In this work, a comprehensive theoretical and numerical study on the cyclic elastic–plastic notch stress and strain distributions is carried out. In more detail, the incremental cyclic plasticity theory, already proposed by other authors to determine the actual stress and strain state arising in two-dimensional or axisymmetric notched components, is extended to the study of three-dimensional effects at the tip of rounded notches in plates of finite thickness. The analytical frame is initially validated considering a number of plane problems and later modified to consider three-dimensional effects. Different notch geometries are investigated, such as plane specimens with finite thickness weakened by circular holes, U-notches and rounded V-notches subjected to cyclic mode-I loading. Theoretical results based on the incremental cyclic plasticity theory are compared with time-consuming elastic–plastic finite element analyses carried out with a commercial finite element code showing a very satisfactory agreement. Finally, a link between the averaged strain energy density criterion and the area subtended by the hysteresis loops tied to the different stress and strain components acting at the notch tip has been investigated.
NeuberH. Theory of stress concentration for shear-strained prismatical bodies with arbitrary non-linear stress–strain law. J Appl Mech1961; 28(4): 544–550.
2.
MolskiKGlinkaG. A method of elastic–plastic stress and strain calculation at a notch root. Mater Sci Eng1981; 50(1): 93–100.
3.
TopperTHWetzelRMMorrowJ. Neuber’s rule applied to fatigue of notched specimens. J Mater1969; 4(1): 200–209.
4.
HoffmannMSeegerT. A generalized method for estimating multiaxial elastic–plastic notch stresses and strains. Part 1: theory. J Eng Mater Technol1985; 107(4): 250–254.
5.
MoftakharABuczynskiAGlinkaG. Calculation of elasto-plastic strains and stresses in notches under multiaxial loading. Int J Fracture1995; 70(4): 357–373.
6.
EllyinFKujawskiD. Generalization of notch analysis and its extension to cyclic loading. Eng Fract Mech1989; 32(5): 819–826.
7.
SinghMNKGlinkaGDubeyRN. Elastic–plastic stress–strain calculation in notched bodies subjected to non-proportional loading. Int J Fracture1986; 76(1): 39–60.
8.
YeD. Further investigation of Neuber’s rule and the equivalent strain energy density (ESED) method. Int J Fatigue2004; 26(5): 447–455.
9.
YeDHertelOVormwaldM. A unified expression of elastic–plastic notch stress–strain calculation in bodies subjected to multiaxial cyclic loading. Int J Solids Struct2008; 45(24): 6177–6189.
10.
ZappalortoMLazzarinP. A unified approach to the analysis of non-linear stress and strain fields ahead of mode III-loaded notches and cracks. Int J Solids Struct2010; 47(6): 851–864.
11.
InceAGlinkaG. A numerical method for elasto-plastic notch-root stress–strain analysis. J Strain Anal Eng2013; 48(4): 229–244.
12.
DruckerDC. Plasticity in structural mechanics. In: HoffG (ed.) Proceedings of the first symposium on naval structural mechanics. New York: Macmillan; London: Pergamon Press, 1960, pp.331–350.
13.
PragerW. The theory of plasticity: a survey of recent achievements. Proc Inst Mech Eng1955; 169: 41–57.
14.
ArmstrongPFrederickCO. A mathematical representation of the multiaxial Bauschinger effect. Rep. RD/B/N 731, 1966. Berkeley: Central Electricity Generating Board and Research and Development Department, Berkeley Nuclear Laboratories.
15.
BowerAF. Some aspects of plastic flow, residual stress and fatigue cracks due to rolling and sliding contact. PhD Dissertation, University of Cambridge, Cambridge, 1987.
16.
BowerAFJohnsonKL. The influence of strain hardening on cumulative plastic deformation in rolling and sliding line contact. J Mech Phys Solids1989; 37(4): 471–493.
17.
OhnoNWangJD. Non linear kinematic hardening rule: proposition and application to ratchetting problems. In: Transaction of the 11th international conference on structural mechanics in reactor technology, Tokyo, Japan, 18–23 August 1991. Shibata Publishing Co.
18.
ChabocheJLDang VanKCordierG. Modelization of the strain memory effect on the cyclic hardening of the 316 stainless steel. In: Transaction of the 5th international conference on structural mechanics in reactor technology, Berlin, 13–17 August 1979. Amsterdam, Netherlands: North-Holland Publishing Co.
19.
ChabocheJL. Cyclic plasticity modelling and ratchetting effects. In: Proceedings of the second international conference on constitutive laws for engineering materials: theory and applications, Tucson, AZ, 5–8 January 1987, 47–58. Tucson, AZ: Elsevier.
20.
JiangYSehitogluH. Modeling of cyclic ratchetting plasticity. Part I: development of constitutive relations. J Appl Mech1996; 63(3): 720–725.
21.
JiangYSehitogluH. Modeling of cyclic ratchetting plasticity. Part II: comparison of model simulations with experiments. J Appl Mech1996; 63(3): 726–733.
22.
JiangYXuB. Deformation analysis of notched components and assessment of approximate methods. Fatigue Fract Eng M2001; 24(11): 729–740.
23.
FolettiSDesimoneHJ. A semi-analytical approach for two-dimensional rolling/sliding contact with applications to shakedown analysis. Wear2007; 262(7–8): 850–857.
24.
QiuBGaoZWangX. Fatigue life prediction of notched components based on multiaxial local stress–strain approach. In: ASME 2009 pressure vessels and piping conference. Volume 6: materials and fabrication, parts A and B, Prague, 26–30 July 2009, 183–189. New York: ASME.
25.
FiratMKozanROzsoyM. Numerical modeling and simulation of wheel radial fatigue tests. Eng Fail Anal2009; 16(5): 1533–1541.
26.
FiratM. A computer simulation of four-point bending fatigue of a rear axle assembly. Eng Fail Anal2011; 18(8): 2137–2148.
27.
GaoZQiuBWangX. An investigation of fatigue of a notched member. Int J Fatigue2010; 32(12): 1960–1969.
28.
PookLP. A 50-year retrospective review of three-dimensional effects at cracks and sharp notches. Fatigue Fract Eng M2013; 36(8): 699–723.
29.
LazzarinPZambardiR. A finite-volume-energy based approach to predict the static and fatigue behavior of components with sharp V-shaped notches. Int J Fracture2001; 112: 275–298.
30.
LazzarinPBertoF. Some expressions for the strain energy in a finite volume surrounding the root of blunt V-notches. Int J Fracture2005; 135: 161–185.
31.
BertoFLazzarinP. Recent developments in brittle and quasi-brittle failure assessment of engineering materials by means of local approaches. Mater Sci Eng R Rep2014; 75: 1–48.
32.
Noori-AzghadFKhademizadehHBaratiE. Some analytical and numerical expressions for evaluation of the critical J-integral in plates with blunt V-notches under Mode I loading. J Strain Anal Eng2013; 49(5): 352–360.
33.
LazzarinPBertoFElicesM. Brittle failures from U- and V-notches in mode I and mixed, I+II, mode: a synthesis based on the strain energy density averaged on finite-size volumes. Fatigue Fract Eng M2009; 32(8): 671–684.
34.
LazzarinPCampagnoloABertoF. A comparison among some recent energy- and stress-based criteria for the fracture assessment of sharp V-notched components under Mode I loading. Theor Appl Fract Mech2014; 71: 21–30.
35.
TorabiARCampagnoloABertoF. Mode II brittle fracture assessment of key-hole notches by means of the local energy. J Test Eval2014; 44(3). DOI: 10.1520/JTE20140295.
36.
TorabiARCampagnoloABertoF. Local strain energy density to predict mode II brittle fracture in Brazilian disk specimens weakened by V-notches with end holes. Mater Des2014; 69: 22–29.
37.
LivieriPLazzarinP. Fatigue strength of steel and aluminium welded joints based on generalised stress intensity factors and local strain energy values. Int J Fracture2005; 133(3): 247–276.
38.
BertoFLazzarinPYatesJR. Multiaxial fatigue of V-notched steel specimens: a non-conventional application of the local energy method. Fatigue Fract Eng M2011; 34(11): 921–943.
39.
BertoFCampagnoloALazzarinP. Fatigue strength of severely notched specimens made of Ti-6Al-4V under multiaxial loading. Fatigue Fract Eng M. 2015; 38(5): 503–517. DOI: 10.1111/ffe.12272.
40.
EllyinFKujawskiD. Plastic strain energy in fatigue failure. J Press Vess: T ASME1984; 106(4): 342.
41.
FeltnerCEMorrowJD. Microplastic strain hysteresis energy as a criterion for fatigue fracture. J Basic Eng: T ASME1961; 83(1): 15.
42.
EllyinF. A criterion for fatigue under multiaxial states of stress. Mech Res Commun1974; 1(4): 219–224.
43.
ChuC-C. Incremental multiaxial Neuber correction for fatigue analysis. SAE technical paper 950705, 1995.
44.
MacaulayWH. A note on the deflection of beams. Messenger Math1919; 48: 129.
45.
JiangYKurathP. Characteristics of the Armstrong-Frederick type plasticity models. Int J Plast1996; 12(3): 387–415.
46.
JiangY. Cyclic plasticity with an emphasis on ratchetting. PhD Dissertation, Department of Mechanical Engineering, University of Illinois at Urbana–Champaign, Champaign, IL, 1994.
47.
SethuramanRViswanadha GuptaS. Evaluation of notch root elasto-plastic stress–strain state for general loadings using an elastic solution. Int J Press Ves Pip2004; 81(4): 313–325.
48.
LiZGuoW. Three-dimensional elastic stress fields ahead of blunt V-notches in finite thickness plates. Int J Fracture2001; 107(1): 53–71.
49.
LiZGuoWKuangZ. Three-dimensional elastic stress fields near notches in finite thickness plates. Int J Solids Struct2000; 37(51): 7617–7632.
50.
MeneghettiGRicottaM. The use of the specific heat loss to analyse the low- and high-cycle fatigue behaviour of plain and notched specimens made of a stainless steel. Eng Fract Mech2012; 81: 2–16.
51.
JiangYKurathP. Nonproportional cyclic deformation: critical experiments and analytical modeling. Int J Plast1997; 13(8–9): 743–763.
KulkarniSDesaiYKantT. Uniaxial and biaxial ratchetting in piping materials – experiments and analysis. Int J Press Ves Pip2004; 81(7): 609–617.
54.
BertoFLazzarinP. A review of the volume-based strain energy density approach applied to V-notches and welded structures. Theor Appl Fract Mech2009; 52: 183–194.
55.
PookLPBertoFCampagnoloA. Coupled fracture mode of a cracked disc under anti-plane loading. Eng Fract Mech2014; 128: 22–36.
56.
CampagnoloABertoFLazzarinP. The effects of different boundary conditions on three-dimensional cracked discs under anti-plane loading. Eur J Mech A: Solid2015; 50: 76–86.
57.
PookLPCampagnoloABertoF. Coupled fracture mode of a cracked plate under anti-plane loading. Eng Fract Mech2015; 134: 391–403.
58.
LazzarinPBertoFZappalortoM. Rapid calculations of notch stress intensity factors based on averaged strain energy density from coarse meshes: theoretical bases and applications. Int J Fatigue2010; 32(10): 1559–1567.
