Strain energy density prediction of fatigue crack growth from hole of aging aircraft structures

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1 Theoretial and Applied Frature Mehanis 38 (2002) Strain energy density predition of fatigue rak growth from hole of aging airraft strutures J.Z. Zuo a, *, Al.Th. Kermanidis b, Sp.G. Pantelakis b a Department of Engineering Mehanis, Xi an Jiaotong University, Xi an, , China b Laboratory of Tehnology and Strength of Materials, Department of Mehanial Engineering and Aeronautis, University of Patras, Panepistimoupolis Rio, Patras, Greee Abstrat Fatigue rak growth rate depends not only on the load amplitude, but also on the morphology of rak path. The strain energy density theory has the ability to analyze rak growth rate. A strain energy density rak growth model is proposed. It an predit the lifetime of fatigue rak growth for mixed mode raks while an equation for mode I rak is also obtained. The validity of the model is established with two ases: a enter-rak panel and raks emanating from the edge of a hole. The stress intensity fator expression for the former ase is analytial while that of the latter is alulated numerially using finite elements. The results are ompared with the testing data. Good agreement shows that the proposed model is useful. Ó 2002 Elsevier Siene Ltd. All rights reserved. 1. Introdution Failure by fatigue ontinues to prevail despite the exhaustive amount of past researh. The problem is partiularly serious for airrafts [1,2]. The fatigue lifetime of strutures is generally divided into rak initiation and rak growth. The rak growth lifetime, suh as aging airraft has reeived ontinual attention [3 7]. Fatigue analyses depend largely on empirial data [5], the obtainment of whih an be time onsuming and ostly. Moreover, no information * Corresponding author. Present address: Department of Mehanial Engineering, University of South Carolina, 300 Main Street, Columbia, SC 29208, USA. address: zuo@engr.s.edu (J.Z. Zuo). ould be gained on the umulative damage harater of the fatigue proess. Even more unertain is the estimate of the remaining life of used airrafts that may have exeeded [8] their design life expetation. Through-thikness fatigue raks are known to prevail near rivet holes [9] and lap joints [10]. It is therefore useful to develop methods for prediting fatigue rak growth in aluminum omponents for aging airrafts [11,12]. The appliation of frature mehanis to fatigue rak growth was first made in [13] using a twoparameter model: da dn ¼ CðDKÞn ð1þ where da=dn is fatigue rak growth rate, DK the stress intensity fator range while C and n are experimentally determined parameters. Eq. (1) /02/$ - see front matter Ó 2002 Elsevier Siene Ltd. All rights reserved. PII: S (02)

2 38 J.Z. Zuo et al. / Theoretial and Applied Frature Mehanis 38 (2002) applies to the intermediate range of stress intensity fators; it, however, does not take mean stress effets into aount. Up to this date, there are a variety of empirial relations similar to Eq. (1) that has been proposed. They an be easily found in the open literature. Most of them are limited to mode I rak growth where the diretion of rak growth is known. When the load is applied symmetrially with referene to the rak plane, the rak path may hange diretion during fatigue. More than one of stress intensity fator would be involved. The interation of the opening and sliding mode of stress intensity fators was proposed in [14] by using the strain energy density fator suh that da=dn in Eq. (1) would depend on the range of the stress energy density fator DS. The parameter C and n would then have to be reevaluated when DK in Eq. (1) is replaed by DS. To inlude damage aumulated, it was proposed in [15 17] to use the full expression dw =dv rather than just the oeffiient of 1=r, namely S, whih represents the singular term in a series expansion for dw =dv in terms r, the distane from the rak tip. This approah was suessfully applied to raked and unraked speimens. It was able to predit raks initiating from the surfae in fatigue [16] in ontrast to stati loading where raks initiate in the speimen (round bar) enter by using the same riterion. This provides onfidene in the strain energy density dw =dv approah. More referenes an be found in [18]. Appliation of this model to the fatigue failure of airraft strutural omponents has also been made [19]. The use of dw =dv aounts for damage that depends on distane from the rak tip without invoking the limit r! 0 whih is inherent in the use of the stress intensity fator. This was pointed out in [20] and more reently in onnetion with the raking of piezoeramis [21 23]. In what follows, a reoniliation of the fatigue rak growth model in [3] with that in the strain energy density model [15 17] is made where the growth inrement of the plasti zone size an be related to the rak growth inrement Da sine for every inrement rak growth there would prevail a orresponding growth inrement of the plasti zone. Fatigue rak growth retardation effet for 2024-T3 and 6061-T6 aluminum speimens an be found in [3]. To be emphasized is that fatigue umulative damage effets are inluded in the omputation for DS [15] or dw =dv [16,17]. In fat, the ritial value of dw =dv after many yles of fatigue loading for a bar has also been estimated analytially by alulating the hysteresis loops for 6061-T6 aluminum by appliation of the isoenergy density theory whih is a more advaned form of the strain energy density [24] requiring no empirial data even in fatigue. Eletron mirosopy would show that the distane between two adjaent striations on the fatigue frature surfae is smaller near the origin of raking in omparison with that further away. This implies that the inrement of rak growth would deay with the distane from the rak tip as in the ase of the strain energy density funtion elaborated by the riterion in [20]. Moreover, the strain energy density hysteresis loop [16 20] represents the energy dissipated in one full yle. They an be sum to yield the aumulation. The strain energy density theory seems to fit naturally into the physis of fatigue rak growth damage. It will be used to analyze two rak onfigurations. They onsist of a enter-raked panel and ollinear raks emanating from a hole. Finite element results are made available for the latter. The predited results agree well with the test data obtained in [9,25]. 2. Earlier fatigue rak growth model Proposed in [3] is a fatigue rak growth model where the rak growth inrement is assumed to orrespond with that of the plasti zone growth inrement x s. For low yle fatigue, damage in an element is assumed to be aused by plasti deformation and disloation aumulation. Failure ours after a ertain number of yles DN determined by a ritial value. The proposed relation is da dn ¼ x s ð2þ DN Two problems arise with referene to the alulation of x s. First, how ould the diretion of rak growth be determined? Next, how ould the inrement x s in the diretion of rak growth be

3 J.Z. Zuo et al. / Theoretial and Applied Frature Mehanis 38 (2002) Fig. 1. Illustration of fatigue rak propagation. estimated? It is known that the size of the plasti zone depends on the diretion of rak growth. The inrement x s in [3] is assumed to be equal to the plasti zone size. This requires the rak growth diretion to be known a priori. It has been shown in [18] and the referenes therein that the fatigue rak an be assumed to grow in the diretion of minimum strain energy density. Moreover, initiation of the fatigue rak is assumed to oinide with the strain energy density funtion reahing its ritial yli value. Hene, the inrement x s would be dependent not only on the material properties suh as the ritial strain energy density funtion but also on the amplitude of fatigue loading. For yli loading, a relation between DN and onstant plasti strain amplitude De p has been proposed [26,27]: DeðDNÞ m ¼ C ð3þ where m and C are empirial onstants determined in [23,27]. Sine the strain energy density fator was proposed in the early 1970s [18], the theory has been employed suessfully to analyze numerous problems in different field [14 20]. In general, the strain energy density funtion flutuates near the loations where abrupt hanges of material properties and/or geometry take plae. The peaks and valleys of the flutuation an be identified with failure by material yielding and frature. The onept applies to the predition of fatigue rak growth without any limitation imposed on the physial behavior of the material. Based on the strain energy density, the sequene of fatigue rak growth in strutures an be illustrated in Fig. 1. During fatigue, the inrement of rak growth an be assumed as dw ¼ S 1 ¼ S 2 ¼¼ S j ¼¼ S ð4þ dv r 1 r 2 r j r where S j is the strain energy density fator and S its ritial value. When S j reahes its ritial value S, global unstable frature will our. r j is the inrement of rak propagation at jth step. It implies in Eq. (4) that the inrement is proportional to the loal strain energy density fator. ðdw =dv Þ is the ritial strain energy density funtion. The rak is assumed to grow an inrement of r j when the ðdw =dv Þ min reahes its ritial value ðdw = dv Þ where ðdw =dv Þ min is the minimum of strain energy density whose position an be used to determine the diretion of fatigue rak propagation. The ritial value ðdw =dv Þ an be alulated from the area under the true stress and true strain urve at frature as shown in Fig. 2. From Eq. (4), the inrement r j an be expressed as 3. Strain energy density fatigue rak growth model Fig. 2. Shematis of the definition of ritial value ðdw =dv Þ.

4 40 J.Z. Zuo et al. / Theoretial and Applied Frature Mehanis 38 (2002) dw r j ¼ S j dv 1 ð5þ The relationship between strain energy density fator S and stress intensity fator an be found in [18]. For example, for a mode I rak, Eq. (5) an also be expressed in the form of stress intensity fator K I r j ¼ K 2 I 2pE dw dv 1 ð6þ where E is the Young s modulus and m the Poisson s ratio of material. The damage aused in one yle is not onstant. The rak is assumed to propagate when the total damage aumulated in DN yles beomes ritial. Therefore, it is not neessary to aurately alulate the damage aused in eah yle if the number of yles DN an be evaluated. Based on this view, the rak is assumed to propagate when a length of r j is reahed after DN j yles under the amplitude of stain energy density fator S j. Hene, Eq. (2) an be rewritten as da dn ¼ r j ð7þ DN j For unraked speimens under onstant strain amplitude yli loading, Eq. (3) an be rewritten as DeðDNÞ m ¼ 2e 0 f ð8þ where m and e 0 f are material onstants whih an be determined experimentally. For the problem of fatigue rak growth, the strain gradient harateristis at the tip of a rak an be aounted for. The effetive strain range near the rak tip an thus be written as Z rj De ¼ A De r b eff dr ð9þ j 0 where A and b are material onstants suh that A > 0 and b > 0. They an be determined experimentally with De eff being the effetive strain. For a mode I rak, De eff has the form De eff ¼ 1 2m E DK I pffiffiffiffiffiffiffi 2pr ð10þ Substituting Eqs. (6) and (10) into Eq. (9), it gives 2ð1 2mÞ De ¼ p E ffiffiffiffiffi 2p A DK 2ðb 1Þ I 2pE dw dv! 1=2 b ð11þ Substituting Eq. (11) into Eq. (8), an expression is obtained for the number of yles DN: 2 DN ¼ 4 e 0 f E p ffiffiffiffiffi 2p Að1 2mÞDK 2ð1 bþ I 2pE dw dv! b 1= =m ð12þ Substituting Eqs. (6) and (12) into Eq. (7), the rate of fatigue rak growth an be expressed as 2 da ¼ 4 Að1 2mÞ dn 2pE dw dv e 0 f E p ffiffiffiffiffi 2p! 1=2 b 31=m 5 2pE dw dv DK 2þ2ð1 bþ=m I ð13þ Integrating Eq. (13), the lifetime N f of fatigue rak growth from its initial length a 0 to the ritial rak length a r is given as 2 dw 2pE dv N f ¼ 4 Að1 2mÞ e 0 f E p ffiffiffiffiffi 2p! 1=2 b 3 1=m 5 2pE dw dv Z ar DK 2ðb 1Þ=m 2 I da ð14þ a 0 Denoting 2 g ¼ 4 Að1 2mÞ 2pE dw dv e 0 f E p ffiffiffiffiffi 2p! 1=2 b 3 5 2pE dw dv 1=m then Eqs. (13) and (14) an be expressed as ð15þ

5 J.Z. Zuo et al. / Theoretial and Applied Frature Mehanis 38 (2002) da dn ¼ gdk2þ2ð1 bþ=m I N f ¼ 1 g Z ar a 0 ð16þ DK 2ðb 1Þ=m 2 I da ð17þ Eqs. (16) and (17) are dedued under the ondition of a mode I rak situation. For mixed mode rak problems, similar expressions an be derived without diffiulty. Of ourse, Eqs. (16) and (17) an also be expressed in an universal form for mixed mode raks by use of the strain energy density fator. For example for mode I raks under the ondition of R r ¼ 0, Eqs. (16) and (17) may be rewritten as da dn ¼ g0 DS 1þð1 bþ=m ð18þ and Case 3: when b ¼ 1, it an be said that the number of yles DN in one step of fatigue growth is independent of the strain energy density. 4. Center-rak panel The proposed model is used to predit fatigue rak growth in a enter-rak panel. The speimen is 152 mm wide, 2.54 mm thik, as shown in Fig. 3. The material of the panel is 2024-T3 aluminum alloy whih is widely used as the fuselage skin sheet. The material properties of aluminum 2024-T3 are listed in Table 1. The tests run under onstant load amplitude ondition with the maximal stress r max ¼ 60:45 MPa, the stress ratio R r ¼ 0:2. The initial half-rak length was a 0 ¼ 9:0 mm. The testing data orrespond to those in [25,28]. N f ¼ 1 Z ar DS ðb 1Þ=m 1 da ð19þ g 0 a 0 where 1þð1 bþ=m g 0 2pE ¼ g For problems enountered in engineering pratie, the stress intensity fators or the strain energy density fators are diffiult to be expressed in analytial form. Numerial methods would have to be used. It an also be found from Eq. (18) that the fatigue rak growth rate will inrease with inreasing of the strain energy density fator. On the other hand, the number of yles DN in one step of fatigue rak growth is dependent not only on the strain energy density, but also on the material properties via the parameter b. This harateristi of fatigue rak growth was also investigated numerially in [15]. The following possible situations should be onsidered: Case 1: when 0 < b < 1, the larger the strain energy density, the smaller the number of yles DN in one step of fatigue growth. Case 2: when b > 1, it an be said that the larger the strain energy density, the bigger the number of yles DN in one step of fatigue growth. Fig. 3. Central rak in a finite width plate.

6 42 J.Z. Zuo et al. / Theoretial and Applied Frature Mehanis 38 (2002) Table 1 Material properties of aluminum 2024-T3 p E (GPa) m K (MPa ffiffiffiffi m ) S (kn/m) m e 0 f For a enter-rak panel, the stress intensity fator is known: p K I ¼ r ffiffiffiffiffi pa se pa 1=2 ð20þ W Substituting Eq. (20) into Eqs. (16) and (17), the fatigue rak growth rate da=dn and total yles N f for a enter rak growing from a 0 to a an be expressed as da dn ¼ g Dr p ffiffiffiffiffi pa se pa 1=2 2þ2ð1 bþ=m ð21þ W and N f ¼ 1 Z a p Dr ffiffiffiffiffi pa se pa 1=2 2ðb 1Þ=m 2 da ð22þ g a 0 W The ritial value ðdw =dv Þ for aluminum 2024-T3 an be alulated by use of the true stress and strain urve. The parameter A and b an be evaluated from the testing data for fatigue rak propagation. For example, for aluminum 2024-T3, ðdw =dv Þ ¼ 56:64 MPa, A ¼ 14:29 and b ¼ 0:685. The predited result by use of the present model is shown in Fig. 4. They are ompared with the testing data in [25]. Figs. 5 7 show the urves of the stress intensity fator, strain energy density fator and fatigue rak growth rate versus half enter rak length, respetively. Figs. 8 and 9 give the predited result of rak growth rate da=dn versus strain energy density fator, stress intensity fator, and ompared with the testing data [25]. 5. Through-thikness raks from a hole Fatigue rak growth in aluminum 2024-T3 is analyzed for the onfiguration of irular hole Stress intensity fator distribution The geometry of the speimen used in testing is shown in Fig. 10. It is 80 mm wide, 218 mm long and 1.6 mm thik. The hole in the plate is 5 mm in diameter. The rak initial length is that a 1 ¼ 0, Fig. 4. Half rak length versus the number of yles.

7 J.Z. Zuo et al. / Theoretial and Applied Frature Mehanis 38 (2002) Fig. 5. Stress intensity fator versus half rak length. Fig. 6. Strain energy density fator versus half rak length. and a 2 ¼ 0:8 mm. The material of the speimen is 2024-T3 aluminum alloy. Fore-loading ontrolled method was used in the fatigue test where r max ¼ 100 MPa, R r ¼ 0:1 and f ¼ 5 Hz.

8 44 J.Z. Zuo et al. / Theoretial and Applied Frature Mehanis 38 (2002) Fig. 7. Crak growth rate versus half rak length. Fig. 8. Crak growth rate versus strain energy density fator. For an infinite plate, the stress intensity fator solution for radial through-thikness raks emanating from open holes an be found in [29]. In this work, FEM was used to determine the stress in-

9 J.Z. Zuo et al. / Theoretial and Applied Frature Mehanis 38 (2002) Fig. 9. Crak growth rate versus stress intensity fator. Fig. 10. Shematis of raks emanating at the edge of a hole. tensity fator for different rak length a 1 and a 2. K I at the rak tip a 2 is shown in Fig. 11. The urves of K I versus a 2 for different rak length a 1 are shown in Fig. 12. The effet of rak length a 1 on the stress intensity fator K I at the rak tip a 2 is shown in Fig. 13. The stress intensity fator K I at the rak tip a 2 is expressed as p K I ¼ Y r ffiffiffiffiffiffiffi pa 2 ð23þ The oeffiient Y is shown in Figs. 14 and 15 for different rak length a 1 and a 2. The omparison with the solution in [29] for an infinite plate is made and shown in Fig. 16. The values of oeffiient Y agree only when the ratio a 2 =D is muh smaller than one. Here D is the diameter of the hole Predition of fatigue rak growth FEM solution of stress intensity fator for a finite plate is used to predit the fatigue rak growth numerially. The material properties for

10 46 J.Z. Zuo et al. / Theoretial and Applied Frature Mehanis 38 (2002) The predited results for rak a 2 are shown in Fig. 17. They are ompared with the test data [9]. The stress intensity fator, strain energy density fator and rak growth rate versus the rak length are shown in Figs , respetively. When the test reahes about yles, the rak a 1 was nuleated at the edge of the hole. The effet of rak length a 1 was taken into aount in the alulation from then on. 6. Conlusions Fig. 11. SIF 3-D distribution for rak a 2 under different length of a 1 and a 2. aluminum alloy 2024-T3 have given earlier. The initial rak lengths are a 1 ¼ 0, and a 2 ¼ 0:8 mm. Fatigue rak growth rate depends on not only the load amplitude, but also the diretion of rak propagation. The diretion of rak growth is affeted by many fators suh as loading type, struture geometry, and material properties. Based on the strain energy density theory, a fatigue rak growth model is developed to predit the lifetime of fatigue rak growth for single or mixed mode raks. The equations for mode I rak are established. The validity of the proposed model is established from two examples. The first is for a Fig. 12. K I at rak tip a 2 versus rak length a 2 under different a 1.

11 J.Z. Zuo et al. / Theoretial and Applied Frature Mehanis 38 (2002) Fig. 13. K I at rak tip a 2 versus rak length a 1 under different a 2. Fig. 14. Normalized SIF of rak a 2 versus rak length a 2 under different a 1.

12 48 J.Z. Zuo et al. / Theoretial and Applied Frature Mehanis 38 (2002) Fig. 15. Normalized SIF of rak a 2 versus rak length a 1 under different a 2. Fig. 16. Normalized SIF of rak a 2 versus a 2 =D.

13 J.Z. Zuo et al. / Theoretial and Applied Frature Mehanis 38 (2002) Fig. 17. Crak growth omparison between experiment result and present model. Fig. 18. Stress intensity fator versus rak length a 2.

14 50 J.Z. Zuo et al. / Theoretial and Applied Frature Mehanis 38 (2002) Fig. 19. Strain energy density fator versus rak length a 2. Fig. 20. Crak growth rate da=dn versus rak length a 2.

15 J.Z. Zuo et al. / Theoretial and Applied Frature Mehanis 38 (2002) enter-rak panel. The other is for raks emanating from the edge of a hole. The predited results for the two problems agree with the test data for fatigue lifetime preditions. Aknowledgements The authors wish to take this opportunity to thank Prof. George C. Sih for his helpful omments related to this work. The first author is grateful to Prof. Th.B. Kermanidis for the private onversation. Referenes [1] M. Jani, An assessment of risk and safety in ivil aviation, J. Air Transport Manage. 6 (2000) [2] D. Owen, Air Aident Investigation: How Siene is Making Flying Safer, Patrik Stephens Limited, Yeovil, [3] Sp.G. Pantelakis, Th.B. Kermanidis, D.G. Pavlou, Fatigue rak growth retardation assessment of of 2024-T3 and 6061-T6 aluminum speimens, J. Theor. Appl. Frat. Meh. 22 (1995) [4] Th.B. Kermanidis, Sp.G. Pantelakis, D.G. Pavlou, Inrease in hardening of 2024-T42 aluminum with fatigue stress amplitude, J. Theor. Appl. Frat. Meh. 14 (1990) [5] A.F. Blom, Fatigue and damage tolerane verifiation of airraft strutures, in: Proeedings of the 4th International Conferene on Fatigue and Fatigue Thresholds, Honolulu, Hawaii, vol. I, 1990, pp [6] A.F. Blom, Fatigue related strutural integrity issues, in: Proeedings of the 6th International Fatigue Congress, Berlin, Germany, vol. I, 1996, pp [7] J.P. Gallagher, Estimating fatigue rak lives for airraft tehniques, Exp. Meh. 16 (11) (1976) [8] R. Jones, L. Molent, S. Pitt, Study of multi-site damage of fuselage lap joints, J. Theor. Appl. Frat. Meh. 32 (1999) [9] Assessment of the Multi-Side Damage Aumulation of Airraft-Components Subjeted to Fatigue, Final Tehnial Report, Laboratory of Tehnology and Strength of Materials, University of Patras, Greee, July, [10] L.F.M. Silva, J.P.M. Gonalves, F.M.F. Oliveira, P.M.S.T. de Castro, Multiple-site damage in riveted lapjoints experimental simulation and finite element predition, Int. J. Fatigue 22 (2000) [11] V. Laara, D.J. Smith, M.J. Pavier, M. Priest, Fatigue rak growth from plain and old expanded holes in aluminum alloys, Int. J. Fatigue 22 (2000) [12] P.J. Golden, A.F. Grandt, G.H. Bray, A omparison of fatigue rak formation at holes in 2024-T3 and 2524-T3 aluminum alloy speimens, Int. J. Fatigue 21 (1999) S211 S219. [13] P.C. Paris, F. Erdogan, A ritial analysis of rak propagation laws, ASME J. Basi Eng. 85 (1963) [14] G.C. Sih, B.M. Barthelemy, Mixed mode fatigue rak growth preditions, Eng. Frat. Meh. 3 (1980) [15] G.C. Sih, E.T. Moyer, Path dependent nature of fatigue rak growth, Engng. Frat. Meh. 17 (3) (1983) [16] G.C. Sih, C.K. Chao, Fatigue initiation in unnothed speimens subjeted to monotoni and yli loading, J. Theor. Appl. Frat. Meh. 1 (2) (1984) [17] G.C. Sih, C.K. Chao, Size effet of ylindrial speimens with fatigue raks, J. Theor. Appl. Frat. Meh. 3 (1) (1984) [18] G.C. Sih, Mehanis of Frature Initiation and Propagation, Kluwer Aademi Publishers, Boston, [19] G.C. Sih, C.K. Chao, Fatigue initiation analysis of wing/ fuselage bolt assembly, J. Theor. Appl. Frat. Meh. 11 (1989) [20] J.Z. Zuo, Z.B. Kuang, Z.W. Lou, Fatigue damage and lifetime preditions of welded strutures under alternative loadings, in: The Chinese 8th Symposium on Strutural Strength and Vibration of Aeroengines, 1996, pp [21] J.Z. Zuo, Z.W. Lou, Z.B. Kuang, Fatigue damage and lifetime predition of aeronauti welded strutures under high temperature, Chinese J. Aeronautis (English Edition) 10 (3) (1997). [22] G.C. Sih, J.Z. Zuo, Multisale behavior of rak initiation and growth in piezoeletri eramis, J. Theor. Appl. Frat. Meh. 34 (2) (2000) [23] G.C. Sih, Z.F. Song, Damage analysis of tetragonal perovskite struture eramis impliated by asymptoti field solutions and boundary onditions, J. Theor. Appl. Frat. Meh. 38 (2002) [24] D.Y. Jeong, Non-equilibrium thermal/mehanial interation of uniaxial, speimens under yli loading, Ph.D. Thesis, Department of Mehanial Engineering and Mehanis, Lehigh University, [25] D.A. Virkler, B.M. Hillberry and P.K. Goel, The statistial nature of fatigue rak propagation, AFFDL-TR-78-43, Air Fore Flight Dynamis Laboratory, April, [26] S.S. Manson, Fatigue: A omplex subjet-some simple approximation, Exp. Meh. 5 (1965) [27] L.F. Coffin, J.F. Tavernelli, Experimental support for generalized equation predited low yle fatigue, Trans. ASME J. Basi Eng. 84 (1962) [28] D.F. Ostergaard, J.R. Thomas, B.M. Hillberry, Effet of Da-inrement on alulating da=dn from a versus N data, in: S.J. Huda, R.J. Bui (Eds.), Fatigue Crak Growth Measurement and Data Analysis ASTM STP738, 1981, pp [29] O.L. Bowie, Analysis of an infinite plate ontaining radial raks originating at the boundary of an internal irular hole, J. Math. Phys. 25 (1956)