THE ANALYSIS OF THE ITERATIONS PROCESS IN THE ELASTO-PLASTIC STRESS MODEL

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1 Plas cit this articl as: Joanna Wróbl, Adam Kulawik, Th analysis of th itrations procss in th lasto-plastic strss modl, Scintific Rsarch of th Institut of Mathmatics and Computr Scinc, 2012, Volum 11, Issu 4, pags Th wbsit: Scintific Rsarch of th Institut of Mathmatics and Computr Scinc 4(11) 2012, THE ANALYSIS OF THE ITERATIONS PROCESS IN THE ELASTO-PLASTIC STRESS MOEL Joanna Wróbl 1 2, Adam Kulawik Institut of Computr and Information Scincs, Czstochowa Univrsity of Tchnology Czstochowa, Poland joanna.wrobl@icis.pcz.pl, adam.kulawik@icis.pcz.pl Abstract. In th papr a numrical modl of mchanical phnomna in th lastic-plastic stat built on th basis of th finit lmnt mthod is prsntd. This modl givs on th possibility to simulat th strss and strain in th lastic-plastic stat with isotropic and kinmatic hardning (two-dimnsional tasks). Th analysis of th influnc mthod for dtrmining th plastic multiplir on th calculation rat is prsnt in th xampl. Introduction Taking into account th lastic-plastic proprtis of matrial is mor and mor common in dsigning of mchanical systms. Gnrating plastic strains in th mchanical construction is gnrally disadvantagous, spcially if th aras of occurrnc ar larg. Th idal solution would b to dsign a structur in which thr ar no plastic strains. Unfortunatly, bcaus of construction and tchnological limitations, it is impossibl. u to th dvlopmnt of numrical mthods it is possibl to analyz a structur which taks into account th non-linar natur of th matrial. Such an analysis allows on to dtrmin th valu of plastic strain and aras of thir occurrnc. 1. Mathmatical and numrical modl In th mathmatical modl th quilibrium quation is usd in th form σ + f =0 (1) whr σ is th tnsor of strss, f is th vctor of volum forcs. Equation (1) is complmntd by th constitutiv rlations in th form of pl ( ε ε ) σ = (2) pl whr ε is th tnsor of total strains, ε is th tnsor of plastic strains, is th tnsor of matrial constants (lastic matrix).

2 152 J. Wróbl, A. Kulawik Th wak form of th quilibrium quations (1) is as follows [1] whr Φ( ) Φ σ fαφdω (3) x dω = pαφdγ + Ω β Γ Ω Φ= x α ar th wight functions, p α [MPa] is th strss boundary. Th quation (3) is supplmntd by appropriat boundary conditions and taks th following form whr K U = R, K = B B dω (4) K is th lmnt stiffnss matrix, drivativ matrix of th wight functions sid vctor, whrin Ω T U th displacmnt vctor, B is th Φ B = B and x β T pl B ε dω + Φ pdγ + Φ R is th right hand R = fdω (5) Ω Γ Ω 2. Plastic strain, hardning of matrial Modling strsss in th lastic-plastic stat, whn th simulatd strss xcds th yild point, it is ncssary to tak into account th plastic strain in th constitutiv rlationships (2) [2-5]. In th prsntd modl to dtrmin th plastic strain th associatd plastic flow law was usd [2, 3] ε pl F = λ σ ( σ ) whr λ is a scalar plastic multiplir (ffctiv plastic strain), F is th function of plastic flow. Th hardning phnomnon is th chang of th plastic surfac in th strss spac. Classical forms of plastic surfac changs ar: isotropic, kinmatic (anisotropic) and combind hardning [2-6]. In this papr th combind hardning has not bn considrd. Isotropic hardning is strngthning of matrial, in which th surfac of plastic flow is symmtrically incrasd (Fig. 1a), whras th hypothsis of kinmatic hardning taks into account th Bauschingr ffct. This hypothsis also assums that th surfac of plastic flow movs as a rigid objct without changing its shap (Fig. 1b) [2, 4]. (6)

3 Th analysis of th itrations procss in th lasto-plastic strss modl 153 a) b) Fig. 1. Graphical intrprtation of th hardning: a) isotropic, b) kinmatic Isotropic hardning Th Hubr-Miss-Hncky (HMH) condition for a matrial with isotropic hardning taks th form whrσ is th ffctiv strss, Y is th flow strss. ff Th ffctiv strss is dfind as [2, 3, 5] F= σ Y=0 (7) ff σ 3 2 σ σ ff= (8) whr σ is th dviatoric strss tnsor( σ = σ σ δ /3) γγ, δ is known as th Kronckr dlta. Th drivation of th quation (7) of strss tnsor coordinats and th (8) quation taks th form Substituting (9) into (6) F 3σ 3σ = = (9) σ 2σ Y f 2 pl σ ε = λ (10) 2Y Introducing (10) into constitutiv rlationship (2) and appropriat diffrntiation of th plasticity condition (7) lads to th scalar multiplir of th plasticity quation. For isotropic hardning this factor is as follows 3

4 154 J. Wróbl, A. Kulawik λ = 2Y 9σ 3σ σ ε + 4Yκσ ff (11) whr κ [MPa] is th hardning modulus [2] Kinmatic hardning In th hypothsis of kinmatic hardning th offst plastic surfac is dscribd by th following quation [2, 5, 6] ( σ ( ), Y 0 ) = 0 F ff σ α (12) whr α is th tnsor rprsnting th displacmnt of th cntr of th plastic surfac (kinmatics of hardning). Th HMH condition in this cas has th form (comp. (7)) ( σ α )( σ α )/ 2 Y = (13) whr Y 0 is th yild point. Th tnsor of displacmnt of th cntr th plastic surfac ( α ) is writtn as 2 pl α= κε (14) 3 Analogously to th cas of isotropic hardning, in th kinmatics hardning is ( σ α) pl 3 ε = (15) 2Y Aftr all, th quation for a scalar plastic multiplir for kinmatic hardning has th form 0 ( σ α) ε α( σ α) ( σ α) ( σ α) 2 λ = Y (16) 3 Th scalar plastic multiplir in th modl of an isotropic and kinmatic hardning can b dtrmind by using th Nwton-Raphson mthod [1, 7]. 3. Numrical simulation Two tasks of th numrical simulation of strsss in th lastic-plastic stat wr ralizd. For th calculation th two-dimnsional ara with dimnsions of m (squard) was takn into account, which was dividd into a rgular

5 Th analysis of th itrations procss in th lasto-plastic strss modl 155 gomtry consisting of triangular lmnts (1444 lmnts, 761 nods, Fig. 2). It 5 was assumd that th Young's modulus E = 2 10 MPa, th hardning modulus 4 κ = MPa and th Poisson s ratio ν= 0.3. Th calculations hav bn don for matrial with isotropic and kinmatic linar hardning. In th subsqunt itrations th scalar plastic multiplir was dtrmind by th dirct mthod (formulas (11) and (16)) and th Nwton-Raphson mthod. In th scond xampl it is assumd that th accurat solution is found through calculations obtaind using th mthod of Nwton-Raphson. Th approximation rror of th nd of th itrations it l l Err = σ Y < was assumd as ( ) 8 ff Th accuracy of th calculation for th rsults of numrical solutions, and th numbr of itrations for th isolatd cass ar compard. Fig. 2. Th considrd gomtry with th finit lmnt msh Th tasks wr solvd with th assumption of plan strss stat [1, 2]. Exampl 1 It was assumd, that th displacmnt of th dgs U x ( 0, y) = 0, ( x, 0) = 0 U y. On th boundary 0 Γ x and 0 Γ y ar qual 1 Γ x th condition of th constant incrmnt of boundary load was assumd for th nxt stps ( x= 40 MPa) stps th sign of load was changd on ( p = 40 MPa) p. Aftr 10 ( ) 4 ff x. Th rror of th nd of th itrations in th sarch of th surfac of th plastic flow was assumd as it l l Err = σ Y < Th obtaind rsults for th stps of growth strsss, in which th plastic flow occurrd, ar shown in Figurs 3 and 4.

6 156 J. Wróbl, A. Kulawik Th dirct mthod Th Ntwon-Raphson mthod Th numbr of itrations Th stp of load growth Th dirct mthod (bfor th first itration) Th Ntwon-Raphson mthod (bfor th first itration) Th dirct mthod (aftr itrations) Th Ntwon-Raphson mthod (aftr itrations) σ ff /Y Th stp of load growth Fig. 3. Th rsults of calculations for th isotropic hardning

7 Th analysis of th itrations procss in th lasto-plastic strss modl 157 Fig. 4. Th rsults of calculations for th kinmatic hardning Exampl 2 In th prsntd simulation of strsss, it was assumd that th displacmnt at 0 1 th dg Γ x was U x ( 0, y) = 0, U y ( 0, y) = 0 (fixd). On th boundary Γ x it was assumd that th growth of displacmnt in vry stp of th load was 5 U x 0.1, y= 2 10 Aftr 10 stps th sign of displacmnt growth was changd ( ) m.

8 158 J. Wróbl, A. Kulawik 5 on ( 0.1, y) = 2 10 m. U x Th rror of th nd of th itrations in this cas was it l l 2 ( Y ) < Err = σ Th obtaind rsults ar shown in Figurs 5 and 6. ff Th dirct mthod Th Ntwon-Raphson mthod Th numbr of itrations Th stp of displacmnt growth Th dirct mthod (bfor th first itration) Th Ntwon-Raphson mthod (bfor th first itration) Th dirct mthod (aftr itrations) Th Ntwon-Raphson mthod (aftr itrations) σ ff /Y Th stp of displacmnt growth Fig. 5. Th rsults of calculations for th isotropic hardning

9 Th analysis of th itrations procss in th lasto-plastic strss modl 159 Fig. 6. Th rsults of calculations for th kinmatic hardning Aftr solving tasks, th filds of ffctiv strsss (Figs. 7 and 8) wr mad. In Figurs 7b, 7c, and 8b, 8c th absolut diffrncs btwn th xact and th approximat solution wr prsntd.

10 160 J. Wróbl, A. Kulawik a) b) c) Fig. 7. Th ffctiv strss (xact solution) (a) and th absolut diffrnc, btwn th ffctiv strss, obtaind from th assumption th isotropic hardning: b) th dirct mthod, c) th Nwton-Raphson mthod a) b) c) Fig. 8. Th ffctiv strss (xact solution) (a) and th absolut diffrnc, btwn th ffctiv strss, obtaind from th assumption th kinmatic hardning: b) th dirct mthod, c) th Nwton-Raphson mthod

11 Th analysis of th itrations procss in th lasto-plastic strss modl 161 Conclusions Th papr prsnts an application of finit lmnt mthods for solving mchanics problms. Th numrical simulation of strsss was prformd in th lastic-plastic stat for matrial with isotropic and kinmatic hardning. Th scalar plastic multiplir was dtrmind using th dirct mthod and th Nwton-Raphson mthod, thn th rsults of th calculations wr compard (Figs. 3-6). For absolut diffrncs concrning th ffctiv strss (Figs. 7 and 8) th normalizd man squar rror (NMSE) has bn assumd [8] NMSE= M N x= 1 y= 1 2 M N [ f( x y) fˆ( x, y) ] / f( x, y) [ ], (17) x= 1 y= 1 NMSE rror valus for ach of th chosn mthods of dtrmining th plastic multiplir and for diffrnt typs of hardning ar qual to: 5 for th isotropic hardning (th dirct mthod) NMSE = , for th isotropic hardning (th Nwton-Raphson mthod) NMSE = 5 = , for th kinmatic hardning (th dirct mthod) NMSE = , for th kinmatic hardning (th Nwton-Raphson mthod) NMSE = 5 = Analyzing th rsults of numrical tsts it can b sn that th us of th Nwton-Raphson mthod givs lss divrsification of rrors (Figs. 7 and 8), than whn using th dirct mthod. In th cas of load on th boundary by th strss (Exampl 1) th numbr of itrations is many tims gratr in Nwton-Raphson mthod, than in th dirct mthod (Figs. 3 and 4). In th cas of th load by th displacmnt of th boundary (Exampl 2) th diffrnc btwn tmporal complxity (th numbr of itrations) for ths two mthods is ngligibly small (Figs. 5 and 6). Th us of Nwton-Raphson mthod lads to gratr accuracy of th calculations, although in som cass this mthod causs an unaccptabl tim of calculations (Figs. 3 and 4). 2 5 Rfrncs [1] Zinkiwicz O.C., Taylor R.L., Th finit lmnt mthod, Buttrworth-Hinmann, Fifth dition, vol. 1-3, Nw York [2] Bdnarski T., Mchanika plastyczngo płynięcia w zarysi, WN PWN, Warszawa [3] Gabryszwski Z., Gronostajski J., Mchanika procsów obróbki plastycznj, PWN, Warszawa [4] Ganczarski A., Skrzypk J., Plastyczność matriałów inżynirskich. Podstawy, modl, mtody i zastosowania komputrow, Wydawnictwo Politchniki Krakowskij, Kraków [5] Klibr M., Mtoda lmntów skończonych w niliniowj mchanic kontinuum, PWN, Warszawa-Poznań 1985.

12 162 J. Wróbl, A. Kulawik [6] Bokota A., Modlowani hartowania stali narzędziowych. Zjawiska cipln, przmiany fazow, zjawiska mchaniczn, Wydawnictwo Politchniki Częstochowskij, Częstochowa [7] Caddmi S., Martin J.B., Convrgnc of th Nwton-Raphson algorithm in lastic-plastic incrmntal analysis, Int. J. Numr. Mth. Eng. 1991, 31, [8] Skarbk W., Multimdia. Algorytmy i standardy komprsji, Akadmicka Oficyna Wydawnicza PLJ, Warszawa 1998.