A fatigue driving energy approach to high-cycle fatigue life estimation under variable amplitude loading

Transcription

1 ORIGINL CONTRIBUTION doi: 0./e.2347 atigue driving energy approach to high-cycle atigue lie estimation under variable amplitude loading Z. PENG, H-Z. HUNG, S-P. ZHU, H. GO and Z. LV Institute o Reliability Engineering, University o Electronic Science and Technology o China, No. 2006, Xiyuan venue, West High-ech Zone, Chengdu, Sichuan 673, China Received Date: 3 February 205; ccepted Date: 5 July 205; Published Online: 26 ugust 205 BSTRCT In this paper, a concept o atigue driving energy is ormulated to describe the process o atigue ailure. The parameter is taken as a combination o the atigue driving stress and strain energy density. By assessing the change o this parameter, a new non-linear damage model is proposed or residual lie estimation within high-cycle atigue regime under variable amplitude loading. In order to consider the eects o loading histories on damage accumulation under such condition, the load interaction eects are incorporated into the new model, and a modiied version is thus developed. Lie predictions by these two models and Miner rule are compared using experimental data rom literature. The results show that the proposed model gives lower deviations than the Miner rule, while the modiied model shows better prediction perormances than the others. Moreover, the proposed model and its modiications are ease o implementation with the use o S N curve. Keywords high-cycle atigue; lie estimation; non-linear damage model; strain energy density; variable amplitude loading. NOMENCLTURE = atigue strength constant b = material constant D = damage variable E = Young s modulus h = atigue strength exponent n = number o cycles at a given stress level N = number o cycles to ailure W = elastic strain energy density = amplitude o elastic strain energy density = atigue driving energy = initial atigue driving energy = critical atigue driving energy σ = applied load stress level σ D = atigue driving stress = equivalent atigue driving stress σ a = applied stress amplitude σ = atigue strength coeicient ε = elastic strain ε a = elastic strain amplitude ω = interaction actor W a W D W D0 W Dc σ equiv INTRODUCTION In engineering applications, atigue ailure is prevalent and prone to be one o the most common ailure modes Correspondence: H.-Z. Huang. hzhuanguestc.edu.cn or most engineering components, which are usually subjected to cyclic loading. During their service operation, atigue loading oten involves variable amplitude loading (VL), yet seldom constant amplitude loading (CL). The concept o damage is crucial to describe 80

2 FTIGUE DRIVING ENERGY PPROCH TO HIGH-CYCLE FTIGUE LIFE ESTIMTION 8 the process o atigue ailure, and it also plays an important role in atigue lie estimation. Owing to the complex loading histories, assessing the atigue damage behavior o these components depends on special testing under actual service conditions, which is costly and time consuming. In contrast, testing under CL is more readily implemented and very cost-eective, and these available CL data contribute to evaluate the atigue damage under VL. However, loading histories eects, such as load sequences, load interactions, etc., have a signiicant inluence on damage evolution. Thereore, there is a strong need to develop a reliable model to deal with the process o damage accumulation under variable atigue loading, aiming to enable a better estimation o atigue lie. Until now, many damage accumulation models have been developed or atigue lie estimation. One o the most commonly used models is the Palmgren Miner rule (Miner rule or linear damage rule) because o its simplicity and ease o implementation in design. Fatigue ailure is expected to occur when the cumulative damage is equal to the critical cumulative damage, i.e. unity. Unortunately, the model is acknowledged to be overshadowed by its intrinsic shortcomings, 2,3 shown as: () In the case o CL, the Miner rule is load-level independent, and the damage evolution process remains the same or any given stress level. (2) In the case o VL, each stress level keeps the relative independence, regardless o load interactions and load sequences accountability. (3) The model is exclusive o the atigue damage induced by small loads below the atigue limit. In addition, many studies 4 8 suggest that Miner rule is inaccurate or gives a large deviation with the reality or certain combinations o variable loading. Some researchers endeavour a lot to modiy it, but lie estimates using these extended versions are still ound to be unsatisactory. 3 The inadequacy may be typically attributed to the model itsel. Subsequently, to address the intrinsic deiciencies associated with the Miner rule, researchers have resorted to the non-linear cumulative damage theories, and various non-linear versions are developed. 3,9,0 Essentially, atigue damage is a process o irreversibility, material properties degradation and energy dissipation. It is essential to the deinition o a reliable damage model when quantiying and accumulating atigue damage. In general, atigue damage can be measured by assessing the changes in mechanical properties or physical properties. For mechanical properties, a series o state variables, such as modulus o elasticity, static toughness, stiness, ductility, reduction o area, tensile strength, etc., are oten used to characterize atigue damage. Belaadi et al. 2 and Zhou et al. 3 suggested that the modulus o elasticity or dierent materials gradually decreased with the accumulation o atigue damage. Ye and Wang 4 demonstrated that the reduction in static toughness o materials was a useul parameter to present the damage variable. Devulder et al. 5 presented that the stiness degradation could be used as a damage measure. Cheng and Plumtree 6 and Zhu et al. 7 showed that the reduction in ductility was suitable or assessing atigue damage. Zhu et al. 8 and Yuan et al. 9 reported that the strength degradation o materials could be deined as atigue damage. For physical properties, some state variables, such as electric, magnetic and thermal properties, are also used to characterize atigue damage accumulation behavior. Lemaitre and Duailly 20 presented that damage could be measured by the variation in the electrical potential or conductive materials. Sun et al. 2 used the relative increment in electrical resistance to describe damage. Khonsari et al.,22 24 proved that the thermodynamic entropy generation could be treated as a degradation process or damage prediction. Ultimately, as aorementioned, these changes o material properties can be taken as the response o stress or strain indirectly.,25 To a certain extent, they may be extremely attributed to the change o energy in materials. zadi et al. 26 also suggested that atigue damage monotonously increased with the accumulation o the plastic strain energy. Recently, the strain energy density parameter has oten been used or atigue analysis. Łagoda 27,28 gave an elaboration o the strain energy density models, which were used to estimate atigue lie under uniaxial random loading. Park et al. 29 proposed a strain energy density based model to describe the atigue behavior o rolled Z3 magnesium alloy, which shows an agreement between the predictions and the experimental data. Djebli et al. 30 used the strain energy density parameter to present damaged stress, while also obtaining a new version o the damaged stress model previously by id et al. 7 or high-cycle atigue lie prediction. Zhu et al. 3 developed a generalized atigue creep damage model on the basis o plastic strain energy density parameter, providing a better lie prediction o turbine disk alloys (GH433). In this regard, the strain energy density parameter oers an eective method to track damage accumulation behavior. Currently, Kwoie and Rahbar 32 proposed a concept o atigue driving stress, and residual atigue lie can be estimated by determining the equivalent atigue driving stress as the previous stress levels under VL. However, the equivalent atigue driving stress does not mean the equivalent atigue damage because o the idea o equal damage introduced by Richart and Newmark. 33

3 82 Z. PENG et al. In this paper, a concept o atigue driving energy (FDE) is introduced to present atigue damage evolution within the high-cycle atigue regime. It is considered that the FDE is taken as a combination o the atigue driving stress and strain energy density. By assessing the change o FDE parameter, a new nonlinear damage model, as well as its modiied version considering load interactions, is ormulated to predict residual atigue lie under VL. Five categories o experimental data in literature are used to investigate the applicability and capability o new models. Furthermore, a comparison with the commonly used Miner rule is also made. NEW NON-LINER DMGE MODEL BSED ON FDE PRMETER For high-cycle atigue, the controlling parameter is signiicantly dependent on elastic strain or stress level. The constitutive relation between stress and lie can be presented by the Wöhler curve (S N curve), and its exponential notation can be expressed as: σn b ¼ () where N is the number o cycles to ailure at a given stress level σ; b is a constant associated with material property (b < 0) and is the atigue strength constant, which presents the inherent nature o the material independent o applied loads. Once the stress (the let side o Eq. () in the material with the accumulated cycles reaches the constant value, ailure or racture occurs. Lately, a concept o atigue driving stress was proposed by Kwoie and Rahbar 32 on the basis o S N curve. It can be expressed as a unction o applied stress, consumed lie raction and atigue lie o the applied stress, shown as: ccording to Eq. (2), the estimation o residual lie raction at σ 2 can be obtained by n 2 ¼ n ln N : (3) N 2 N ln N 2 In Eq. (2), or the initial state (n/n = 0), note that the atigue driving stress depends on the applied stress only; or the critical state (n/n = ), the atigue driving stress is a constant equal to the atigue strength o the material, i.e.. Plots o σ D versus n/n or two dierent stress amplitudes (σ and σ 2, σ > σ 2 ) under CL are shown in Fig.. However, particularly or a low high loading sequence (the irst applied stress is σ 2 or n 2 cycles, ollowed by σ or n cycles to ailure), suppose that the consumed lie raction at the lower stress level is small enough, it will cause a near impossibility o inding an available equivalent atigue driving stress corresponding to the higher stress level (see Fig., σ equiv denotes the equivalent atigue driving stress), hindering its implementation or residual lie prediction. Early in 948, Richart and Newmark 33 introduced the concept o equal damage based on S N approach. Up to now, residual lie assessment is oten implemented by the equivalent damage rule. Despite the availability o σ equiv, it is always unable to guarantee that the applied stresses can yield the equivalent atigue damage in accordance with the equivalent driving stress (except when σ equiv = ). Thus, it is necessary to develop an available damage model, in terms o the equivalent damage rule, to assess residual atigue lie. Recently, the strain energy density, generally expressed as a unction o stress σ and strain ε, is oten used as a damage parameter or damage calculation and σ D ¼ σn b n N (2) where n is the number o loading cycles at a given stress level σ, n/n is the consumed lie raction and σ D is atigue driving stress, which increases with the accumulated cycles until a critical value is attained, where the inal racture occurs. In the case o VL, let the previous stress levels yield the equivalent driving stress equal to the last one, the residual lie or lie raction can be achieved. For a two-level block loading, let the stress level σ be applied n cycles irst, ollowed by the stress level σ 2 applied n 2 cycles up to ailure, and let N and N 2 be the atigue lives o applied stress levels, respectively. Fig. Variation o atigue driving stress under CL or two dierent stress amplitudes (σ and σ 2 ).

4 FTIGUE DRIVING ENERGY PPROCH TO HIGH-CYCLE FTIGUE LIFE ESTIMTION 83 atigue analysis. In the elastic range, the strain energy density parameter can be calculated in a mathematical orm, 27 shown as: W ¼ σε: (4) 2 Then, the amplitude o this parameter can be derived as a unction o stress amplitude and strain amplitude, i.e. σ a and ε a respectively, 27,28,30 shown as: W a ¼ 2 σ aε a : (5) For high-cycle atigue, located in the elastic range, there is a signiicant linear correlation between the stress and strain, expressed by the Basquin equation and Manson Coin equation (regardless o plastic strain) respectively, shown as: σ a ¼ σ h 2N (6) ε a ¼ σ E 2N h (7) where σ is the atigue strength coeicient, h is atigue strength exponent and E is Young s modulus. Substituting Eqs. (6) and (7) into Eq. (5) leads to W a ¼ 2E σ2 a : (8) Combining Eqs. (2) and (8), a new parameter o atigue driving energy, W D, is expressed as ollows: W D ¼ 2E σ2 n N : (9) For the initial state without damage, i.e. n/n = 0, the FDE is W D0 ¼ 2E σ2 : (0) For the critical state, i.e. n/n =, it is W Dc ¼ 2 2E : () Note that the initial value W D0, depending on the applied stress level, denotes the initial FDE, which attempts to cause damage in the material, while the critical value W Dc, a constant with respect to the atigue strength o the material, can be taken as the inherent energy stored in material as well as a threshold or preventing the material rom ailure. With regard to Eq. (9), the intermediate variable W D, driving atigue damage, will increase non-linearly with the accumulated cycles until the threshold W Dc is reached when the inal racture occurs. dmittedly, atigue damage accumulation arises rom the energy dissipation in materials, which can be equal to the continuous increase o FDE up to the threshold W Dc. Hence, the increment o W D can be used to present the process o atigue damage evolution, and a new damage accumulation model is deined as: D ¼ W D W D0 W Dc W D0 ¼ N 2bN n : (2) In Eq. (2), at the irst cycle, the instantaneous FDE is identical to the initial value W D0, i.e. D = 0; at the last cycle, it equals the critical value W Dc and thus D =. The model relates to three itting parameters, i.e. atigue lie at the applied load (N ), consumed lie raction (n/n ) and material constant (b). In addition, it shows a nonlinear (exponential) correlation between D and n/n, and the damage unction relies on the atigue lie at the applied load, that is load-level dependence. For a simply two-level block loading (at the irst stress level σ or n cycles, then at the second stress level σ 2 or n 2 cycles to ailure), according to the equivalent damage rule, the estimation o residual lie raction or σ 2 can be obtained as: n 2 N 2 0 lnb pp ¼ 2b ln N 2 n 2b N 2 N þ C (3) where the subscript pp reers to the prediction by the proposed model. Interestingly, suppose that >>, Eq. (2) can be rewritten as: D 2b -N n : (4) N ccording to the equivalent damage rule, the predicted residual lie raction at σ 2 can be derived as:

5 84 Z. PENG et al. n 2 ¼ n ln N : (5) N 2 N ln N 2 The result is the same as Eq. (3). It indicates that the atigue driving stress model, as shown in Eq. (2), is just a particular case o the proposed model Eq. (2) when >>. Similarly, or a three-level block loading, let the third stress level σ 3 yield the equivalent damage equal to the sum o those by the previous stress levels. Then, the estimation o n 3 / N 3 at σ 3 can be given by: n 3 N 3 ¼ 2b ln N pp n 2b 2 N þ n 2 2 N 2 pp 3 B N 2 C ln þ 2 : B C (6) It is worth noting that Eq. (6) can be easily generalized into multi-level block loading; let the stresses σ, σ 2, σ 3,, σ i orderly be applied the number cycles o n, n 2, n 3,, n i, and let N, N 2, N 3,, N i be the ailure lives o these applied stresses, respectively. Let the cumulative damage caused by all o the previous stresses be equal to that by the last one. Thus, the estimation o n i / N i at σ i can be derived as: ln n i N i 0 B pp ¼ 2b ln N i 0 n 2b i N þ n i ði Þ i N N ð i Þ pp B ði Þ C ði Þ þ : C (7) Note that Eqs. (3), (6) and (7) relate to the parameters, that is atigue lives o applied stresses and material constant b in Eq. (). Hence, residual lie raction can be easily obtained using S N curve. MODIFICTIONS ND TYPICL BEHVIORS OF THE PROPOSED NON-LINER RULES modiied model accounting or load interactions Generally, in the case o VL, loading histories, such as load sequences and load interactions, have a signiicant eect on atigue damage evolution. Fatigue damage variable D with a non-linear atigue behavior is commonly ormulated regarding the applied stress and the consumed lie raction. Thus, a general description o D can be presented as a unction: n D ¼ σ; N (8) where D should be limited to 0 D. Many studies implied that atigue damage evolution under VL diered rom that under CL, the prior loading histories at a certain stress could aect the damage accumulation law by the subsequent stress. Speciically, or high low loading sequence, it tends to accelerate the process o damage evolution at the subsequent stress, on the contrary or low high loading sequence. Simultaneously, the bigger dierence between the applied stress levels, the greater eect it shows. This phenomenon (load interactions) is commonly described in a manner o load ratio (interaction actor), reported by Corten and Dolan, 39 Freudenthal and Heller, 34 Morrow 35 and Huang et al. 9,38,40 s aorementioned, considering a two-level loading (n cycles at σ ollowed by n 2 cycles at σ 2 to ailure), the phase o damage evolution at the irst stress σ remains the same as its original evolution law under CL, while the phase at the second stress σ 2 will change. In this study, assuming that the load ratio between two successive stress levels is used as an interaction actor to present load interactions. For simplicity, the interaction actor, which is taken as a parameter ω or later convenience, can be postulated as ω = σ / σ 2. Despite the change o damage accumulation law at σ 2, the damage variable D with respect to n / N 2 should still be limited to 0 D during the entire lie independently. s a consequence, a modiied damage variable considering load interactions or σ 2 can be deined as ollows: n ω n σ σ 2: D ¼ σ 2 ; ¼ σ 2 ; (9) N 2 N 2 For the proposed model, according to the equivalent damage rule, a new prediction o residual lie raction at σ 2 can be calculated by:

6 FTIGUE DRIVING ENERGY PPROCH TO HIGH-CYCLE FTIGUE LIFE ESTIMTION 85 n 2 N 2 mp ¼ 2b ln N 2 0 ln 2 n σ 0 2 σ N B C B þ C (20) where the subscript mp reers to the prediction by the modiied model. Similarly, or three-level block loading, a new estimation o n 3 / N 3 at σ 3 is n 3 ¼ N 3 2b ln N mp n 2b 2 N þ n 2 2 N 2 mp N ln 3 2 B B 2 C σ σ σ 3 2 σ 2 þ : C (2) lso, or multi-level block loading, a new estimation o n i / N i at σ i is n i N i ¼ mp 2b ln N i 0 0 n 2b i N þ n i ði Þ N ði Þ mp N ln ði Þ i B ði Þ B C σ i 2 σ σ i i σ i þ : C (22) Notice that in Eqs. (20), (2) and (22), the prediction o residual lie raction relates to the parameters, i.e. the material constant b in Eq. (), the applied stresses and their atigue lives. These parameters can also be determined rom the S N curve. Typical behaviors o the proposed non-linear rules under two-level atigue loading Two-level atigue loading is commonly used as a representative to analyse the typical behaviours o atigue damage rules. In order to present the signiicance o the proposed non-linear damage rules, three dierent applied loads are chosen as a demonstration o high-cycle atigue regime, their atigue lives are N = , N = and N = , respectively. Besides, our typical values o material constant b are adopted, i.e , 0.250, and Consider that the initial lie raction and residual lie raction at the irst and second stress loading are n / N and n 2 / N 2, respectively. Six combinations o atigue lives (in the orm o N / N 2 ) are 0.3, and 0. or high low loading and 3.333, 3 and 0 or low high loading, respectively. Plots o n 2 / N 2 versus n / N or the selected values o b and all combinations o atigue lives are shown in Fig. 2. In the igure, the ollowing non-linear behaviours can be noted: () The proposed method Eq. (3) and its modiication Eq. (20) yield dierent atigue envelopes dependent on the loading sequences, while Miner rule always gives the same envelope (a black straight line in Fig. 2). It suggests that Miner rule is load sequence independent; these two models developed consider the load sequence eects. (2) The high low and low high atigue envelopes lie in the let and right side o Miner s envelope, respectively. These two envelopes exhibit some symmetry with respect to the Miner s envelope. (3) Either the high low envelope or low high envelope deviates rom the Miner s envelope with the decrease o the material constant b. Ib tends to 0, these two envelopes become very close to the Miner s envelope. s the absolute value o b increases, the atigue envelopes using Eq. (20) show a signiicant dierence with the Miner s envelope gradually. However, all the envelopes o these two proposed non-linear rules are conined by the box rom (0, 0) to (, ). EXPERIMENTS ND DISCUSSIONS In this section, ive categories o experimental data, including aluminium alloy l-2024, 7,4 welded aluminium alloy joints o Electric Multiple Units, 42 spheroidal graphite cast-iron (GS6), 7 aluminium alloy 6082T6, 7,30 and titanium alloy Ti 6l 4V, 43 are used to veriy the eectiveness o the proposed model, discussed in the Section on New Non-linear Damage Model Based on FDE Parameter, and the modiied model, discussed in the Section on Modiied Model ccounting or Load Interactions. Comparisons among the Miner rule, the proposed model and the modiied model are made to show their prediction perormances. Case : aluminium alloy l-2024 The material in this case is the aluminium alloy l ,4 The experiments were carried out under two-level block loading or both low high and

7 86 Z. PENG et al. Fig. 2 Plots o residual lie raction n 2 / N 2 versus initial lie raction n / N or our typical values o b and all combinations o atigue lives. high low sequences, whose load spectrums are Mpa and Mpa, respectively. Fatigue lives or these two applied stresses σ = 50 Mpa and σ = 200 Mpa are N = cycles and N = cycles, respectively. In Table, it shows the experimental data and predicted results o the residual lie raction at the second stress level σ 2. It can be seen that all the predictions obtained by the proposed model and the modiied model agree well with the experimental data; these two presented models predict n / N + n 2 / N 2 > or low high loading sequence and n / N + n 2 / N 2 < or high low loading sequence, while Miner rule predicts n / N + n 2 / N 2 = or each loading sequence. In Fig. 3, it also unolds a clear comparison between the experimental data and predicted results using these three models. It is worth noting that the proposed model predicts better results than the Miner rule, while the modiied model shows a signiicant improvement in prediction perormance over other two models. Case 2: butt weld joint and illet weld joint The welded aluminium alloy joints o Electric Multiple Units are used in this study, 42 including butt weld joint and illet weld joint. For butt weld joint, the experimental results were obtained by two-level block loading up to the ailure, three stress levels were applied or both low high and high low loading sequences; our conigurations o load spectrums are Mpa and Mpa or

8 FTIGUE DRIVING ENERGY PPROCH TO HIGH-CYCLE FTIGUE LIFE ESTIMTION 87 Table Predicted results obtained by Miner rule, the proposed model and the modiied model or Case : aluminium alloy l-2024 Load sequence Low high High low Load stress σ = 50 Mpa σ 2 = 200 Mpa σ = 200 Mpa σ 2 = 50 Mpa Experimental data Miner rule Proposed model Modiied model n n / N n 2 n 2 / N 2 n 2 / N 2 n 2 / N 2 n 2 / N Fig. 3 Comparison o predicted results obtained by Miner rule, the proposed model and the modiied model or aluminium alloy l high low loading sequence, and Mpa and Mpa or low high loading sequence; atigue lives or these three applied stresses σ = 04 Mpa, σ = 89 Mpa and σ = 74 Mpa are N = cycles, N = cycles and N = cycles, respectively. For illet weld joint, the tests were also carried out under two-level block loading, including low high and high low loading sequences; three applied stress levels are σ = 93 Mpa, σ = 83 Mpa, and σ = 73 Mpa and their atigue lives are N = cycles, N = cycles and N = cycles, respectively; our conigurations o load spectrums are Mpa and Mpa under high low loading, and Mpa and Mpa under low high loading, respectively. The original data rom experiments and theoretical results obtained by Miner rule, the proposed model and the modiied model are listed in Table 2. It shows that all the predictions rom the proposed model and the modiied model are consistent with the experimental results. For Table 2 Predicted results obtained by three models or Case 2: butt weld joint and illet weld joint Type o weld joint Load sequence Load stress Butt weld joint High low σ = 04 Mpa σ 2 = 74 Mpa σ = 89 Mpa σ 2 = 74 Mpa Low high σ = 74 Mpa σ 2 = 89 Mpa σ = 74 Mpa σ 2 = 04 Mpa Fillet weld joint High low σ = 93 Mpa σ 2 = 73 Mpa σ = 83 Mpa σ 2 = 73 Mpa Low high σ = 73 Mpa σ 2 = 83 Mpa σ = 73 Mpa σ 2 = 93 Mpa Experimental data Miner rule Proposed model Modiied model n /0 3 n / N n 2 /0 3 n 2 / N 2 n 2 / N 2 n 2 / N 2 n 2 / N

9 88 Z. PENG et al. each weld joint, both o the presented models always give n / N + n 2 / N 2 > or low high loading sequence and n / N + n 2 / N 2 < or high low loading sequence. In addition, a graph comparison o the experimental data and predictions are illustrated in Fig. 4. Note that the proposed model gives better predictions than the Miner rule, while the modiied model shows a better correlation between the theoretical and experimental results than the others. Case 3: spheroidal graphite cast-iron (GS6) The material studied here in this case is spheroidal graphite cast-iron (GS6). 7 Two types o loading tests, plane bending and torsion, were also perormed under twolevel block loading or both low high and high low loading sequences. For plane bending loading, three applied stress levels are σ = 352 Mpa, σ = 320 Mpa, and σ = 303 Mpa and their atigue lives are N = cycles, N = cycles, N = cycles, respectively; our combinations o load spectrums are Mpa and Mpa or low high loading sequence, and Mpa and Mpa or high low loading sequence, respectively. For torsion loading, two applied stress levels are σ = 249 Mpa and σ = 233 Mpa, and their atigue lives are N = cycles, and N = cycles, respectively; load spectrums under low high and high low loading sequence are Mpa and Mpa, respectively. In Table 3, it shows the details o experimental data and predicted results using Miner rule, the proposed model and the modiied model. The proposed model, as well as the modiied model, again ollows the trend in that n / N + n 2 / N 2 > or low high loading sequence and n / N + n 2 / N 2 < or high low loading sequence. lso, most predictions rom these two presented models are close to the experimental results, only one observation deviates rom experimental value, which occurs in the torsion test under low high loading sequence. Besides, in Fig. 5, it clearly displays the dierences between the experimental data and predictions. It is worth noting that the calculation results using the proposed model are better than those using the Miner rule, while the modiied model gives reasonably accurate predictions in contrast with the others. Case 4: aluminium alloy 6082T6 Fig. 4 Comparison o predicted results obtained by Miner rule, the proposed model and the modiied model or butt weld joint and illet weld joint. The material used in the current study is aluminium alloy 6082T6. 7,30 The experiments were conducted under our-level block loading with three dierent Table 3 Predicted results o three models or Case 3: spheroidal graphite cast-iron (GS6) Type o loading Load sequence Load stress Plane bending Low high σ = 320 Mpa σ 2 = 352 Mpa σ = 303 Mpa σ 2 = 352 Mpa High low σ = 352 Mpa σ 2 = 320 Mpa σ = 352 Mpa σ 2 = 303 Mpa Torsion Low high σ = 233 Mpa σ 2 = 249 Mpa High low σ = 249 Mpa σ 2 = 233 Mpa Experimental data Miner rule Proposed model Modiied model n n / N n 2 n 2 / N 2 n 2 / N 2 n 2 / N 2 n 2 / N

10 FTIGUE DRIVING ENERGY PPROCH TO HIGH-CYCLE FTIGUE LIFE ESTIMTION 89 and n / N + n 2 / N 2 + n 3 / N 3 + n 4 / N 4 < or high low loading sequence. In addition, a comparison between the experimental data and models predictions is represented in Fig. 6. Obviously, the predicted lie ractions by the proposed model match well with the experimental results, compared with those o the Miner rule, while the modiied model yields better results than the others. Case 5: titanium alloy Ti 6l 4V Fig. 5 Comparison o predicted results obtained by Miner rule, the proposed model and the modiied model or spheroidal graphite cast-iron (GS6). The material o titanium alloy Ti 6l 4V is studied in this case. 43 The experiments were perormed with three conigurations o two-level block loading, i.e. high low, low high and repeated block loading. For these three loading tests, the applied higher and lower stress levels are σ = 27 Mpa and σ = 2 Mpa, and their average a- loading sequences (increasing, decreasing and irregular loading sequence). Four applied stress levels were considered, i.e. σ = 240 Mpa, σ = 260 Mpa, σ = 280 Mpa and σ = 305 Mpa, and their atigue lives are N = cycles, N = cycles, N = cycles and N = cycles, respectively. Load spectrums or increasing, decreasing and irregular loading sequence are Mpa, Mpa and Mpa, respectively. detailed overview o experimental data and predicted results rom Miner rule, the proposed model and the modiied model is shown in Table 4. It seems that the predictions o these two presented models closely accord with the experimental data; once again they still ollow the same trend in that n / N + n 2 / N 2 + n 3 / N 3 + n 4 / N 4 > or low high loading sequence Fig. 6 Comparison o predicted results obtained by Miner rule, the proposed model and the modiied model or aluminium alloy 6082T6. Table 4 Predicted results o three models or Case 4: aluminium alloy 6082T6 Experimental data Block Miner rule Proposed model Modiied model 0 Increasing loading Load stress (Mpa) n cycles n / N Decreasing loading Load stress (Mpa) n cycles n / N Irregular loading Load stress (Mpa) n cycles n / N

11 90 Z. PENG et al. tigue lives are N = cycles and N = cycles, respectively. Load spectrums under high low and low high loading are 27 2 Mpa and 2 27 Mpa, respectively. In the repeated two-level block loading, each block is taken as a high low loading (27 2 Mpa) with various combinations o speciied atigue cycles or these two stress levels. Loading conditions, experimental data and model predictions are listed in Table 5 (M denotes the number o repeated blocks to racture, M = or the two-level high low and low high block loading). In the table, it can be ound that the calculation results using the proposed rules conorm to the experimental data; both o them yield M n / N + M n 2 / N 2 > orlow high loading and M n / N + M n 2 / N 2 < orhigh low loading and repeated block loading. To clearly show these results, plots o the lie raction M n 2 / N 2 versusthelieractionm n / N are shown in Fig. 7 (Exp. and Pre. denote the experimental data and predictions, respectively). It is observed that the predictions o the Miner rule show a signiicant discrepancy with the experimental data; the modiied model yields a better correlation between the theoretical and experimental results than the proposed model. loading is greater than, while the decreasing loading is less than. This phenomenon agrees with many experimental evidences, suggesting that the damage mechanisms responsible or variable atigue loading have a strong dependence o load sequences. The proposed non-linear rules are capable o producing dierent CL damage unctions (the cumulative damage D in relation to the consumed lie raction n/n ), which depends on the applied loads. Using the equivalent damage rule, it DISCUSSIONS ccording to the ive above-mentioned case studies, it shows that the proposed model and the modiied model predict that the sum o lie ractions with the increasing Fig. 7 Plots o the lie raction M n 2 / N 2 versus the lie raction M n / N under three conigurations o two-level block loading or titanium alloy Ti 6l 4V. Table 5 Predicted results obtained by three models or Case 5: titanium alloy Ti 6l 4V Load stress n N 2 2 Experimental data Miner rule Proposed model Modiied model n 2 N N N N N 2 N N 2 M n / M n 2 / M n / M n 2 / M n / M n 2 / M n / M n 2 / Two-level high low block loading σ = 27 Mpa σ 2 = 2 Mpa Two-level low high block loading σ = 2 Mpa σ 2 = 27 Mpa Repeated two-level block loading σ = 27 Mpa σ 2 = 2 Mpa

12 FTIGUE DRIVING ENERGY PPROCH TO HIGH-CYCLE FTIGUE LIFE ESTIMTION 9 leads to dierent Miner s damage sums or dierent loading sequences. The proposed model, as well as its modiications, shows a better agreement between the predicted and experimental results than the Miner rule whatever the loading coniguration. mong these three models, Miner rule still dominantly used has the simplest orm and is easy to implement in design. However, the model does not take into account loading histories (such as load sequences and load interactions). The unction ormulating the CL atigue damage with respect to n/n is independent o applied loads, while it has been veriied that atigue ailure or many metallic materials commonly exhibits highly non-linear damage behaviours with load dependency. 44,45 Thereore, the predictions using Miner rule do not strongly correlate with the experimental results. Compared with the Miner rule, the proposed model o Eq. 2 yields a non-linear damage behaviour dependent on the atigue lie o applied load, showing the eects o load sequences under VL. Thus, the model is expected to give lower deviations than Miner rule. In the present work, the proposed model is developed based on the concept o atigue driving energy. The parameter is exhibited as a combination o the atigue driving stress and strain energy density. For the proposed model, it assumes that atigue damage variable D is deined by assessing the change o FDE. The model can not only remedy the insuiciency associated with atigue driving stress model (σ equiv should be available in Fig. ) but also provide clear physical connotations. It is the FDE driving atigue damage with the consumed lie raction n / N gradually. t n / N = 0, the initial FDE is just dependent on the applied stress, attempting to cause damage in the material, and the cumulative damage D = 0; thereater, the FDE, as well as the cumulative damage, will increase non-linearly with the accumulated cycles, and D > 0, indicating that the material is damaged progressively; when n/n =, the FDE reaches a threshold, treated as the inherent energy stored in material or preventing the material rom ailure, the complete racture occurs, and D =. s a consequence, the proposed model oers an appropriate method to characterize the complete process o atigue ailure. However, it is ound that the predictions o this model are slightly better than the counterparts by Miner rule. This suggests that the damage evolution under VL may become very complicated. lthough the proposed model can deal with the load sequence eects, the damage evolution law o variable atigue loading can be dominated by other eects because o the complex loading histories, such as load interactions. In addition, the modiied model exhibits a better prediction perormance than the others. On the basis o the proposed model, its modiications are designed to consider the eects o load interaction. In Eq. 9, it suggests a possible way to assess the eects between two successive stress levels on damage accumulation. Owing to incorporating the eects o load sequence and load interaction, the predictions using these modiications can correctly ollow the experimental results and are more representative than those o the proposed model and Miner rule. Moreover, or titanium alloy Ti 6l 4V, as illustrated in Fig. 7, the proposed model, as well as its modiication, yields dierent atigue envelopes or dierent loading sequences, while Miner rule yields the same envelope (a straight line). This once again suggests that Miner rule is load order independent and ails to consider this eect on damage accumulation; the proposed rules have the capability o producing load sequences. The high low envelope and low high envelope are symmetric with respect to the Miner s envelope, and all o envelopes are conined by the box rom (0, 0) to (, ). Hence, this case can demonstrate the typical behaviours o the proposed non-linear rules. For the case o two-level block loadings, as shown in Table 5, the atigue envelopes using the modiied model are in a better agreement with the experimental data than those o the proposed rule and Miner rule, because o the consideration o load interaction eects. In the repeated block loading, note that the modiied rule achieves better prediction accuracy than the others. This modiication allows us to consider more load histories inormation and to give reasonable predictions, in spite o the complex atigue behaviour under such repeated loading. Consequently, the modiied rule can be used to estimate the atigue lie under repeated block loading conditions. Furthermore, the newly proposed models still maintain a simple orm and are easy to be implemented based on the S N curve. Both o them are only applicable to high-cycle atigue regime under uniaxial loading and can be extended to multi-axial loading. t present, the new rules developed are based on deterministic methodology, while variable atigue loading is oten involved in various uncertainties and exhibits highly complexities. Hence, lie prediction associated with these uncertainties remains to be investigated, aiming to provide an accurate assessment. CONCLUSIONS In this study, a concept o atigue driving energy is presented as a combination o the atigue driving stress and strain energy density to characterize the atigue damage. By assessing the change o FDE parameter, a non-linear damage model is developed or high-cycle atigue lie

13 92 Z. PENG et al. prediction under VL. This new model oers the possibility o describing the complete process o atigue ailure. Through comparing it with the Miner rule, the model shows lesser deviations with the reality and can correctly describe the eects o load sequence responsible or variable atigue loading. Through taking the load interaction eects into account, a modiication based on the proposed model is also developed and urther improves its prediction accuracy. The typical behaviours o these two presented non-linear rules under two-level block loading are demonstrated by the case o titanium alloy Ti 6l 4V. The models properties are also veriied and the modiied rule can be applied to the repeated block loading. Moreover, both o the new rules maintain a simple orm and are ease o application by using the S N curve only. cknowledgements The present study was partially supported by the National Natural Science Foundation o China under the contract number REFERENCES Miner, M.. 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