Obtaining Constants of Johnson-Cook Material Model Using a Combined Experimental, Numerical Simulation and Optimization Method

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1 Vol:0, No:9, 06 Obtaiig Costats of Johso-Cook Material Model Usig a Combied Experimetal, Numerical Simulatio ad Optimizatio Method F. Rahimi Dehgola, M. Behzadi, J. Fathi Sola Iteratioal Sciece Idex, Mechaical ad Mechatroics Egieerig Vol:0, No:9, 06 waset.org/publicatio/0005 Abstract I this article, the Johso-Cook material model s costats for structural steel ST.7 have bee determied by a method which itegrates experimetal tests, umerical simulatio, ad optimizatio. I the first step, a quasi-static test was carried out o a plai specime. Next, the costats were calculated for it by miimizig the differece betwee the results acquired from the experimet ad umerical simulatio. The, a quasi-static tesio test was performed o three otched specimes with differet otch radii. At last, i order to verify the results, they were used i umerical simulatio of otched specimes ad it was observed that experimetal ad simulatio results are i good agreemet. Chagig the diameter size of the plai specime i the eckig area was set as the objective fuctio i the optimizatio step. For fial validatio of the proposed method, diameter variatio was cosidered as a parameter ad its sesitivity to a chage i ay of the model costats was examied ad the results were completely corroboratig. Keywords Costats, Johso-Cook material model, otched specimes, quasi-static test, sesitivity. I. INTRODUCTION HERE are umerous material models to estimate the Tstress-strai curve i plastic regio. Most of these models have bee obtaied by experimets ad each model gives differet costats for various materials. To obtai these costats i earlier methods, experimets were desiged i three groups to cosider the effect of work hardeig, strai rate ad temperature. However, experimets are always expesive ad time cosumig ad their results are ot always reliable which ca be due to the type of the testig machie, difficulty of performig the procedures uder costat strai rate almost i all testig machies ad the eed for usig sophisticated istrumetatio, etc. []-[4]. Beallal ad Berstad [5] used experimetal data ad optimizatio methods to calculate the model costats. Their experimets were doe uder three differet strai rates ad three differet temperatures. I aother study, Zhao ad Lee [6] used a similar method to determie the work hardeig behaviour of the material (isotropic or kiematic). I their work, specimes F. Rahimi Dehgola was with the Turbotec-co, Tehra, Ira (correspodig author, phoe: ; f.rahimi.dehgolaa@gmail.com). M. Behzadi is with Turbotec-co, Tehra, Ira ( mh.behzadi@gmail.com). J. Fathi Sola is with the Mechaical Egieerig Departmet, Uiversity of Texas at Arligto, Arligto, TX 7609 USA ( jalal.fathisola@mavs.uta.edu). were put uder tesio ad pressure usig three poit bedig test method. Recetly, combied methods such as umerical simulatio plus optimizatio, are used to obtai the costats of material models. For example, Sasso et al. [7] used Hopkiso pressure bar to carry out their experimets ad the umerically simulated their process. They used specimes with three differet legths ad cosidered the temperature variatio due to plastic deformatio to make the results more accurate. I aother study, Majzoobi et al. [8] used a combied experimetal, umerical, ad optimizatio techique to determie the costats of Zerilli-Armstrog material model. I aother research, Majzoobi ad Rahimi Dehgola [9] used the same method but this time to obtai the costats of Johso-Cook damage model. They used the diameter decreasig as the objective fuctio i optimizatio. I this study, static costats of Johso-Cook material model for a plai specime of structural steel, ST.7, have bee obtaied by usig a experimetal, umerical simulatio ad optimizatio method. I optimizatio process, easily measurable geometrical parameters were used to defie the objective fuctio. To validate the procedure, obtaied material model costats were used to umerically simulate a simple tesio test performed o three otched specimes with three differet otch radii. II. EFFECTIVE PARAMETERS ON MATERIAL BEHAVIOR Stress-strai behaviour of materials i plastic regio is geerally affected by three parameters: strai, strai rate, ad temperature. Plastic deformatio is a irreversible process which meas that the behaviour of a material uder specified stress ad strai is iflueced by deformatio history i additio to the three parameters metioed before (): f,, T, Deformatio history () Uder low ad fixed strai rates, metals show work hardeig behaviour, meaig that stregth of the material icreases by icreasig the strai value. This pheomeo ca be formulated by (): k. () 0 where σ 0 is the yield stress, k is the coefficiet of work hardeig, ad represets the work hardeig's power. Metals usually show differet behaviour uder various strai Iteratioal Scholarly ad Scietific Research & Iovatio 0(9) 06 6 scholar.waset.org/07-689/0005

2 Vol:0, No:9, 06 Iteratioal Sciece Idex, Mechaical ad Mechatroics Egieerig Vol:0, No:9, 06 waset.org/publicatio/0005 rates. Effect of strai rate o stress is show by (). However, it is ot valid for high strai rates. L. () Most of the eergy i a plastic deformatio is trasformed ito heat. I dyamic deformatios, ot eough time is usually available for heat trasfer betwee the specime ad eviromet which eds i icreasig the temperature of the specime. The temperature icremet ca be calculated by (4): d T (4) Cp where T is temperatures icrease, ρ is desity, ad C p is specific heat. Effect of temperature o stress is formulated as (5): m T T r r (5) Tmelt Tr where T melt ad T are meltig ad eviromet temperatures, T r is referece temperature used to obtai σ r, ad m is a experimetal parameter. III. REVIEW OF SOME WELL-KNOWN MATERIAL MODELS A. Power Law Model This model which is defied by (6) oly icludes the effect of strai rate but the coefficiets ca be cosidered as a fuctio of temperature. m k (6) B. Zerillie-Armestrog Model Zerillie ad Armestrog [0] preseted equatios to model the behaviour of materials with FCC (Face Cetered Cubic) ad BCC (Body Cetered Cubic) crystal structures. These equatios are give i (7) ad (8): C T C L C for BCC C C exp 4 (7) 5 C T C L for FCC C C exp (8) 4 where C is to cosider the effect of residual stress, C is used for curve fittig, C ad C 4 are the coefficiets of thermal softeig ad strai rate, respectively. Also C 5 ad are to accout the strai hardeig behaviour of the BCC metals. C. Johso-Cook Model Johso ad Cook [] preseted (9) to model the behaviour of material cosiderig the ifluece of work hardeig, strai rate ad temperature. * A B C L T m (9) 0 where A ad B are strai hardeig coefficiets, C is odimesioal sesitivity coefficiet of strai rate, m ad represet power of thermal softeig ad strai hardeig respectively ad T * is defied by (0). T T r T * (0) Tmelt Tr IV. EXPERIMENTS A. Experimetal Procedure I this study, quasi-static tesio tests were doe o oe plai ad three otched specimes by usig a Istro tesile testig machie. Specimes were chose accordig to ASTM stadard (Fig. ). Fig. Geometrical specificatio of otched specimes (i millimeter) Tesio test of plai specimes was carried out by usig a extesometer of 50 millimeters log. Sice for a plai specime, the locatio where eckig begis, depeds o the locatio where structural faults are cocetrated ad caot be determied before the experimet, the extesometer was set i a positio symmetric with respect to the ceter of specime. However, i otched specimes, plastic deformatio is cocetrated at the locatio of the otch ad displacemet at two eds of specime after fracture is much less tha the plai specime, therefore the tesio tests of otched specimes Iteratioal Scholarly ad Scietific Research & Iovatio 0(9) 06 6 scholar.waset.org/07-689/0005

3 Vol:0, No:9, 06 were doe by usig a extesometer of 5 millimeters log. Fig. shows the extesometer set o the plai specime durig the tesio test. measuremets ad simulatio results obtaied for eckig diameter (): OBJ d exp () erimetal d umerical For defiig the objective fuctio o the basis of material model costats used i simulatio, it is approximated by a polyomial of secod order as (): OBJ x a0 ai xi i j i i i, j b x x j () Iteratioal Sciece Idex, Mechaical ad Mechatroics Egieerig Vol:0, No:9, 06 waset.org/publicatio/0005 Fig. Extesometer set o the plai specime durig the tesio test B. Experimetal Result Experimetal result obtaied from the otched specimes validates that icreasig the otch radius icreases time to fracture, which is due to strai rate reductio caused by a slight icrease i the legth of the specime. Furthermore, it has bee show that icreasig the otch radius decreases the force to fracture, which is due to the iverse proportioality of strai rate ad otch radius. Table I icludes experimetal result after ultimate fracture. Type of specime TABLE I EXPERIMENTAL RESULT AFTER ULTIMATE FRACTURE Time to fracture (S) Fial displacemet at two eds (mm) Maximum force to fracture (N) Diameter reductio at fracture locatio (mm) Plai otched R= Notched R= Notched R= V. DERIVING THE CONSTANTS A. Defiig a Objective Fuctio I this paper, a combied experimetal, umerical simulatio, ad optimizatio method has bee used to obtai the costats of Johso-Cook material model. At first, a quasi-static tesio test was carried out o a specime by usig a Istro tesile testig machie. I order to obtai the costats, the value of fracture strai which is eeded ca be calculated by accurately measurig the miimum fracture diameter to be used i (): d 0 f L () d f where d 0 is the iitial diameter, ad d f is the diameter measured after fracture. Neckig diameter was chose as the optimizatio parameter sice it is the oly parameter that the value of fracture strai depeds o. Therefore, i order to determie the costats, a objective fuctio was defied to miimize the differece betwee the experimetal where x i is the desig parameter (costat of material model), is the umber of desig parameters ad a 0, a i ad b ij are the coefficiets of the objective fuctio. The umber of equatios required to be solved i order to obtai these coefficiets, equals to the umber of polyomial coefficiets. The equatios usually come from umerical simulatio with differet values for material model costats. I this paper it is iteded to obtai the first three material parameters of Johso-Cook relatio (A, B ad ), show as x, x ad x i the objective fuctio. Expadig the polyomial for these parameters we have (4) OBJ a 6 x x a 7 0 a x x a x a x 8 a x x a a 9 x x x a 4 x a 5 x (4) The above equatio has 0 coefficiets therefore 0 equatios are required to calculate the coefficiets. As metioed before these equatios come from umerical simulatio with differet values for material model costats. So experimets were simulated usig fiite elemet. Table II shows the results from simulatio with 0 differet sets of material model costats. By solvig the equatios, the objective fuctio is rewritte as (5) x OBJ e x 5.66 e x 7.45 x x 0.00 x 6.5 x 6.58 e 5 x x (5) 7.0e x x.40 e x x B. Usig Geetic Algorithm I the ext step, the objective fuctio was optimized usig GA (Geetic Algorithm). The GA is the most widely used optimizatio method to tackle egieerig problems. The GA was popularized by Hollad [] ad has emerged as a global search method to simulate the evolutio i complex physical ad biological systems. At each iteratio, GA geerates a populatio of poits that approach the optimal solutio by usig stochastic ad ot determiistic operators. It starts by iitializig a set of idividuals that form the first populatio. The, the first populatio is submitted to geetic operators, resultig i the evolutio of populatios through geeratios (iteratio cycles). I each geeratio, the best idividuals are chose by evaluatio accordig to the objective fuctio. The idividuals that are selected as better, have a higher possibility Iteratioal Scholarly ad Scietific Research & Iovatio 0(9) scholar.waset.org/07-689/0005

4 Vol:0, No:9, 06 of beig icluded i the recombiatio procedure. Mutatio, which periodically chages the parts of idividuals, is the mai operator to protect the algorithm from permaetly losig geetic material through the evolutio of geeratios. Crossover is used for the recombiatio of geetic exchage betwee idividuals. Aother operator is migratio which is the movemet of idividuals amog sub-populatios of existig idividuals, with the best idividuals from oe subpopulatio replacig the worst idividuals i aother subpopulatio. GA proposes the best idividual as the solutio to the problem. A flowchart of a basic GA is show i Fig.. Iteratioal Sciece Idex, Mechaical ad Mechatroics Egieerig Vol:0, No:9, 06 waset.org/publicatio/0005 TABLE II REDUCED DIAMETER OBTAINED FROM SIMULATION WITH 0 DIFFERENT SETS OF CONSTANTS (A,B,) Reduced diameter Obj. (A,B,) Reduced diameter Obj. 90,00, ,05, ,0, ,05, ,00, ,05, ,00, ,00, ,00, ,00, Fig. Flowchart of a basic GA [] C. Results Fial values obtaied for Johso-Cook material models costats (A, B ad ) are 80, 87.7, ad 0.8, respectively. The result of plai experimets was used to drive costats. The calculated A, B, ad were used to simulate the experimets doe for plai specimes i Ls-Dya ad the result for diameter reductio was completely reasoable. Fig. 4 shows the eckig regios obtaied after experimets ad simulatios for plai specimes. Fially, these costats were used to simulate the experimets doe for the otched specimes. Fig. 5 shows the eckig regios obtaied after experimets ad simulatio for the otched specimes. Also, stress-strai curves from experimets ad simulatio have bee compared i Figs Table III compares betwee the values obtaied for diameter reductio from experimetal process ad simulatio. VI. SENSITIVITY ANALYSIS I this study, variatios of geometrical parameters resulted from utilizig differet material model costats i simulatio were used to derive the Johso-Cook material model costats. I order to validate the method, sesitivity of diameter icremet (Δd) toward variatios of obtaied costats, A, B, ad was ivestigated ad the results were illustrated i Figs. 9-. It ca be see that chagig the value of each Johso- Cook material model costats varies the value of Δd cosiderably. Therefore, it is acceptable to use diameter icremet (Δd) as a parameter to obtai the costats. Fig. shows percetage of variatio i d parameter by exertig a 0% chage i each of the three costats. It ca be see that d shows the maximum ad miimum sesitivity toward A ad, respectively. TABLE III COMPARES BETWEEN THE VALUES OBTAINED FOR DIAMETER REDUCTION FROM EXPERIMENTS AND SIMULATIONS Type of specime reductio i diameter obtaied from experimet(mm) reductio i diameter obtaied from simulatio(mm) Error % R= R= R= VII. CONCLUSION I this study, the costats of Johso-Cook material model for a plai specime of structural steel ST.7 were obtaied by usig a combied experimetal, umerical simulatio, ad optimizatio method. It was show that these costats could be used to simulate the tesio behaviour of otched specime. Calculatio process started with a quasi-static tesio test doe for a plai specime. After simulatig the test, a secod degree polyomial error fuctio was defied as the differece betwee experimetal measuremet ad umerical simulatio results of fracture diameter ad optimized by usig geetic algorithm. The, simple tesio tests were doe o three otched specimes with differet otch radii, ad by usig the costats obtaied for plai specime experimet, tesio tests for otched specimes were umerically simulated. Results from experimets ad umerical simulatio were i good agreemet. Fially, a sesitivity aalysis was performed to cofirm the method used i this paper which is validated by usig diameter icremet (Δd) as a parameter to obtai the costats, ad it was show that the diameter icremet (Δd) is cosiderably sesitive to chagig the A costat. Iteratioal Scholarly ad Scietific Research & Iovatio 0(9) scholar.waset.org/07-689/0005

5 Vol:0, No:9, 06 Iteratioal Sciece Idex, Mechaical ad Mechatroics Egieerig Vol:0, No:9, 06 waset.org/publicatio/0005 Fig. 4 Neckig regios obtaied after experimets ad simulatios for plai specimes (a) Iteratioal Scholarly ad Scietific Research & Iovatio 0(9) scholar.waset.org/07-689/0005

6 Vol:0, No:9, 06 Iteratioal Sciece Idex, Mechaical ad Mechatroics Egieerig Vol:0, No:9, 06 waset.org/publicatio/0005 (b) Fig. 5 Comparig the eckig regios obtaied after experimets ad simulatios for otched specimes Fig. 6 Stress-strai curves obtaied from experimets ad simulatios for otched specime r =7 mm Iteratioal Scholarly ad Scietific Research & Iovatio 0(9) scholar.waset.org/07-689/0005

7 Vol:0, No:9, 06 Iteratioal Sciece Idex, Mechaical ad Mechatroics Egieerig Vol:0, No:9, 06 waset.org/publicatio/0005 Fig. 7 Stress-strai curves obtaied from experimets ad simulatios for otched specime r =0mm Fig. 8 Stress-strai curves obtaied from experimets ad simulatios for otched specime r = mm Fig. 9 Sesitivity of d (i percet) toward costat A Fig. Sesitivity of d (i percet) toward costat Fig. 0 Sesitivity of d (i percet) toward costat B Fig. Variatio i d (i per cet) parameter by exertig a 0 per cet chage i A, B, ad Iteratioal Scholarly ad Scietific Research & Iovatio 0(9) scholar.waset.org/07-689/0005

8 Vol:0, No:9, 06 Iteratioal Sciece Idex, Mechaical ad Mechatroics Egieerig Vol:0, No:9, 06 waset.org/publicatio/0005 REFERENCES [] C. J. Maide ad S. J. Gree, Compressive strai-rate tests o six selected materials at strai rates from 0-5 to0 5 s -, Joural of applied mechaics, Vol. 966 pp [] U. S. Lidholm, A. Nagy, G. R. Johso ad J. M. Hoegfedt, Large Strai, High Strai Rate Testig of Copper, J. Eg. Mater. Techol 0(4), 76-8 Oct [] F. E. Hauser, Techiques for measurig stress-strai relatios at high strai rates, Exp. Mech., 966 pp [4] A. Nadia ad M. J. Majoie, High speed tesio tests at elevated temperature-part I ad II, Tras. ASME, 94 6, A77. [5] A. Beallal, T. Berstad, A experimetal ad umerical ivestigatio of the behaviour of AA508 alumiium alloy i presece of the Portevi Le Chatelier effect, Iteratioal Joural of Plasticity [6] K. M.Zhao, J.K. Lee, fiite elemet aalysis of the three-poit sheet metals, Joural of material processig techology, [7] M. Sasso, G. Newaz, ad D. Amodioa, Material characterizatio at high strai rate by Hopkiso bar tests ad fiite elemet optimizatio, Joural Materials Sciece ad Egieerig A [8] G. H. Majzoobi, S. F. Z. Khosroshahi, H. B. Mohammadloo, Determiatio of the Costats of Zerilli-Armstrog Costitutive Relatio Usig Geetic Algorithm, Advaced Materials Research, Vols , pp , Ju. 0. [9] G. H. Majzoobi, F. Rahimi Dehgola, Determiatio of the costats of damage models, Procedia Egieerig [0] F. J. Zerilli ad R. W.Armstrog, Dislocatio-mechaics-based costitutive relatio for material dyamic calculatio, joural of applied physics, Vol pp [] G. R. Johso, W. H. Cook, Fracture characteristics of three metals subjected to various strais, strai rates, temperatures ad pressure, EggFractMech, Vol. () 985 pp [] J. H. Hollad, Adaptatio i Natural ad Artificial Systems, Uiversity of Michiga Press, A Arbor, Michiga, USA 975. Iteratioal Scholarly ad Scietific Research & Iovatio 0(9) scholar.waset.org/07-689/0005