Modelling of Fatigue life of 6082 T6 Al-alloy based on Genetic Programming

Transcription

1 Modellng of Fatgue lfe of 6082 T6 Al-alloy based on Genetc Programmng J. R. Mohanty Department of Mechancal Engneerng, Veer Surendra Sa Unversty of Technology, Burla, Sambalpur (Odsha), In ABSTRACT: The current work presents a smple model to estmate the constant ampltude fatgue crack growth lfe of 6082 T6 Al-alloy under the nfluence of load ratos. The model produced by GEP s constructed from the set of expermental results conducted n the laboratory. The model result has been compared wth the expermental fndngs. It s found that the GEP model slghtly underestmates the fatgue lfe wth percentage devaton of from expermental results. KEYWORDS: Genetc programmng; fatgue crack growth lfe; correlaton coeffcent (R) and mean squared error (MSE) I. INTRODUCTION All most all engneerng structures and components contan cracks or crack lke flaws. Durng the servce perod, these cracks grow under cyclc loadng whch eventually leads to catastrophc falure causng loss of human lfe. Therefore, fatgue crack growth study must be consdered n desgn and n the analyss of falure. Fatgue crack growth rate (/dn) not only depends on stress ntensty factor range ( K), but also strongly dependent on load rato (R) whch s the rato of mnmum load to maxmum load. It s well known that an ncrease n load rato results n an ncrease n fatgue crack growth rate at a gven cyclc stress ntensty factor range. Earler several nvestgators [1-5] have analyzed the nfluence of load rato on fatgue crack growth rate. However, predcton of constant ampltude fatgue crack growth lfe under the effect of load rato has been rarely studed. Further, to determne fatgue lfe one has to ntegrate growth rate equaton (n the form of dfferental equaton), whch s qute complex as t nvolves several parameters. Recently, softcomputng technques have been used n those complex stuatons. In the present study, genetc programmng approach has been used to predct the constant ampltude fatgue lfe of 6082 T6 Al-alloy under the nfluence of load rato usng expermental ta. It has been observed that the proposed model predcts the fatgue lfe wth reasonable accuracy. Paper s organzed as follows. Secton II descrbes automatc text detecton usng morphologcal operatons, connected component analyss and set of selecton or reecton crtera. The flow gram represents the step of the algorthm. After detecton of text, how text regon s flled usng an Inpantng technque that s gven n Secton III. Secton IV presents expermental results showng results of mages tested. Fnally, Secton V presents concluson. II. MATERIALS AND METHODS The materal used n the present nvestgaton s 6082 T6 Al alloy whose chemcal composton and the mechancal propertes are gven n Tables 1 and 2 respectvely. Sngle-edge notched tenson (SENT) specmens have been machned n the longtudnal transverse drecton from a 6.5 mm plate whose geometry s shown n Fg 1. The notch preparaton has been made by electrcal-dscharge machnng. Both the sdes of the specmen have been mrror polshed n order to facltate the observaton of crack growth. Copyrght to IJIRSET DOI: /IJIRSET

2 Table 1 Chemcal compostons of 6082-T6 Al-alloy (wt%) Fe Cu Mn Mg Cr T Zn S Other Tensle strength MPa Table 2 Mechancal propertes of 6082-T6 Al-alloy Yeld strength MPa Young s modulus GPa Brnell Hardness (HB) Elongaton ε r (%) Fg. 1 Sngle Edge Notch Tenson (SENT) Specmen geometry The crack growth tests have been performed n ar at room temperature on a servo-hydraulc test machne havng a load capacty of 100 kn wth a frequency of 5 Hz. Pre-crackng has been ntroduced under mode-i loadng wth a snusol waveform to an a/w (.e. crack length to wdth rato) rato of 0.3 and then subected to constant load test mantanng dfferent load ratos (R) of 0.1, 0.2, 0.3, 0.4, 0.5, and 0.6 respectvely. The crack growth has been montored wth the help of a COD gauge mounted on the face of the machned notch. The followng equatons have been used to determne stress ntensty factor K [6]. F a K f ( g). wb (1) where, f ( g) ( a / w) 10.55( a / w) 21.72( a / w) 30.39( a / w) (2) Copyrght to IJIRSET DOI: /IJIRSET

3 Usually, the crack length and number of cycles (a ~ N) ta obtaned from the fatgue tests contan much scatter. Hence, several technques [7-10] have been proposed to determne the fatgue crack growth rate (/dn) from raw laboratory ta. In the present work, author s earler proposed the exponental equaton method [11] has been appled to determne the fatgue crack growth rate at dfferent load ratos. The raw a ~ N ta have been ftted by usng followng exponental equaton. m ( N N ) a a e (3) where, a and a = crack length n th step and th step n mm respectvely, N and N = No. of cycles n th step and th step respectvely, m = specfc growth rate n the nterval -, = No. of expermental steps, and = +1 The specfc growth rate m, whch s an mportant parameter of the exponental equaton, has been calculated by takng the logarthm of equaton (3) as follows: a ln a m (4) N N The specfc growth rate calculated from the above equaton stll contans scatter whch has been subsequently refned by curve fttng wth calculated a values (.e. crack lengths from ntal to fnal wth an ncrement of 0.005mm). The smoothened values of the number of cycles have been calculated n the excel sheet from the refned m values as per the followng equaton. a ln a N N (5) m The crack growth rates (/dn) have been calculated drectly from the above calculated values of N as follows: d a a a dn N N (6) Fgs. 2 and 3 depct the supermposed a ~ N and log (/dn) log ( K) curves under dfferent load ratos. Crack length (a),mm R=0.1 R=0.2 R=0.3 R=0.4 R=0.5 R= E E E E E+04 No. of cycle (N) Fg. 2 Smoothened values of a N curves for dfferent load ratos Copyrght to IJIRSET DOI: /IJIRSET

4 log (ΔK), MPa. m log (/dn), mm/cycle R=0.1 R=0.2 R=0.3 R=0.4 R=0.5 R=0.6 Fg. 3 Smoothened values of log (/dn) log ( K) curves for dfferent load ratos III. GENETIC PROGRAMMING APPROACH Genetc programmng (GP), proposed by Koza [12] s a generalzaton of genetc algorthms (GAs) [13]. GP creates computer programs to solve the specfc problem by executng the followng steps: An ntal populaton (generaton 0) of random compostons of the functons and termnals of the problem s generated. The followng sub-steps are performed teratvely tll the termnaton crteron s satsfed: () Each program n the populaton s executed by assgnng some ftness measures lke measure lke Mean Square Error, Mean Relatve Error and so on, that can measure the capablty of the model to solve the problem wth respect to the expermental ta. () A new populaton of computer programs s created by applyng the followng operatons descrbed below. Reproducton: It copes an exstng program to the new populaton. Crossover: It creates new offsprng program(s) for the new populaton by recombnng randomly chosen parts of two exstng programs. Mutaton: It produces one new offsprng program for the new populaton by mutatng a randomly chosen part one exstng program. The program that s dentfed by the method of result desgnaton (e.g., the best-so-far ndvdual) s desgnated as the result of GP system for the run. Ths result may be a soluton (or approxmate soluton) to the problem [14, 15]. IV. GENE EXPRESSION PROGRAMMING APPROACH Gene expresson programmng (GEP) s a new algorthm proposed by Ferrera [16] whch s based on genetc algorthm (GA) and genetc programmng (GP). It develops a computer program encoded n lnear chromosomes of fxed length whch performs the symbolc regresson usng the most of the genetc operators of GA and funmentally ams to fnd a mathematcal functon prncpal usng a set of ta presented [17, 18]. Once the problem s encoded and the ftness functon s specfed, an ntal populaton of vable ndvduals (chromosomes) s randomly created. Then, the algorthm converts each chromosome nto an expresson tree (ET) correspondng to a mathematcal expresson usng functon set, termnal set, ftness functon, control parameters and stop condton. Copyrght to IJIRSET DOI: /IJIRSET

5 Start Intal populaton creaton Chromosome expresson as ET ET executon Ftness evaluaton Termnate? Yes Stop No Chromosome selecton Reproducton New generaton creaton Fg. 4 Flowchart of gene expresson programmng [19] After that the predcted target s compared wth the actual one based on the ftness score for each ndvdual. If t satsfes the requred ftness measure, then the algorthm stops. If the condton does not fulfl, then some of the chromosomes are selected usng roulette wheel samplng and mutated to obtan the new generatons. Ths process s contnued untl the desred ftness measure s acheved and then the chromosomes are decoded for the best soluton of the problem [19, 20] as shown n Fg. 4. IV. APPLICATION OF GEP FOR CRACK GROWTH RATE DETERMINATION In the present study, three man parameters.e. stress ntensty factor range (ΔK), maxmum stress ntensty factor (K max ) and load rato (R) that affect the fatgue crack growth rate of 6082 T6 Al-alloy have been selected for nput varables, whle crack growth rate (/dn) has been selected as one output varable. The expermental ta base contans sx sets of ta for dfferent load ratos such as R = 0.1, 0.2, 0.3, 0.4, 0.5, and 0.6. Each set conssts of crack growth rate (/dn) ta along wth ther correspondng crack drvng parameters.e. stress ntensty factor range (ΔK) and maxmum stress ntensty factor (K max ), responsble for crack growth whch ncludes 1200 (300 4) ta ponts. Out of those, fve sets.e. R = 0.1, 0.2, 0.3, 0.5, and 0.6 have been used as tranng sets and the ta correspondng to R = 0.4 has been left for testng the generalzaton capacty of the proposed model. Copyrght to IJIRSET DOI: /IJIRSET

6 Gven the number of nput and output parameters n the tranng set, the process s characterzed as a non-lnear stochastc regresson analyss. Durng the tranng phase the genetc programmng tool establshed several relatons (by regresson analyss) n the form of computer programs between the nput and output varables.e. output = f(nput), or /dn = f(r, K max, ΔK). Parameters of the GEP models are presented n Table 3. Usng an teratve process the parameters of the establshed relatons were adusted n order to mnmze the error between targeted output and selected program outputs. The same model (the selected evolved program) can be stored and potentally be used to predct other output values for a new appled nput ta set (.e. R = 0.4). Table 3 Parameters of GEP model Parameter defnton GEP model P 1 : Functon set -, *, power P 2 : Chromosomes 30 P 3 : Head sze 6 P 4 : Number of genes 2 P 5 : Lnkng functon Multplcaton P 6 : Mutaton rate P 7 : Inverson rate 0.1 P 8 : One-pont recombnaton rate 0.3 P 9 : Two-pont recombnaton rate 0.3 P 10 : Gene recombnaton rate 0.1 P 11 : Gene transposton rate 0.1 V. RESULT AND DISCUSSION In the present study, GEP model was appled on the tranng ta sets for modelng fatgue crack growth rates for load rato R = 0.4 whch has not been ncluded n tranng set. The ta contanng n the tranng fle have been used for learnng by applyng the ftness functon. Subsequently, the new nputs of the test ta set (.e. R = 0.4) have been fed to the traned GEP model to predct the correspondng predcted outputs. The overall performances of both sets have been evaluated by the correlaton coeffcent (R) and mean squared error (MSE) gven by: m dn expermental dn expermental dn predcted dn predcted 1 R (7) 2 2 dn expermental dn expermental dn predcted dn predcted MSE m dn ermental dn exp 1 n predcted Where, dnexp ermental and dn predcted are the expermental and predcted crack growth rates, dn expermental and dn predcted are ther correspondng mean values and n s the number of observatons. The GEP estmates are compared to the expermental ta for tranng and testng sets. The statstcal performance of the model has been presented n Table 4. (8) \ Copyrght to IJIRSET DOI: /IJIRSET

7 Table 4 Statstcal results of GP for tranng and testng Set MSE Corr. Coff. (R) Tran Test The tranng results proved that the proposed GEP models have learned well the nonlnear relatonshp between the nput and output varables wth hgh correlaton (R = ) and relatvely low error (MSE = ) values. Comparng the GEP predctons wth the expermental ta for the test stage (Fg. 5) demonstrates a hgh generalzaton capacty of the proposed model (R = ) and relatvely low error (MSE = ) values. Crack growth rate, /dn (GEP) 3.00E E E E E E E+00 Perfect ft GP 0.00E E E E E E E-03 Crack growth rate, /dn (Exp.) Fg. 5 Modellng ablty of genetc programmng for the test set All these fndngs show a successful performance of the GEP model for estmatng fatgue crack growth rates n tranng and testng stages. The testng results of /dn vs. K n lnear as well as n log scale have been presented n Fgs. 6 and 7 respectvely for 6082 T6 Al-alloy. Crack growth rate (/dn), mm/cycle 1.74E-01 Expermental 1.54E-01 Predcted 1.34E E E E E E E Stress ntensty factor range ( K), MPa m Fg. 6 Comparson of predcted and expermental /dn K curves Copyrght to IJIRSET DOI: /IJIRSET

8 Log(/dN), mm/cycle Log ( K), MPa m Expermenta l Fg. 7 Comparson of predcted and expermental /dn K curves n log scale The numbers of cycles have been calculated from predcted and expermental results n the excel sheet (Fg. 8) as per the followng equaton: a a N 1 N (9) 1 dn From the a N plot t has been observed that the fatgue lfe (at R = 0.4) of 6082 T6 Al-alloy from GEP model s cycles wth an error of 2.04% n comparson to ts expermental value whch s cycles. Crack length (a), mm Expermental Predcted E E E E E+04 No. of cycle (N) Fg. 8 Comparson of predcted and expermental a N curves VI. CONCLUSION The present study reports an effcent approach for the predcton of constant ampltude fatgue crack growth lfe of 6082 T6 Al-alloy under the nfluence of load rato usng GEP. Fatgue crack growth test results are used to buld and Copyrght to IJIRSET DOI: /IJIRSET

9 valte the model. From the model result t s found that the proposed GEP model predcts the fatgue lfe wth -2.04% devatons from the expermental results. The proposed model s so smple that t can be used by any one not famlar wth GEP. The model also gves a practcal way for the predcton of fatgue lfe of 6082 T6 Al-alloy under load rato effect to obtan reasonable results and encourages the use of GEP n other aspects of mechancal engneerng felds. REFERENCES [1] Stofanak, R. J., Hertzberg, R. W., Mller, G., Jaccard, R., and Donald, K., On the cyclc behavour of cast and extruded alumnum alloys, Part A: Fatgue crack propagaton, Engneerng Fracture Mechancs, vol.17, pp , [2] Vazquez, J. A., Morrone, A., and Ernst, H., Expermental results on fatgue crack closure for two alumnum alloys, Engneerng Fracture Mechancs, vol.12, pp , [3] Kardomateas, G. A., and Carlson, R. L., Predctng the effects of load rato on the fatgue crack growth rate and fatgue threshold, Fatgue Fracture Engneerng Materal Structure, vol.21, pp , [4] Huang, X., and Moan, T., Improved modellng of the effect of R-rato on crack growth Rate, Internatonal Journal of Fatgue, vol.29, pp , [5] Kumar, R., Investgaton of Fatgue Crack Growth Under Constant Ampltude Loadng, Internatonal Journal of Pressure Vessels and Ppng, vol.41, pp , [6] Brown, W. F., and Srawley, J. E., Plane stran crack toughness testng of hgh strength metallc materals, ASTM STP, Phladelpha, USA, vol. 410, pp. 1, [7] Mukheree, B., A note on the analyss of fatgue crack growth ta, Internatonal Journal of Fracture, vol.8, pp , [8] Smth, R. A., The determnaton of fatgue crack growth rates from expermental ta, Internatonal Journal of Fracture, vol.9, pp , [9] Daves, K. B., and Feddersen, C. E., Evaluaton of fatgue-crack growth rates by polynomal curve fttng, Internatonal Journal of Fracture, vol.9, pp , [10] Munro, H. G., The determnaton of fatgue crack growth rates by ta smoothng Technque, Internatonal Journal of Fracture, vol.9, pp , [11] Mohanty, J. R., Verma, B. B., and Ray, P. K., Determnaton of fatgue crack growth rate from expermental ta: A new approach, Internatonal Journal of Mcrostructure & Materal Propertes, vol.5, pp.79 87, [12] Koza, J. R., Genetc programmng: on the programmng of computers by means of natural Selecton, MIT Press, London, [13] Gen, M., and Cheng, R., Genetc algorthms and engneerng desgn, USA, Wley, [14] Koza, J. R., Bennett, F. H., Andre, D., and Keane, M. A., Four problems for whch a computer programmng performance program evolved by genetc programmng s compettve wth human, In: Proceedngs of the IEEE nternatonal conference on evolutonary computaton, pp.1 10, [15] Ashour, A. F., Alvarez, L. F., and Toropov, V. V., Emprcal modellng of shear strength of RC deep beams by genetc programmng, Computer Structure, vol.1, no.5, pp , [16] Ferrera, C., Gene expresson programmng: a new aptve algorthm for solvng problems, Complex Systems, vol.13, no.2, pp , [17] Munoz, D. G., Dscoverng unknown equatons that descrbe large ta sets usng genetc programmng technques, Master s Thess, Lnkopng Insttute of Technology, [18] Kose, M., and Kayadelen, C., Modelng of transfer length of pre-stressng strands usng genetc programmng and neuro-fuzzy, Advanced Engneerng Software, vol.41, pp , [19] Teodorescu, L., and Sherwood, D., Hgh energy physcs event selecton wth gene expresson programmng, Computer Physcs Communcatons, vol.178, pp , [20] Kayadelen, C., Gunaydın, O., Fener, M., Demr, A., and Ozvan, A., Modellng of the angle of shearng resstance of sols usng soft computng systems, Expert Systems wth Applcatons, vol.36, pp , Copyrght to IJIRSET DOI: /IJIRSET