Numerical simulation of sheet metal forming using anisotropic strain-rate potentials

Transcription

1 Numerical simulation of sheet metal forming using anisotroic strain-rate otentials Meziane Rabahallah, Salima Bouvier, Tudor Balan, Brigitte Bacroix To cite this version: Meziane Rabahallah, Salima Bouvier, Tudor Balan, Brigitte Bacroix. Numerical simulation of sheet metal forming using anisotroic strain-rate otentials. Materials Science and Engineering: A, Elsevier, 009, 57 (-), <0.06/j.msea >. <hal-09743> HAL Id: hal htts://hal.archives-ouvertes.fr/hal Submitted on 3 Se 05 HAL is a multi-discilinary oen access archive for the deosit and dissemination of scientific research documents, whether they are ublished or not. The documents may come from teaching and research institutions in France or abroad, or from ublic or rivate research centers. L archive ouverte luridiscilinaire HAL, est destinée au déôt et à la diffusion de documents scientifiques de niveau recherche, ubliés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires ublics ou rivés.

2 Science Arts & Métiers (SAM) is an oen access reository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where ossible. This is an author-deosited version ublished in: htt://sam.ensam.eu Handle ID:.htt://hdl.handle.net/0985/9907 To cite this version : Meziane RABAHALLAH, Salima BOUVIER, Tudor BALAN, Brigitte BACROIX - Numerical simulation of sheet metal forming using anisotroic strain-rate otentials - Materials Science & Engineering A - Vol. 57, n -, Any corresondence concerning this service should be sent to the reository Administrator : archiveouverte@ensam.eu

3 Numerical simulation of sheet metal forming using anisotroic strain-rate otentials MEZIANE RABAHALLAH,, SALIMA BOUVIER, TUDOR BALAN *, BRIGITTE BACROIX LPMTM-CNRS, Université Paris 3, Institut Galilée, 99 rue J.-B. Clément, Villetaneuse, France LPMM, Arts et Métiers ParisTech Metz Camus, 4 rue A. Fresnel, Metz Cedex 03, France Abstract For numerical simulation of sheet metal forming, more and more advanced henomenological functions are used to model the anisotroic yielding. The latter can be described by an adjustment of the coefficients of the yield function or the strain rate otential to the olycrystalline yield surface determined using crystal lasticity and X-ray measurements. Several strain rate otentials were examined by the resent authors and comared in order to analyse their ability to model the anisotroic behaviour of materials using the methods described above to determine the material arameters. Following that, a secific elastic-lastic time integration scheme was develoed and the strain rate otentials were imlemented in the FE code. Comarison of the reviously investigated otentials is continued in this aer in terms of numerical redictions of cu drawing, for different bcc and fcc materials. The identification rocedure is shown to have an imortant imact on the accuracy of the FE redictions. Keywords: sheet metal forming; anisotroy; finite element simulation; cu drawing; earing; strain-rate otential. * Corresonding author. Tel.: +(33) , Fax: +(33) ; address: tudor.balan@ensam.fr (T. Balan).

4 Introduction The role of numerical simulation in automotive sheet metal forming alications is continuously increasing, as new materials are considered in order to answer environmental and security issues. Plastic anisotroic was shown to have an imortant imact on the strain distribution during sheet forming as well as the formability of sheet materials. Thus, its accurate mathematical descrition, along with convenient arameter identification, is a key factor for realistic numerical simulations. This is articularly true when forming limits or sringback must be redicted. Phenomenological yield functions are commonly used do describe lastic anisotroy in finite element codes. Many such functions can be found in the literature (see e.g. reviews [,]). Usually associated with isotroic hardening for industrial alications, they can also be combined with kinematic or more advanced anisotroic hardening [3-7]. The simulation of cu drawing has been often used to address the accuracy of anisotroic yield criteria in FE simulations; Gotoh and Ishise [8] used Gotoh s yield criterion for simulations on a mild steel, Chung and Shah [9] alied the Yld9 criterion for a 008-T4 aluminum alloy sheet, Andersson et al. [0] emloyed the criterion of Karafillis and Boyce for the limiting dome height (LDH) test, and Yoon et al. [] imlemented Yld004-8 and demonstrated its ability to redict six- and eight-ear cu rofiles. Another aroach to erform dee drawing simulations for anisotroic materials is to imlement and aly comutationally-efficient crystal lasticity models in finite element codes, e.g., [-4]. This aroach has the advantage of taking into account texture evolution and its associated deformation-induced anisotroy. However, this method remains very time consuming and has not been roven to rovide better results when comared to the henomenological aroach. Instead of an anisotroic yield function, a strain rate otential can be used in FE codes to simulate forming oerations. Arminjon et al. [5,6] and Van Houtte et al. [7] roosed fourth-order and sixth-order strain rate functions, resectively. Barlat and Chung [8], Chung et al. [9], Barlat and Chung [0], and Kim et al. [] introduced strain rate otentials that were seudo-conjugate of the yield functions ublished earlier. Strain-rate otentials have roven very useful for inverse analysis and blank design alications [-6]. Also, efficient rigid-lastic finite element imlementations have been develoed based on strain-rate otentials [7], allowing for large time increments and simle state udate rocedure. In order to redict unloading and sringback henomena, more exensive elastic-lastic finite element imlementations of strain-rate otentials have been erformed by Van Houtte et al. [8], Bacroix and Gilormini [9], Zhou et al. [30], and Li et al. [3]. Recently, Kim et al. [3] and Rabahallah et al. [33] roosed generic comuter imlementations for strain rate otentials (using shell and solid elements, resectively) and imlemented several models, including the Sr004-8 otential []. One of the most interesting features of strain rate otentials is related to the secific identification techniques based on texture [5,7,34]. In this framework, very flexible models have been roosed with a considerably large number of arameters that cannot be identified by classical methods [35,36]. The main drawback of this identification method is that it relies on the accuracy of the micromechanical model used to construct the reference data.

5 Alternatively, the more classical identification method based on mechanical tests is currently alied to otentials (stress or strain-rate formulation) with numerous arameters (tyically twenty) using a number of exerimental oints equal to the number of arameters. Given the large number of arameters, automatic identification is erformed by minimization of an error function. On one hand, the exerimental task becomes more and more laborious and errorrone. On the other hand, until now, the imact of the identification technique on the finite element results has not been investigated. In addition to the efforts devoted to the rediction of the earing rofile during cu drawing using FE simulations, there have been some analytical attemts at this rediction. The main difficulty in analytical aroaches is determination of the major mechanical arameters that are directly related to the earing rofile. Bourne and Hill [37] observed that the angular ositions, α, of the ears of various mild steel sheets corresond to the maxima of r(π/-α) and to the minima of σ(π/-α). Hosford and Caddell [38] and Chung et al. [4] rovided a quantitative trend between r-value anisotroy and ossible earing rofiles in a mild steel and an aluminum alloy, resectively. In a different aroach, Barlat et al. [,39] attemted to correlate the yield stress anisotroy (not r-values) to the earing trends. More quantitative evaluations of the cu heights have been roosed by Yoon et al. [], based on an analytical aroach considering the r-value directionality as a main contributor to the earing rofile derived from the work of Chung et al. [4]. The aim of this aer is to evaluate the finite element redictions of several strain rate otentials, in articular the recent Sr004-8 model. Five different b.c.c. and f.c.c. sheet materials that exhibit strong dissimilar initial anisotroy have been selected for this urose. Particular attention is given to the imact of the arameter identification method on the FE redictions. For this urose, Sr004-8 has been imlemented in the FE code Abaqus, together with other models used in the comarison [33]. The modeling and comutational issues of this work are summarized in Section, and the FE simulations are analyzed in Section 3, which is the main art of the aer. Constitutive modeling, comuter imlementation, and arameter identification This section summarizes the theoretical modeling and numerical methodologies used for the numerical simulations. For more details, the reader may refer to Kim et al. [] and Rabahallah et al. [36] for strain-rate otential models, Rabahallah et al. [33] for the comlete elastic-lasticity model and its numerical imlementation, and to Rabahallah et al. [34] for the arameter identification of the strain-rate otentials.. Strain-rate-otential-based elastic-lasticity model In order to reresent the rate-insensitive lastic behavior of materials henomenologically, it is tyical to use a yield function, a flow rule, and a hardening law. The yield function Φ gives the stress level at which yielding occurs for a given stress mode, and its gradient (the normal to the yield surface at the loading oint) gives the direction of the lastic strain rate D, while 3

6 the lastic multilier λ ɺ defines its intensity. Alternatively, a dual strain rate otential Ψ [40,4] can be associated with any convex stress otential (or yield surface): Ψ ( D ) = λɺ, () which is exressed as a function of the traceless lastic strain rate tensor, while its gradient with resect to D leads to the direction of the stress deviator σ ; i.e., Ψ σ =. () D τ In Eq. (), τ is a roortionality factor necessary to scale the stress deviator. Its value is related to the reference stress, such as the uniaxial stress in the rolling direction, or to the crystallograhic critical shear stress. When kinematic hardening is taken into account in the modeling, Eq. () takes the more general form: Ψ σ = X + τ D, (3) where X is the second order backstress tensor used to describe the translation of the yield surface in the stress sace. In addition to the lastic otential definition and the flow rule, the hyo-elastic resonse of the material is modeled by: :( ) σɺ C D D, (4) e = where σɺ is the rate of the Cauchy stress tensor σ, D is the total strain rate tensor, and the fourth-order elasticity tensor. A classical nonlinear hardening model is considered, involving both isotroic and kinematic hardening: τ = τ0 + R ; (5) R = CR ( Rsat R) ɺ λ, R(0) = 0 (6) Ψ Xɺ = Cx X sat X ɺ λ, ( 0 X ) = 0. (7) D e C is where τ 0, CR, Rsat, Cx and X sat are material arameters.. Plastic otentials considered in this work Five strain-rate otentials are comared in this work by means of several f.c.c. and b.c.c. sheet materials. The von Mises and Hill otentials are derived as the analytical duals of the corresonding widely known yield functions. The fourth-order otential Quartus [5,6] can be considered as conjugate to the three-dimension extension of the fourth-order yield function given by Gotoh [4]. Finally, two strain-rate otentials with noninteger exonents are considered, as roosed by Barlat and co-workers: Sr93 [8] and Sr004-8 [0,]. The quadratic strain-rate otential can be written as: 4

7 4F 4G 4H Ψ ( D ) = ( D ) + ( D ) + ( D33 ) + ( D3 ) + ( D3 ) + ( D ) L M N (8) where = FH + FG + HG and the three axes,, and 3 are the rolling, transverse, and normal directions, resectively, in the case of rolled orthotroic sheet metal. For an isotroic material, the von Mises otential is recovered by setting the Hill s arameter values to F = G = H = and L = M = N = 3. The fourth-order Quartus otential reads [5]: ( D ) ψ = α k= k X k D ( D ) 3, (9) where α k, k =,..., are material (anisotroy) arameters and X k are terms deending on the lastic strain rate tensor comonents: X = ( D ) X = ( D ) X 3 = ( D3 ) X 4 = ( D3 ) X 5 = ( D ) X 6 = ( D ) D 3 7 = ( ) 8 = ( ) ( ) 9 = ( ) ( 3 ) = ( ) ( ) = ( ) ( ) = ( ) ( ) = ( ) ( ) = ( ) ( ) = ( ) ( ) = ( ) ( ) = ( ) ( ) X8 = D D ( D3 ) = ( ) = ( ) = X D D X D D X D D X D D X D D X D D X D D X D D X D D X D D X D D X D D D X D D D X D D D D X = D D D D 3 3 Finally, the Sr004-8 otential takes the mathematical form []: ( ) ( ) b b b b b b b D E E E 3 E E3 E3 E E E. (0) Ψ = ɶ + ɶ + ɶ + ɶ + ɶ + ɶ + ɶ + ɶ + ɶ b +, () where Eɶ i and Eɶ i are the rincial values of D ɶ and D ɶ, defined resectively by the two linear transformations hereafter: ˆ D ɶ = A I s4 D, () ˆ D ɶ = A I s4 D. (3) In Eqs. () and (3), I s 4 designates the unit tensor in the sace of fourth-order deviatoric and symmetric tensors while the fourth-order arrays A and A contain anisotroy coefficients. For the case of orthotroic symmetry, they can be reresented as the following 6 6 arrays: 5

8 0 a a a0 a a 0 a a 0 a a a a a A = and A =. (4) a a a a a a 9 8 where a i are the material arameters. In order to use these comact notations, the D -like tensors are written here as 6-comonent vectors; e.g., D = [ D D D D D D ] T, with comonents in the frame of material symmetry. The isotroic case is obtained for a = a =... = a8 = and b = 4/3 or 3/ for b.c.c. or f.c.c. materials, resectively. The Sr93 otential [8] can be recovered from the revious model by enforcing which is to say: A I = A I = A, (5) s4 s4 a = a0 = ( a 3 + a ) 4 3 ( 3) 3 a = a = a + a 3 a7 = a6 = a 4 a = a = ( a + a 3) a5 = a4 = ( a + a ) a8 = a7 = a 5. (6) 3 3 a 9 = a8 = a 6 a3 = a = ( a 3 + a ) a6 = a5 = ( a + a ) 3 3 Recently, a generalization of Sr93 and Sr004-8 has been roosed by Rabahallah et al. [36], involving an arbitrary number of linear transformations..3 Comuter imlementation An imlicit backward Euler state udate algorithm has been develoed for the aforementioned elastic-lastic model in the framework of finite strains; this algorithm is described in detail in [33]. The flow rule (3) and of the deviatoric art of the hyo-elastic law (4) take the following discrete (incremental) forms Ψ σ n+ = Xn+ + τ n+. (7) ( ε ) e n+ n C σ = σ + ε ε, (8) : ( ) where X n+ and τ n+ are the internal variables at the end of the increment, obtained as functions of the lastic strain increment ε by time integration of equations (5)-(7); σ and n σ n+ denote the deviatoric stress at the beginning and at the end of the time increment, resectively, while ε is the (given) deviatoric strain increment. By eliminating σ n+ from 6

9 these two equations, the following nonlinear equation is obtained, with the traceless lastic strain rate increment ε as the main unknown: Ψ( ε ) e Xn+ ( ε ) + τ n+ ( ε ) C : ( ε ε ) σ n = 0. (9) ( ε ) The resolution of this nonlinear system is erformed with a Newton-Rahson method. A articularity of the strain rate otential aroach is the lack of an exlicit yield condition, which allows one to decide whether or not the elastic trial violates the yield surface. This roblem is overcome by solving a minimization roblem roosed by Bacroix and Gilormini [9], which states that for a given stress tensor σ < 0 if σ lays outside the yield surface, Min τψ ( N) ( σ X) : N = 0 if σ lays on the yield surface, N > 0 if σ lays inside the yield surface. where N is a unit-length deviatoric tensor designating the lastic strain rate direction. In summary, the constitutive algorithm makes use of two nonlinear numerical rocedures, namely the resolution of the nonlinear system and of the minimization roblem for the yield condition. These rocedures may diverge in case of too large strain increments, esecially when highly nonlinear anisotroic otentials are used. In order to avoid early divergence in the constitutive algorithm, a sub-steing technique has also been develoed and imlemented. This is an extension to strain-rate otentials of the sub-steing rocedure introduced by Yoon et al. [,43] and it is described in detail in [33]. The resulting constitutive algorithm has been imlemented in Abaqus/Standard and has been used to erform all the numerical simulations resented in the current aer. Several numerical validations [33,44] have shown that this imlementation of the strain-rate-otentialbased aroach induces no extra comuting time as comared to the Abaqus built-in udate algorithm that makes use of the yield-function aroach..4 Parameter identification Parameter identification is an imortant issue, esecially when a large number of arameters must be determined. The advanced lastic otentials adoted in this work require about twenty arameters (8 and for the Sr004-8 and Quartus otentials, resectively). Classically, the arameter identification makes use of material data issued from mechanical tests, such as yield stress values and/or r-values from tensile tests, biaxial tests, shear tests, lane strain tests, etc. An alternative method uses a micromechanical model and crystallograhic texture data for the identification and has been extensively used for the arameter identification of lastic otentials [,5,7,34,45]. Both identification techniques have been used in this work, and the imact of the identification rocedure on the results of the FE simulations is one of its main objectives. The two identification methods are briefly discussed hereafter; the reader should refer to [34,44] for more details on the identification rocedure and for a detailed resentation of the arameter identification results and comarison. 7

10 .4. Parameter identification using exerimental mechanical tests The exerimental data considered for identification is a combination of in-lane uniaxial tensile strength and r values along various directions, as well as the balanced biaxial strength ( ) σ b = σ = σ and strain rate ratio r b( = D D ). Whether the number of exerimental data is equal to or larger than the number of coefficients of the otential considered, it is necessary to aly the least-squares method based on an objective function for the identification. Outof-lane roerty data, such as ure shear or uniaxial tension at 45 from symmetry axes, were assumed to be isotroic in this work in order to calculate the out-of-lane anisotroy coefficients. However, more generally, any other convenient deformation state could be considered for the out-of lane roerties. When all of the inut data are selected, the coefficients are obtained by minimizing the following objective function (see []): F Mech m ψ ψ σ ψ ψ wm m m + wm m m + τ m ε ε33 τ ε ε 33 =. (0) σ n ψ ψ σ b ψ ψ ψ τ ij wr + wr w m m + n n ε ε33 τ ε ε n εij τ Here, m reresents the number of uniaxial yield stresses and r values available. The first term under the first summation sign corresonds to the (arbitrary) longitudinal uniaxial tensile stress (direction ) when the imosed strain rate state is calculated with the associated r value. The second term under the first summation sign corresonds to the (vanishing) stress transverse (direction ) to the reviously calculated longitudinal direction. The third and fourth terms corresond to balanced biaxial stress conditions when the imosed strain rate state is calculated with the associated r b value. Finally, n reresents the number of exerimental ure shear yield stresses available (from out of lane roerties in this work). Each term in the objective function is multilied by a weight w. In Eq.(0), the otential is defined with resect to the strain comonents instead of the strain rate comonents, since the otential can be redefined simly by relacing the strain rate with true (or logarithmic) strain when the deformation is monotonously roortional [46]..4. Texture-based arameter identification using a micromechanical model For any given lastic strain rate direction N, the micromechanical Taylor-Bisho-Hill model (TBH) is used to calculate the normalized lastic work rate P P Π TBH ( N) = W ɺ P TBH ( N) τ c = ( σ τc ) : N, where Wɺ TBH ( N) is the lastic work rate and τ c is the crystallograhic critical shear stress. This quantity corresonds to the lastic otential since it can be easily shown (see [34]) that, in the framework of the adoted henomenological modelling, ( N) ψ = Wɺ ( N) τ. () 8

11 P In other words, for any strain rate direction N i, the functions TBH ( i ) Π N and ψ( N ) corresond to the lastic ower associated to a unit-norm strain rate tensor and normalized by τ c. The coefficients of the lastic otential ψ are thus identified by minimizing the objective function: i F Tex = i w Π ( N ) ψ ( N ) P i TBH i i i Π P TBH ( Ni ), () with resect to the coefficients of the chosen otential. The sum is erformed over P Π N are selected 3D strain rate directions [5]. Then, the micromechanical values TBH ( i ) comuted for all of these directions. This is a lengthy task, but has to be erformed only once for a given initial crystallograhic texture. The nonlinear least-squares-roblem is solved using a Levenberg-Marquardt minimization algorithm [47]. This algorithm requires the calculation of the objective function and its firstorder derivatives with resect to the arameters subject to identification. 3 Finite element simulation of cu drawing rocesses Cu drawing is a simle and tyical sheet forming rocess, which is widely used to illustrate sheet metal anisotroy and to evaluate the rediction caabilities of yield functions. Although more comlex sheet metal forming rocesses can be simulated with the existing imlementation [33], cylindrical cu drawing is adoted here for several reasons: This test has been widely used to evaluate different materials and corresonding data is available in the literature; Sheet metal roviders systematically erform cu drawing tests for all sheet materials; Although simle, this test is relevant for industrial forming of beverage cans and other ackaging roducts. It also has ractical relevance for more comlex sheet forming rocesses, esecially in terms of in-lane anisotroy; Cu drawing involves large strains, together with contact evolution, thus requiring a comlete numerical simulation; Most imortantly, the cu rofile has been shown to be sensitive to the initial anisotroy (and to the corresonding models and arameters) and to illustrate it in a very intuitive way. On the other hand, it is almost insensitive to the hardening model [48-50], at least as long as texture evolution is not taken into account in the modeling [5,5]. As a consequence, the imact of the anisotroic otential can be decouled from the imact of the hardening model. Several sheet metals have been used in [34] for a detailed comarison of advanced strain-rate otentials. The imact of the identification rocedure has been articularly addressed. The aim of the current investigation is to ursue this comarison in terms of finite element redictions of cu heights for cylindrical cu drawing. This section starts with an overview of the material roerties and cu drawing test geometry. A detailed mesh sensitivity analysis has been erformed in order to investigate the accuracy and limitations of the numerical results. Then, cu heights are redicted for five materials using all lastic otentials resented 9

12 in Section.. These results are used to investigate the ability of the recent Sr004-8 otential to redict various tyes of anisotroy; exerimental validation data is rovided and discussed whenever available, together with a comarison of the strain-rate-otential aroach and the yield-criterion aroach. Eventually, the effect of the arameter identification technique is evaluated in terms of finite element redictions. 3. Materials and test configurations In order to erform a detailed comarison and to enhance the generality of the discussion and conclusions, five different materials have been selected for this study. These are both f.c.c. (several aluminum alloys) and b.c.c. (mild ferritic steel DC06, dual hase steel DP600) sheet metals, exhibiting various kinds of anisotroic behaviors. The study required material data (exerimental data from mechanical tests and/or texture data), together with cu drawing test data and cu height results. This information has been exerimentally generated in revious work or collected from literature [,,53,54]. The geometries of the cu drawing tests are given in Table and Figure, together with the hardening arameters, friction coefficient, and an outline of the material data available for the anisotroy arameter identification and validation. Table. Material data and cu drawing test descrition for the five materials used in the resent work. Aluminum alloys Steels Materials: AA60 AA008 AA090 DP600 DC06 Cu drawing test geometry; dimensions in [mm] Sheet thickness Samle diameter Die radius (r die ) Punch radius (r unch ) Die diameter (D die ) Punch diameter (D unch ) Blank-holder force [N] Material hardening and friction arameters Yield stress (rolling dir.) [MPa] C r R sat [MPa] C x X sat [MPa] Coulomb friction coefficient Available data for anisotroy identification and validation Texture X X X Mechanical tests (r, σ ) X X X X Cu height rofile X X 0

13 D die Die Samle Blank holder Punch D unch D samle Figure. Cu drawing test geometry. 3. Mesh sensitivity analysis Finite element simulation of cu drawing is known to deend on the descrition of the lastic anisotroy in the modeling and on the corresonding arameters. However, when quantitative results are considered, several numerical features and arameters may also have a considerable influence [44,49,55]. For the choice of the finite element and mesh density, a sensitivity analysis has been erformed that comares two linear solid elements, three inlane mesh densities, and different numbers of element layers through the thickness. The comuter imlementation of the material model [33,44] is restricted to solid elements (all the lastic otentials being fully three-dimensional). Moreover, linear elements are recommended whenever nonlinear roblems are solved which involve contact evolution. In this category, reduced integration elements are often referred in industrial alications due to their time efficiency and good overall accuracy. In this investigation, the reduced integration element C3D8R available in Abaqus/Standard is comared to the incomatible-modes enriched, hybrid dislacement-ressure element C3D8IH. Three degrees of mesh refinement are comared (see Figure ) that are designated in the following as coarse, fine and very fine, resectively. As for the thickness direction, u to three layers of elements have been tested, thus generating meshes containing between 600 to 5500 elements for ¼ of the sheet. The material model used throughout the mesh sensitivity analysis is Hill s quadratic lastic otential corresonding to the DC06 mild steel (with anisotroy coefficients r 0 =.53, r 45 =.84, r 90 =.7). Obviously, these designations have no absolute meaning since a mesh can be considered either coarse or fine deending on the aim of the simulation.

14 a) Very fine mesh 850 elements (40x45) Fine mesh 950 elements (30x30) Active zone; structured mesh (radius x circumf erence) Inactive zone; f ree mesh (coarse) Coarse mesh 600 elements (0x4) b) c) Figure. Finite element simulation of cu drawing: a) meshes used for the mesh sensitivity analysis; b) intermediate and c) final configurations redicted with the fine mesh. Figure 3 shows the mesh sensitivity of the cu height redictions with different elements and mesh densities. It is obvious from Figure 3a that the redictions with the reduced integration element are very sensitive to mesh density. Finer meshes bring the redictions closer to the converged solution. Moreover, buckling aeared during the simulations with one layer of elements through the thickness (thus not reresented on the lot). In contrast, the hybrid element exhibits almost no mesh sensitivity, even when the coarse mesh is used with only one layer of elements. In articular, the number of layers aears to have no influence at all on the cu height, which will be the main simulation result exloited in the following. The mesh sensitivity of the strain distributions is illustrated by Figure 4. The logarithmic strain in the radial direction is reresented as a function of the curvilinear coordinate measured from the center of the cu to its rim, along its outer surface (facing the die), in the rolling direction. The strains are very small on the bottom of the cu and they rogressively increase along the cu wall, with a local increase of the radial strain in the unch radius zone due to bending. Since bending induces a strain gradient through the thickness, the values

15 reresented in Figure 4 corresond to the outer surface of the sheet. Indeed, this is the surface where exerimental strains are usually measured in sheet metal forming. Cu height (mm) layers - coarse mesh 3 layers - coarse mesh layers - fine mesh 3 layers - fine mesh Cu height (mm) layer - coarse mesh layers - coarse mesh 3 layers - coarse mesh layer - fine mesh layers - fine mesh 3 layers - fine mesh layer - very fine mesh a) b) Angle from the rolling direction ( ) Angle from the rolling direction ( ) Figure 3. Cu height redictions with the different meshes using a) the reduced integration element C3D8R, and b) the hybrid element with incomatible modes C3D8IH. Radial logarithmic strain radius bottom wall layer - coarse mesh layers - coarse mesh 3 layers - coarse mesh layers - fine mesh 3 layers - fine mesh a) Radial logarithmic strain layer - coarse mesh layers - coarse mesh 3 layers - coarse mesh layers - fine mesh 3 layers - fine mesh bottom radius wall b) Radial curvilinear coordinate Radial curvilinear coordinate Figure 4. Radial strain distribution along the rolling direction redicted with the different meshes using: a) the reduced integration element C3D8R, and b) the hybrid element with incomatible modes C3D8IH. 3

16 The imortant mesh sensitivity of the reduced integration element is also observed in terms of strain distribution. The one-layer simulations fail to redict the strain increase due to bending, since only one integration oint is resent through the thickness. Finer meshes redict strains closer to the converged solution. Again, the hybrid element exhibits almost no mesh sensitivity. These conclusions corroborate well with the cu height redictions, since these two quantities are related. It is noteworthy that the radial strain resents a considerable jum close to the rim of the cu. Indeed, a strain variation as large as takes lace along the last element of the mesh in the radial direction. The exlanation for this henomenon is found at the end of the blank holding hase, when the rim of the flange loses contact with the blank-holder. As shown in Figure 5a and b, the last element in contact with the blank-holder is strongly inched between it and the die, as it conveys the entire holding force. It is noteworthy that contact on a single node or a line of nodes is not recommended and its effect strongly deends on the contact surface definition in the FE code. This inching effect is also visible in the thickness distribution reresented in Figure 5c; indeed, the thickness strongly decreases along the last element, in the rolling and transverse directions, while it increases continuously in the 45 direction. This henomenon corresonds to the real inching of the samle in dee drawing rocesses with constant holding force. Figure 5d shows exerimental evidence of this henomenon, which often exaggerates any misalignment between the samle and the tools (here, the right-hand side of the cus has been inched more strongly). Deending on the strength of the material and the holding force, the occurrence of inching can modify the shae of the ears, which do not deend only on material anisotroy but also on such rocess arameters. Finally, the mesh sensitivity of the redicted stresses is also investigated. Figure 6 shows the equivalent stress distributions on the inner surface of the cu for the different meshes and finite elements. When buckling has occurred, the stress distribution has been strongly deviated and could not be comared to the remaining ones. While the reduced integration element exhibits relatively small mesh sensitivity in terms of redicted stresses, a much larger sensitivity is observed for the hybrid element. The in-lane refinement of the mesh does not really influence the stresses, yet the number of layers has a very imortant imact since the maximum stress is reduced by 5% between the simulations with three layers and only one layer, resectively. One ossible exlanation for this behavior could be the technique used in the FE code to extraolate stresses to the surface of the art. Indeed, when one layer of elements is used, the stresses are calculated at two integration oints along the thickness and then extraolated to the two surfaces. A non-negligible stress gradient is observed between the inner and the outer surface of the cu even when one single hybrid element is used. In summary, the cu height redictions are mesh-insensitive whenever the hybrid element is used, and one single layer of elements is sufficient in any case. The fine mesh will be used for all the following simulations, with one layer of elements through the thickness. While the (radial) strain distributions are fairly insensitive to the mesh, the redicted stresses are much more sensitive and should not be used for the comarison of different models. It has also been observed that rocess arameters such as inching, holding force, clearance between the unch and the die as well as numerical arameters like the value of friction coefficient or contact algorithm arameters have a non-negligible influence on the cu heights, which 4

17 cannot be considered as a ure effect of material anisotroy (although this is their main origin). a) b) blank-holder die Thickness (mm) art c) Initial Rolling direction Diagonal direction Transverse direction 0.75 d) Normalized initial coordinate Figure 5. Pinching effect at the end of the blank holding ste: a) as redicted by Abaqus; b) detail of the one-element inched zone; c) radial thickness distribution along the rolling direction, the transverse direction, and at 45 with resect to the rolling direction; d) exerimental cus in mild steel which exhibit inched ears (to right). 5

18 Figure 6. Mesh sensitivity of the equivalent stress distribution for the cu drawing simulations with reduced integration elements (left) and hybrid dislacement-ressure elements (right), for different mesh densities and one, two, or three layers of elements through the thickness. 6

19 3.3 Results and analysis of the cu drawing simulations The correctness of the numerical imlementation of the constitutive model and its state udate algorithm have been validated [33] in the case of Hill s otential. The aim here is to investigate the accuracy of the finite element redictions for the other strain-rate otentials, esecially Sr The results are comared to exeriments and to the redictions if the Yld004-8 stress otential. The comutational efficiency of the roosed models is also addressed, together with their accuracy and their ability to describe the lastic anisotroy of various materials Comarison to exeriments and to the Yld004 criterion redictions Chung and Shah [9] rovided the exerimental cu rofile and the r-value in-lane variation for the cu drawing of the AA008 material. Figure 7 shows these results together with the redictions obtained in the current investigation with three otentials. The in-lane variation of the anisotroy coefficient is very well redicted by the three otentials, the three redictions being very close to each other. However, the cu rofiles redicted by Quartus and Sr004-8 are different from those of Hill s otential, in site of their close redictions of r-values. This is not surrising in view of the more detailed comarison given in [34]. There, it has been observed that when mechanical tests are used for arameter identification, the Quartus and Sr004-8 yield surfaces are very close to each other (Figure 8 in [34]) and the corresonding error functions are almost identical (Figure in [34]), while the one corresonding to Hill is much larger, indicating that its redictions of the stress distribution and ossibly the biaxial stress oint are less accurate. Hill s redictions of cu height overestimate the amlitude of the ears, while they better locate the valley between them. Figure 7 seems to confirm the (loose) similarity between the r-values r(α) and the cu heights h(π/-α), reorted in several aers. Based on this assumtion, Yoon et al. [] have develoed an analytical formula for the cu height distribution, whose redictions are also given in Figure 7. With this analytical formula, the ear locations are well redicted but the cu height is not accurate. A second material is used for the exerimental confrontation; this is the AA090,aluminum alloy which exhibits six ears []. The corresonding cu height redictions are given in Figure 8, along with the r-values. As exected, Hill s otential fails in redicting more than four ears. However, the height redictions of the quadratic otential are surrisingly accurate (excet for the two missing ears, which haen to be relatively small), in site of a oor rediction of the r-values. Sr004-8 redicts six ears at the good locations, however the years at 0 and 80 are underestimated and hardly distinguishable; its redictions lie close to the ones obtained by Yoon et al. [] with Yld Quartus redicts eight ears, although two of them are almost negligible (at 90 and 70 ) and its overall redictions are very similar to the ones of Sr004-8 and Yld Finally, the analytical formula of Yoon et al. [] redicts the number and the location of the ears but their amlitude is underestimated. 7

20 Cu height (mm) Sr004-8 Quartus Hill Exerimental data Analytical result Angle from the rolling direction ( ) r(α) Sr004-8 Quartus 0.4 Hill Exerimental data Angle from the rolling direction ( ) Figure 7. Cu height and r-value redictions for the AA008 aluminum alloy Cu height (mm) Sr004-8 Quartus 4 Yld004-8 Hill Exerimental data Analytical result Angle from the rolling direction ( ) r(α) Sr Quartus Hill 0. Exerimental Angle from the rolling direction ( ) Figure 8. Cu height and r-value redictions for the AA090 aluminum alloy. 8

21 3.3. Ability of Sr004 to describe various tyes of anisotroy The Sr004-8 model is further investigated in this section since it has been seldom alied in the literature to address its redictive caabilities in FE simulations. Figure 9 reviews the cu drawing results obtained for some of the materials investigated in this study. As commonly acceted, the shae of the ears aears to follow loosely the shae of the r( α ) grah and several four-ear cus are correctly simulated, as well as the six-ear cu already discussed. Two more simulations are used to further illustrate the flexibility of the Sr004-8 otential. A virtual material has been generated by enforcing the r-value variation deicted in Figure 0a. After arameter identification based on this data, the cu drawing simulation allowed for the rediction of eight ears. Finally, the tensile stress variation of an AA58 aluminum alloy [34] deicted in Figure 0b has been used to identify the arameters of Sr As shown in Figure 0b, the simulation leads to a ten-ear cu rofile. Therefore, u to ten ear cu rofiles can be redicted with Sr AA60-T43 AA090-T3 AA008-T4 DC06 DC06 AA090-T3 r(α).5 AA60-T Angle from the rolling direction ( ) AA008-T4 Figure 9. Exerimental r-values and redicted cu heights for the different materials, using Sr004-8 with material arameters identified by mechanical tests. 9

22 a) b) (a). Virtuel Material c.95 σ /τ r(α) Sr004-8 Exerimental data Angle from the rolling direction ( ) α( ) 0 ears 8 ears Figure 0. Cu drawing simulations with Sr004-8, for a) a virtual material, exhibiting eight years and b) the AA58 aluminum alloy, exhibiting ten ears Comarison of CPU time The revious comarisons have shown the suerior ability of Sr004-8 and Quartus to redict six, eight, or more cu ears and to reroduce the r-values more accurately. However, the cu height redictions of the quadratic otential are surrisingly good for the analyzed tests and materials. In order to ut this comarison into ersective, Table shows the comuting time required by the cu drawing simulations for two materials when various otentials are used (the calculations have been run on a PC comuter with a Pentium.8 GHz dual core rocessor). The quadratic otential is also comared to the Abaqus built-in Hill 48 yield criterion, which is taken as reference. The quadratic otentials give not only identical redictions [33], but the comuting time is almost the same (less than 5% larger in the strainrate otential case). However, the use of the Quartus otentials increases the comuting time by a factor of two. The cost of the Sr004-8 otential is even higher, going from a factor of 9 u to 4, deending on the nonlinearity of the otential. Indeed, the shae of the yield surface for the AA60 material requires a systematic activation of the sub-steing rocedure, thus increasing the comuting time considerably. This is definitely a art of the comuter code that might be imroved in the future. 0

23 Table. Comuting time imact of the different strain-rate otentials. Potential CPU time Number of DP600 AA60-T43 increments Hill48 (Abaqus) 3h7 (00%) 3h3 (00%) 40 Hill48 (Umat) 3h37 (05%) 3h39 (04%) 40 Quartus 8h4 (5%) 5h50 (66%) 38 Sr h49 (9%) 84h3 (390%) 3 Figure recalls the gains in accuracy obtained with each of these otentials, as calculated in [34] during the arameter identification rocedure. This theoretical accuracy gain is balanced by its cost (Table ) as well as its ractical relevance e.g., in cu drawing simulations, as reorted in the revious sections..e-03 Hill Quartus Sr004-8.E-04.E-05 AA58 AA606 DC mm DP mm DP600.0 mm HSLA 0.7 mm HSLA.0 mm Figure. Texture-based error functions, Eq. (9), for the three strain rate otentials and for several sheet materials investigated in [3].

24 3.4 Imact of arameter identification on FE simulation results The revious simulations have shown that the choice of the strain rate otential may have an imortant imact on the redicted cu rofile. It has been also recognized for a long time that the data used for the arameter identification may lay an imortant role and may be more imortant than the model itself [5,5,54,56]. In the case of Hill s yield criterion, for examle, the arameter identifications based on the r-values or based on the uniaxial yield stresses, resectively, are known to lead to different results. However, for more flexible otentials, the number of arameters becomes so large that all the exerimental data available is used for the identification. Moreover, there is no analytical link between measured quantities and model arameters (as is the case for the quadratic model); the arameters are usually automatically identified by an inverse method. In [34], the identification method based on mechanical tests has been comared to the identification method based on texture data and a micromechanical model. The latter has roven to be valuable tool for classifying in a rigorous way different otentials, in terms of their mathematical ability to fit different tyes of anisotroy. It also rovides the only means available to identify material arameters related to out-of-lane comonents of the lastic strain rate tensor. It has been noticed however that, as their accuracy solely deends on the redictive caacity of the micromechanical model, in some situations the results may oorly fit mechanical test results. Similar conclusions have been ublished recently by Grytten et al. [57]. Therefore, the imact of the identification technique on the FE results is addressed in the following. Figure comares the r-values of the dual-hase steel redicted by Sr004-8, for each of the two identification techniques. The r-values redicted by the TBH model are different from the exerimental ones, although both show maxima in the rolling and transverse directions. It is clear from these lots that when the identification is erformed with the TBH model, the henomenological redictions lay closer to the r-values redicted by the TBH model, even though in this case the r-values are not used as data for identification (see [34]) for more details on the texture-based identification rocedure). When mechanical tests are used, the redictions fit closer to the exeriments. In this articular case, one can note that the otimal fit obtained by the automatic identification caused slight deviation of one maximum of the curve, which lies at 7 instead of 90 with resect to the exerimental data. In terms of curve fitting, this is the otimal solution that could be achieved with the given mathematical model.

25 ... TBH Sr004-8 Exerimental data. TBH Sr004-8 Exerimental data r(α) r(α) Angle from the rolling direction ( ) Angle from the rolling direction ( ) Figure. Imact of the identification technique on the redicted r-values for the Sr004-8 otential and comarison to exerimental and micromechanical results; a) identification using texture data, and b) identification using exerimental data from mechanical tests Sr004-8 (Texture) Quartus (Texture) Sr004-8 (Mechanical tests) Quartus (Mechanical tests) b) Cu height (mm) a) (a) Angle from the rolling direction ( ) Figure 3. Numerical simulations for the DP600 steel sheet: a) cu height redicted by Sr004-8 and Quartus with the two identification techniques; b) eight-ears rofile redicted with Sr004-8 and mechanical tests identification (arrows indicate the four unrealistic ears). 3

26 In order to further investigate this issue, Figure 3 comares the cu rofiles for the dual hase steel sheet, as redicted by Quartus and Sr004-8 when both identification techniques are used. The two redictions with Quartus are very close to each other, redicting four ears as suggested by the in-lane anisotroy grah. The texture-based Sr004-8 is slightly different, yet has the same characteristics. In turn, the exeriment-based Sr004-8 redicts sulementary, unrealistic ears at aroximately 4 with resect to the rolling direction. The height of these ears is very small since the anisotroy of dual hase steels is not very large; however, it is surrising that the number of ears is not correctly redicted. As one can see in Figure 4 (a and b), this is not an excetional case, and it is not secific to the Sr004-8 otential only. Indeed, the exeriment-based Quartus redictions for the AA60 aluminum alloy sheet exhibit eight ears, while the texture-based Quartus and the two Sr004-8 versions correctly redict only four ears. The four-ear rofile is doubtlessly suggested by the r-value distribution (Figure 4c), which is moreover fairly classical and easy to fit with almost any anisotroic model. However, the yield stress distribution (Figure 4d) associated to it is different in shae and is almost a straight line. The automatic identification rocedure generated a best-fit solution for which the uniaxial yield stresses oscillate in the neighborhood of the exerimental values, thus minimizing the objective function at most. As a consequence, the redicted cu rofile lies relatively close to the correct one but with two surious oscillations at 0 and 90. 4

27 Normalized cu height..05 Sr004-8 (Texture) Quartus (Texture) Sr004-8 (Mechanical tests) Quartus (Mechanical tests) b) 0.95 a) r(α) Angle from the rolling direction ( ) α( ).04 c).03 d).0 Quartus Exerimental data σ /σ b Quartus 0.96 Exerimental data α ( ) Figure 4. Numerical simulations for the AA60 aluminum alloy sheet: a) cu height redicted by Sr004-8 and Quartus with the two identification techniques; b) eight-ears rofile redicted with Quartus and the mechanical tests identification; c) r-values and d) uniaxial stresses (normalized by ) as redicted by the exeriment-based Quartus model. 5

28 In site of these two extreme examles, the exeriment-based identification technique consistently rovides arameter sets whose redictions better reroduce the exerimental results (as comared to the TBH-based arameters). Therefore, these examles do not reresent a criticism against the exeriment-based identification, but rather a warning regarding the automatic arameter identification aroaches (which are more and more oular), whatever reference data is used (either texture or exeriments). As one can see in Figure 5, the choice of the identification method may have considerably more imact on the redicted results (here, the logarithmic radial strain) than the choice of the otential. This is esecially true for the more flexible models (e.g., Quartus, Sr004-8), while Hill s model exhibits small sensitivity to the identification method. In this framework, the very large number of data used by the texture-based identification seems to revent the solution from aberrant oscillations. For the Quartus otential, it has already been shown to lead less often to arameter sets that violate the convexity conditions [34]. However, in the case of the exeriments-based identification of advanced otentials, the number of reference data is very often equal to the number of arameters to identify; this may lead to oscillating solutions between the exerimental oints. Similar conclusions have been drawn by Bouvier et al. [56] for the identification of yield criteria with reduced sets of exerimental data. The authors suggest increasing the number of reference data-oints with seudo-exeriments generated by the Hill model. While these data are given a smaller weight so that they do not deviate the solution significantly from the real exeriments, they revent aberrant results from being acceted. In our case, the combined exeriments-texture identification method roosed in [34,58] rovides a consistent and accurate method to overcome such aberrant solutions; in the meantime, it allows for the identification of the otential s arameters corresonding to the out-of-lane terms of the lastic strain rate tensor which cannot be identified by exeriments alone. The smoothing effect of the texture-based identification is obvious from the resulting yield surfaces [34]. Figure 6 shows the redicted yield surfaces for the AA60 aluminum alloy when one or the other identification methods are used. The exeriments-based models lay close to the exerimental uniaxial stress oints and to the exerimental biaxial stress oint (at least for Quartus and Sr004-8). They also redict the exerimental r-values much better than the texture-based versions. On the other hand, however, the corresonding yield surfaces are fairly different from each other since very few data oints are available for the σ σ. In contrast, the texture-based identification makes use of identification in the lane ( ) numerous, uniformly sread data oints and thus generates yield surfaces that are much closer to each other. It is likely that the same conclusions aly to other D reresentations of the yield surface, including shear terms, where almost no exerimental oint is resent. 6