Algorithmic Formulations of Evolutionary Anisotropic Plasticity Models Based on Non-Associated Flow Rule

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1 853 Algorithmic Formulations of Evolutionar Anisotroic Plasticit Models Based on Non-Associated Flow Rule Abstract In the resent aer, orthotroic elasto-lastic constitutive formulations for sheet metal forming based on non-associated flow rule that assume distortion of ield function/lastic otential with ongoing deformation rocess are analzed. The ield function/lastic otential are considered as two different functions with functional form as orthotroic quadratic Hill or non-quadratic Karafillis-Boce stress function. Based on the rincile of lastic work equivalence, anisotro arameters of the utilized ield function/lastic otential are set as functions of the equivalent lastic strain. In the constitutive formulation, for this internal variable, evolution equation consistent with the same rincile of lastic work equivalence is introduced. For DC06 sheet samle with reorted significant variation of the incremental r-values with straining, redictions of the evolution of the ield stress and r-value directional deendences with straining obtained b the analzed models are resented. The algorithmic formulations of the analzed constitutive models are derived b alication of the imlicit return maing algorithm. For the derived stress integration rocedures the accurac is investigated b calculating iso-error mas. The mas are comared according to the flow rule and involved orthotroic stress functions. It has been revealed that although there is a difference in mas configuration there is no rominent difference in error magnitudes. Vedrana Cvitanić a Maja Kovačić b, * a,b Deartment of Mechanical Engineering and Naval Architecture, Facult of Electrical Engineering, Mechanical Engineering and Naval Architecture, Universit of Slit, Ruđera Boškovića 3, 000 Slit, Croatia. vcvit@fesb.hr, majkovac@fesb.hr * Corresonding author htt://dx.doi.org/0.590/ Received In revised form Acceted Available online Kewords constitutive modeling; sheet metals; anisotro evolution; nonassociated flow rule; imlicit return maing INTRODUCTION Sheet metals are roduced b comlex thermo-mechanical rocessing routes among which the roduction rolling stes introduce a referential orientation to the grains called texture. This referential orientation results in anisotroic material behavior that is closel related to the formabilit of the material. The finite element codes that are widel used to design sheet forming rocesses most

2 854 V. Cvitanić and M. Kovačić / Algorithmic Formulations of Evolutionar Anisotroic Plasticit Models Based on Non-Associated Flow Rule frequentl utilize the henomenological lasticit constitutive theories that are based on ield surface concet and lastic otential theor. The basis of these henomenological constitutive models are: a ield criterion reresenting surface that searates the elastic and lastic regions in the stress sace, a lastic otential whose gradient reresents direction of the lastic strain rate and a hardening rule b which evolution of the ield surface is described. The simlest and the most utilized aroach in henomenological lasticit theories that are intended for the metallic materials is the concet of isotroic hardening and use of the assumtion known as associated flow rule or normalit condition. Associated flow rule assumtion imlies that the ield function and lastic otential function are identical and it is often regarded as the necessar condition for the metal lasticit fundamentals (stabilit of lastic flow and uniqueness). However, according to some theoretical observations (Mroz, 963; Runesson and Mroz, 989; Stoughton, 00; Stoughton and Yoon, 004, 006; Cvitanić et al., 008), an aroriate lastic flow descrition can be achieved b using less restrictive constrains over the constitutive equations than the associated flow rule. B non-associated flow rule, ielding and lastic flow are described b two different functions. In the last few ears, several metal lasticit non-associated formulations were develoed and this aroach became articularl engaging in sheet metal lasticit. An accetable constitutive model for a sheet material based on associated flow rule requires a ield function that can simultaneousl redict anisotro of ield stresses and anisotro of lastic flow. Under the assumtion of associated flow rule, various anisotroic henomenological ield functions have been roosed for metallic sheets (Hill, 948, 979, 990, 993; Barlat and Lian, 989; Karafillis and Boce, 993; Barlat et al., 99, 997, 003, 005; Banabic et al., 003, 005; Aretz, 005; and others). Anisotro is introduced into these stress functions b the arameters determined b the data related to the directional deendences of the ield stresses and Lankford arameters as well as data related to the biaxial tensile tests. Lankford arameter, also known as r-value or lastic strain ratio, is used as the measure of lastic flow. It is defined as the ratio of the sheet secimen transverse and thickness true lastic strain increments in uniaxial tensile testing. According to the standards, r-value is calculated b linear regression of the transverse versus longitudinal lastic strain lot between certain limits of the measured strains. Hence, it is considered as a constant value. In orthotroic sheet metal models based on nonassociated flow rule, ield function and lastic otential function might have an identical functional form but their arameters can be defined b different exerimental data. That means that the ield function can be adjusted to the ield stresses and the lastic otential can be adjusted to r-values. Such formulation can address sufficient number of exerimental data and can result in accetable redictions of the uniaxial material behavior even if ield/lastic otential stress functions with a low number of anisotroic arameters are utilized (Lademo et al., 999; Stougthon, 00; Stoughton and Yoon, 004; Cvitanić et al., 008; Safaei et al., 04). Isotroic hardening concet imlies a roortional exansion of the ield surface with ongoing deformation rocess, without an change of its shae and osition. In constitutive formulations intended for sheet metals, b alication of isotroic hardening and standard calculation rocedure for r-value, anisotro arameters are calculated using the initial ield stress ratios and/or constant r-values. Such aroach results in fixed anisotro arameters and accordingl an ossible distortion of ield function/lastic otential with continuation of the lastic deformation rocess is ne-

3 C. Cvitanić and M. Kovačić / Algorithmic Formulations of Evolutionar Anisotroic Plasticit Models Based on Non-Associated Flow Rule 855 glected. In addition, uniform size change of the orthotroic ield function in the stress sace is controlled b single scalar hardening variable and mostl flow stress curve for the rolling direction is adoted as the exansion rule. However, some recent exerimental studies reort alternation of the ield stress ratios and/or instantaneous r-values with evolution of sheet texture during deformation rocess (Zamiri and Pourboghrat, 007; An et al., 03; Safaei et al., 04). Furthermore, numerous studies related to the alication of the orthotroic lasticit formulations with constant anisotro arameters in redicting comlex forming rocesses indicate that ossible model imrovements could be achieved b incororating the evolution of ield stress ratios and r-values into the model. This is articularl evident in the simulations of the clindrical cu drawing roblem. In this forming rocess, the lastic anisotro has imortant effect on the formation of undulative to edge of the finall drawn cu often termed as earing. Numerous henomenological orthotroic ield functions with constant (fixed) anisotro arameters under associated or non-associated flow rule have been tested in this forming rocess (Yoon et al., 998, 999, 000, 004, 006; Cvitanić et al., 008; Taherizadeh et al., 00, 0; Park and Chung, 0; Safaei et al., 03; Vrh et al., 04; and others). These extensive numerical studies clearl indicate that the redicted earing trend (location of eaks and valles) is the mirror image of the redicted r-value trend with the resect to the transverse direction, and that the rediction of the earing rofile amlitude (the maximum difference in cu heights) is correlated to the rediction of the ield stress anisotro amlitude. Considering redictions of the comlex henomenological ield functions that are caable to accuratel redict directional deendence of the r-values, it can be observed that there is still discreanc between their redictions of earing rofile and exerimental cu heights. Although the descrition of the material behavior is not the onl factor that influences the finite element redictions, due to the exressed correlation between redicted r-values and redicted earing rofile, it can be concluded that reliable material model should take into account not onl directional deendences of the uniaxial material data determined at the initial ield state or their averaged values but also their ossible evolution with continuation of the deformation rocess. That means that the anisotro arameters of the ield function/lastic otential should be altered b a certain measure of lastic flow. In this aer, constitutive formulations based on non-associated flow rule and distortional evolution of the orthotroic ield function/lastic otential with ongoing deformation rocess are considered. As ield function/lastic otential orthotroic four arametric quadratic stress function roosed b Hill (948) or non-quadratic stress function roosed b Karafillis-Boce (993) are utilized. Anisotro arameters of the utilized ield function/lastic otential are set as olnomial functions of the equivalent lastic strain and evolution equation for this hardening arameter consistent with the rincile of the lastic work equivalence is introduced. Previousl develoed algorithmic formulations of the evolutionar anisotroic model based on the considered stress functions and associated flow rule (Cvitanić et al., 06) are now extended for the case of non-associated flow rule. Considered algorithmic formulations are derived b alication of the imlicit return maing algorithm. Numerical analsis of accurac of the roosed stress integration rocedures is erformed and comared according to the flow rule and the functional form of the involved stress functions b calculating iso-error mas. The constitutive descritions are develoed and analzed considering exerimental data for DC06 steel sheet reorted b Safaei et al. (04).

4 856 V. Cvitanić and M. Kovačić / Algorithmic Formulations of Evolutionar Anisotroic Plasticit Models Based on Non-Associated Flow Rule EVOLUTIONARY ANISOTROPIC PLASTICITY FORMULATION BASED ON NON-ASSOCIATED FLOW RULE Assuming isotroic linear elasticit and additive decomosition of the strain tensor increment d ε into elastic d ε e and lastic art d ε, the stress tensor increment d σ reads e e e dσ = C : dε = C : ( dε -dε ) () e where C is the tensor of elastic module. Considering sheet material with anisotro evolution, the ield criterion is stated as follows where F = f (, σ e )- k( e ) = 0 () f (, σ e ) is an orthotroic ield function with orthotroic arameters introduced as functions of hardening arameter e and ke ( ) is a scalar function reresenting stress-strain relation for the referent direction. According to the lastic otential theor, the lastic art of the strain tensor increment d ε is roortional to the gradient of the stress function named lastic otential function f (, σ e ) dε dl = σ (3) where dl is a non-negative scalar called lastic multilier or consistenc arameter. Considering anisotro evolution, lastic otential f (, σ e ) is also introduced as an orthotroic stress function with orthotroic arameters stated as functions of the arameter e. If the lastic otential and ield function are identical f (, σ e ) º f (, σ e ), ielding and lastic flow are described b the same function and Eq. (3) becomes the so-called associated flow rule. In the resent formulation, the arameter e is considered as equivalent lastic strain that obes the following form of the rincile of lastic work equivalence f (, σ e ) de = σ : dε (4) If the lastic otential function fulfils identit f (, σ e ) = σ : f (, σ e )/ σ, using Eqs. (3) and (4), following evolution equation for the arameter e is obtained σ : f( σ, e )/ σ f( σ, e ) de = dl = dl (5) f (, σ e ) f (, σ e ) where for the associated flow rule de = dl. If deformation rocess is elastic, the incremental changes of the internal variables (lastic strain tensor d ε and arameter e ) vanish and dl = 0. Therefore, for the hardening material the lastic multilier obes the comlementar conditions dl ³ 0, F 0, dlf = 0 and consistenc conditiondl df = 0.

5 C. Cvitanić and M. Kovačić / Algorithmic Formulations of Evolutionar Anisotroic Plasticit Models Based on Non-Associated Flow Rule ANISOTROPIC STRESS FUNCTIONS WITH ANISOTROPY EVOLUTION The analzed constitutive formulations utilize ield function and lastic otential with functional form of orthotroic quadratic Hill (948) stress function or non-quadratic Karafillis-Boce (993) stress function. In sheet metal forming it is common ractice to assume that the sheet is aroximatel subjected to lane stress conditions and that material exhibits orthotroic smmetr in lastic roerties. Accordingl, the utilized ield functions/lastic otentials are stated as functions of in-lane stress comonents s xx, s and s zz, where x-axis denotes the original sheet rolling direction and -axis denotes the direction in sheet lane transverse to the rolling direction. The z-axis denotes the sheet normal direction. The utilized Hill stress function and Karafillis-Boce stress function inclose four arameters b which material lastic anisotro can be described. Under the associated flow rule, the utilized stress function acts as ield function as well as lastic otential, therefore, its anisotro arameters can be adjusted to the ield stresses or exerimental data indicating lastic flow. In the nonassociated formulation, anisotro arameters of the Hill or Karafillis-Boce ield function are defined in terms of three directional ield stresses obtained in the uniaxial tension of the secimens oriented at 0, 45 and 90 to the rolling direction and equibiaxial ield stress. The associated ield stresses are denoted as s 0, s 45, s 90 and s b. The anisotro arameters of the Hill or Karafillis- Boce lastic otential are defined in terms of exerimental data indicating lastic flow such as Lankford arameter that reads r 33 de = (6) de where de and de 33 are width and thickness lastic strain increments, resectivel, obtained in uniaxial sheet secimen tension. In calculating arameters of the lastic otential, three lastic strain ratios obtained in the uniaxial tensions along 0, 45 and 90 to the rolling direction and the ield stress for the referent direction s 0 are used. The associated lastic strain ratios are denoted as r 0, r 45 and r Hill Stress Function and Karafillis-Boce Stress Function The orthotroic quadratic Hill (948) stress function is derived as an extension of the isotroic von Mises ield function. For lane stress condition Hill stress function can be written in the following form xx xx x f = ls + ls - ns s + rs = s (7) where l, l, n and r are anisotroic material arameters that can be adjusted to exerimental data and s is the ield stress for the referent direction. For Hill stress function calculation rocedure results in the exlicit exressions for anisotro arameters. The arameters of the Hill stress function adjusted to the ield stresses read

6 858 V. Cvitanić and M. Kovačić / Algorithmic Formulations of Evolutionar Anisotroic Plasticit Models Based on Non-Associated Flow Rule = = 0 90 = b = ( 0 / 45) -0.5( 0 / b) l, l ( s / s ), n 0.5( ( s / s ) ( s / s ) ), r s s s s (8) If the Hill stress function is adjusted to the lastic strain ratios, the calculation rocedure results in following exlicit exressions + / r90 (+ r45)(/ r0 + / r90) l =, l =, n =, r = + / r + / r (+ / r ) (9) The orthotroic Karafillis-Boce (993) stress function is a linear combination of two convex non-quadratic functions æ c 3 m f = s s s s3 s3 s s s s = s m- (0) ç + è ø ( - c ) m ( ) ( ) ( ) ö / m m m m m m where c is a weighting arameter and exonent m is an even number. In the above equation, s, s and s 3 are the rincial values of the so-called isotroic lasticit equivalent stress tensor. For lane stress conditions, these values can be calculated as s, s xx + s æ sxx s ö - = + sx, s3 = szz ç () çè ø where ì s ü é xx b b 0ùì s ü xx s b a b3 0 s ï í ï ý = C ï ï s zz b b3 a 0 í ý szz = 0 s x g ê 3 ú s ïî ïþ ë ûî ï x ïþ () and a - a - a - a - - a - a b b b,, 3 = = = (3) In Karafillis-Boce stress function material anisotro is defined b the arameters C, a, a and g 3. For isotroic material, these arameters have the values C = /3, a = a = and g 3 = 3/ and stress comonents defined b Eq. () reduce to the comonents of the stress deviator tensor.

7 C. Cvitanić and M. Kovačić / Algorithmic Formulations of Evolutionar Anisotroic Plasticit Models Based on Non-Associated Flow Rule 859 For Karafillis-Boce stress function, calibration rocedure i.e. calculation of anisotro arameters, leads to the sstem of non-linear equations that can be solved using a numerical iterative rocedure. 3. Evolution of Hill and Karafillis-Boce Yield Function/Plastic Potential In this aer, exerimentall determined directional deendences of the uniaxial material roerties and their evolution with ongoing lastic deformation rocess for 0.8 mm thick DC06 steel sheet reorted b Safaei et al. (04) are used to develo and analze evolutionar anisotroic constitutive models. The utilized data are related to the uniaxial straining tests of the seven secimens with orientations 0, 5, 30, 45, 60, 75 and 90 to the rolling direction. The reorted data utilized in model develoment are: ) arameters of the combined Swift-Voce hardening law b which the exerimental true stress and longitudinal true lastic strain are aroximated; ) arameters of the 3 rd order olnomial fit b which the exerimental transverse and longitudinal true lastic strains are aroximated. In the revious stud conducted b the authors (Cvitanić et al., 06), using these data, arameters of the Hill and Karafillis-Boce ield function/lastic otential are introduced as olnomial functions of the equivalent lastic strain. In the first ste, following rocedure based on the rincile of lastic work equivalence resented in Safaei et al. (04), for each secimen orientation ield stresses s and instantaneous r-values corresonding to the certain amount of the equivalent true lastic strain are calculated. In the adoted aroach, longitudinal true lastic strain in the rolling direction is used as the equivalent true lastic strain. In calculating instantaneous r-values incomressibilit hothesis is alied, thus Eq. (6) is considered in the following form r q deq+ 90 q mq =- =- de de m + q q (4) where d q d q e and e + 90 are the increments of true longitudinal and transverse lastic strains corresonding to the loading direction q and direction q + 90, resectivel. According to Eq. (4), instantaneous r-values corresonding to the certain longitudinal true lastic strain can be calculated using the sloe m q of the available 3 rd order olnomial fit b which the exerimental transverse and longitudinal true lastic strains are aroximated. According to the adoted calculation rocedure, ield stresses and r-values corresonding to the seven orientations (0, 5, 30, 45, 60, 75 and 90 ) and amounts of the equivalent lastic strain starting from 0.00 to 0.30 at each 0.00 increment are calculated. The calculated ield stress ratios (ield stresses normalized with ield stress for the rolling direction) and r-values corresonding to the selected values of the equivalent lastic strain are resented in Figure and Figure, resectivel. From Figure it can be observed that the directional deendence trend of the ield stress ratios at the start of lastic deformation rocess ( e = 0.00 ) is rather distorted with further straining. Furthermore, as resented in Figure, for the considered material, there is a significant decrease of the r-values and certain alternation of the r-value directional deendence with ongoing deformation.

8 860 V. Cvitanić and M. Kovačić / Algorithmic Formulations of Evolutionar Anisotroic Plasticit Models Based on Non-Associated Flow Rule Figure : Yield stress ratio directional deendences corresonding to several values of the equivalent lastic strain. Predictions of ield stress ratios obtained b Hill ield function. Figure : Lankford arameter directional deendences corresonding to several values of the equivalent lastic strain. Predictions of Lankford arameters obtained b Hill lastic otential. These observations clearl indicate the suitabilit of the non-associated flow rule aroach, b which the evolution of ield stress and r-value directional deendences can be indeendentl considered and described. In Figures and, redictions of the ield stress and r-value directional deendences corresonding to the selected equivalent lastic strains obtained b the Hill ield function/lastic otential are also resented. The redictions of the Karafillis-Boce ield function/lastic otential are almost identical to the redictions obtained b the Hill functions and therefore the are not searatel resented. The anisotro arameters of Hill and Karafillis-Boce ield functions are calculated using the ield stresses corresonding to the orientations 0, 45 and 90 ( s 0, s 45, s 90 ) and assuming that the

9 C. Cvitanić and M. Kovačić / Algorithmic Formulations of Evolutionar Anisotroic Plasticit Models Based on Non-Associated Flow Rule 86 ield stress at balanced biaxial stress state is the averaged value of the ield stresses corresonding to the rolling and transverse direction sb = ( s0 + s90)/. The arameters of the Hill and Karafillis-Boce lastic otentials are calculated using r-values corresonding to the orientations 0, 45 and 90 ( r 0, r 45, r 90 ). The arameters of the analzed Hill and Karafillis-Boce ield function/lastic otential are calculated considering calculated ield stresses and r-values corresonding to the equivalent lastic strain values starting from 0.00 to 0.30 at each 0.00 increment. From Figures and, it can be observed that the analzed ield functions oorl redict the ronounced directional deendence at the onset of the deformation rocess while the analzed lastic otentials result in good redictions of the r-value directional deendences, articularl for greater strain levels. In Figure 3 redictions of ield stress ratios obtained b the models based on the associated flow rule and Hill or Karafillis-Boce stress function are resented. In these models, utilized orthotroic stress function adjusted to r-values acts as lastic otential and as ield function. It can be observed that the analzed associated models fail in redicting directional deendence of the ield stress ratios for the considered deformation levels. These results in addition suort the use of the non-associated flow rule aroach couled with analzed four arametric stress functions in describing lastic anisotro and its evolution for the considered material. In order to relate the anisotro arameters with the equivalent lastic strain, the fourth order olnomial fit is used Pol4( e ) = a + b e + b ( e ) + b ( e ) + b ( e ) (5) The calculated olnomial arameters for the analzed ield functions/lastic otentials are obtained using the least square method and are resented in Table. Figures 4 and 5 show values of each anisotro arameter corresonding to the selected values of the equivalent lastic strain and related olnomial fit. From these figures it can be seen that there is a good correlation between adoted fit and lastic otential arameters, while there is a certain discreanc for ield function arameters for the lower lastic levels. a) b) Figure 3: Yield stress ratio directional deendences redicted b associated model and a) Hill ; b) Karafillis-Boce function adjusted to r-values corresonding to several values of the equivalent lastic strain.

10 86 V. Cvitanić and M. Kovačić / Algorithmic Formulations of Evolutionar Anisotroic Plasticit Models Based on Non-Associated Flow Rule Hill ield function l l n r Hill lastic otential function l l n r a b b b b Karafillis-Boce ield function Karafillis-Boce lastic otential function C a a g 3 C a a g 3 a b b b b Table : Parameters of Pol4 fit for Hill and Karafillis-Boce ield function/lastic otential. Contours of the analzed ield functions and lastic otentials in normalized stress sace for zero shear stress corresonding to the selected values of the equivalent lastic strain are resented in Figure 6 and Figure 7, resectivel. For material that obes the isotroic hardening concet, ield contours that corresond to the different amount of lastic strain should coincide if resented in normalized stress sace. Considering obtained contours, it can be observed that there is a significant discreanc between initial and subsequent ield contours and notable evolution of lastic otential contour articularl for the quadratic otential function. a) b) Figure 4: Pol4 function fit for the anisotro arameters of a) Hill; b) Karafillis-Boce ield function.

11 C. Cvitanić and M. Kovačić / Algorithmic Formulations of Evolutionar Anisotroic Plasticit Models Based on Non-Associated Flow Rule 863 a) b) Figure 5: Pol4 function fit for the anisotro arameters of a) Hill; b) Karafillis-Boce lastic otential. a) b) Figure 6: Contours of a) Hill; b) Karafillis-Boce ield function corresonding to e = 0.00; 0.0; 0.0; a) b) Figure 7: Contours of a) Hill; b) Karafillis-Boce lastic otential corresonding to e = 0.00; 0.0; 0.0; 0.30.

12 864 V. Cvitanić and M. Kovačić / Algorithmic Formulations of Evolutionar Anisotroic Plasticit Models Based on Non-Associated Flow Rule 4 STRESS INTEGRATION PROCEDURE In this section, for the analzed evolutionar anisotroic lasticit models based on non-associated flow rule, comutational rocedure for calculating state variables at time t n+ ( σ n+, e n+ ) based on the known state variables at time t n ( σ, e ) and known increment of total deformation Dε is n n derived. The derived rocedure is based on the imlicit return maing (Simo and Hughes, 988; Yoon et al., 999; Cvitanić et al., 008) and resents an extension of the rocedure reviousl derived for evolutionar anisotroic model based on the associated flow rule (Cvitanić et al., 06). B the alication of the imlicit return maing rocedure, the stress solution is obtained in two stes. In the elastic redictor ste, the strain increment Dε is assumed to be elastic and trial elastic stress tensor trial σ is calculated based on the reviousl converged solution σ n trial σ = σ + : Dε (6) n C e If the trial state violates the ield condition, the lastic correction ste is erformed assuming trial state as initial condition. In this ste the final stress σ n+ is stated as e trial e σ + = σ + C :( D ε -D ε ) = σ - C : D ε (7) n n and evolution equations for the lastic strain tensor and hardening arameter are integrated to restore the consistenc condition. B alication of imlicit Euler backward integration rocedure and assuming non-associated flow rule, increment of lastic strain tensor is aroximated as follows D ε = D l( f ( σ, e )/ σ ) n + (8) Consistentl, the increment of the hardening arameter defined b Eq. (5) is aroximated as follows D e = Dl( f( σ, e )/ f( σ, e )) n+ (9) In above aroximations, D l is an incremental consistenc arameter that obes discrete form of the comlementar conditions n+ n+ F n+ n+ D l ³ 0, F( σ, e ) 0, D l ( σ, e ) = 0 (0) B using Eqs. (6)-(9), incremental form of the constitutive model can be stated b the following sstem of four non-linear algebraic equations (tensorial and scalar) f n+ en e rn+ F º ( σ, +D )- = 0 () e - trial n+ l n+ Φ º ( C ) :( σ - σ ) + D n = 0 () n F 3 = r + - k ( e +D e ) = 0 (3) n

13 C. Cvitanić and M. Kovačić / Algorithmic Formulations of Evolutionar Anisotroic Plasticit Models Based on Non-Associated Flow Rule f n+ n f n+ n F = ( σ, e +De ) De - ( σ, e +De ) D l = 0 (4) where n = f (, σ e )/ σ. The solution of above equation sstem (four unknowns n n σ +, r +, D e, D l) can be obtained using iterative Newton-Rahson rocedure. At each iteration, denoted as k, above four equations are linearized around the current values of state variables to obtain ( k) ( k) ( k) ( k) ( k) ( k) f e d e r F + m : D σ + ( / ) D -D = 0 (5) ( k) e - ( k) ( k) ( k) ( k) ( k) ( k) ( k) ( k) ( k) Φ + ( C ) : D σ +D l ( n / σ) : D σ +D l ( n / e ) dd e + dd l n = 0 (6) ( k) ( k) ( k) ( k) F 3 + Dr -( dk / de ) dd e = 0 (7) ( k) ( k) ( k) ( k) ( k) ( k) ( k) ( k) ( k) 4 e m : σ e ( f / e ) d e f d e ( k) ( k) ( k) ( k) ( k) ( k) ( k) ( k) l n σ l f e d e f d l F + D D + D D + D -D : D -D ( / ) D - D = 0 (8) where m = f (, σ e )/ σ. B solving above linearized sstem, exlicit exressions for the increments of state variables are obtained k ( ) ( k) ( k) ( k) - ( k) ( k) ( k) ( k) - ( k) ( k) ( k) F - m :( M ) : Φ + F3 - m :( M ) : n ( F4 / f ) ( k) ( k) æ ( k) f ö ( k) ( k) - ( k) ( k) ( k) ( k) - ( k) - + :( ) : R + :( ) : ç n Dl ç m M n m M dd e = æ ö æ dk ö æ ö ç çèd e ø èç e ø çè e ø çè ø (9) ( k) ( k) - ( k) ( k) ( k) ( k) ( k) ( k) ( k) ( k) ( k) 4 D σ = -( M ) : ( n ( F / f ) + Φ + ( n R + D l ( n / e ) ) dde ) (30) () k () k () k () k () k () k () k () k (/ f ) : ( 4 / f ) R dd l = G D σ + F + dde (3) () k () k () k () k D r =-F 3 + ( dk/ de ) dd e (3) where ( k) e - ( k) ( k) ( ) ( ) ( ) ( ) ( ) C = ( C ) + D l ( n / σ ), k =De k k -Dl k k () k () () () () = k Ä k / f k k + M n G C, G m n, () k () k () k () k () k () k () k () k ( / ) ( e ( / e ) l ( / e ) )/ R = f f + D f -D f f. Finall, udated state variables for non-associated formulation are defined as ( k+ ) ( k) ( k) ì e ü ì e ü ìd e ü D D D r r r ï D í ï ý = ï í ï ý + ï í ï ý σ σ Dσ Dl Dl ddl ïî ïþ ïî ïþ ïî ïþ ( n+ ) ( n+ ) ( n+ ) (33)

14 866 V. Cvitanić and M. Kovačić / Algorithmic Formulations of Evolutionar Anisotroic Plasticit Models Based on Non-Associated Flow Rule If the associated flow rule is assumed, according to Eq. (9) D e = Dl holds and integration scheme roosed b Cvitanić et al. (06) is reroduced. On the other hand, if anisotro evolution is neglected orthotroic arameters of ield function/lastic otential are not altered b hardening arameter and integration scheme conform the one develoed b Cvitanić et al. (008). 5 NUMERICAL ANALYSIS OF ACCURACY. ISO-ERROR MAPS In order to estimate the accurac of the roosed algorithm based on non-associated flow rule and distortional evolution of the ield function and the lastic otential function, iso-error mas (Simo and Hughes, 988) are calculated. The iso-error mas are generall acceted as an effective and reliable tool for assessing the accurac of the constitutive integration algorithms under realistic strain stes. For two-dimensional imlementation of the develoed algorithms, iso-error mas are generated following known rocedure. For the selected stress oint on the ield surface sequence of two- comonent strain increments is alied and for each strain increment numerical solution is obtained according to the tested algorithm. The solution obtained b the same algorithm b dividing the considered strain increment into large number of sub-increments of the same size is adoted as exact solution. Furthermore, for each considered strain increment, based on the obtained numerical and exact solution, the error is calculated as the ercentage of the relative root mean square of errors between the comuted stress tensor σ and exact stress tensor σ *. This error measure is defined as * * ( σ -σ ):( σ -σ ) d(%) = 00 * * σ : σ (34) Finall, for the considered stress oint iso-error ma is drawn as the contour lot of the error field resented over the two-comonent strain sace. B insection of these lots obtained for the reresentative stress oints, deendence of the algorithm accurac on the strain increment size can be assessed. For the analzed formulations, iso-error mas are calculated at three reresentative stress oints on the ield surface: A-uniaxial, B-balanced biaxial and C- ure shear as shown in Figure 8. Figure 8: Plane stress ield surface and oints A, B and C for iso-error mas.

15 C. Cvitanić and M. Kovačić / Algorithmic Formulations of Evolutionar Anisotroic Plasticit Models Based on Non-Associated Flow Rule 867 The strain increments along sheet rolling and transverse direction ranging from zero to six times of the ield strain e are alied to the considered stress oints. The calculations are erformed assuming D e x = 0 thus s x = 0 alies. As the exact solution, the stresses obtained b 00 substes of each strain increment are utilized for the Hill formulations. For the Karafillis-Boce formulations solutions obtained b 50 sub-stes are adoted. In the tested algorithms, the olnomial relations for the anisotroic arameters of the Hill stress functions and Karafillis-Boce stress functions corresonding to DC06 steel sheet that are resented in Section 3 are utilized. Furthermore, combined Swift-Voce stress-strain relation for the rolling direction reorted b Safaei et al. (04) for the considered material is used to govern the ield surface exansion - ke ( ) = c ( k( e + e ) ) + ( - c )( R + Q( - e e )) (35) n b 0 where c = 0.848; k = ; e0 = 0.0; n = 0.36; R = 9.47; Q = 557.3; b = For the elastic constants, Young s modulus E = 00 GPa and Poisson s coefficient n = 0.3 are utilized. Figures 9 and 0 show calculated iso-error mas for three reresentative stress oints obtained b the Hill and Karafillis-Boce formulations, resectivel. Calculated iso-error mas are comared to the mas obtained b the algorithm based on associated flow rule and distortional hardening concet couled with the ield function adjusted to r-values (Cvitanić et al., 06). For the analzed formulations, the magnitudes of errors are comarable to the results reorted in literature obtained b the associated/non-associated formulations based on the utilized orthotroic stress functions and isotroic hardening concet (Cvitanić et al. 008). Commonl, considering oints B and C, for the isotroic formulations the exact solutions are obtained for the loading directed along the ield surface smmetr axis. As shown in Figures 9 and 0, considering these oints, for the analzed orthotroic formulations the axes of exact solution are shifted to the stress smmetr axes. From the comarison of the iso-error mas for different stress oints, it can be seen that the errors are relativel smaller for the balanced biaxial stress state for all analzed formulations. Considering different flow rule aroach, for the Hill formulations there is no significant difference between isoerror mas obtained b the associated or non-associated flow rule. For the Karafillis-Boce formulations, the errors are relativel smaller if associated flow rule is utilized. Furthermore, from the comarison of the iso-error mas for different orthotroic stress functions it can be seen that although there is a difference in mas configuration there is no rominent difference in error magnitudes. Finall, it can be stated that for the analzed formulations, a reduction in error magnitude can be exected when the strain size increment is reduced.

16 868 V. Cvitanić and M. Kovačić / Algorithmic Formulations of Evolutionar Anisotroic Plasticit Models Based on Non-Associated Flow Rule a) b) c) d) e) f) Figure 9: Iso-error mas based on Hill associated formulation for oints: a) A; b) B; c) C and on Hill non-associated formulation for oints: d) A; e) B; f) C. a) b) c) d) e) f) Figure 0: Iso-error mas based on Karafillis-Boce associated formulation for oints: a) A; b) B; c) C and on Karafillis-Boce non-associated formulation for oints: d) A; e) B; f) C.

17 C. Cvitanić and M. Kovačić / Algorithmic Formulations of Evolutionar Anisotroic Plasticit Models Based on Non-Associated Flow Rule CONCLUSIONS In the resent aer, constitutive formulations for sheet metal forming based on non-associated flow rule that enable distortion of the ield function/lastic otential with on-going deformation rocess are analzed. The formulations are develoed considering evolution of the ield stress ratios and r- values with straining obtained in uniaxial tensile tests for DC06 steel sheet samles with different alignment to the rolling direction. As ield function/lastic otential, simle orthotroic quadratic Hill (948) or non-quadratic Karafillis-Boce (993) stress functions are utilized. According to the non-associated flow rule aroach, the anisotro arameters of the ield function are adjusted to the data related to the evolution of the ield stress ratios while the anisotro arameters of the lastic otential are adjusted to the data related to evolution of the r-values. Considering these ield stress ratio and r-value evolutions, based on the rincile of the lastic work equivalence, the anisotro arameters of the utilized stress functions are set as 4 th order olnomial functions of the equivalent lastic strain. For the considered DC06 sheet samle, the analzed non-associated formulations result in accetable redictions of the ield stress and r-value directional deendences and their evolutions with on-going deformation. Besides, considering ield function/lastic otential contours, evolution of the lastic otential is more ronounced eseciall for the quadratic lastic otential function. Therefore, for the considered material, the resented results clearl indicate advantage of the non-associated over the associated flow rule aroach. Furthermore, in the constitutive formulation, for the equivalent lastic strain as the internal variable that governs the evolution of the ield function/lastic otential, evolution equation consistent with the same rincile of lastic work equivalence is utilized. In the resent aer, the algorithmic formulations of the analzed non-associated constitutive descritions with ield function/lastic otential evolution are develoed based on the imlicit Euler backward integration rocedure. Alication of this rocedure results in the sstem of four nonlinear algebraic equations (tensorial and scalar) that resent incremental form of the constitutive equations. For the obtained sstem, the solution rocedure based on the iterative Newton-Rahson rocedure is develoed. Considering data for DC06 sheet samle, accurac of the derived stress integration rocedures has been estimated b calculating iso-error mas for three reresentative stress oints. For the analzed non-associated formulations with ield function/lastic otential evolution, the error magnitudes are comarable to the error magnitudes obtained for the associated/non-associated formulations based on the same orthotroic stress functions and the isotroic hardening concet. It has been revealed that although there is a difference in mas configuration there is no considerable difference in error magnitudes for the formulations based on the different flow rule aroach and the quadratic or the non-quadratic stress function. Furthermore, based on the obtained results it can be stated that for the analzed formulations, a reduction of error magnitude can be exected when the strain size increment is reduced. References An, Y.G., Vegter, H., Melzer, S., Triguero, P.R. (03). Evolution of the lastic anisotro with straining and its imlication on formabilit for sheet metals, Journal of Materials Processing Technolog 3: Aretz, H. (005). A non-quadratic lane stress ield function for orthotroic sheet metals, Journal of Materials Processing Technolog 68: -9.

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