variable amplitude fatigue crack growth modelling - Universidade de ...

VARIABLE AMPLITUDE FATIGUE CRACKGROWTH MODELLINGRibeiro A.S. 1,2Jesus A.P. 1,2Costa J.M. 3Borrego L.P. LP 4Maeiro J.C. 11 Departamento de Engenharias, Escola de Ciências e Tecnologia, Universidade de Trás-os-Montes e Alto Douro2 IDMEC-UCVE, Pólo FEUP, Rua Roberto Frias3 Departamento de Engenharia Mecânica, Faculdade de Ciências e Tecnologia da Universidade de Coimbra4 Departamento de Engenharia Mecânica, Instituto Superior de Engenharia de Coimbra

INTRODUCTION Fatigue crack propagation may occur under random, variableamplitude or in constant amplitude loading;Under variable amplitude loading, the load sequence effect is a majorfactor in determining the fatigue life;Due to the random nature of variable loading, significant accelerationsand/or retardations in crack growth rate may occur;

INTRODUCTIONOverloads are known to retard crack growth, while underloads generallyaccelerate crack growth relatively to the baseline crack growth rate;The overloads and underloads, which are highly dependent upon theloading sequence, make the prediction of fatigue life under variableamplitude loading more complex than under constant amplitude loading;Many models have been developed to predict the fatigue life ofcomponents under variable amplitude loading, namely by Wheeler,Willemborg , Elber and Newman .

OBJECTIVESThe goal of this work is to describe models dl available to predict crackgrowth, using variable amplitude loading sequences.

LOAD INTERACTIONS EFFECTS UNDER SIMPLEVARIABLE AMPLITUDE LOAD SEQUENCES

LOAD INTERACTIONS EFFECTS UNDER SIMPLEVARIABLE AMPLITUDE LOAD SEQUENCESTypical effect of periodically applied overloads for several numbersof baseline cycles between overloads for 2024-T3 aluminium alloy.

LOAD INTERACTIONS EFFECTS UNDER SIMPLEVARIABLE AMPLITUDE LOAD SEQUENCESBehaviour generally observed in load interaction phenomenon highlow(Hi-Lo), low-high (Lo-Hi) following an underload??rever!.(a)(b)da/dNda/dNK 1K 2Crack length, aK 12 1K 2Crack length, a

LOAD INTERACTIONS EFFECTS UNDER SIMPLEVARIABLE AMPLITUDE LOAD SEQUENCESLoad interaction effects under simple variable amplitude loadsequences.length, aCrack lload variationApplied cycles, N

MODELS FOR VARIABLE AMPLITUDE FATIGUECRACK GROWTHRetardation models may be classified into two maincategories:• Crack tip plasticity models• Crack Cac closure cosuemodels

WHEELER MODELThis model dlcalculates l the fti fatigue crack growth retardationtiwhen a single tensile overload is applied.The Wheeler model is characterized by the introduction ofa retardation parameter, R , which prescribes a reducedgrowth rate for fatigue cracks.Applying the Wheeler model to the Paris’s law results thefollowing equation:dadNm CKR

WHEELER MODELThe retardation parameter, R , is defined as:R a1,OLrp,i rOL ain, ai raip,i rp,i< aOL a rOLOL rOL• a OL is the crack length when the overload is applied• a i is the crack length at each load cycle i;• r OL is the size of the plastic zone produced by the overload at a OL ;• r p,i is the size of plastic zone produced by the post-overload constantamplitude loading at current crack length a i ;• n is an adjustable experiment-based shaping exponent, depends on thetype of material, geometry and overload magnitude.

WILLEMBORG MODELThe Willemborg model suggest the use of an effective stressintensity factor, when an overload is applied, defined by thefollowing expression:(K ) K ieffiKRed (K i ) eff and K i is the effective and apparent (under constant amplitude) stressintensity factors in each load cycle i, respectively.K Red , is given by the following equation:KRe d KOL1 ai arOLOL Kmax,i K OL is the overload stress intensity factor and K max,i the maximum stressintensity factor in each load cycle i.

WILLEMBORG MODEL Willemborg proposed the Forman equation to predict thecrack growth propagation when a overload is applied:dadNC (K(1 R ) Keffeffc)n KeffK c isthecritical ii stress intensityi factor.In this model the retardation effect, in the crack growth isdefined using the effective stress ratio in each load cycle i,(R i ) eff , given by:( K( Ri) eff ( Kmin,imax,i))effeff

CRACK CLOSURE MODELThe crack closure approach permits to correlate the majorityof the crack c growth gow transients se sobserved under ude simple variableamplitude loading sequences using experimental crack closuremeasurements.However the K op must be obtained for the specific materialand loading condition or by an approximated expression of theevolution of the opening stress proposed from experimentalcrack closure measurements, which is very complex and timeconsuming.

CRACK CLOSURE MODELElber suggest that the effective stress intensity range, K eff , shouldbe obtain by the following expression:KeffKmaxKopK op represents the stress intensity factor corresponding to the opening stress ofthe crack, op.The retardation effect in the crack growth can be obtained by:dadNf(K)eff

WHEELER MODEL ENHANCEMENTSimilaril to the original i Wheeler model, dl Yuen and Th Taheri appliedsome modifications to the Wheeler model using the equation in thefollowing form:dadN RDICKacm• R is the retardation parameter as defined in the original Wheelermodel;• D is the delay parameter;• I is the overload interaction parameter• K ac is the accelerated stress intensity factor range Under constant amplitude stable crack growth, R =1, D =1, I =1 andK ac =K.

WHEELER MODEL ENHANCEMENTThe delayed-retardation parameter D is defined as:Da 1,OL rrd,OLd,i aiq,a ria rid,id,i< a aOLOL r rd,OLd,OLr d,OL is the size of the overload effective delay zone;r d,i is the size of the current effective delay zone;q is the shaping exponent for the modified model.

WHEELER MODEL ENHANCEMENTThesizeoftheoverload effective delay zone and of the currenteffective delayed zone can be assumed to be calculated using thefollowing formula:rd K YS 2 is the effective delay zone size constant

WHEELER MODEL ENHANCEMENTThe overload interaction parameter I is defined as presented inthese formulas:1 1 min,i I 1 1 min,i1,q rp,iaOL rd ,OL ai1 , ai rd ,i aOL rd ,OLaOL rp,OL aird ,i q rp,i1 , aOL rd ,OL ai rd ,i ,ai rp,i aOL rp,OLaOL rp,OL aiaOL rp,OL ai rp,i

WHEELER MODEL ENHANCEMENTThe accelerated stress intensity factor range, K ac, is defined in themodified Wheeler model as presented in equation:KacK Kii,KOL Ki1aOLr rd ,id ,OL aiq,aia rOLd ,ir ap,OLOLa rid ,OLrp, iK OL is the overload stress intensity factor range and K i the current stressintensity factor in each load cycle i.

WHEELER MODEL ENHANCEMENTCrack growth modelling proposed by Xiaoping Huang, introducethe retardation parameter R of the original Wheeler model, withsmall modification to consider underload effects.dadN CmK eq K0th0mKeq0MRMPK• K th0 is the threshold at R=0, C and m are those corresponding to R=0 andparameters M R and M P are correction factors for the equivalent stressintensity factor range.

WHEELER MODEL ENHANCEMENTM R is the correction factor of the stress ratio effect:MR 1R1 , 1 R,1.05 1.4R 0.6R2- 5R00 R 0.5 ,0.5 R 1 where and 1 are parameters depending on material properties .

WHEELER MODEL ENHANCEMENT The loading sequenceinteraction correction factor M Pis defined as:MP a1,OLr rPOL a rn,a ra rPP< a aOLOL r rOLOL rn is a shaping exponent determinedby fitting to the experimental data,set to the same value for the samematerial;r

WHEELER MODEL ENHANCEMENT The r can be quantitatively calculated using the followingequations:Kur Y K auYSi1min2 imini1i YS is the compressive yield stress and minmin the minimum stresses up to andfollowing the ith overload, respectively.

CONCLUSIONSSome relevant fti fatigue crack growth models dl under variable iblamplitude loading have been presented.There are merits and limitations of each model.From the literature it is observed that most of the modelsrequire one or more calibration parameters or constants toconduct crack growth analysis.It is necessary the use of curve fitting as an importedprocedure to correlate the predictions with the experimentaldata.Due to the number and complexity of the mechanismsinvolved in this problem, no universal model exists yet.The selection of the appropriate model is usually based onthe researcher experience and persona1 preference, thusaccurate predictions remain problematic.

THANKS FOR YOUR ATTENTIONRibeiro A.S.Jesus A.P.Costa J.M.Borrego L.P.Maeiro J.C.