Kinematic Hardening Parameters Identification with Respect to Objective Function

Transcription

1 Vol:8, No:4, 04 Knematc Hardenng Parameters Identfcaton wth Respect to Objectve Functon Marna Franulovc, Robert Basan, Bozdar Krzan Internatonal Scence Index, Mathematcal and Computatonal Scences Vol:8, No:4, 04 waset.org/publcaton/ Abstract Consttutve modelng of materal behavor s becomng ncreasngly mportant n predcton of possble falures n hghly loaded engneerng components, and consequently, optmzaton of ther desgn. In order to account for large number of phenomena that occur n the materal durng operaton, such as knematc hardenng effect n low cycle fatgue behavor of steels, complex nonlnear materal models are used ever more frequently, despte of the complexty of determnaton of ther parameters. As a method for the determnaton of these parameters, genetc algorthm s good choce because of ts capablty to provde very good approxmaton of the soluton n systems wth large number of unknown varables. For the applcaton of genetc algorthm to parameter dentfcaton, nverse analyss must be prmarly defned. It s used as a tool to fne-tune calculated stress-stran values wth expermental ones. In order to choose proper objectve functon for nverse analyss among already exstent and newly developed functons, the research s performed to nvestgate ts nfluence on materal behavor modelng. Keywords Genetc algorthm, knematc hardenng, materal model, objectve functon. I. INTRODUCTION ATERIAL behavor modelng plays very mportant role Mn structural components desgn and ther fatgue analyss. Materal models dffer n the range of materal propertes they can descrbe and proportonally, n complexty of ther defnton. Complex materal models are characterzed by numerous materal parameters that have to be carefully dentfed to follow materal behavor as accurately as possble. Due to the complexty of chosen Chaboche s materal model [], [], t s necessary to use complex numercal procedures to dentfy materal parameters. The usage of evolutonary algorthms s proposed because of ther advantageous characterstcs, manly consderng nsenstvty to errors n measured data, relablty n achevng convergence to accurate results, mprobablty for convergence to local mnma and t s robustness regardng the choce of objectve functon [], [4]. Genetc algorthm s stochastc search method for obtanng good approxmate solutons for complex problems [5]. It s based on mechansms of natural evoluton and genetc prncples. The genetc algorthm creates a populaton of solutons and apples genetc operators, such as scalng, selecton, mutaton and crossover to evolve the solutons n order to fnd the best ones. The proper evoluton of populaton s assured by selecton of adequate genetc M. Franulovc, R. Basan, and B. Krzan are wth the Faculty of Engneerng, Unversty of Rjeka, HR-5000, Croata (e-mal: marna.franulovc@rteh.hr, robert.basan@rteh.hr, bozdar.krzan@rteh.hr). operators n order to acheve fast convergence to global optma. One of the man premses n genetc algorthm applcaton for parameter dentfcaton s the choce of objectve functon for nverse problem soluton. There are numerous publshed papers that suggest dfferent objectve functons for the problem soluton. In order to evaluate these suggestons and the nfluence of objectve functon on smulaton of materal behavor by parameter dentfcaton wth genetc algorthm usage, the most common ones are nvestgated [6]-[8], and also ther modfed versons that are proposed. II. CONSTITUTIVE MATERIAL MODEL The materal model consdered n ths paper s based on contnuum mechancs theory [], [], [9], [0]. Low-cycle fatgue materal behavour s descrbed by means of models of knematc and sotropc hardenng accordng to Chaboche materal model [], []-[4]. The nonlnearty n knematc hardenng of the model makes t superor n relaton to some smpler models [], [5], but t also makes t very complcated and tme consumng to defne. In order to account for materal behavour usng Chaboche s materal model, stran doman s observed through ts elastc and plastc part. Elastc stran tensor corresponds to Hooke s law of lnear elastcty, whle the von Mses yeld functon for plastcty crtera descrpton s gven by ( S X )( S X ) R 0 f = j j j j σ y = () where S j s devatorc stress tensor j s back stress tensor that defnes the centre of the yeld surface, R s the sotropc hardenng varable and σ y s ntal yeld stress. The flow rule [] can be wrtten as df ε = dλ () dσ d p Non-lnear knematc hardenng behavor s descrbed by followng three-decomposton rule of Armstrong-Frederck model [], [] dx j = () p () () ( C d γ X dp) j j = ε () where C and γ are the characterstc coeffcents of the materal. The ntegraton of ths equaton leads to exponental Internatonal Scholarly and Scentfc Research & Innovaton 8(4) scholar.waset.org/07-689/

2 Vol:8, No:4, 04 expresson sutable for dentfcaton of knematc hardenng materal parameters Intal populaton Objectve functon Internatonal Scence Index, Mathematcal and Computatonal Scences Vol:8, No:4, 04 waset.org/publcaton/ p Δσ () Δε = R + X tanh γ (4) = () () () () () () where X, γ, γ, γ are knematc hardenng materal parameters (knematc hardenng coeffcents and rates of knematc hardenng), whle R s the boundary of sotropc hardenng. Consderng hgh nonlnearty n (4), dentfcaton of sx knematc hardenng parameters s made part of genetc algorthm procedure. III. MATERIAL PARAMETERS IDENTIFICATION A. Genetc Algorthm for Parameter Identfcaton Based on proposed materal model and t s mechancal prncple, the parameters of knematc hardenng () () () () () () ( X, γ, γ, γ ) are obtaned on the bass of materal response n the fully reversed tensle compressve cyclc tests, from the recorded cyclc stress stran curves. The calculaton procedure s automated by usng genetc algorthm for materal parameters dentfcaton and fnte element method materal behavor smulaton. The genetc algorthm based procedure conssts of three man parts. The frst part s system characterzaton, whch means determnaton of parameters that can completely characterze the system. In the second part, forward modelng, mechancal prncples and physcal laws are defned to enable predcton of system behavor. The thrd part s backward or nverse modelng. Inverse analyss plays an mportant role n problems where the cause has to be defned from the results. It conssts of defnng the search methods of unknown sample characterstcs by observng sample s response to a probng sgnal. Defnton of objectve functon represents the soluton of nverse problem. The mathematcal structure of the model ( ) σ = σˆ ε ; (5) s defned by mappng functon whch defnes the dependence among stress and stran values and the materal parameter () () () () () () values a = [ X, γ, γ, γ ] that are consdered wthn the chosen doman. Parameter R s calculated as the dfference between ntal yeld stress and yeld stress n stable cycle and therefore sn t part of genetc algorthm calculaton procedure. B. Genetc Operators The genetc algorthm creates a populaton of solutons and apples genetc operators, such as scalng, selecton, mutaton and crossover to evolve the solutons n order to fnd the best ones (Fg. ). a Fg. Genetc algorthm procedure The proper evoluton of populaton s assured by choosng adequate genetc operators n order to acheve fast convergence to global optma [6]. Wthn selecton procedure, 4-tournament method s used, whle crossover s accomplshed through ntermedate recombnaton wth 0% dsperson, characterzed by chldren s values Chld _ K Chldren M Mutaton Chldren K = 0,9 PK + Rato _ K (, PK 0, PK ) (6) 9 where PK and PK are parents values, acheved through selecton procedure, whle Rato_K s random number between 0 and. In order to mprove chldrens characterstcs, correctons of ther values are performed n case of unrealstc parameter values Chld _ K Scalng Ftness estmaton Crossover Elte chldren = PK + Rato K ( PK PK ) (7) _ Chld _ K = 0,5 PK PK (8) + 0, 5 Elte parents Selecton PM, PK,PK Another mprovement of recombnaton process s made by assurng mpossblty of two equal parents exstence. In case of two dentcal parents selecton, one of them s mutated, usng (9), wth mutaton doman that equals 0.5 nstead of 0., as t s n orgnal expresson. All procedures of the proposed genetc algorthm have the same mutaton routne. The possblty of mutaton s set to, whch means each varable s changng durng mutaton. Chldren s values are calculated by Chld _ M = PM + Rato _ M Change_ M (9) where PM s parent value, acheved through selecton procedure, whle Change_M s random number between 0 and. The mutaton rato s decreasng through generatons, accordng to Current _ generaton Rato _ M = (0) Last _ generaton Internatonal Scholarly and Scentfc Research & Innovaton 8(4) scholar.waset.org/07-689/

3 Vol:8, No:4, 04 Internatonal Scence Index, Mathematcal and Computatonal Scences Vol:8, No:4, 04 waset.org/publcaton/ IV. OBJECTIVE FUNCTION Scalng of populaton s based on the ftness values of the ndvduals, whch s the soluton of chosen objectve functon. In general f = n = y yˆ ( x ; parameters) y () where astersk refers to expermental value, whle mark ^ refers to value calculated by usng set of parameters. The best ndvduals have low ftness value and the possblty of ther selecton s hgh. In order to assure convergence of solutons to global optma, the bad ndvduals are also nvolved n evoluton process, just wth the lower possblty and expectancy of selecton. Snce evolutonary algorthm for parameter dentfcaton s used, the soluton of the problem s searched n the global doman. It s not necessary to localze soluton doman n order to acheve more accurate data. The chosen objectve functons used for comparson n ths research are taken n the form publshed by some authors and also n modfed form of each of them as shown n Table I. TABLE I OBJECTIVE FUNCTIONS FOR GENETIC ALGORITHM Source Functon Equaton [6] ; ; () [6] ; ; 00 () [7] ; [7] ; [8] ; [8] ; (4) (5) (6) (7) V. RESULTS ANALYSES The procedure for determnaton of materal parameters of the steel 4CrMo4 n normalzed state wth hardness of 96 HV s presented n ths paper. The chemcal composton of the materal s gven n Table II. TABLE II CHEMICAL COMPOSITION OF TESTED MATERIAL (%) Element Percent C 0,4 S 0,6 Mn 0,65 P 0,05 S 0,0 Cr,07 N 0,9 Mo 0,6 Cu 0,6 Al 0,0 Sn 0,006 Detaled response of the materal to the cyclc loadng was recorded durng own experments and t serves as a bass for modelng of ts behavor. Stran-controlled low-cycle fatgue testng [7] has been performed. Specmens (Fg. ) used for the testng have sold crcular cross secton. The test s performed tll total fracture of the specmen n two parts. The stran ampltude ε a for cyclc testng s mantaned at value.5%. M6 o 0 Fg. Geometry of the specmen Materal parameters for modelng of materal behavor of the steel 4CrMo4 n normalzed state wth hardness of 96 HV have been dentfed and are gven n Table III. For ths purpose, genetc algorthm procedure was performed wth the appled objectve functons () to (7). Equa tons TABLE III MATERIAL PARAMETERS FOR PRESENTED OBJECTIVE FUNCTIONS (N/mm ) R0 (N/mm ) 7.7±0,0 o (N/mm ) () () (4) (5) (6) (7) R0 A A The sets of materal parameters for modelng materal behavor of chosen steel show no smlarty among themselves or notable tendency to any value. In order to understand ths, all three components of Chaboche s model for knematc hardenng descrpton are presented n Fgs. to 8, along wth Internatonal Scholarly and Scentfc Research & Innovaton 8(4) scholar.waset.org/07-689/

4 Vol:8, No:4, 04 ther total value (grey full lne). Internatonal Scence Index, Mathematcal and Computatonal Scences Vol:8, No:4, 04 waset.org/publcaton/ Fg. Stress stran materal behavor calculated usng () Fg. 4 Stress stran materal behavor calculated usng () Fg. 5 Stress stran materal behavor calculated usng (4) Fg. 6 Stress stran materal behavor calculated usng (5) Fg. 7 Stress stran materal behavor calculated usng (6) Fg. 8 Stress stran materal behavor calculated usng (7) Although parameter values as well as components of Chaboche s model (every component s n fact smple Armstrong-Frederck model) dffer consderably among themselves, each group of parameters gves very good soluton. Smulated knematc hardenng behavor of materal follows real materal behavor extremely well. The stressplastc stran relatonshp n all smulatons s completely acceptable for the materal behavor smulaton, as s shown n Fg. 9 (all curves concde very closely wth the expermental one). Internatonal Scholarly and Scentfc Research & Innovaton 8(4) scholar.waset.org/07-689/

5 Vol:8, No:4, 04 Internatonal Scence Index, Mathematcal and Computatonal Scences Vol:8, No:4, 04 waset.org/publcaton/ The dfference among expermental response of the materal and smulated behavor s barely dscernble. Devatons of smulated stress values from expermental ones are calculated. The bggest dfference appears n dentfcaton process that has appled objectve functon (5), but even n ths case, the calculated value dffers from the expermental one for only.57%, whch s neglgble. Fg. 9 Stress stran materal behavor VI. CONCLUSION Generally, when referrng to the functonal nverse problems for the parameter dentfcaton, approprate objectve functon must be used n the most calculaton procedures. The choce of the functon depends on the numercal procedure n materal behavor modelng that wll be used. In genetc algorthm for parameter dentfcaton random applcatons were used to solve complex problem. In order to evaluate robustness of such calculaton procedure, regardng the choce of objectve functon, the most commonly used functons and ther modfed versons were examned. The calculatons showed extremely good compatblty n results and only very small devatons of smulated from real materal s response. Therefore, t can be concluded that genetc algorthm n parameter dentfcaton for knematc hardenng behavor n low-cycle fatgue problems s robust enough to gve relable results wthout the need to consder the choce of the objectve functon for nverse problem. The probablty of convergence to the accurate results s very hgh and there s no need for the mprovement n the calculaton procedure by usng specfcally orented objectve functon. [4] X. T. Feng,C. Yang, Genetc evoluton of nonlnear materal consttutve models, Comp. Meth. Appl. Mech. Eng. Vol. 90, 00., p [5] M. Franulovc, R. Basan, I. Prebl, Genetc algorthm n materal model parameters dentfcaton for low-cycle fatgue, Comp. Mat. Sc., vol. 45 no, 009, p [6] T. Furukawa, T. Sugata, S. Yoshmura, M. Hoffman, An automated system for smulaton and parameter dentfcaton of nelastc consttutve models, Comp. Meth. Appl. Mech. Eng., vol 9, 00, p [7] R. Fedele, M. Flppn, G. Maer, Consttutve Model for Ralway Wheel Steel through Tenson-torson Tests, Comp. Struct. vol 8, 005, p [8] D. Szelga, J. Gawad, M. Petrzyk, Parameter Identfcaton of Materal Model Based on the Inverse Analyss, Int. J. Appl. Math. Comp. Sc. vol 4, 004, p [9] J. Lematre, A Course on Damage Mechancs, Sprnger, 996. [0] D. Krajcnovc, J. Lematre, Contnuum Damage Mechancs Theory and applcatons, Sprnger Verlag, 987. [] S. Bar, T. Hassan, Anatomy of coupled consttutve models for rachetng smulaton, Int. J. Plast., vol. 6, 000, p [] R. Kunc, Low cycle carryng capacty for bearng raceway wth hardened rollng surface, Ph.D. Thess, Unversty of Ljubljana, 00. [] R. Kunc, I. Prebl, Low-cycle fatgue propertes of steel 4CrMo4, Mat. Sc. Eng. A, vol. 45, 00, p [4] T. O. Pedersen, Cyclc plastcty and low cycle fatgue n tool materals, Ph.D. Thess, Techncal Unversty n Denmark, 998. [5] N. E. Dowlng, Mechancal Behavour of Materals Engneerng methods for deformaton, fracture and fatgue, Prentce Hall Internatonal, 99. [6] D. E. Goldberg Genetc Algorthms n Search, Optmtzaton, and Machne Learnng, Addson-Wesley, 989. [7] E606 9, Standard Practce for Stran Controlled Fatgue Testng, ASTM Internatonal standard, 99, reapproved 998. REFERENCES [] J. L. Chaboche, A revew of some plastcty and vscoplastcty consttutve theores, Int. J. Plast. vol. 4 no. 0, 008, p [] J. Lematre, J. L. Chaboche, Mechancs of Sold Materals, Cambrdge Unversty Press, 990. [] T. Furukawa, G. Yagawa, Inelastc Consttutve Parameter Identfcaton usng an Evolutonary Algorthm wth Contnuous Indvduals, Int. J. Num. Meth. Engng.vol. 40, 997, p Internatonal Scholarly and Scentfc Research & Innovaton 8(4) scholar.waset.org/07-689/