© Strojni{ki vestnik 48(2002)5,257-266 © Journal of Mechanical Engineering 48(2002)5,257-266 ISSN 0039-2480 ISSN 0039-2480 UDK 621.833:539.388.1:620.178.3 UDC 621.833:539.388.1:620.178.3 Izvrini znanstveni ~lanek (1.01) Original scientific paper (1.01) Ra~unski model za prera~un upogibne trdnosti zobnikov A Computational Model for Calculating the Bending-Load Capacity of Gears Sre~ko Glode` - Jo`e Fla{ker - Damir Jelaska - Janez Kramberger V prispevku je predstavljen računski model za določitev dobe trajanja zobnikov glede na trdnost v zobnem korenu. Potek utrujanja, ki vodi do zloma zoba v korenu, sestoji iz nastanka in širjenja utrujenostne razpoke. Za določitev potrebnega števila obremenitvenih ponovitev Ni za nastanek utrujenostne razpoke je uporabljen Coffin-Mansonov zakon, pri katerem je predpostavljeno, da je začetna razpoka locirana na mestu največjih napetosti v zobnem korenu. Za nadaljnje širjene razpoke je uporabljen znani Parisov zakon, pri katerem so potrebne materialne veličine določene poprej na podlagi ustreznih testnih preskušancev. Funkcijska odvisnost med faktorjem intenzivnosti napetosti in dolžino razpoke K=f(a), ki je potrebna za določitev potrebnega števila obremenitvenih ciklov Np za razširitev razpoke od začetne do kritične dolžine, preračunih niso upoštevani, je predstavljen računski model zelo primeren za določitev dobe trajanja zobnikov, saj so tukaj predstavljeni numerični postopki hitrejši in cenejši v primerjavi z eksperimentalnim delom. © 2002 Strojniški vestnik. Vse pravice pridržane. (Ključne besede: zobniki, procesi utrujanja, doba življenjska, širjenje razpok) A computational model for determining the service life of gears with regard to bending fatigue in a gear-tooth root is presented. The fatigue process leading to tooth breakage is divided into the crack-initiation and crack-propagation periods. The Coffin-Manson relationship is used to determine the number of stress cycles, N, required for the fatigue crack initiation, where it is assumed that the initial crack is located at the point of i the largest stresses in a gear-tooth root. The simple Paris equation is then used for the further simulation of the fatigue crack growth, where the required material parameters have been determined previously with appropriate test specimens. The functional relationship between the stress-intensity factor and the crack length, K=f(a), which is needed for determining the required number of loading cycles, N, for a crack propagation from the initial to the critical length, is obtained numerically in the framework of the finite-element method. The total number of stress cycles, N, for the final failure to occur is then a sum N = N +N. Although some influences (non-homogeneous material, travelling of dislocations, etc.) were not taken i into p account in the computational simulations, the presented model seems to be very suitable for determining the service life of gears because the numerical procedures used here are much faster and cheaper than experimental testing. © 2002 Journal of Mechanical Engineering. All rights reserved. (Keywords: gears, bending fatigue, service life, crack propagation) 0 UVOD Pri zobnikih se lahko zaradi ponavljajoče se obremenitve pri utrujanju pojavita dve poškodbi; jamičenja zobnih bokov in zlom zoba v korenu [1]. V tem prispevku je obravnavan samo zlom zoba v korenu. Predstavljen računski model je uporabljen za preračun upogibne trdnosti v korenu zoba oziroma dobe trajanja zobnika. Za trdnostni nadzor zobnikov glede na napetost v zobnem korenu se običajno uporabljajo 0 INTRODUCTION Two kinds of tooth damage can occur on gears under repeated loading due to fatigue: the pitting of gear-tooth flanks and tooth breakage in the tooth root [1]. In this paper only tooth breakage is addressed and the developed computational model is used for calculating the tooth bending strength i.e. the service life of the gear tooth root. Several classical, standardised procedures (DIN, AGMA, ISO, etc.) can be used for the approximate gfin^OtJJlMlSCSD 02-5 stran 257 |^BSSITIMIGC Glode` S. - Fla{ker J. - Jelaska D. - Kramberger J.: Ra~unski model - A Computational Model postopki po različnih standardih (DIN, AGMA, ISO itn.). Ti so zasnovani na primerjavi največje upogibne napetosti v zobnem korenu z dopustno upogibno napetostjo [1]. Obe napetosti sta odvisni od številnih vplivnih koeficientov, s katerimi upoštevamo dejanske razmere obratovanja zobnika (dodatne notranje in zunanje dinamične obremenitve, slika nošenja, material zobnika, površinska hrapavost itn.). Omenjeni standardni postopki so zasnovani samo na preskusih na testnih zobniških dvojicah in upoštevajo le zadnjo fazo utrujanja zobnega korena zobnika, to je pojav končne poškodbe. V splošnem lahko potek nastanka utrujenostnih poškodb na strojnih elementih razdelimo v naslednje faze ([2] do [5]): (1) nastanek mikrorazpoke; (2) širjenje kratke razpoke; (3) širjenje dolge razpoke; in (4) nastanek končne poškodbe. Pri inženirskih analizah sta prvi dve fazi običajno obravnavani kot “faza nastanka razpoke”, širjenje dolgih razpok pa kot “faza širjenja razpoke”. Čeprav je natančna meja prehoda med nastankom in širjenjem razpoke običajno neznana, v splošnem velja, da pomeni nastanek razpoke pogosto večji delež dobe trajanja, predvsem pri obremenitvi blizu trajne trdnosti materiala (sl. 1). Skupno število obremenitvenih ponovitev N do pojava poškodbe potem določimo kot vsoto obremenitvenih ponovitev Ni za nastanek razpoke in obremenitvenih ponovitev N za razširitev razpoke od začetne do kritične dolžine, ko se utrujenostna poškodba tudi dejansko pojavi: determination of the load capacity of the gear-tooth root. They are commonly based on a comparison of the maximum tooth-root stress with the permissible bending stress [1]. Their determination depends on a number of different coefficients that allow for the appropriate consideration of real working conditions (additional internal and external dynamic forces, contact area of the engaging gears, gear material, surface roughness, etc.). The classical procedures are exclusively based on the experimental testing of the reference gears, and they consider only the final stage of the fatigue process in the gear tooth root, i.e. the occurrence of final failure. However, the complete process of fatigue failure of mechanical elements may be divided into the following stages ([2] to [5]): (1) microcrack nucleation; (2) short crack growth; (3) long crack growth; and (4) occurrence of final failure. In engineering applications the first two stages are usually termed as the “crack-initiation period”, while long crack growth is termed as the “crack-propagation period”. An exact definition of the transition from the initiation to the propagation period is usually not possible. However, the crack-initiation period generally accounts for most of the service life, especially in high-cycle fatigue (HCF), see Figure 1. The total number of stress cycles N, can then be determined from the number of stress cycles Ni, required for the fatigue crack initiation and the number of stress cycles, Np, required for a crack to propagate from the initial to the critical crack length, when the final failure can be expected to occur: N=N +N logs f sU (1). Wöhlerjeva krivulja Wöhler curve nastanek razpoke crack initiation 1/4 Nq Nfl DsFL log N Sl. 1. Shematska predstavitev dobe trajanja strojnih elementov Fig 1. Schematic representation of the service life of machine elements 1 NASTANEK UTRUJENOSTNE RAZPOKE Predstavljen model nastanka utrujenostne razpoke je zasnovan na teoriji mehanike trdnin, pri kateri je predpostavljeno, da je material homogen in izotropen ter brez poškodb in nepravilnosti. V tem primeru so postopki za analize utrujanja materiala običajno zasnovani na Coffin-Mansonovem zakonu, ki podaja razmerje med deformacijami (e), napetostmi (s) ter številu ponovitev (N) v obliki ([6] in [7]): 1 FATIGUE CRACK INITIATION The presented model for the fatigue crack initiation is based on the continuum mechanics approach, were it is assumed that the material is homogeneous and isotropic, i.e. without imperfections or damage. Methods for the fatigue analyses are, in that case, usually based on the Coffin-Manson relation between deformations (e), stresses (s) and the number of cycles (Ni), which can be described as follows ([6] and [7]): VBgfFMK stran 258 Glode` S. - Fla{ker J. - Jelaska D. - Kramberger J.: Ra~unski model - A Computational Model De = De + De s fNb+e,Nc (2), kjer so De prirastek skupne ter Deel in Depl prirastek elastične ter plastične deformacije priWujanju, E modul elastičnosti materiala, s’ koeficient trdnosti, e’ koeficient žilavosti, b eksponent trdnosti in c eksponent žilavosti materiala. Prirastek deformacije pri utrujanju lahko določimo numerično (običajno po MKE), ali eksperimentalno z merilnimi lističi, ki jih namestimo v zobnem korenu na mestu, kjer pričakujemo nastanek razpoke. Materialne veličine s’ e’ b in c se določijo z ustreznimi preskusi, ločeno za vsak material v odvisnosti od razmerja napetost/deformacija. V področju obremenitev blizu trajne trdnosti, kateri so v večini primerov izpostavljeni tudi zobniki, je delež plastične deformacije zanemarljivo majhen, tako da se Coffin-Mansonov zakon omeji le na elastični del ([8] in [9]): where De is the strain increment, Deel and Depl are the elastic and plastic strain increment, E is the Young’s modulus of the material and s’ e’ b and c are the strength coefficient, ductility coefficient, strength exponent and ductility exponent for crack initiation, respectively. The strain icrement can be obtained numerically (usually by FEM), or by strain-gauge measuring in the area of tooth root, where the crack initiation is expected. The material constants s’ e’ b and c are obtained for each material and stress/ strain ratio, from strain controlled tests. In the HCF region commonly applied for gears, where the plastic strain can be neglected, the Coffin-Manson relation reduces only to the elastic part and so transforms to an equation of the Basquin type ([8] and [9]): (Ds ) ki-Ni = Ci (3), kjer so Ds prirastek napetosti ter ki in C materialni veličini. Če predpostavimo, da se ujema v točki (NF ; DsFJ Wöhlerjeva krivulja s krivuljo nastanka razpoke (celotno dobo trajanja predstavlja v tej točki faza nastanka razpoke), lahko po enačbi (3) določimo število obremenitvenih ponovitev za nastanek razpoke pri poljubni napetosti Ds: Ni=NF kjer je NF število obremenitvenih ponovitev pri trajni dinamični trdnosti DsFL (sl. 1). Na podlagi enake predpostavke lahko določimo eksponent ki po enačbi: where Ds is the applied stress increment and ki and C are the material constants. It is easy to obtain the crack initiation life, Ni, using this relation, if we assume that the crack initiation curve passes the same point (N; DsF J as the Wöhler curve, it means at the fatigue limit level the whole fatigue life consists of the crack-initiation period: Ds (4), where N is the number of cycles at the knee of the Wöhler curve, see Figure 1. On the basis of the same assumption, the exponent ki can be obtained as: log(4NFL ) log(sU/DsF (5), kjer je sU natezna trdnost materiala (sl. 1). Enačba (5) kaže dobro ujemanje z razpoložljivimi eksperimentalnimi rezultati [9]. Pri določevanju števila obremenitvenih ponovitev za nastanek razpoke Ni po enačbi (4) je najpomembnejša veličina trajna dinamična trdnost DsFL, ki je tipična materialna veličina in jo določimo z ustreznimi preskusi. Pri zobnikih določimo trajno dinamično trdnost običajno z ustreznimi referenčnimi zobniki. Po standardu ISO [1] so to valjasti zobniki z ravnim ozobjem, modulom m = 3 do 5 mm, širino zob b = 10 do 50 mm, površinsko hrapavostjo Rz « 10 mm, faktorjem koncentracije napetosti Y = 2,0 itn., ki so obremenjeni z utripno obremenitvijo. T e geometrijska oblika, površinska hrapavost, velikost zobnika in obratovalne razmere zobnikov v praksi odstopajo od veličin pri referenčnih zobnikih, se vrednost trajne dinamične trdnosti DsFL spremeni in jo določimo z ustreznimi vplivnimi koeficienti. where sU is the ultimate strength, see Figure 1. This relation was found to be in good agreement with available experimental results [9]. The most important parameter when determining the crack-initiation life, Ni, according to equation (4), is the fatigue limit D s FL, which is a typical material parameter and is determined using an appropriate test specimen. When determining the fatigue limit for gears, the reference test gears are usually used as the test specimens. According to the ISO standard [1], they are spur gears with a normal module m = 3 to 5 mm, face width b = 10 to 50 mm, surface roughness R * 10 mm, stress concentration factor YS = 2.0, etc, which are loaded with repeated pulsating tooth loading. If the geometry, surface roughness, gear size and loading conditions of the real gears in practice deviate from the reference testing, the previously determined fatigue limit DsFL must be modified by the appropriate correlation factors. | IgfinHŽslbJlIMlIgiCšD I stran 259 glTMDDC Glode` S. - Fla{ker J. - Jelaska D. - Kramberger J.: Ra~unski model - A Computational Model 2 ŠIRJENJE UTRUJENOSTNE RAZPOKE Pri analizi širjenja utrujenostne razpoke z uporabo linearno elastične mehanike loma (LELM) je hitrost širjenja razpoke da/dN funkcija faktorja intenzivnosti napetosti pri utrujanju AK=K -K , kjer sta a dolžina razpoke in Nštevilo obremenitvenih ponovitev. V prispevku je za analizo širjenja razpoke uporabljen Parisov zakon [10]: da dN kjer sta C in m materialni veličini. Z integracijo enačbe (6) sledi ob upoštevanju potrebnega števila ponovitev za širjenje razpoke N po enačbi (1): 2 FATIGUE CRACK PROPAGATION The application of linear elastic fracture mechanics (LEFM) to fatigue is based upon the assumption that the fatigue-crack growth rate, da/dN, is a function of the stress intensity range DK=Kmax-Kmin, where a is the crack length and N is the number of load cycles. In this study the simple Paris equation is used to describe of the crack growth rate [10]: C[AK(a)]m (6), where C and m are the material parameters. In respect to the crack propagation period, Np, according to eq. (1), and with the integration of eq. (6) one can obtain: jdN 0 1 C da (7). Enačba (7) pove, da lahko eksplicitno določimo potrebno število obremenitvenih ponovitev N za razširitev razpoke od začetne dolžine a do kritične dolžine a , če so znane veličine C, m in AK(a). C in m sta materialni veličini in ju določimo eksperimentalno, običajno s tritočkovnimi preskušanci po standardu ASTM E 399-80 [11]. Odvisnost med faktorjem intenzivnosti napetosti in dolžino razpoke K = f(a) lahko za preproste primere razpok določimo po postopkih, navedenih v [10] in [11], pri zahtevnejših geometrijskih oblikah elementov in obremenitvenih primerih pa moramo uporabiti druge metode. V tem prispevku je za analizo širjenja utrujenostne razpoke uporabljena metoda končnih elementov v okviru programskega paketa FRANC2D [12]. Z uporabo te metode je določitev faktorja intenzivnosti napetosti zasnovana na načelu odvisnosti pomikov z uporabo singularnih četrtinskih končnih elementov (sl. 2). Za ravninsko deformacijsko stanje in kombinirano obremenitev sledita faktorja intenzivnosti napetosti: c Equation (7) indicates that the required number of loading cycles, N, for a crack to propagate from the initial length a to the critical crack length a can be explicitly determined if C, m and AK(a) are known. C and m are material parameters and can be obtained experimentally, usually by means of a three-point bending test, in accordance with the standard procedure ASTM E 399-80 [11]. For simple cases the dependence between the stress-intensity factor and the crack length K = f(a) can be determined using the methodology given in [10] and [11]. For more complicated geometry and loading cases it is necessary to use alternative methods. In this paper the finite-element method in the framework of the programme package FRANC2D [12] was used for the simulation of the fatigue crack growth. In this paper the determination of the stress-intensity factor is based on the displacement-correlation method using singular quarter-point elements, Figure 2. The stress-intensity factor in the mixed-mode plane-strain condition can then be determined as: KI K 2G (3- 4n ) 2G -----J— -[4vd -v -4vb +vc] J2L-[ 4ud-ue-4ub+uc ] (8), (3-4n) + 1 kj er so G strižni modul materiala, n Poisonovo število, where G is the shear modulus of the material, n is the L dolžina končnih elementov okrog razpoke ter u in v Poisson ratio, L is the finite-element length on the crack vozliščni pomiki elementov okrog razpoke. Skupni face, and u and v are displacements of the crack-tip faktor intenzivni napetosti je potem: elements. The combined stress-intensity factor is then: K^(KI 2+KII 2 )(1-n2) (9). Opisan računski postopek temelji na majhnih prirastkih razpoke, pri katerih je velikost prirastka v posameznem koraku vnaprej predpisana. Za določitev smeri širjenja razpoke je uporabljen kriterij največjih nateznih napetosti. Pri tem je predpostavljeno, da se razpoka razširi radialno glede na vrh razpoke v ravnini, ki je pravokotna na smer The computational procedure is based on in-cremental crack extensions, where the size of the crack increment is prescribed in advance. In order to predict the crack extension angle the maximum tensile stress criterion (MTS) is used. In this crite-rion it is proposed that the crack propagates from the crack tip in a radial direction in the plane per- VH^tTPsDDIK stran 260 Glode` S. - Fla{ker J. - Jelaska D. - Kramberger J.: Ra~unski model - A Computational Model novi vrh razpoke new crack tip razpoka crack vrh razpoke crack tip Sl. 2. Trikotni četrtinski elementi okrog vrha razpoke Fig. 2. Triangular quarter-point elements around crack tip največjih nateznih napetosti. Kot širjenja razpoke lahko na temelju te predpostavke izračunamo po enačbi (sl. 2): q = 2tan- 4 K Za vsak korak je treba okrog novega vrha razpoke načrtovati novo mrežo končnih elementov. Postopek ponavljamo toliko časa, dokler faktor intenzivnosti napetosti ne doseže kritične vrednosti K, ko se pojavi tudi zlom zoba v korenu. Na ta način lahko določimo funkcijsko odvisnost K = f(a). 3 PRAKTIČNI PRIMER Predstavljen model je uporabljen za določitev dobe trajanja dejanskega zobnika z ravnim ozobjem in osnovnimi podatki iz preglednice 1. Material zobnika je zelo trdno legirano jeklo 42CrMo4 (0,43 %C, 0,22 %Si, 0,59 %Mn, 1,04 %Cr, 0,17 %Mo) z modulom elastičnosti E=2,1 • 105 MPa in Poissonovim številom n = 0,3. Toplotna obdelava zobnika je naslednja: segrevanje na 810°C; 2 min, kaljeno v olju; 3 min in popuščano na 180°C; 2 h. pendicular to the direction of greatest tension. The predicted crack-propagation angle can be calculated by, see Figure 2: (10). A new local remeshing around the new crack tip is then required. The procedure is repeated until the stress-intensity factor reaches the critical value K, when the complete tooth fracture is expected. Following the above procedure, one can numerically determine the functional relationship K = f(a). 3 PRACTICAL EXAMPLE The presented model has been used for the computational determination of the service life of a real spur gear with the complete data set given in Table 1. The gear is made of high-strength alloy steel 42CrMo4 (0.43 %C, 0.22 %Si, 0.59 %Mn, 1.04 %Cr, 0.17 %Mo) with a Young’s modulus E = 2.1 • 105 MPa and a Poison’s ratio n = 0.3. The gear material is thermally treated as follows: flame heated at 810°C, 2 min; hardened in oil, 3 min, and tempered at 180°C, 2 h. Preglednica 1. Poglavitni podatki obravnavanega zobnika z ravnim ozobjem Table 1. Basic data of a treated spur gear modul module število zob number of teeth ubirni kot na razdelnem krogu pressure angle on pitch circle koeficient profilne premaknitve coefficient of profile displacement širina zoba face width material zobnika gear material površinska hrapavost surface roughness mn = 4,5 mm z = 39 an = 24o x = 0,06 b = 28 mm 42CrMo4 Rz = 10 mm gfin^OtJJIMISCSD 02-5 stran 261 |^BSSITTMIGC Glode` S. - Fla{ker J. - Jelaska D. - Kramberger J.: Ra~unski model - A Computational Model 3.1 Nastanek utrujenostne razpoke Računski postopek, opisan v poglavju 1 je uporabljen za določitev števila obremenitvenih ponovitev Ni za nastanek utrujenostne razpoke. Natezna trdnost su=1100 MPa, trajna trdnostAsF =550 MPa in število obremenitvenih ponovitev na pregibu Wöhlerjeve krivulje N=3-106 so uporabljeni iz virov [1], [13] in [14] za enak material kakor je material obravnavanega zobnika. Izračuni so izvedeni za različne vrednosti normalne utripne obremenitve F, ki deluje v zunanji točki enojnega ubiranja zobnika (sl. 3). Največja glavna napetost v zobnem korenu As kot posledica delovanja obremenitve F je izračunana numerično po metodi končnih elementov, pri čemer je uporabljen numerični model na sliki 3. Rezultati izračunov so prikazani v preglednici 2. 3.2 Širjenje utrujenostne razpoke Za analizo širjenja utrujenostne razpoke je uporabljena metoda končnih elementov v okviru programskega paketa FRANC2D (poglavje 2). Začetna razpoka je locirana pravokotno na površino na mestu največje glavne napetosti na natezni strani zobnega korena (sl. 4). 3.1 Fatigue crack initiation The procedure as described in Section 1 has been used to determine the number of stress cycles, N, required for the fatigue crack initiation. The ultimate tensile strength 269 MPaVmm available in [13] and [14] the threshold crack length is equal to a h* 0.1 mm. The fracture toughness KIc * 2620 MPaVmm, and the material parameters C = 3.31-10-17 mm/cycl/(MPaVmm)m and m = 4.16 were determined previously using three-point bending samples according to the ASTM E 399-80 standard and for the same material as used in this paper[14]. The tooth loading was equal to the computational analysis of the fatigue crack initiation, see Section 3.1. During numerical simulations the crack increment size, Da, was 0.2 mm, up to the crack length a = 4 mm, and after this, 0.4 mm up to the critical crack length a, see Figure 2. To be able to determine the number of loading cycles, N, required for the crack to propagate from the initial crack length, ah, to the critical crack length, a , according to equation (7), it is necessary to determine the dependence DK = f(a) first. Figure 5 shows the functional relationship between the combined stress-intensity factor, K, and crack length, a, where K is obtained with equation (9) using numerically determined values of KI and KII. Numerical analysis has shown that the K stress-intensity factor is much higher than KII (KII was less than 5 % of KI for all load cases and crack lengths ). Therefore, the fracture toughness, K , can be considered as the critical value of K and the appropriate crack length can be taken as the critical crack length, a . The loading cycles, N, for the crack propagation to the critical crack length can Glode` S. - Fla{ker J. - Jelaska D. - Kramberger J.: Ra~unski model - A Computational Model razširitev razpoke do kritične dolžine potem then be estimated using equation (7), see Table 3. izračunamo po enačbi (7), (preglednica 3). Slika 6 Figure 6 shows the numerically determined crack- prikazuje pot širjenja razpoke v korenu zoba. propagation path in a gear-tooth root. 3000 , MPaVmm 2500 : K 2000 1500 1000 500: 0 01234 56789 a mm Sl. 5. Funkcijska odvisnost med faktorjem intenzivnosti napetosti in dolžino razpoke Fig. 5. Functional relationship between the stress intensity factor and crack length Sl. 6. Pot širjenja razpoke v zobnem korenu zobnika Fig. 6. Crack propagation path in a gear tooth root Preglednica 3. Računski rezultati širjenja utrujenostne razpoke Table 3. Computational results for the fatigue crack propagation Obremenitev Loading F N/mm Kritična dolžina razpoke Critical crack length ac mm Število ponovitev Number of cycles Np 800 8,6 9,473.105 900 8,4 5,845.105 1000 8,2 3,768.105 1100 7,9 2,534.105 1200 7,7 1,773.105 1300 7,5 1,264.105 1400 7,3 9,322.104 1500 7,1 6,993.104 VBgfFMK stran 264 Glode` S. - Fla{ker J. - Jelaska D. - Kramberger J.: Ra~unski model - A Computational Model Na temelju dobljenih rezultatov za nastanek razpoke (N) in širjenje razpoke (N) iz preglednic 2 in 3 lahko določimo skupno dobo trajanja zoba zobnika po enačbi (1) (sl. 7). S slike 7 je razvidno, da je razmerje med fazo nastanka razpoke in koncem širjenja razpoke (zlomom zoba) odvisno od ravni napetosti v korenu zoba. Pri majhnih napetostih blizu trajne trdnosti pomeni faza nastanka razpoke skoraj celotno dobo trajanja zobnika, pri večjih napetostih je zelo pomembna predvsem faza širjenja razpoke. Dobljeni računski rezultati dobe trajanja kažejo dobro ujemanje z razpoložljivimi eksperimentalnimi rezultati, uporabljenimi iz vira [13]. On the basis of the computational results for the crack-initiation (Ni) and crack-propagation (Np) periods in Tables 2 and 3 the total service life of the gear-tooth root can be obtained according to equation (1), see Figure 7. It is clear from Figure 7 that the ratio among the periods of initiation and of the end of propagation (i.e. final breakage) depends on the stress level. At low stress levels almost all the service life is spent in crack initiation, but at high stress levels a significant part of the life is spent in the crack propagation. The computational results for total service life are in a good agreement with the available experimental results, which are taken from [13]. MPa Ds 1100 a 1000 900 800 700 600 500 400 1,E+00 1,E+01 1,E+02 1,E+03 1,E+04 1,E+05 1,E+06 1,E+07 1,E+08 N ponovitve/cycles Sl. 7. Izračun dobe trajanja obravnavanega zobnika Fig. 7. The computed service life of treated gear 4 SKLEPI V prispeveku je predstavljen računski model za določitev dobe trajanja zobnikov glede na trdnost v zobnem korenu. Potek utrujanja zobnika, ki vodi do končnega zloma zoba v korenu, je razdeljen v fazo nastanka razpoke (N) in fazo širjenja razpoke (N). Skupna doba trajanja zobnika je definirana kot N = Ni+Np. Za izračun števila ponovitev za nastanek razpoke Ni je uporabljena preprosta Basquinova enačba. V modelu je predpostavljeno, da je začetna razpoka na mestu največje glavne napetosti v korenu zoba, ki je določena numerično z MKE. Funkcijska odvisnost med faktorjem intenzivnosti napetosti in dolžino razpoke K = f(a), potrebna za analizo širjenja razpoke in določitev števila ponovitev N, je določena numerično z metodo odvisnosti pomikov v okviru metode končnih elementov. Predstavljen model je uporabljen za določitev dobe trajanja realnega zobnika iz zelo trdnega legiranega jekla 42CrMo4. Rezultate računskih analiz prikazuje slika 7, kjer sta predstavljeni dve krivulji: krivulja nastanka razpoke in krivulja zloma zoba, ki hkrati pomeni skupno dobo trajanja zobnika. Rezultati kažejo, da pomeni faza 4 CONCLUSIONS This paper presents a computational model for determining the service life of gears in regard to bending fatigue in a gear-tooth root. The fatigue process leading to tooth breakage in a tooth root is divided into the crack-initiation (Ni) and crack-propagation (Np) periods, which enables the determination of total service life as N = Ni+Np. The simple Basquin equation is used to determine the number of stress cycles, Ni. In the model it is assumed that the crack is initiated at the point of the maximum principal stress in a gear-tooth root, which is calculated numerically using the FEM. The displacement-correlation method is then used for the numerical determination of the functional relationship between the stress-intensity factor and crack length K = f(a), which is necessary for a subsequent analysis of fatigue crack growth, i.e. determination of the stress cycles Np. The model is used to determine the complete service life of a spur gear made from high-strength alloy steel 42CrMo4. The final results of the computational analysis are shown in Figure 7, where two curves are presented: the crack-initiation curve and the curve of tooth breakage, which at the same time represents the Glode` S. - Fla{ker J. - Jelaska D. - Kramberger J.: Ra~unski model - A Computational Model nastanka razpoke pri majhnih napetostih blizu trajne trdnosti skoraj celotno dobo trajanja zobnika. To je zelo pomembna ugotovitev pri izračunavanju dobe trajanja zobniških dvojic v praksi, ki v večini primerov dejansko obratujejo v obremenitvenih razmerah blizu trajne dinamične trdnosti. Dobljeni računski rezultati za skupno dobo trajanja zobnika se dobro ujemajo z razpoložljivimi eksperimentalnimi rezultati. Kljub temu velja model v prihodnje še izboljšati na področju teoretičnih spoznanj in numeričnih analiz, prav tako pa bodo potrebne dodatne eksperimentalne raziskave za določitev potrebnih materialnih veličin. total service life. The results show that at low stress levels near the fatigue limit, almost all service life is spent in crack initiation. It is a very important knowledge for the determination of the service life of real gear drives in practice, because the majority of them really operate with loading conditions close to the fatigue limit. The computational results for total service life are in a good agreement with the available experimental results. However, the model can be further improved with additional theoretical and numerical research, although additional experimental results will be required to provide the required material parameters. 5 LITERATURA 5 REFERENCES [I] ISO 6336, Calculation of load capacity of spur and helical gears, International Standard, 1993 [2] Shang, D.G., W.X. Yao, D.J. Wang (1998) A new approach to the determination of fatigue crack initiation size, Int. J. Fatigue, 20, 683-687. [3] Glodež, S., J. Flašker, Z. Ren (1997) A new model for the numerical determination of pitting resistance of gear teeth flanks, Fatigue Fract. Engng Mater. Struct, 71-83. [4] Glodež, S., H. Winter, H.P. Stüwe (1997) A fracture mechanics model for the wear of gear flanks by pitting, Wear, 208, 177-183. [5] Cheng, W, H. S. Cheng, T Mura, L. M. Keer (1994) Micromechanics modeling of crack initiation under contact fatigue, ASME J. Tribology, 116, 2-8. [6] Manson, S. (1953) “Proc.Heat Transfer Symp.”, Univ. Michigan, Eng. Res. Inst., 9-75. [7] Tavernelli, J.F., L. F. Coffin (1959) A compilation and interpretation of cyclic strain fatigue tests on metals, Trans. Amer Soc. of Metals, 51, 438-450. [8] Nicholas, T J. R, J. R. Zuiker (1996) On the use of the Goodman diagram for high cycle fatigue design, Int. J. Fracture, 80, 219-235 [9] Jelaska, D. (2000) Crack initiation life at combined HCF/LCF loading, Proc. Int. Conf. Life Assessment and Management for Structural Components, Kiev 2000, 239-246. [10] Ewalds, H.L., R.J. Wanhill (1989) Fracture Mechanics, Edward Arnold Publication, London. [II] ASTM E 399-80, American standard. [12] FRANC2D, User’s Guide, Version 2.7, Cornell University. [13] Niemann, G, H. Winter (1983) Maschinenelemente - Band II, Springer Verlag. [14] Aberšek, B. (1993) Analysis of short fatigue crack on gear teeth, Ph.D. thesis, University of Maribor, Faculty of Mechanical Engineering, Maribor. [15] Bhattacharya, B., B. Ellingwood (1998) Continuum damage mechanics analysis of fatigue crack initiation, Int. J. Fatigue, 20, 631-639. Naslova avtorjev: prof. dr. Srečko Glodež prof. dr. Jože Flašker dr. Janez Kramberger Fakulteta za strojništvo Univerza v Mariboru Smetanova 17 2000 Maribor joze.flasker@uni-mb.si jkramberger@uni-mb.si prof dr. Damir Jelaska Fakulteta za elektrotehniko, strojništvo in ladjarstvo Univerza v Splitu R Boškoviča b.b. 21000 Split, Hrvaška damir.jelaska@fesb.hr Authors’ Addresses:Prof Dr. Srečko Glodež Prof. Dr. Jože Flašker Dr. Janez Kramberger Faculty of Mechanical Eng. University of Maribor Smetanova 17 2000 Maribor, Slovenia joze.flasker@uni-mb.si jkramberger@uni-mb.si Prof Dr. Damir Jelaska Faculty of Electrical Eng., Mech. Eng. and Naval Architecture University of Split R Boškoviča b.b. 21000 Split, Croatia damir.jelaska@fesb.hr Prejeto: Received: 13.12.2001 Sprejeto: Accepted: 23.5.2002 VH^tTPsDDIK stran 266