Finite Element Analysis of J-Integral for Surface Cracks in Round Bars under Combined Mode I Loading

Transcription

1 nternational Journal of ntegrated Engineering, Vol. 9 No. 2 (207) p. -8 Finite Element Analysis of J-ntegral for Surface Cracks in Round Bars under Combined Mode Loading A.E smail, A.K Ariffin 2, S. Abdulla 2, M.J Gazali 2 Department of Engineering Mecanics, Faculty of Mecanical and Manufacturing Engineering, Batu Paat, Joor, MALAYSA. 2 Department of Mecanical and Materials Engineering, Faculty of Engineering, Universiti Kebangsaan Malaysia, Bangi, Selangor, MALAYSA. Received 5 Marc 207; accepted 2 April 207, available online 2 April 207 Abstract: Tis paper numerically discusses te role of J-integral along te surface crack front in cylindrical bar under combined mode loading. t is also verified te analytical model derived from te first part of tis paper by comparing te results obtained numerically using ANSYS finite element program. t is found tat te proposed model capable to predict te J-integral successfully along te crack front but not for te area away from te deepest crack dept. Tis is probably due to te fact tat te problem of singularity. Keywords: FEA, J-integral, combined limit load, surface crack, stress intensity factors.. ntroduction n modern engineering, saft is generally used to transfer mecanical power from one component to anoter. During in-service task, te saft is exposed to te environmental arsness suc as corrosion and material defects suc as voids and pores. Tese defects will grow if no appropriate action is taken. According to Lin & Smit [], any arbitrary sapes of cracks take semielliptical sape during growing processes. Ten, linear elastic fracture mecanics approac is used to analyze te crack driving force for example stress intensity factor (SF) [2-4]. Oter solutions of oter types of crack can be found in [5-6]. However, if te plasticity is sufficient, te use of SF is not recommended [7-9]. Ten, J-integral is appropriately implemented [0-2]. Te solutions of SFs for a wide range of geometries ave been reported widely [, 3]. However, it is not for te case of J-integral [4-5]. Te solutions of J-integral is paramount important since mecanical components can be broke down due to excessive plastic deformation [7]. However, it is limited for te surface crack embedded in plates [6-9] and tubes [5, 20]. n tis present study, surface crack in round bar subjected to combined loading is analyzed and discussed. Firstly, te present model is validated wit te previous model using SFs approac since limited solutions of J- integral are available. After, J-integral is calculated along te crack front for various types of crack geometries. Considering te first part of tis paper, te analytical model is developed and te predicted values of J-integral are ten compared. Recently, an elastic-plastic analysis of surface crack become an important work especially wen te cracked components are subjected to combine loading [2, 3]. Corresponding autor: emran@utm.edu.my / al_emran@otmail.com 20 UTHM Publiser. All rigt reserved. penerbit.utm.edu.my/ojs/index.pp/ijie 2. Numerical Modelling Te geometry of te crack sown in Fig. can be described by te dimensionless a/d and a/b, te so-called relative crack dept and crack aspect ratios, were D, a and b are te diameter of te bar, te crack dept and te major diameter of te ellipse, respectively. Any arbitrary points on te crack front can also be normalized as x/, were is te crack widt, and x is te arbitrary distance of P from te symmetry axis. Te outer diameter of te cylinder is 50 mm and te total lengt is 200 mm. Due to te symmetrical analysis involved, a quarter finite element model is constructed, in wic te surface crack was situated at te center of te cylinder. A finite element model is developed wit special attention given to te crack tip by employing 20-node isoparametric quadratic brick elements. Te square-root singularities of stresses and strains are modelled by sifting te mid-point nodes to te quarter-point locations around te crack-tip region. Te detail of te finite element model is sown in Fig. 2 wit te associated singular finite elements around te crack tip. n order to remotely apply loadings to te structural component, a rigid element or multi-point constraint (MPC) elements was used to connect te nodes at a circumferential line at te end of te component, to an independent node. Fig. 3 sows a tecnique for constructing te independent node connected to te model using rigid beam elements. Te bending moment, M y is directly applied to tis node, wereas te axial force is directly applied in te direction-x on te cross-sectional area of te bar. At te oter end, te component is constrained appropriately. n order to obtain a suitable finite element model, it is necessary to compare te proposed model wit oter publised models [, 6, 7]. n tis work SFs results are used for te validation purposes. Since, it is ard to find te result of J-integral results for tese particular

2 A.E smail et al., nt. J. of ntegrated Engineering Vol. 9 No. 2 (207) p. -8 crack geometries. Fig. 4 sows a comparison of te dimensionless SFs under bending moment, F,b and axial force, F,a. Te findings of tis study are in good agreement wit tose of previous models. For modelling plastic beavior of te component, multilinear isotropic ardening (MSO) is used. MSO used von Mises criterion associated wit isotropic ardening wit a flow rule. Te material stress-strain followed te Ramberg-Osgood relation as te following expression: o o o n were o = E o is a 0.2% of proof stress, is a material constant and n is a strain ardening exponent. Two values of n are used, 5 and 0 represent te iger and lower strain ardening material models, respectively. All te model construction, linear and non-linear analyses are programmed into ANSYS APDL (Ansys Parametric Design Language). () Fig. 3 Remotely applied moments using an MPC84 element. Fig. Nomenclature of a semi-elliptical surface crack. Fig. 4 Validation of finite element model, bending and (c) tension loadings. 3. Results and Discussion SFs under bending and tension loadings involved only mode failure mecanisms. Terefore, a superposition metod can be explicitly used to combine SFs as te following expression [2]: Fig. 2 Quarter finite element model wit associated singular element at te crack tip., a, b K K K (2) 2

3 A.E smail et al., nt. J. of ntegrated Engineering Vol. 9 No. 2 (207) p. -8 Substituting te SFs stated in te first part of tis to yield te following expression:, a a, b b K a a (3) obtained in smail et al. [28], respectively. Results of combined SFs calculated using Eq. (8) are presented in Fig. 8 for different loading ratio, at te deepest crack dept, x/ = 0.0. Given tat: b a (4) were is te ratio between bending and tension stresses. Substituting Eq. (4) into Eq. (3) produces te following expression: a, a, b K a F (5) Rearrange Eq. (5) as te expression below: K, a, b a a F (6) Eq. (6) can be divided into two different expressions:, a, b, EQ F (7) F K, FE F, FE a a (8) were a is a tension stress. Eq. (7) is used explicitly to combine te SF from bending and tension loadings and it is called as F,EQ. Ten, Eq. (8) is used to determine combined SF directly from FEA and it is called F,FE. n ANSYS, it is ard to ave combined SFs directly because te SFs are given in terms of K, K and K. Terefore, an elastic J-integral was used by assuming tat a single value of J-integral under te combined loading represented an unified SFs consisting of K, K and K. Tis is because in ANSYS, if J-integral is used in te elastic or plastic regions, it calculates only a single value of J-integral even under combined loadings. Te elastic J- integral, J e. Rearrange it into te term of SF, K for plain strain condition yields te following expression: K FE E Je 2 Eq. (9) is used to convert te J-integral into combined SF, K FE, under combined loadings using FEA, and it was ten substituted into Eq. (8). 4. Results and Discussion Combination of F,b and F,a is conducted using Eq. (7) were it is formulated analytically using a superposition metod proposed by Newmann and Raju [3]. Te dimensionless SFs, F,b and F,a can also be (9) Fig. 5 Beaviour of F,FE against a/d, = 0.5 and =.0. Fig. 5 sows tat for te SFs dominated by te bending moment, all te SFs seem to converge at a/d = 0.. However, wen te tension stress plays an important role te dispersion of te curves increased as sown in Fig. 5. Tis is indicated tat is an important factor in determining te evolution of crack propagation processes. Te comparisons between te SFs combined explicitly and from FEA are sowed in Fig. 6. Bot results produce an excellent agreement to eac oter and te developed SFs metodology can be successfully used to combine SF for a similar type of failure mode. Fig. 7 sows a linear relationsip between Jp-FE and Jp-normal obtained from six points along te crack front under combined loadings. Relative crack dept, a/d = 0.2 is considered in tis work because te pattern of te curves are almost identical to eac oter for different a/d except different in magnitudes. For combined loadings dominated by tension force ( = 0.5) as sown in Fig. 8, function is lower tan if = 2.0 is used as compared wit te Fig. 8(c). t is also sowed tat te is almost flattened along te crack front until x/ < 0.6 3

4 A.E smail et al., nt. J. of ntegrated Engineering Vol. 9 No. 2 (207) p. -8 before as turned down wen it is reaced x/ 0.7. Te decrement of in tat region become significant if > 2.0 is used as revealed in Fig. 8. lower tan wen n = 5 is used. Tis is due to te fact tat n = 5 is a material assumed to beave lower strain beaviour. Up to tis date, no suc works available on tis similar analysis to compare wit. Terefore, no comparison is conducted to validate te present results. Fig. 6 Comparison of F, a/b = 0.2 and a/b = 0.6. Fig. 7 Relationsip between J p-fe and J p-normal for a/b = 0.6 and a/d = 0.2 subjected to combined loadings. Tis is related to te reduction of crack widt wit te increment of a/d. t meant tat te deeper te cracks wit sorter crack widt are capable to reduce te propagating rate of te crack. Wen n = 0 is used instead of 5, te curve pattern of is almost similar to eac oter as sown in Fig. 9. However, obtained using n = 0 is Fig. 8 Effect of against x/ for a/d = 0.2 and n = 5 wit varied a/b subjected to different loading ratio, = 0.5 and =.0. Te caracteristics of limit load, a-b under combined loadings are presented in Figs. 0 and for n = 5 and 0, respectively. n general, te limit load reduced as te a/d is increased. Tis is due to te fact tat wen a/d increased, te cross-sectional area of te bar is decreased. Consequently, it is affected te resistant capability of te bar terefore reduced te limit load. Normalised load, eqv/ 0 is also played an important role in determining te limit load were it is reduced asymptotically as te normalised load increased. Te curve patterns of te limit loads are typically observed for all crack geometries tat ave considered. Terefore for tis reason, te crack wit a/b = 0.6 is considered to be discussed in tis work. t is found tat te limit load distributions can be divided into two distinct regions, eqv/ 0 <.0 (low load level) and eqv/ 0 >.0 (ig load level). For te case eqv/ 0 <.0, te limit load distributions are relatively ig wic is 4

5 A.E smail et al., nt. J. of ntegrated Engineering Vol. 9 No. 2 (207) p. -8 indicated tat te elastic J-integral is not suitable to be used in calculating te limit load. Fig. 9 Effect of against x/ for a/d = 0.2 and n = 0 wit varied a/b subjected to different loading ratio, = 0.5 and =.0. Te effect of J e is still existed even it is omitted from te calculation. n order to eliminate te effect of J e, it sould be minimised as possible. Compared wit te region of eqv/ 0 >.0, te plastic J-integral as dominated around te crack tip. Tis condition produced insignificant limit load fluctuations. Tis is also indicated tat, plastic J-integral alone must be used in order to ave accurate limit load of any cracked structures. Wen a/d is increased causing te limit load reduction. Tis is true for te fact tat wen a/d increased, it will reduce te crack ligament area. Consequently, increasing te plastic J- integral along te crack fronts. Te effect of loading ratio, sown in Fig. 2 on te combined limit load is significant and found tat by increasing te loading ratio as dispersed te limit load distribution. Te beaviour of combined limit load can be described by observing te J/J e pattern along te crack front. Tis expression is derived as functions bar geometry, loading and material properties as follows: Fig. 0 Effect of eqv/ o on te a-b for a/b = 0.6 and n = 5 wen a/d are varied a/d = 0. and a/d = 0.2. were: n x a 2 x J o x a 2 2 x Je F R J = J e + J p, cos F = F,a + F,b. (0) n Eq. (0), parameter x/ is assumed to be varied and oters parameters are kept constant trougout te analysis. Terefore, J/J e is determined by 2 variety crack geometries under considerations. Te beaviour of 2 for against x/ for n = 5 and 0 are sown in Figs. 3 and 4, respectively using different loading ratios. Fig. 3 sows te 2 for a/d 5

6 A.E smail et al., nt. J. of ntegrated Engineering Vol. 9 No. 2 (207) p. -8 = 0. wit a/b are varied. t is found tat te flattened curves of 2 occurred in te region x/ < 0.4. Tis is indicated tat a single value of limit load capable to predict J-integral. However, te predictions are limited witin te specified region. Te effects of on te curves are minimal. By increasing a/d produced te region of constancy sorter compared wit lower value of a/d. Fig. 2 Effect of eqv/ o on te a-b for a/b = 0.6 and n = 5 using different loading ratios, = 0.5, =.0 dan = 2.0 for n = 5. Fig. Effect of eqv/ o on te a-b for a/b = 0.6 and n = 0 wen a/d are varied a/d = 0. and a/d = 0.2. Te distribution of 2 is observed to diverge significantly if = 2.0 is used sowing te tensile stress dominated te stress condition in te bar. Terefore, it is induced lower plasticity effect and consequently, it is reduced te capability of te combined limit load to predict J-integral efficiently as sown in Figs. 3. However, te influence of become significant wit te increment of a/d more tan 0.2 especially for = 0.5. Fig. 4 sows te beaviour of 2 wic is plotted against x/ using n = 0. t is found tat te magnitude of 2 is iger tan if n = 5 is used. However, it is obviously revealed tat te patterns of curves are almost te same as in te Fig. 3. t is also found tat te constancy of 2 can be observed clearly mainly for.0. n te same time, te constancy for a/d = 0.3 is limited witin te region of x/ < 0.3 compared wit te x/ < 0.6 for a/d 0.2. Tese caracteristics are paramount important in order to predict J-integral using te proposed limit load. n general, for te combined bending and tension loadings, different limit load must be used to predict te J-integral for different points on te crack front. Tis is due to te fact tat te constancy of te 2 difficult to occur and it is limited to te certain region of te x/ on te crack front. 5. Summary Linear and non-linear finite element analyses (FEA) ave been performed to investigate te fracture response of te surface cracks in round bars under combined tension and bending loadings. Two fracture parameters are used namely stress intensity factors (SF) and J- integral. Combined SFs from FEA are compared wit te explicitly combined SFs troug te use of a superposition metod. Te results sow an excellent is 6

7 A.E smail et al., nt. J. of ntegrated Engineering Vol. 9 No. 2 (207) p. -8 agreement to eac oter. For elastic-plastic analysis, J- integral is used as te fracture driving force and te solutions are calculated along te crack front for various crack geometries. Plastic influence function, under combined loadings are determined according to te EPR formulation using different loading ratio,. t is sowed tat is strongly related to te x/, a/b, a/d, n and. Since no available solutions of under combined loadings are available in te literature. Fig. 4 Beaviour 2 against x/ for, a/d = 0. and a/d = 0.2 for n = 0 using tree different loading ratios. Fig. 3 Beaviour 2 against x/ for, a/d = 0. and a/d = 0.2 for n = 5 using tree different loading ratios. Terefore, it is assumed tat te model ave produced acceptable results. Te limit load in tis work is based on te reference stress metod. Ten, te relation between J- integral and limit is establised to investigate te J- integral prediction along te crack fronts. t is found tat, te present limit load is not fully satisfied to predict te J- integral for all crack geometries considered in tis work. Different limit loads sould be used for different points along te crack front to predict J-integral. However, te prediction of J-integral can be performed for limited points on te crack fronts and it is strongly affected by a/d and. References [] Findley, K.O., Ko S.W., Saxena, A., J-integral expressions for semi-elliptical cracks in round bars. nternational Journal of Fatigue, Vol. 29, (2007), pp [2] smail, A.E, Arrifin, A.K., Abdulla, S., Gazali, M.J., Stress intensity factors for surface cracks in round bar under single and combined loadings. Meccanica, Vol. 47, (202), pp [3] smail, A.E., Mode stress intensity factors for slanted cracks in round bars. nternational Review of Mecanical Engineering, Vol. 8, (204a), pp [4] smail, A.E., Multiple crack interaction in bi-material plates under mode tension loading. Applied Mecanics and Materials, Vol. 629, (204b), pp [5] smail, A.E., Ariffin, A.K., Abdulla, S., Gazali, M.J., Off-set crack propagation analysis under mixed mode loadings. nternational Journal of Automotive Tecnology, Vol. 2, (20a), pp [6] smail, A.E., Ariffin, A.K., Abdulla, S., Gazali, M.J., J-integral evaluation of surface cracks in round 7

8 A.E smail et al., nt. J. of ntegrated Engineering Vol. 9 No. 2 (207) p. -8 bar under mode loadings. Researc Journal of Applied Science, Engineering and Tecnology, Vol. 7, (204a), pp [7] smail, A.E., Ariffin, A.K., Abdulla, S., Gazali, M.J., Ungkapan kamiran-j retak permukaan pada bar silinder padu kenaan beban ragam. Jurnal Teknologi, Vol. 68, (204b), pp [8] smail, A.E., Ariffin, A.K., Abdulla, S., Gazali, M.J., Daud, R., J-ntegral Analysis of Surface Cracks in Round Bars under Combined Loadings. Advanced Material Researc, Vol. 24, (20b), pp [9] Kim, Y.J., Sim, D.J., Coi, J.B., Kim, Y.J., Approximate J estimates for tension-loaded plates wit semi-elliptical surface cracks. Engineering Fracture Mecanics, Vol. 69, (2002a), pp [0] Lei, Y., J-integral and limit load analysis of semielliptical surface cracks in plates under combined tension and bending. nternational Journal of Pressure Vessel and Piping, Vol. 8, (2004a), pp [] Lei, Y., J-integral and limit load analysis of semielliptical surface cracks in plates under tension. nternational Journal of Pressure Vessel and Piping, Vol. 8, (2004b), pp [2] Lei, Y., J-integral and limit load analysis of semielliptical surface cracks in plates under bending. nternational Journal of Pressure Vessel and Piping, Vol. 8, (2004c), pp [3] Lei, Y., A review of limit load solutions for cylinders wit axial cracks and development of new solutions. nternational Journal of Pressure Vessel and Piping, Vol. 85, (2008), pp [4] Lei, Y., Budden, P.J., Limit load solutions for tinwalled cylinders wit circumferential cracks under combined internal pressure, axial tension and bending. Journal of Strain Analysis, Vol. 39, (2004), pp [5] Lin X.B, Smit. R.A., Sape growt simulation of surface cracks in tension fatigue round bars. nternational Journal of Fatigue, Vol. 9, (997), pp [6] Newman, Jr. J.C., Raju,.S., An empirical stressintensity factor equation for te surface crack. Engineering Fracture Mecanics, Vol. 5, (98), pp [7] Raju.S., Newman, J.C., Stress intensity factors for circumferential surface cracks in pipes and rods under tension and bending loads. Fracture Mecanics: ASTM Special Tecnical Publication Vol. 905, (986), pp [8] Rice, J.R., A Pat ndependent ntegral and te Approximate Analysis of Strain Concentration by Notces and Cracks. Journal of Applied Mecanics, Vol. 35, (968), pp [9] Sattari-Far,., Dillstrom, P., Local limit load solutions for surface cracks in plates and cylinders using finite element analysis. nternational Journal of Pressure Vessel and Piping, Vol. 8, (2004), pp [20] Si, C.F., Moran, B., Nakamura, T., Energy release rate along a tree-dimensional crack front in a termally stressed body. nternational Journal of Fracture, Vol. 30, (986), pp [2] Sin, C.S., Cai, C.Q., Experimental and finite element analyses on stress intensity factors of an elliptical surface crack in a circular saft under tension and bending. nternational Journal of Fracture, Vol. 29, (2004), pp