresses applied normal to the maximum shear planes in multiaxial fatigue. In: Biaxial and Multiaxial fatigue and Fracture, Edited b A. Carpinteri, M. de Freitas and A. Spagnoli, Elsevier and ESIS, 3, pp [33] Benasciutti D, ovo R. Spectral methods for lifetime prediction under wide-band stationar random processes. Int J Fatigue 5;7:

20 Please, cite this paper as: Wang, Y., Susmel, L. he Modified Manson Coffin Curve Method to estimate Journal of Fatigue 83 (), pp , 6. [34] Benasciutti D, ovo R. Ccle distribution and fatigue damage assessment in broad-band non- Gaussian random processes. Probab Eng Mech 5;:5-7. [35] Susmel L, ovo R, Benasciutti D. A novel engineering method based on the critical plane concept to estimate lifetime of weldments subjected to variable amplitude multiaxial fatigue loading. Fatigue Fract Engng Mater Struct 9;3: [36] Susmel L. Estimating fatigue lifetime of steel weldments locall damaged b variable amplitude multiaxial stress fields. Int J Fatigue ;3:57 8. [37] Susmel L, ovo R. Estimating Fatigue Damage under Variable Amplitude Multiaxial Fatigue Loading. Fatigue Fract Engng Mater Struct ;34: [38] Susmel L, alor D. A critical distance/plane method to estimate finite life of notched components under variable amplitude uniaxial/multiaxial fatigue loading. Int J Fatigue ;38:7-4. [39] Louks R, Gerin B, Draper J, Askes H, Susmel L. On the multiaxial fatigue assessment of complex three-dimensional stress concentrators. Int J Fatigue 4;63:-4. [4] Susmel L. Four stress analsis strategies to use the Modified Wöhler Curve Method to perform the fatigue assessment of weldments subjected to constant and variable amplitude multiaxial fatigue loading. Int J Fatigue 4;64: [4] Zhao, Jiang Y. Fatigue of aluminium allo. Int J Fatigue 8;3: [4] Hoffmeer J, Döring R, Seeger, Vormwald M. Deformation behaviour, short crack growth and fatigue lives under multiaxial nonproportional loading. Int J Fatigue 6;8:58 5. [43] Lin H, Naeb-Hashemi H, Pelloux RM. Constitutive relations and fatigue life prediction for anisotropic al-66 t6 rods under biaxial proportional loadings. Int J Fatigue 99;4: [44] Shang DG, Sun GQ, Yan CL. Multiaxial fatigue damage parameter and life prediction for medium-carbon steel based on the critical plane approach. Int J Fatigue 7;9: 7. [45] Hua C, Socie DF. Fatigue damage in 45 steel under constant amplitude biaxial loading. Fatigue Fract Eng Mater Struct 984;7: [46] Kurath P, Downing SD, Galliart DR. Summar of non-hardened notched shaft-round robin program. multiaxial fatigue-analsis and experiments. G. E. Leese and D. F. Socie, eds., SAE, Warrendale, PA, 989:AE-4:3 3. [47] Nelson DV, Rostami A. Biaxial fatigue of A533B pressure vessel steel. ASME J. Pressure Vessel echnol. 997;9: [48] Jiang Y, Hertel O, Vormwald M. An experimental evaluation of three critical plane multiaxial fatigue criteria. Int J Fatigue 7;9:49 5. [49] Hertel O, Vormwald M. Short-crack-growth-based fatigue assessment of notched components under multiaxial variable amplitude loading. Eng Frac Mech ;78: [5] Chen X, An K, Kim KS. Low-Ccle Fatigue of Cr-8Ni-9i Stainless Steel and Related Weld Metal Under Axial, orsional and 9 Out-of-Phase-Loading. Fatigue Fract Eng Mater Struct 4;7: [5] Socie DF, Waill LA, Dittmer DF. Biaxial Fatigue of Inconel 78 Including Mean Stress Effects. In: Multiaxial Fatigue, ASM SP 853, K. J. Miller and M. W. Brown, eds., American Societ for esting and Materials, Philadelphia, PA, 985: [5] Socie DF, Kurath P, Koch J. A multiaxial fatigue damage parameter. Biaxial and Multiaxial Fatigue, EGF 3, M. W. Brown and K. J. Miller, eds., Mechanical Engineering, London, 989:

21 Please, cite this paper as: Wang, Y., Susmel, L. he Modified Manson Coffin Curve Method to estimate Journal of Fatigue 83 (), pp , 6. [53] Kim KS, Lee BL, Park JC. Biaxial Fatigue of Stainless Steel 34 under irregular loading. Fatigue and Fracture Mechanics: 3 st Volume, ASM SP 389, G. R. Halford and J. P. Gallagher, Eds., ASM, West Conshohocken, PA, : [54] Chen X, Jin D, Kim KS. Fatigue life prediction of tpe 34 stainless steel under sequential biaxial loading. Int J Fatigue 6;8: [55] Lee BL, Kim KS, Nam KM. Fatigue analsis under variable amplitude loading using an energ parameter. Int J Fatigue 3;5:6-63. [56] Han C, Chen X, Kim KS. Evaluation of multiaxial fatigue criteria under irregular loading. Int J Fatigue ;4:93-9. [57] Shamsaei N, Fatemi A. Effect of hardness on multiaxial fatigue behaviour and some simple approximations for steels. Fatigue Fract Engng Mater Struct 9;3: [58] Lee Y-L, Pan J, Hathawa RB, Barke ME. Fatigue esting and Analsis. Elsevier Butterworth Heinemann, Oxford, UK, 5 (ISBN: ISBN ). [59] Jiang, Y., and Sehitoglu, H., 996, Modelling of Cclic Ratchetting Plasticit Part I: Development and Constitutive Relations, ASME rans. J. Appl. Mech., 63, pp [6] Jiang, Y., and Sehitoglu, H., 996, Modelling of Cclic Ratchetting Plasticit Part II: Comparison of Model Simulations With Experiments, ASME rans. J. Appl. Mech., 63, pp

22 List of Captions able : Figure : Figure : Figure 3: Figure 4: Figure 5: Figure 6: Figure 7: Figure 8: Static and fatigue properties of the investigated materials Static and fatigue properties of the investigated materials (values in bold indicate the material constants being estimated; values in italic indicate the material constant being determined b post-processing the provided experimental results generated under full-reversed axial loading and full-reversed torsion). Fatigue damage model (a) and modified Manson-Coffin diagram (b). Adopted definitions to calculate the amplitude and the mean value of the stress components relative to the critical plane under both constant and variable amplitude fatigue loading. In-field use of the MMCCM applied along with the -MVM to estimate fatigue lifetime under constant amplitude fatigue loading. In-field use of the MMCCM applied along with the-mvm to estimate fatigue lifetime under variable amplitude fatigue loading. Reference loading paths (IPh=in-phase; OoPh=out-of-phase, ZMS=zero mean strain; N-ZMS=non-zero mean strain). Determination of the reference error band (a); accurac of the MMCCM in predicting fatigue lifetime under CA nominal loading paths (b) see also Figure 5. Specimens of S46N [48], Al [4] and 34SS [54] tested under combined axial and torsional CA strain signals of different frequencies. Specimens of 34SS [48], pure titanium [4], titanium B9 [4], SNCM439 [55], SNCM63 (A) [56] and SNCM63 (B) [55] tested under full-reversed sequential loading (A=Axial cclic loading; =torsional cclic loading: I=inphase axial loading and torsion: O=9 out-of-phase axial loading and torsion). Figure 9: Specimens of S45C [4], 34SS [53], SNCM439 [55], SNCM63 (A) [56] and SNCM63 (B) [55] subjected to short variable amplitude load histories. Figure : Specimens of 45 Q [5] and 34L [5] subjected to discriminating strain paths. Figure : Specimens of Al583 [49] and S46N [49] subjected to short variable amplitude load histories. Figure A: Definition of angles,, and.

23 ables Material Ref. E G K' ' f ' f n' ' f b c [GPa] [GPa] [MPa] [MPa] [MPa] ' f b c Al [4] Al583 [4] (A) [43] (B) [43] S45C [4] Steel [44] SAE 45 [45, 46] A533B [47] S46N [48, 49] AISI 34 [9] Cr-8Ni-9i [5] Inconel 78 [5, 5] SS [53-55] SNCM63 (A) [56] Q [5, 57] SNCM439 [55] SNCM63 (B) [55] L Stainless Steel [5] Pure itanium [4] itanium B9 [4] able : Static and fatigue properties of the investigated materials Static and fatigue properties of the investigated materials (values in bold indicate the material constants being estimated; values in italic indicate the material constant being determined b post-processing the provided experimental results generated under full-reversed axial loading and full-reversed torsion).

24 Figures (t) x (t) Critical plane a, a x (t) n,m n,a x (t) x (t) x (t) n,m n,a a, a x (t) (a) (t) a Full-Reversed orsional fatigue curve < i << j = i Increasing j = Full-reversed uniaxial fatigue curve N f (b) Figure : Fatigue damage model (a) and modified Manson-Coffin diagram (b).

25 (b) Critical Plane O MV (t) MV (t) n (t) MV (t), (t) F j (t) z O x F k (t) F i (t) (a) Constant Amplitude Variable Amplitude MV (t) MV,max MV,min a m t m a MV,max MV,max (c) MV,min MV,min MV (t) a m t Var m a (t) (t) MV MV Var (t) dt MV (d) (t) MV m dt MV (t) n (t) MV,max MV,min n,max n,min a m t n,a n,m t m a n,m n,a MV,max MV,max n,max n,max MV,min MV,min n,min n,min MV (t) n (t) a m t n,a n,m t m Var a Var (t) (t) n,m n,a MV MV Var (t) dt MV (t) (t) dt MV (t) (t) n n Var n n (t) n,m m dt dt Figure : Adopted definitions to calculate the amplitude and the mean value of the stress components relative to the critical plane under both constant and variable amplitude fatigue loading.