International Journal of Fatigue

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1 International Journal of Fatigue () 88 9 Contents lists available at SieneDiret International Journal of Fatigue journal homepage: Analysis of fatigue rak growth in an attahment lug based on the weight funtion tehnique and the UniGrow fatigue rak growth model S. Mikheevskiy a,, G. Glinka a, D. Algera b a University of aterloo, Department of Mehanial Engineering, aterloo, ON, Canada NL 3G b Tehnial Data Analysis, In., 7A Leesburg Pike, Suite, Falls Churh, VA 3, United States artile info abstrat Artile history: Reeived Otober Reeived in revised form April Aepted July Available online 3 July Keywords: Fatigue rak growth Corner rak Load shedding eight funtion A generalised step-by-step proedure for fatigue rak growth analysis of strutural omponents subjeted to variable amplitude loading spetra has been presented. The method has been illustrated by analysing fatigue growth of planar orner rak in an attahment lug made of Al7-T7 alloy. Stress intensity fators required for the fatigue rak growth analysis were alulated using the weight funtion method. In addition, so-alled load-shedding effet was aounted for in order to determine appropriate magnitudes of the applied stress intensity fators. The rate of the load shedding was determined with the help of the finite element (FE) method by finding the amount of the load transferred through the raked ligament. The UniGrow fatigue rak growth model, based on the material stress strain behaviour near the rak tip, has been used to simulate the fatigue rak growth under two variable amplitude loading spetra. The omparison between theoretial preditions and experimental data proved the ability of the UniGrow model to orretly predit fatigue rak growth behaviour of twodimensional planar raks under omplex stress field and subjeted to arbitrary variable amplitude loading. Ó Elsevier Ltd. All rights reserved.. Introdution Lug-type joints are often used to onnet omponents in various mehanial systems. Sine the lug is usually attahed to the fork by a single bolt or pin without lamping, it is easy to mount and does not produe any additional loal bending moments. However, raks or other disontinuities an be initiated due to orrosion, material imperfetion, manufature defets, or fatigue. Despite the improved damage tolerane design proedures and usage of non-destrutive inspetion tehniques, unsafe raks may still be present in lugs and go unheked during regular maintenane inspetions. If suh a rak or flow ours in the region of high stress onentration near the lug hole it an quikly propagate and signifiantly redue the operational life of a omponent. In order to ensure operational safety of a struture it is neessary to perform fatigue rak growth analysis assuming the possibility of fatigue rak initiation and growth. A number of analytial and experimental studies have been performed over last three deades, aiming at investigating the fatigue behaviour of attahment lugs under both onstant and variable amplitude loading [ 3]. The approah disussed below is based on the weight funtion method [] and the UniGrow fatigue rak growth model proposed Corresponding author. address: volhishka@gmail.om (S. Mikheevskiy). by Noroozi et al. [] and extended by Mikheevskiy and Glinka []. The list of main steps required for the fatigue analysis of an attahment lug is given below: Collet material information (modulus of elastiity, yli stress strain urve, onstant amplitude fatigue rak growth data). Determine required stress field in an un-raked lug indued by the applied load (with the help of the FE-finite element analysis or any other method). Calulate the load shedding parameter for several different rak sizes. Determine the rak shape and alulate stress intensity fators using the weight funtion method and the load shedding orretion fator for eah yle of the loading spetrum. Determine the instantaneous fatigue rak growth rate using the UniGrow fatigue model.. Geometry of the omponent and material data The attahment lug [7] was made of Al 7-T7 aluminium alloy material. The onstant amplitude fatigue rak growth data for this material was olleted from several literature soures [7 9]. Eight onstant amplitude (CA) fatigue rak growth (FCG) data sets, obtained at six different stress ratios, were seleted as -3/$ - see front matter Ó Elsevier Ltd. All rights reserved. doi:./j.ijfatigue..7.

2 S. Mikheevskiy et al. / International Journal of Fatigue () the referene for subsequent fatigue rak growth and fatigue life analysis (Fig. ). The geometrial onfiguration and dimensions of the lug tested [7] under variable amplitude loading spetra are shown in Fig.. The lug was mm thik (t = mm) with the hole radius of 3 mm (R i = 3) and the outer radius of 3 mm (R o = 3). A beryllium opper bushing was installed with. mm diametrial interferene between the lug and the pin. A quarter-irular orner rak (Fig. ) with initial dimensions of a = = mm was artifiially made in the lug whose growth monitored later. As soon as the orner rak propagated through the entire thikness of the speimen it quikly transformed into the lassial edge rak emanating from a hole. Therefore speial stress intensity fator solution was needed in order to appropriately model the fatigue rak shape evolution. A more detailed desription of the speimen and the testing proedure an be found in the original work of Kim et al. [7]. 3. The stress field in the ritial ross setion of the attahment lug The stress field in the un-raked attahment lug was determined with help of the finite element method. The stress field was used to determine the stress intensity fator by using the weight funtion method. The stress field in the potential rak plane of an un-raked lug is shown in Fig. 3. The variation of through thikness stress was assumed to be small due to relatively large diameter of the lug-hole ompared to the lug thikness. The same lug was also analysed by Kim et al. [7] using the boundary element method. As one an see the results are almost idential and the elasti gross stress onentration fator was around five. Fig.. Geometry of the attahment lug with an initial quarter-irular orner rak ( = mm, R i = 3 mm, R o = 3 mm and t = mm). Stress/Gross Stress a/(w/) R i R o Fig. 3. Stress distribution in un-raked lug. a t Algera (FE) Kim (BEM). Determination of the stress intensity fator The geometrial shape of the growing fatigue rak, as was mentioned earlier, was hanging from quarter-irular to an edge one. Therefore, the stress intensity fator analysis onsisted of several stages appropriate for various rak shapes as shown in Fig.... The stress intensity fator (SIF) for quarter-elliptial rak The quarter-elliptial orner rak was analysed first using twodimensional generalised weight funtion proposed by Glinka and FCG rate, da/dn [mm/y].e-.e-.e-.e-7.e-8.e-9.e-.e- CA FCG data: Al 7 T7 R=. Sharp R= Sharp R=-.3 Sharp R=.3 Lee R=, Lee R=-., Lee R=.8 Sadananda R=. Sadananda Applied Stress Intensity Fator Range, ΔK appl [MPa m] Fig.. Experimental onstant amplitude fatigue rak growth data for the Al 7 T7 alloy in terms of the applied stress intensity range. Shen []. Stritly speaking, the fatigue growth of a orner or any other planar rak depends on the value of stress intensity fator at eah point along the rak front. In the analysis below it is assumed that the rak shape remained quarter-elliptial, and therefore only two surfae points A and B (Fig. ) needed to be onsidered. It has been shown [] that stress intensity fators at point A and point B an be determined using the integrals involving the stress field and appropriate weight funtions. K A ¼ K B ¼ Z Z Fig.. Evolution of the fatigue rak in attahment lug. rðxþm A ðx; ; =aþdx rðxþm B ðx; ; =aþdx where r(y) is the stress distribution through the width of the speimen, and a are the rak dimensions as shown in Fig.. The weight funtion expressions m A and m B are given in referene []. ðþ

3 9 S. Mikheevskiy et al. / International Journal of Fatigue () 88 9 hile the fatigue rak propagates point B moves into the region of lower stresses as shown in Fig. 3. On the other side, stresses for point A stay the same sine they are uniformly distributed through the thikness. Therefore, it is reasonable to expet higher stress intensity fators and fatigue rak growth rates at point A resulting in the hange of the initial rak shape from quarter-irular to quarter-elliptial and elongated along the thikness. The variation of stress intensity fators at points A and B obtained as a funtion of the rak length a for the aspet ratio a/ = and the applied stress S = 3 MPa is shown in Fig.... The stress intensity fator for the transition rak hen the orner rak breaks through the whole thikness of the lug it turns into an edge rak. Unfortunately, the weight funtion enabling smooth transition from orner to edge rak as shown Fig. (Region 3) has not been derived yet. Thus, the following approah has been adopted. The two-dimensional weight funtions (Eq. ()) have been derived for stress fields whih are uniform through the thikness. Therefore, despite the fat that they depend on both rak dimensions a and, they do not depend on the through thikness stress variation. Thus, the rak was analysed first as a orner rak until the length a exeeded (see Figs. and ) the lug thikness t by %, i.e. when a >.t. After that stage the rak was modelled as an edge rak with its initial depth oiniding with the loation of point B (Fig. )..3. Thestress intensity fator for an edge rak emanating from a hole An edge rak in a lug is less diffiult to analyze than a orner one, and therefore, several attempts have been made in the past aiming to investigate the stress intensity fator for only an edge rak in a lug. Aberson and Anderson [] used finite element analysis with a rak tip singularity elements, Impellizeri and Rih [] employed the weight funtion method, and Shijve and Hoeymaker [3] found the empirial solution based on the fatigue rak growth data. Glinka and Shen [] have derived the general form of onedimensional weight funtion (Eq. (3)) whih was used to alulate the stress intensity fator (Eq. ()) for the edge rak in the lug based on the atual stress field r(x) obtained from finite element analysis (Fig. 3) of un-raked lug. The stress distribution r(x) in the prospetive rak plane was obtained from the finite element analysis. The load fore P in the hole was replaed by a pressure distribution resulting from the ontat between the pin and the hole in the lug and it was simulated by imposing restrains on radial displaement in the region of the ontat between the pin and the lug. mðx; Þ ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ M x pð xþ þ M x þ M3 x K ¼ Fig.. Edge rak in attahment lug and the weight funtion model. Z rðxþmðx; Þdx 3 Coeffiients M, M, and M 3 were derived for an edge rak in a finite plate width with fixed one edge as shown in Fig.. The stress intensity fator solution obtained from the weight funtion has been ompared to the solution alulated by Kim et al. [7] by the using boundary element method and both sets of results are shown in Fig. 7. The weight funtion based stress intensity fators seem to be slightly onservative while ompared with the boundary element method.. The load shedding The weight funtion method desribed in Setion requires using the stress field from un-raked body indued by the applied load in the prospetive rak plane (Fig. 3). This approah is valid as long as the raked setion is taking all the time the same amount of the resultant load while the rak propagates through the ross setion. However, in the ase of a single rak in a lug (Fig. 8), the raked setion beomes less rigid than ðþ ð3þ ðþ SIF, MPa m 8 Ka, a/= Kb, a/=....8 a/t SIF [MPam. ] Glinka (F) Kim (BEM) Crak Depth a [mm] Fig.. SIF for quarter-irular rak obtained using generalised two-dimensional weight funtion. Fig. 7. Comparison of stress intensity fators obtained from the weight funtion and the boundary element method (nominal gross stress S = 3 MPa).

4 S. Mikheevskiy et al. / International Journal of Fatigue () P P P / P / P P,initial Hole Fig. 8. Shemati illustration of the load shedding effet. un-raked setion, and part of the applied load is transferred to setion. This situation would not happen in the ase of two symmetri raks beause both setions have to hold the same amount of load equal to the half of the load applied to the lug. It has been shown using finite element analysis that this effet is relatively small as long as rak stays quarter-elliptial, but beomes signifiant when rak breaks through the entire thikness of the lug. Introdution of the load shedding oeffiient, LS(/), enables to estimate the amount of load taken by the raked setion and estimate the atual load (Fig. 8). P ¼ LS P where LSðÞ ¼ ðþ Sine the magnitude of the stress field in a ligament depends only on the magnitude of the load transferred through the ligament the instantaneous stress distribution an be written as rðxþ ¼LS P r nðxþ where r n (x) is the stress field/distribution normalised obtained for the unit load. Thus, the stress intensity fator for a single edge rak in a lug an be subsequently written in the form: KðÞ ¼ Z LS P r nðxþmð; xþdx Taken into aount the fat that the load shedding oeffiient ontrols only the magnitude of the stress field and does not depend on the x oordinate, the final equation for the resultant stress intensity fator an be written as: KðÞ ¼LS K F ðþ ð8þ where K F () is the standard solution obtained by using the weight funtion method as desribed in Setion. The main hallenge is to determine the load shedding parameter, LS(/), as a funtion of the rak size. Therefore, omplete 3D finite element analysis was arried out for three different edge raks of mm, mm, and mm deep. Aurate modelling of the stress field near the rak tip was not neessary in those ases, and therefore, oarse finite element mesh ould be used over the entire ligament. The stress distributions obtained from the FE analyses are presented in Fig. 9. The resultant fore P transferred by the raked ligament was determined by integrating appropriate stress field ating in the remaining ross setion. ZZ Z P ¼ rðx; yþdx dy ¼ t rðxþdx ð9þ s As one an see, stresses in the un-raked setion went up as the rak propagated through the other ligament. This means that the resultant fore P in the raked ligament must have got dereased and the amount of the derease ould be measured by the load shedding fator LS. ðþ ð7þ Un-raked The variation of the load in the raked and un-raked ligament of the lug as a funtion of the rak depth is shown in Fig.. The results show that at the rak depth = mm the load transferred by the raked ross-setion was redued by around %, whih resulted in approximately % redution in the fatigue rak growth rate. The load shedding parameter was subsequently fitted into following expression LS ¼ A q ðþ with parameters A =. and q =.3. It has been shown that Eq. () fits well into the data obtained from the FE analysis for relative rak depths < /i <.8. The final expression for the edge rak stress intensity fator was obtained by inluding all the information disussed above into Eq. (3). KðÞ ¼ : a :3 Z rðxþmð; xþdx Craked setion Fig. 9. Stress fields obtained for three different rak lengths. ðþ Based on Eq. () the maximum applied stress intensity fator, K max,appl and the applied stress intensity range, DK appl, ould be determined for eah yle of the applied loading spetrum. As mentioned in Setion, the UniGrow fatigue model was used for the determination of the instantaneous fatigue rak growth rate and rak inrements indued by eah individual loading yle. LS (/) Unraked ligament Craked ligament LS=-A*(a/t) q Load Shedding....8 / Fig.. Load shedding oeffiient.

5 9 S. Mikheevskiy et al. / International Journal of Fatigue () Basis of the UniGrow fatigue model The UniGrow fatigue rak growth model, initially proposed by Noroozi et al. [], is based on the idea that the fatigue proess near raks and nothes is governed by highly onentrated strains and stresses in the noth/rak tip region. Therefore, the fatigue rak growth an be subsequently onsidered as a proess of suessive rak inrements resulting from material damage in this region. Aording to the miro-support onept of Neuber [], the real material an be modelled as a set of elementary partiles or material bloks of a finite dimension, q. The idea of elementary material bloks was also postulated by Forsyth [] based on mirosopi observations of the fatigue rak front advane. The assumption of the elementary material blok implies that the atual stress strain and fatigue response of the material near the rak tip is suh as the rak had a blunt tip with the radius of q. It means that it might be reasonable to model the rak, within the ontinuum mehanis framework, as a sharp noth with the tip radius q. Therefore, the lassial noth stress strain analysis tehniques an be applied in order to determine stresses and strains in the rak tip region. The following assumptions and omputational rules form the base for the UniGrow fatigue rak growth model. The material onsists of elementary bloks of a finite dimension q. The fatigue rak is regarded as a noth with the tip radius q. The analysis is based on the Ramberg-Osgood (yli) [7] and Manson-Coffin (fatigue) [8] material properties. The number of yles to fail the material over the distane q an be obtained using the Smith atson Topper damage parameter [9] and the Manson-Coffin fatigue urve. The instantaneous fatigue rak growth rate an be expressed as da/dn = q /N. Based on the assumptions stated above Noroozi et al. [] have analytially derived the fatigue rak growth expression in the form of da m dn ¼ C ðk max;appl þ K r Þ p ðdk appl þ K r Þ p ðþ where K max,appl and DK appl, are the applied maximum stress intensity fator and the stress intensity range respetively, and K r is the residual stress intensity fator aounting for the effet of rak Applied load, kn Crak length [mm] Number of reversals Fig.. Applied loading spetrum. Fatigue life for 8% and % Clipped Loading History+Shedding 8% Clipped Experimental 8% UniGrow % Clipped Experimental % UniGrow Number of bloks Fig. 3. Fatigue rak growth preditions and the experimental data for the original and the trunated load spetrum. tip residual stresses resulting from reversed plasti deformations. For the simpliity, Eq. () an be rewritten as da dn ¼ CðDjÞm ; Dj ¼ðK max;appl þ K r Þ p ðdk appl þ K r Þ p ð3þ where Dj is total two-parameters driving fore. A very similar fatigue rak growth equations has been proposed by alker [] and Dinda and Kujawski [] based on empirial fitting of observed onstant amplitude fatigue rak growth data. However, alker and Kujawski expressions did not.e- CA FCG data: Al 7 T7.E- FCG rate, da/dn [m/y].e-.e-7.e-8.e-9.e- R=. Sharp R= Sharp R=-.3 Sharp R=.3 Kim R=, Kim R=-., Kim R=.8 Sadananda R=., Sadananda.E-.E+.E+ Total driving fore, Δκ tot [MPa m].e+ Fig.. Fatigue rak growth rate in terms of total two-parameter driving fore.

6 S. Mikheevskiy et al. / International Journal of Fatigue () % Clipping 8% Clipping 8 8 Depth, mm Experiment UniGrow Depth, mm Experiment UniGrow 8 Surfae, mm 8 Surfae, mm Fig.. Experimental and predited rak shape evolution. take into aount the fat that the orrelation between the stress intensity fator and the rak tip stress/strain field is often altered by residual stresses resulting from reversed plasti deformations. The UniGrow fatigue rak growth model has been subsequently modified and extended by Mikheevskiy and Glinka [] in order to make it appliable to arbitrary variable amplitude loading spetra. It was found that the instantaneous fatigue rak growth rate depended not only on the residual stresses produed by the latest loading yle but also on all stress fields generated by preeding yles of the loading history. In order to ombine residual stress fields generated by all preeding loading yles into one resultant stress field ontrolling the urrent fatigue rak growth rate several so-alled memory rules have been established based on the experimental observations of fatigue rak growth under variable amplitude loading. Detailed desription and experimental verifiations of the UniGrow fatigue rak growth model in the ase of variable amplitude loading spetra an be found in referenes [,,]. The two parameter driving fore (Eq. (3)) is valid for any stress ratio and therefore da/dn DK fatigue rak growth urves obtained at various R ratios an be approximated by one master urve (Fig. ) by presenting the fatigue rak growth rate as a funtion of the total two parameter driving fore Dj tot. The master urve was subsequently divided into two segments and approximated by two linear piees in the log log sale by using the linear regression method. 7. Analysis results and disussion The fatigue rak growth analysis was arried out for the lug subjeted to variable loading spetrum desribed in referene [7] and shown in Fig.. The loading spetrum was predominantly tensile with oasional high overloads and underloads. The data onerning theoretial analysis and the experimental fatigue rak growth through the lug ligament are shown in Fig. 3. The first set of data was obtained from the lug tested under the original loading spetrum. The seond loading spetrum, denoted as being 8% lipped, was obtained, aording to the nomenlature of referene [7], from the original loading spetrum with all high peaks redued (trunated) to the 8% of the highest peak in the original (% lipped) spetrum while all lower stress peaks being unhanged. The trunation of the loading spetrum from the top redues residual stresses produed by overloads but also eliminates yles with high stress intensity ranges and high maxima whih may signifiantly ontribute to the fatigue rak propagation. Thus, it was of great interest to find out whih effet was dominating. The retardation effet of multiple overloads an be quantified by omparing the fatigue lives orresponding to the trunated loading spetrum with that one obtained under the original loading spetrum. In this partiular ase the trunation resulted in shorter fatigue life and it was orretly predited by the UniGrow model. Good agreement between omputed and experimentally measured fatigue rak growth ( N data) indiate that the model orretly simulated the effet of overloads, under-loads and their sequene. Kim et al. [7] has additionally investigated the orner rak growth by applying marker yle load with frequeny of times per one flight spetrum. He has shown that the distanes between beah marks were wider for the 8% lipped spetrum than for the original loading spetrum. This led to shorter fatigue life in the ase of the lipped loading spetrum. The experimental beah marks obtained by Kim et al. [7] and the rak shape estimated by the UniGrow fatigue model are shown in Fig.. In both ases the alulated rak shape was very similar to that one observed experimentally. 8. Conlusions The analysis presented in the paper indiates that a variety of effets influening the fatigue rak growth and resulting from the appliation of yli variable amplitude loading an be modelled by onsidering the effet of residual stresses in the rak tip region indued by reversed yli plasti deformation. The analysis needs to be arried out on yle-by-yle basis aounting for eah yle of the applied load/stress history. It has been also shown that the use of appropriate memory rules and the two-parameter driving fore enables relatively aurate predition of fatigue lives of raked bodies subjeted to omplex variable amplitude servie loading spetra. The importane of the load shedding in the lug has been quantified by aounting for the derease of the resultant load in the raked ross-setion. It has been found that the exlusion of this effet in the fatigue rak growth analysis an ause high underestimation of the fatigue rak growth life. It has been also shown that the weight funtion tehnique an be used to obtain stress intensity fator solutions for orner, edge, and transient raks. Referenes [] Shijve J, Hoeymakers AH. Fatigue rak growth in lugs. Fatigue Eng Mat Strut 979;:8.

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