Abstract
Structural components with different scales normally show different fatigue behaviors, which are virtually dominated by defects originated from multiple sources, including manufacturing processes. This paper reviews three types of size effects (statistical, geometrical, technological) as well as their recent advances in metal fatigue, aiming to provide a guide for fatigue strength assessment of engineering components containing defects, inclusions and material inhomogeneity. Firstly, the background of inherent defects and defect-based failure mechanism are briefly outlined, and fatigue failure analysis based on fracture mechanics as well as statistics theory are emphasized. Then, two approaches commonly applied in statistical size effect modeling, i.e. critical defect method and weakest link method, are elaborated. In addition, the highly stressed volume method is introduced for considering the geometrical size effects, and the technological (production and surface) size effect is briefly overviewed. Finally, further directions on size effect in metal fatigue under defects are explored.
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Data Availability
All materials data for model validation used during the study are available from the corresponding author by request.
References
Abroug F, Pessard E, Germain G, Morel F (2018) Fatigue size effect due to defects in an AA7050 alloy. MATEC Web Conf 165:14015
Abroug F, Pessard E, Germain G, Morel F (2018) HCF of AA7050 alloy containing surface defects: study of the statistical size effect. Int J Fatigue 110:81–94
Afferrante L, Ciavarella M, Valenza E (2006) Is Weibull’s modulus really a material constant? Example case with interacting collinear cracks. Int J Solids Struct 43(17):5147–5157
Ai Y, Zhu SP, Liao D, Correia JAFO, De Jesus AMP, Keshtegar B (2019) Probabilistic modelling of notch fatigue and size effect of components using highly stressed volume approach. Int J Fatigue 127:110–119
Ai Y, Zhu SP, Liao D et al (2019) Probabilistic modeling of fatigue life distribution and size effect of components with random defects. Int J Fatigue 126:165–173
Alang NA, Razak NA, Miskam AK (2011) Effect of surface roughness on fatigue life of notched carbon steel. Int J Eng Technol 11:160–163
Alava MJ, Nukala PKVV, Zapperi S (2009) Size effects in statistical fracture. J Phys D Appl Phys 42(21):214012
Al-Owaisi SS, Becker AA, Sun W (2016) Analysis of shape and location effects of closely spaced metal loss defects in pressurised pipes. Eng Fail Anal 68:172–186
Åman M, Okazaki S, Matsunaga H, Marquis GB, Remes H (2017) Interaction effect of adjacent small defects on the fatigue limit of a medium carbon steel. Fatigue Fract Eng Mater Struct 40(1):130–144
Anderson CW, Coles SG (2002) The largest inclusions in a piece of steel. Extremes 5(3):237–252
Anderson CW, De Maré J, Rootzén H (2005) Methods for estimating the sizes of large inclusions in clean steels. Acta Mater 53(8):2295–2304
Atkinson HV, Shi G (2003) Characterization of inclusions in clean steels: a review including the statistics of extremes methods. Prog Mater Sci 48(5):457–520
Bažant ZP (1999) Size effect on structural strength: a review. Arch Appl Mech 69(9–10):703–725
Bažant ZP, Yavari A (2005) Is the cause of size effect on structural strength fractal or energetic-statistical? Eng Fract Mech 72(1):1–31
Ben Ahmed A, Nasr A, Bahloul A, Fathallah R (2017) The impact of defect morphology, defect size, and SDAS on the HCF response of A356-T6 alloy. Int J Adv Manuf Technol 92(1–4):1113–1125
Beretta S, Regazzi D (2016) Probabilistic fatigue assessment for railway axles and derivation of a simple format for damage calculations. Int J Fatigue 86:13–23
Beretta S, Ghidini A, Lombardo F (2005) Fracture mechanics and scale effects in the fatigue of railway axles. Eng Fract Mech 72(2):195–208
Besel M, Brueckner-Foit A (2008) Surface damage evolution of engineering steel. Fatigue Fract Eng Mater Struct 31(10):885–891
Blasón S, Muniz-Calvente M, Koller R, Przybilla C, Fernández-Canteli A (2017) Probabilistic assessment of fatigue data from shape homologous but different scale specimens. Application to an experimental program. Eng Fract Mech 185:193–209
Brückner-Foit A, Ehl W, Munz D, Trolldenier B (1990) The size effect of microstructural implications of the weakest link model. Fatigue Fract Eng Mater Struct 13(3):185–200
Buffière JY, Savelli S, Jouneau PH, Maire E, Fougères R (2001) Experimental study of porosity and its relation to fatigue mechanisms of model Al-Si7-MgO.3 cast Al alloys. Mater Sci Eng A 316(1–2):115–126
Carpinteri A (1989) Decrease of apparent tensile and bending strength with specimen size: two different explanations based on fracture mechanics. Int J Solids Struct 25(4):407–429
Carpinteri A, Spagnoli A (2004) A fractral analysis of size effect on fatigue crack growth. Int J Fatigue 26(2):125–133
Carpinteri A, Spagnoli A, Vantadori S (2002) An approach to size effect in fatigue of metals using fractal theories. Fatigue Fract Eng Mater Struct 25(7):619–627
Carpinteri A, Spagnoli A, Vantadori S (2009) Size effect in SN curves: a fractral approach to finite-life fatigue strength. Int J Fatigue 31(5):927–933
Carpinteri A, Spagnoli A, Vantadori S (2010) A multifractral analysis of fatigue crack growth and its application to concrete. Int J Fatigue 77(6):974–984
Casellas D, Pérez R, Prado JM (2005) Fatigue variability in Al–Si cast alloys. Mater Sci Eng A 398(1–2):171–179
Castillo E, López-Aenlle M, Ramos A, Fernández-Canteli A, Kieselbach R, Esslinger V (2006) Specimen length effect on parameter estimation in modelling fatigue strength by Weibull distribution. Int J Fatigue 28(9):1047–1058
Couper MJ (1993) Casting defects and the fatigue behaviour of an aluminium casting alloy. Fatigue Fract Eng Mater Struct 13:213–227
Cova M, Nanni M, Tovo R (2014) Geometrical size effect in high cycle fatigue strength of heavy-walled ductile cast iron GJS400: weakest link vs defect-based approach. Procedia Eng 74:101–104
Dai DN, Nowell D, Hills DA (1993) Eigenstrain methods in three-dimensional crack problems: an alternative integration procedure. J Mech Phys Solids 41(6):1003–1017
De Jesus AMP, Ramos GFS, Gomes VMG, Marques MJ, de Figueiredo MAV, Marafona JDR (2020) Comparison between EDM and grinding machining on fatigue behaviour of AISI D2 tool steel. Int J Fatigue 139:105742
Editorial Committee China Aeronautical Materials Handbook (2002) China aeronautical materials handbook, vol 4: titanium alloy. China Standards Press, Beijing
Fei CW, Tang WZ, Bai GC (2014) Novel method and model for dynamic reliability optimal design of turbine blade deformation. Aerosp Sci Technol 39:588–595
Fjeldstad A, Wormsen A, Härkegård G (2008) Simulation of fatigue crack growth in components with random defects. Eng Fract Mech 75(5):1184–1203
Flaceliere L, Morel F (2004) Probabilistic approach in high-cycle multiaxial fatigue: volume and surface effects. Fatigue Fract Eng Mater Struct 27(12):1123–1135
Frost NE, Greenan AF (1964) Cyclic stress required to propagate edge cracks in eight materials. J Mech Eng Sci 6(3):203–210
Fry GT, Lawrence FV, Robinson AR (1996) A model for fatigue defect nucleation in thermite rail welds. Fatigue Fract Eng Mater Struct 19(6):655–668
Furushima T, Manabe K, Alexandrov S (2013) Size effects on free surface roughening and necking behavior of metal thin sheets using inhomogeneous finite element material model. AIP Conf Proc 1567:460–463
Furuya Y (2008) Specimen size effects on gigacycle fatigue properties of high-strength steel under ultrasonic fatigue testing. Scr Mater 58(11):1014–1017
Furuya Y (2011) Notable size effects on very high cycle fatigue properties of high-strength steel. Mater Sci Eng A 528(15):5234–5240
Goto M (1991) Statistical investigation of the behaviour of microcracks in carbon steels. Fatigue Fract Eng Mater Struct 14(8):833–845
Grell WA, Niggeler GH, Groskreutz ME, Laz PJ (2007) Evaluation of creep damage accumulation models: considerations of stepped testing and highly stressed volume. Fatigue Fract Eng Mater Struct 30(8):689–697
Griffith A (1921) The phenomena of rupture and flow in solids. Philos Trans R Soc Lond 221:163–198
Gruenberg KM, Craig BA, Hillberry BM (1999) Probabilistic method for predicting the variability in fatigue behavior of 7075–T6 aluminum. AIAA J 37(10):1304–1310
Gurney TR (1979) Influence of thickness on the fatigue strength of welded joints. Acta Inform 1:523–534
Härkegrd G, Halleraker G (2010) Assessment of methods for prediction of notch and size effects at the fatigue limit based on test data by Böhm and Magin. Int J Fatigue 32(10):1701–1709
He JC, Zhu SP, Liao D, Niu XP (2020) Probabilistic fatigue assessment of notched components under size effect using critical distance theory. Eng Fract Mech 235:107150
He JC, Zhu SP, Liao D, Niu XP, Gao JW, Huang HZ (2021) Combined TCD and HSV approach for probabilistic assessment of notch fatigue considering size effect. Eng Fail Anal 120:105093
Hertel O, Vormwald M (2012) Statistical and geometrical size effects in notched members based on weakest-link and short-crack modelling. Eng Fract Mech 95:72–83
Hu D, Mao J, Wang X, Meng F, Song J, Wang R (2018) Probabilistic evaluation on fatigue crack growth behavior in nickel based GH4169 superalloy through experimental data. Eng Fract Mech 196:71–82
Illg W (1956) Fatigue tests on notched and unnotched sheet specimens of 2024-T3 and 7075-T6 aluminum alloys and of SAE 4130 steel with special consideration of the life range from 2 to 10,000 cycles. National Advisory Committee for Aeronautics, Kitty Hawk
Jayatilaka ADS, Trustrum K (1977) Statistical approach to brittle fracture. J Mater Sci 12(7):1426–1430
Kaděrová J, Miroslav V (2013) Experimental testing of statistical size effect in civil engineering structures. Int J Civ Environ Eng 7(10):716–723
Kaffenberger M, Vormwald M (2012) Considering size effects in the notch stress concept for fatigue assessment of welded joints. Comput Mater Sci 64:71–78
Karolczuk A, Palin-Luc T (2013) Modelling of stress gradient effect on fatigue life using Weibull based distribution function. J Theor Appl Mech 51(2):297–311
Khoukhi DEL, Morel F, Saintier N, Bellett D, Osmond P (2018) The effect of microstructural heterogeneities on the high cycle fatigue scatter of cast aluminium alloys? From an elementary volume to the structure. In: MATEC web of conferences
Khoukhi DEL et al (2019) Experimental investigation of the size effect in high cycle fatigue: role of the defect population in cast aluminium alloys. Int J Fatigue 129:105222
Kim JY, Kang SK, Lee JJ, il Jang J, Lee YH, Kwon D (2007) Influence of surface-roughness on indentation size effect. Acta Mater 55(10):3555–3562
Kittl P, Diaz G (1988) Weibull’s fracture statistics, or probabilistic strength of materials: state of the art. Res Mech 24(2):99–207
Kittl P, Díaz G (1990) Size effect on fracture strength in the probabilistic strength of materials. Reliab Eng Syst Saf 28(1):9–21
Kloos KH (1976) Einfluss des Oberflächenzustandes und der Probengröße auf die Schwingfestigkeitseigenschaften. VDI Ber 268(268):63–76
Kloos KH, Buch A, Zankov D (1981) Pure geometrical size effect in fatigue tests with constant stress amplitude and in programme tests. Materwiss Werksttech 12(2):40–50
Ktari A, Haddar N, Rezai-Aria F, Ayedi HF (2016) On the assessment of train crankshafts fatigue life based on LCF tests and 2D-FE evaluation of J-integral. Eng Fail Anal 66:354–364
Kuguel R (1961) A relation between theoretical stress concentration factor and fatigue notch factor deduced from the concept of highly stressed volume. Proc ASTM 61:732–748
Kuhn P, Hardraht HF (1952) An engineering method for estimating the notch-size effect in fatigue tests on steel, NACA TN2805. Langly Aeronautic Laboratory, Washington
Leitner M, Vormwald M, Remes H (2017) Statistical size effect on multiaxial fatigue strength of notched steel components. Int J Fatigue 104:322–333
Li GW, Tang JY, Zhou W, Li L (2018) Fatigue life prediction of workpiece with 3D rough surface topography based on surface reconstruction technology. J Cent South Univ 25(9):2069–2075
Li D, Hu D, Wang R, Ma Q, Liu H (2018) A non-local approach for probabilistic assessment of LCF life based on optimized effective-damage-parameter. Eng Fract Mech 199:188–200
Li W, Chen H, Huang W et al (2021) Effect of laser shock peening on high cycle fatigue properties of aluminized AISI 321 stainless steel. Int J Fatigue 147:106180. https://doi.org/10.1016/j.ijfatigue.2021.106180
Liao D, Zhu SP, Correia JAFO, De Jesus AMP, Berto F (2020) Recent advances on notch effects in metal fatigue: a review. Fatigue Fract Eng Mater Struct 43(4):637–659
Liao D, Zhu SP, Behrooz K, Qian GA, Wang QY (2020) Probabilistic framework for fatigue life assessment of notched components under size effects. Int J Mech Sci 181:105685
Lieb K, Horstman R, Peters K, Meltzer R, Bruce Vieth M, Trantina G (1981) Statistical fatigue failure analysis. J Test Eval 9(1):44–49
Lin CK, Lee WJ (1998) Effects of highly stressed volume on fatigue strength of austempered ductile irons. Int J Fatigue 20(4):301–307
Lipp K, Baumgartner J, Beiss P (2014) Fatigue design of sintered steel components: effect of stress concentrations and mean stresses on local strength using highest stressed volume approach. Powder Metall 56(5):337–341
Lukás P, Kunz L, Weiss B, Stickler R (1989) Notch size effect in fatigue. Fatigue Fract Eng Mater Struct 12(3):175–186
Makkonen M (2001) Statistical size effect in the fatigue limit of steel. Int J Fatigue 23(5):395–402
Makkonen L, Rabb R, Tikanmäki M (2014) Size effect in fatigue based on the extreme value distribution of defects. Mater Sci Eng A 594:68–71
McEvily AJ, Endo M, Yamashita K, Ishihara S, Matsunaga H (2008) Fatigue notch sensitivity and the notch size effect. Int J Fatigue 30(12):2087–2093
Mohammad M, Abdullah S, Jamaludin N, Innayatullah O (2014) Predicting the fatigue life of the SAE 1045 steel using an empirical Weibull-based model associated to acoustic emission parameters. Mater Des 54:1039–1048
Muñiz-Calvente M, Fernández Canteli A, Shlyannikov V, Castillo E (2015) Probabilistic Weibull methodology for fracture prediction of brittle and ductile materials. Appl Mech Mater 784:443–451
Muñiz-Calvente M, Blasón S, Fernández-Canteli A, Jesus A, Correia J (2017) A probabilistic approach for multiaxial fatigue criteria. Frattura Ed Integrità Strutturale J 39:160–165
Muniz-Calvente M, de Jesus AMP, Correia JAFO, Fernández-Canteli A (2017) A methodology for probabilistic prediction of fatigue crack initiation taking into account the scale effect. Eng Fract Mech 185:101–113
Murakami Y (2012) Material defects as the basis of fatigue design. Int J Fatigue 41:2–10
Murakami Y, Endo M (1994) Effects of defects, inclusions and inhomogeneities on fatigue strength. Int J Fatigue 16(3):163–182
Naik DL, Fronk TH (2016) Weibull distribution analysis of the tensile strength of the kenaf bast fiber. Fibers Polym 17(10):1696–1701
Neuber H (1960) Theory of stress concentration for shear-strained prismatical bodies with arbitrary nonlinear stress–strain law. J Appl Mech Trans ASME 28(4):544–550
Nohut S (2014) Influence of sample size on strength distribution of advanced ceramics. Ceram Int 40(3):4285–4295
Norberg S, Olsson M (2007) The effect of loaded volume and stress gradient on the fatigue limit. Int J Fatigue 29(12):2259–2272
Okeyoyin OA, Owolabi GM (2013) Application of weakest link probabilistic framework for fatigue notch factor to turbine engine materials. World J Mech 3(5):237–244
Pavlov VF, Kirpichev VA, Vakulyuk VS, Sazanov VP (2014) Surface hardening influence on the fatigue limit for cylindrical parts of different diameter. Russ Aeronaut 57(3):324–326
Peterson RE (1959) Notch sensitivity. In: Metal fatigue, pp 293–306. McGraw-Hill, New York
Pierce FT (1926) Tensile tests for cotton yarns-‘the weakest link’ theorems on the strength of long and of composite specimens. J Text Inst Trans 17(7):T355–T368
Przybilla C, Fernández-Canteli A, Castillo E (2011) Deriving the primary cumulative distribution function of fracture stress for brittle materials from 3- and 4-point bending tests. J Eur Ceram Soc 31(4):451–460
Przybilla C, Fernández-Canteli A, Castillo E (2013) Maximum likelihood estimation for the three-parameter Weibull cdf of strength in presence of concurrent flaw populations. J Eur Ceram Soc 33(10):1721–1727
Przybilla C, Koller R, Fernández-Canteli A, Castillo E (2013) A model allowing for the influence of geometry and stress in the assessment of fatigue data. In: 13th international conference on fracture 2013 (ICF 2013), vol 3
Qian G, Lei WS, Peng L, Yu Z, Niffenegger M (2018) Statistical assessment of notch toughness against cleavage fracture of ferritic steels. Fatigue Fract Eng Mater Struct 41(5):1120–1131
Rafsanjani HM, Sørensen JD (2015) Effect of defects distribution on fatigue life of wind turbine components. Procedia IUTAM 13:144–150
Rafsanjani HM, Sørensen JD (2015) Reliability analysis of fatigue failure of cast components for wind turbines. Energies 8(4):2908–2923
Ravi Chandran KS (2005) Duality of fatigue failures of materials caused by Poisson defect statistics of competing failure modes. Nat Mater 4(4):303–308
Ravi Chandran KS, Jha SK (2005) Duality of the S–N fatigue curve caused by competing failure modes in a titanium alloy and the role of Poisson defect statistics. Acta Mater 53(7):1867–1881
Ravi Chandran KS, Chang P, Cashman GT (2010) Competing failure modes and complex S-N curves in fatigue of structural materials. Int J Fatigue 32(3):482–491
Saffar S, Gouttebroze S, Zhang ZL (2014) Fracture analysis and distribution of surface cracks in multicrystalline silicon wafers. J Sol Energy Eng 136(2):021024
Shi G, Atkinson HV, Sellars CM, Anderson CW (1999) Application of the generalized Pareto distribution to the estimation of the size of the maximum inclusion in clean steels. Acta Mater 47(5):1455–1468
Shirani M, Härkegård G (2009) Fatigue crack growth with simulation in components random defects. J ASTM Int 6(9):102542
Shirani M, Härkegård G (2011a) Large scale axial fatigue testing of ductile cast iron for heavy section wind turbine components. Eng Fail Anal 18(6):1496–1510
Shirani M, Härkegård G (2011b) Fatigue life distribution and size effect in ductile cast iron for wind turbine components. Eng Fail Anal 18(1):12–24
Shirani M, Härkegård G (2012) Damage tolerant design of cast components based on defects detected by 3D X-ray computed tomography. Int J Fatigue 41:188–198
Shirani M, Härkegård G (2014) A review on fatigue design of heavy section EN-GJS-400-18-LT ductile iron wind turbine castings. Energy Equip Syst 2(1):5–24
Smith EP (2009) An introduction to statistical modeling of extreme values. Technometrics 44(4):397
Song J et al (2013) Study of tower surface crack size effect based on Weibull theory. Intell Autom Soft Comput 19(4):581–588
Song LK, Bai GC, Fei CW (2019) Probabilistic LCF life assessment for turbine discs with DC strategy-based wavelet neural network regression. Int J Fatigue 119:204–219
Sonsino CM, Kaufmann H, Grubišić V (1997) Transferability of material data for the example of a randomly loaded forged truck stub axle. SAE Technical Papers
Sonsino CM, Fischer G (2005) Local assessment concepts for the structural durability of complex loaded components. Materwiss Werksttech 36(11):632–641
Sun C, Song Q (2018) A method for predicting the effects of specimen geometry and loading condition on fatigue strength. Metals (Basel) 8(10):811
Susmel L, Taylor D (2007) A novel formulation of the theory of critical distances to estimate lifetime of notched components in the medium-cycle fatigue regime. Fatigue Fract Eng Mater Struct 30(7):567–581
Ting JC, Lawrence FV Jr (1993) Modeling the long-life fatigue behavior of a cast aluminum alloy. Fatigue Fract Eng Mater Struct 16(6):631–647
Tiryakioglu M (2009) Relationship between defect size and fatigue life distributions in Al-7 Pct Si-Mg alloy castings. Metall Mater Trans A Phys Metall Mater Sci 40(7):1623–1630
Tiryakioğlu M (2015) Weibull analysis of mechanical data for castings II: Weibull mixtures and their interpretation. Metall Mater Trans A Phys Metall Mater Sci 46(1):270–280
Tiryakioglu M, Campbell J (2010) Weibull analysis of mechanical data for castings: a guide to the interpretation of probability plots. Metall Mater Trans A Phys Metall Mater Sci 41(12):3121–3129
Todinov MT (2001) Estimating the probabilities of triggering brittle fracture associated with the defects in the materials. Mater Sci Eng A 302(2):235–245
Todinov MT (2005) Limiting the probability of failure for components containing flaws. Comput Mater Sci 32(2):156–166
Toft HS, Branner K, Berring P, Sørensen JD (2011) Defect distribution and reliability assessment of wind turbine blades. Eng Struct 33(1):171–180
Tomaszewski T, Sempruch J, Piątkowski T (2014) Verification of selected models of the size effect based on high-cycle fatigue testing on mini specimens made of EN AW-6063 aluminum alloy. J Theor Appl Mech 52(4):883–894
Trantina G, Johnson C (2009) Probabilistic defect size analysis using fatigue and cyclic crack growth rate data. In: Probabilistic fracture mechanics and fatigue methods: applications for structural design and maintenance, pp 67–78
Trujillo E, Moesen M, Osorio L, Van Vuure AW, Ivens J, Verpoest I (2014) Bamboo fibres for reinforcement in composite materials: strength Weibull analysis. Compos Part A Appl Sci Manuf 61:115–125
Van Hooreweder B, Moens D, Boonen R, Sas P (2012) Fatigue strength analysis of notched aluminium specimens using the highly stressed volume method. Fatigue Fract Eng Mater Struct 35(2):154–159
Wan Q, Zhao H, Zou C (2014) Effect of micro-porosities on fatigue behavior in aluminum die castings by 3D X-ray tomography inspection. ISIJ Int 54(3):511–515
Wang F, Shao J (2014) Modified Weibull distribution for analyzing the tensile strength of bamboo fibers. Polymers (Basel) 6(12):3005–3018
Wang F, Li L, Chen Z (2012a) Scaling effects on the tensile strength of fibrous composites. Key Eng Mater 525–526:149–152
Wang W, Zhong Y, Lu K, Lu L, McDowell DL, Zhu T (2012b) Size effects and strength fluctuation in nanoscale plasticity. Acta Mater 60(8):3302–3309
Wang R, Li D, Hu D, Meng F, Liu H, Ma Q (2017) A combined critical distance and highly-stressed-volume model to evaluate the statistical size effect of the stress concentrator on low cycle fatigue of TA19 plate. Int J Fatigue 95:8–17
Weibull W (1951) A statistical distribution function of wide applicability. J Appl Mech 13:293–297
Wormsen A, Sjödin B, Härkegård G, Fjeldstad A (2007) Non-local stress approach for fatigue assessment based on weakest-link theory and statistics of extremes. Fatigue Fract Eng Mater Struct 30(12):1214–1227
Wormsen A, Haerkegaard G (2015) A statistical investigation of fatigue behaviour according to Weibullś weakest link theory. In: ESIS-ECF 15, Sweden
Xin P, Hu X, Song Y (2012) LCF life prediction for TC4 alloy notched specimens based on theory of critical distance. J Aerosp Power 27(5):1105–1112
Xu Y, Liu C, Chai L, Luo C (2019) The effect of defect size on the integrity of CFRP-confined concrete column. Constr Build Mater 200:521–529
Yao WX (1993) Stress field intensity approach for predicting fatigue life. Int J Fatigue 15(3):243–246
Yi JZ, Gao YX, Lee PD, Flower HM, Lindley TC (2003) Scatter in fatigue life due to effects of porosity in cast A356-T6 aluminum–silicon alloys. Metall Mater Trans A Phys Metall Mater Sci 34 A(9):1879–1890
Zaitsev J, Wittmann FH (1974) Theoretical study of the behaviour of concrete under short-time uniaxial and biaxial loading. Mater. Test. 16(6):170–174
Zamber JE, Hillberry BM (1999) Probabilistic approach to predicting fatigue lives of corroded 2024–T3. AIAA J 37(10):1311–1317
Zamiri Akhlaghi F, Acevedo C, Nussbaumer A, Krummenacker J (2011) Investigation of technological size effects of welding on the residual stresses and fatigue life of tubular joints made of structural steels S355 and S690. In: Fatigue design CETIM, Senlis, pp 1–8
Zech B, Wittmann FH (1977) A complex study on the reliability assessment of the containment of a PWR, Part II. Probabilistic approach to describe the behavior of materials. Load Cond Struct Anal React Contain 48(2–3):563–574
Zhang X, Liu X, Hong Y (2016) Effects of specimen size on fatigue life of metallic materials in high-cycle and very-high-cycle fatigue regimes. Fatigue Fract Eng Mater Struct 39(6):770–779
Zhao XL, Herion S (2001) CIDECT Design Guide 8—design guide for circular and rectangular hollow section joints under fatigue loading. CIDECT, Cologne
Zhu SP, Huang HZ, Ontiveros V, He LP, Modarres M (2012) Probabilistic low cycle fatigue life prediction using an energy-based damage parameter and accounting for model uncertainty. Int J Damage Mech 21(8):1128–1153
Zhu SP, Huang HZ, Smith R, Ontiveros V, He LP, Modarres M (2013) Bayesian framework for probabilistic low cycle fatigue life prediction and uncertainty modeling of aircraft turbine disk alloys. Probab Eng Mech 34:114–122
Zhu SP, Huang HZ, Li Y, Liu Y, Yang Y (2015) Probabilistic modeling of damage accumulation for time-dependent fatigue reliability analysis of railway axle steels. Proc Inst Mech Eng Part F J Rail Rapid Transit 229(1):23–33
Zhu SP, Huang HZ, Peng W, Wang HK, Mahadevan S (2016) Probabilistic physics of failure-based framework for fatigue life prediction of aircraft gas turbine discs under uncertainty. Reliab Eng Syst Saf 146:1–12
Zhu SP, Foletti S, Beretta S (2017) Probabilistic framework for multiaxial LCF assessment under material variability. Int J Fatigue 103:371–385
Zhu SP, Liu Q, Zhou J, Yu ZY (2018a) Fatigue reliability assessment of turbine discs under multi-source uncertainties. Fatigue Fract Eng Mater Struct 41(6):1291–1305
Zhu SP, Liu Q, Peng W, Zhang XC (2018b) Computational-experimental approaches for fatigue reliability assessment of turbine bladed disks. Int J Mech Sci 142–143:502–517
Zhu SP, Liu Q, Lei Q, Wang Q (2018c) Probabilistic fatigue life prediction and reliability assessment of a high pressure turbine disc considering load variations. Int J Damage Mech 27(10):1569–1588
Zhu SP, Foletti S, Beretta S (2018d) Evaluation of size effect on strain-controlled fatigue behavior of a quench and tempered rotor steel: experimental and numerical study. Mater Sci Eng A 735:423–435
Zhu SP, He JC, Liao D, Wang Q, Liu Y (2020) The effect of notch size on critical distance and fatigue life predictions. Mater Des 196:109095
Zok FW (2017) On weakest link theory and Weibull statistics. J Am Ceram Soc 100(4):1265–1268
Acknowledgements
Financial support of the National Natural Science Foundation of China (No. 11972110), Sichuan Provincial Key Research and Development Program (No. 2019YFG0348), Science and Technology Program of Guangzhou, China (No. 201904010463), Fundamental Research Funds for the Central Universities (No. ZYGX2019J040) and Opening funds of Key Laboratory of Deep Earth Science and Engineering (Sichuan University), Ministry of Education (No. DESE201901) are acknowledged. This research was also supported by base funding—UIDB/04708/2020 and programmatic funding—UIDP/04708/2020 of the CONSTRUCT—Instituto de I&D em Estruturas e Construções—funded by national funds through the FCT/MCTES (PIDDAC). The Add. Strength project entitled “Enhanced Mechanical Properties in Additive Manufactured Components” (Reference PTDC/EME-EME/31307/2017) funded by the Programa Operacional Competitividade e Internacionalização, and Programa Operacional Regional de Lisboa funded by FEDER and National Funds (FCT) is also acknowledged.
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Zhu, SP., Ai, Y., Liao, D. et al. Recent advances on size effect in metal fatigue under defects: a review. Int J Fract 234, 21–43 (2022). https://doi.org/10.1007/s10704-021-00526-x
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DOI: https://doi.org/10.1007/s10704-021-00526-x