PARAMETER IDENTIFICATION OF ADVANCED PLASTIC POTENTIALS AND IMPACT ON PLASTIC ANISOTROPY PREDICTION

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1 PARAMETER IDENTIFICATION OF ADVANCED PLASTIC POTENTIALS AND IMPACT ON PLASTIC ANISOTROPY PREDICTION Meziane Rabahallah, Tudor Balan, Salima Bouvier, Brigitte Bacroix, Frédéric Barlat, Kwansoo Chung, Cristian Teodosiu To cite this version: Meziane Rabahallah, Tudor Balan, Salima Bouvier, Brigitte Bacroix, Frédéric Barlat, et al.. PARAM- ETER IDENTIFICATION OF ADVANCED PLASTIC POTENTIALS AND IMPACT ON PLASTIC ANISOTROPY PREDICTION. International Journal of Plasticity, Elsevier, 009, 5 (3), <0.06/j.ijlas >. <hal > HAL Id: hal htts://hal.archives-ouvertes.fr/hal Submitted on 7 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/0985/9934 To cite this version : Meziane RABAHALLAH, Tudor BALAN, Salima BOUVIER, Brigitte BACROIX, Frédéric BARLAT, Kwansoo CHUNG, Cristian TEODOSIU - PARAMETER IDENTIFICATION OF ADVANCED PLASTIC POTENTIALS AND IMPACT ON PLASTIC ANISOTROPY PREDICTION - International Journal of Plasticity - Vol. 5, n 3, Any corresondence concerning this service should be sent to the reository Administrator : archiveouverte@ensam.eu

3 PARAMETER IDENTIFICATION OF ADVANCED PLASTIC POTENTIALS AND IMPACT ON PLASTIC ANISOTROPY PREDICTION M. Rabahallah *, **, T. Balan **, S. Bouvier *,, B. Bacroix *, F. Barlat +, K. Chung ++ and C. Teodosiu * * LPMTM-CNRS, UPR900, University Paris3, 99 Av. J-B. Clément, Villetaneuse, France ** LPMM, UMR7554, ENSAM Metz, 4 rue A. Fresnel, Metz Cedex 3, France + Alcoa Technical Center, Materials Science Division, 00 Technical Drive, Alcoa Center, PA , USA ++ Deartment of Materials Science and Engineering, Intelligent Textile System Research Center, Seoul National University, 56-, Shillim-dong, Kwanak-gu, Seoul, 5-74, Korea December 0 th, 006 ABSTRACT In the work resented in this aer, several strain rate otentials are examined in order to analyze their ability to model the initial stress and strain anisotroy of several orthotroic sheet materials. Classical quadratic and more advanced non-quadratic strain rate otentials are investigated in the case of FCC and BCC olycrystals. Different identifications rocedures are roosed, which are taking into account the crystallograhic texture and/or a set of mechanical test data in the determination of the material arameters. KEYWORDS: Anisotroic sheet metals, Strain rate otentials, Parameter identification, Plasticity, Micromechanical model.. Introduction Numerical simulation is nowadays commonly used in industry for the otimization of the forming technologies for manufacturing new arts and roducts. Several commercial comuter codes are available for this urose. The accuracy of the simulations deends on the ability of the simulation codes to suitably describe the behaviour of the material during Corresonding author: Tel.: Fax: address: salima.bouvier@lmtm.univ-aris3.fr

4 forming. In sheet metal forming, materials are rimarily characterized by their hardening behaviour and by their initial lastic anisotroy, which is mainly due to the crystallograhic texture. The descrition of this initial anisotroy is one of the key factors that guarantee the reliability of the finite element simulations of forming rocesses. This is articularly true when final art roerties like sringback or forming limits are to be investigated. The initial lastic anisotroy of sheet metals can be assessed by means of micro-mechanical crystal lasticity calculations, considering the material as a collection of grains of different orientations, subjected to a given loading ath, and obeying to the Schmid law. The well known Taylor model (Taylor, 938; Bisho and Hill, 95), as well as other olycrystal schemes such as self-consistent models (e.g. (Berveiller and Zaoui, 979)), have been widely used for this urose. However, the large comuting times associated with this method have revented its wide utilization in an industrial environment. Alternatively, continuum mechanics rovides a general theoretical framework for the socalled henomenological descrition of lastic anisotroy. This aroach is based on the use of lastic otentials and associated flow rules for the comutation of stresses and strain rates. The otential can be defined either as a function of stresses (stress otential or yield function) or lastic strain rates (strain rate otential). The classical quadratic (Hill, 948) criterion, widely imlemented in commercial codes, rovides an aroximate descrition of the real yield locus. Hill himself roosed more comlex yield functions (Hill, 979; Hill, 993), yet restricted to secial loading conditions such as lane stress or assuming that the rincial stress axes are also the orthotroic axes of the material. (Gotoh, 977) introduced a lanestress, fourth-order olynomial yield function. (Budiansky, 984) exressed the yield function in olar coordinates, an aroach that was further develoed by (Ferron et al., 994). (Vegter, 99) roosed the reresentation of the yield function with the hel of a Bezier interolation of selected mechanical test results. An imortant contribution in this area has been made by (Hersey, 954) and (Hosford, 97) who introduced a very accurate yield function for isotroic olycrystals, as comuted with crystal lasticity models. This yield function was generalized to orthotroic materials by (Barlat et al., 99, 997), (Karafillis and Boyce, 993), (Barlat et al., 003). Reviews of anisotroic yield functions can be found in (Życzkowski, 00; Banabic, 00; Yu, 00 and Barlat et al., 004). Recently, (Barlat et al., 005) roosed a new yield criterion based on two linear transformation functions, while

5 (Cazacu and Barlat, 00, 003) roosed anisotroic extensions to Drucker s yield criterion, based on the theory of reresentation of second-order tensors. As shown by (Ziegler, 977 and Hill, 987), two convex otentials dual of each other, exist from which the stress tensor can be derived as a function of the strain rate tensor and viceversa. Yield functions, such as those listed above, act as otential functions for the determination of the lastic strain rate tensor using the normality rule (only associated flow rules are considered in the current work, although the theory at hand is not restricted to this articular case). Any mathematical function used to define yielding could be transformed in order to describe a lastic otential in lastic strain rate sace. However, excet for a few secific cases, their analytical exression is virtually imossible to obtain. Equivalently, lastic otentials can be defined in the sace of lastic strain rates using their gradient to derive the deviatoric stresses. Formally, the two aroaches, stress or strain rate otential, are identical. For some alications such as rigid-lastic finite element (FE) simulations (Barlat et al., 994; Yoon et al., 995; Chung et al. 996, 997), minimum lastic-work ath calculations (Barlat et al., 993), and analytical calculations in forming, the strain rate otential aroach can be comutationally more suitable. (Arminjon et al., 99, 994 and Van Houtte et al., 989) roosed fourth-order and sixth-order strain rate functions resectively. (Barlat and Chung, 993; Barlat et al., 993 and Chung et al., 999) introduced strain rate otentials that were seudo-dual of yield functions ublished earlier. Regardless of the tye of otential, imroved accuracy and versatility are obtained at the exense of a larger number of anisotroy arameters. Consequently, an increased number of mechanical tests are required for the identification of these material coefficients. Since Hill s ioneering work on lastic anisotroy of sheet metals, the most oular exerimental tests in this area are the tensile tests in the rolling, transverse and 45 directions. Either the corresonding Hill coefficients of anisotroy, yield stresses or both are used for the identification, deending on the number of arameters and the comlexity of the otential. Nevertheless, most advanced anisotroic functions require more exerimental data. Although additional tensile tests, along different directions with resect to the rolling direction can be used (e.g. (Gotoh, 977)), it has been clearly shown that the use of the yield stress corresonding to a balanced biaxial stress state significantly imroves the material descrition, e.g., (Lege et al., 989). A corresonding biaxial anisotroy coefficient has also been defined 3

6 (Barlat et al., 003; Pöhlandt et al., 00) and used for identifying the arameters of lastic otentials. Secific exerimental tests have been develoed for the determination of these articular exerimental data, (Kuwabara et al., 998 and Barlat et al., 003). The number of exerimental values used for the identification is tyically equal to the number of arameters. This may lead to oor redictions in areas of the yield locus that are not well reresented in the set of exerimental data used for the identification. For examle, accurate mechanical tests are not easy (or even imossible) to erform in the through-thickness direction (tyically in case of sheet materials). For this reason, even in the most advanced criteria, the coefficients corresonding to out-of-lane stress comonents are usually set to their isotroic values. The use of micromechanical simulation results rovide a very inexensive and still accurate alternative to non-conventional mechanical test data, which require the develoment of comlex and costly exerimental test settings. Several researchers use the redictions of crystal lasticity calculations for the identification of henomenological lastic otentials. This aroach is more suitable for the identification of strain rate otentials than the more classical stress otentials esecially when based on Taylor-like micromechanical models. Consequently, advanced identification rocedures have been roosed for the identification of strain rate otentials (Arminjon and Bacroix, 99; Van Houtte et al., 989). The main interest of this aroach is that the reference oints used for the identification are uniformly distributed in the whole stress or strain rate sace, thus enforcing a uniform accuracy in all ossible loading directions. Nevertheless, subsaces of the stress or strain rate saces that are of articular interest for the alications in mind can be given a referential weight in the identification rocedure. Since crystal lasticity models cature only art of the comlex mechanisms and interactions occurring in a olycrystal, their redictions cannot be considered as accurate as the exeriments. Nevertheless, it is generally acceted that even simle micromechanical models redict the initial yield surface with good accuracy. Thus, the traditional exerimental identification aroach can be used but some of the exerimental values are relaced by their micromechanical counterarts (Kim et al., 006). Alternatively, the available exerimental data can be combined with the micromechanical simulations in a single cost function (Bacroix et al., 003). Both the size and the shae of the yield locus can be determined in this manner, 4

7 ensuring an otimum balance between the exerimental and micromechanical data, according to the available data and the user s desire. In this work, both identification methods (using micromechanical or exerimental test data) are used to investigate a newly develoed lastic strain rate otential, called Sr004-8 (Barlat and Chung, 004). Other models roosed earlier by (Barlat et al., 993 and Arminjon and Bacroix, 99), as well as classical quadratic otentials, are used for comarison urose. The Sr004-8 strain rate otential is based on two linear transformations of the lastic strain rate tensor to account for the orthotroic anisotroy of the material, and involves 8 anisotroy arameters. Its mathematical flexibility allows this model to redict the initial anisotroy better than most of the existing henomenological otentials for a very wide range of materials, as recently shown by (Rabahallah et al., 006). This is an interesting feature since a unique mathematical function could be used for all forming alications, while former mathematical functions were known to better erform for either BCC or FCC sheet materials, but not for both (Bacroix et al., 003). Nevertheless, this theoretical advantage is subjected to the availability of sufficient exerimental data and the accuracy of the arameter identification. In the next section several strain-rate otentials, including some of the most advanced ones, are briefly reviewed. Section 3 rovides a detailed descrition of a arameter identification rocedure based on texture measurements and micromechanical modelling of the reference data. Parameter identification using exerimental data issued from mechanical tests is outlined in section 4. These two identification rocedures are alied in section 5 to a series of anisotroic sheet metals, both FCC and BCC, and the imact of the identification rocedure on the results is highlighted.. Strain rate lastic otentials As shown by (Ziegler, 977 and Hill, 987), for many models of material behaviour, including lasticity, two convex dual otentials exist from which the stress tensor can be derived as a function of the strain rate tensor and vice-versa. In lasticity, the most classical formulation is the one that uses the yield criterion: φ ( ) σ = τ, () where φ( σ ) is a yield function, σ is the stress tensor and τ is a ositive scalar with the 5

8 dimension of stress. The associated flow rule defines the lastic strain rate tensor D as: D s φ = λɺ, () σ Where the suerscrit s denotes the symmetric art, and λ ɺ is a scalar function, which defines the elastic range and scales the lastic strain rate tensor under lastic loading. The dual otential ψ(d ) of this yield criterion is then simly written as ψ ( ) D = λ ɺ, (3) and leads to s ψ S = τ, (4) D where S is the deviatoric art of σ. When the functions φ and ψ are made homogeneous of degree one with resect to their arguments, it is easy to show that the macroscoic lastic ower associated to the strain rate tensor D is ( ) ɺ = = λτ, (5) W D S : D ɺ therefore, Wɺ ψ = ( D ) P ( D ) τ. (6) The function ψ acts as a ower-equivalent measure of the lastic strain rate tensor, since the work rate is the same for all D with a common value of ψ(d ). Using eqs. () and (), or (3) and (4) to describe the lastic behaviour of a material is thus comletely equivalent. However, with the use of the dual otential, it becomes easy to comare an analytical descrition with a crystallograhic aroach since, in the latter case, it is much easier to calculate the macroscoic lastic ower associated with a given strain rate tensor than to derive the macroscoic yield function. In this aer, beside the classical and Hill lastic otentials, three different quadratic and non-quadratic lastic otentials are considered. The mathematical forms of the latter are first briefly reviewed hereafter.. The fourth order non quadratic otential The fourth-order olynomial function of the lastic strain rate tensor D roosed by (Arminjon and Bacroix, 99) for orthotroic symmetry, is used in this work. This otential is 6

9 exressed as a function of only five indeendent comonents of D T 3 3 = [D, D,D,D,D ], due to the isochoric character of the lastic deformation: ( D ) ψ = α k= X ( D ) k k 3 D, (7) where ( ) ( ) ( ) ( ) X = D X = D X = D X = D ( 4 ) ( 3 ) ( 3 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = ( ) ( ) = ( ) ( ) = ( ) ( ) = ( ) ( ) = ( ) ( ) X8 = D D ( D3 ) X9 = D D ( D3 ) X0 = D D ( D ) X = 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 X D D 7 3 X = D D D D X = D D D D In Eq.(7), α k are material arameters, which can be exressed using exlicit functions of the coefficients of the crystallograhic orientation distribution function (ODF). ψ(d ) is firstorder homogeneous with resect to D. It is worth noting that sixth-order homogeneous olynomial functions have been roosed (Van Houtte et al., 99; Savoie and MacEwen, 996) that were able to further imrove the anisotroy descrition. The major drawback of the fourth and sixth order otentials is that they are not convex for all values of their arameters. Van Houtte and Van Bael (004) showed that the convexity is guaranteed when the material arameters satisfy a secial mathematical condition that restricts the admissible ranges of the arameters.. The non quadratic otential «Sr93» The third strain rate otential selected for this work has been introduced by (Barlat et al., 993) in order to describe the behaviour of orthotroic materials. This otential was first develoed for isotroic materials and subsequently extended to orthotroy using a linear transformation of the lastic strain rate tensor D. Moreover, taking into account the assumtion that the lastic flow occurs without volume change, the general form of the otential reduces to ( ) / µ µ µ µ ψ D = D I + D II + D III, (9) k. (8) 7

10 where µ is a non-integer exonent and D I, isotroic-lastically-equivalent (IPE) strain rate D II and D III are the rincial values of the D, which is obtained from the real strain rate tensor by the linear transformation D = L : D involving a fourth-order tensor L that contains the anisotroy coefficients. If orthotroic symmetry is assumed, this linear relationshi involves only six anisotroy coefficients and can be written as D D (c + c 3) / 3 c 3 / 3 c / D c 3 / 3 (c3 c ) /3 c D / D 33 c /3 c / 3 (c + c ) / D33 = c. (0) D D c5 0 D D c 6 D D Based on micromechanical observations, the authors recommend the values µ = 4/3 for FCC materials and µ = 3/ for BCC materials (Barlat and Chung, 993). Nevertheless, considering µ as an adjustable arameter, one can further imrove the accuracy of the rediction esecially for BCC materials (Bacroix et al., 003)..3 The non quadratic otential «Sr004-8» (Barlat and Chung, 005) have extended the flexibility of their former otential by adding a second linear transformation of the lastic strain rate tensor. The resulting function takes the following form: µ µ µ DI DII D III ψ ( D ) = µ + µ µ µ D I + D II + D II + D III + D III + D I / µ, () where D and D are obtained by the linear transformations D = B : T : D and D = B : T : D. The matrix T reresents the fourth-order symmetric, deviatoric unit tensor while the two fourth-order tensors B and B are the material anisotroy tensors. If the sixcomonent vector notation is used for D like in Eq.(0), then the B-tensors take the following matrix form: 8

11 0 b b b3 0 b b5 b B =, () b b b9 0 b0 b b 0 b b4 b B =. (3) b b b8 Therefore, this otential contains 8 arameters lus the exonent µ. Due to their availability in many comuter codes, the classical and Hill quadratic otentials are used in this work as references. The fourth-order otential has been shown to be the best choice to describe the anisotroy of sheet steels, while the otential Sr93 exhibits a better accuracy for aluminium alloys. Thus, in this work they are both comared to the recently roosed otential Sr Texture-based identification rocedure The comarison of different otentials is a articularly difficult task when they are based on exerimental test results, since each of them is accomanied with a distinct arameter identification technique. When the number of arameters increases, additional exerimental tests are needed to determine their values. Thus, the comarison becomes inconsistent since different numbers and tyes of mechanical tests are fitted with the different models. A comletely different situation is rovided when a texture-based identification method is emloyed. In this case a constant, very large number of evenly distributed reference oints (i.e. various D P ; tyically here 80,000 lastic strain rate directions) are generated by means of micro-mechanical calculations, and further used for the arameter identification. Among other interesting advantages, this rovides a consistent way to comare the relative flexibility of various lastic otentials. In the resent work, the well-known Taylor-Bisho-Hill () micro-mechanical model is used in order to generate the flow surface based on crystallograhic texture measurement. 9

12 3. Assessment of the yield locus by the crystallograhic aroach The crystallograhic aroach takes exlicitly into account the texture of the material by considering the olycrystal as a collection of grains, each of them having a secific orientation. Plastic roerties of the aggregate are then calculated from the resonse of each of its constituents to a given loading. This aroach relies on several levels of aroximation concerning the microscoic deformation mechanisms or the link between the imosed boundary conditions and the stress and strain rate in each grain. The (Taylor, 938), Bisho and (Hill, 95) () model belongs to this category and is interesting for several reasons: (i) first, the calculated yield surface is an uer bound of the real resonse of the material, (ii) the agreement between calculated and exerimental roerties is quite satisfactory and (iii) among the olycrystal models, it is one of the simlest to use. The model is based on the assumtion of lastic strain rate homogeneity: D g = D, (4) where Dg designates the lastic strain rate tensor in grain g. It is assumed that the deformation is accommodated by sli, which obeys the Schmid law. The critical resolved shear stress τ c is assumed to be the same for all sli systems in all grains. Knowing the strain rate in each grain, it is ossible to calculate the lastic work rate in each grain: ( ) ɺ g Dg = Sg Dg, g (5) W : where the deviatoric stress S g is determined through the rincile of maximum lastic ower associated to the Schmid law. The average lastic ower for the aggregate can then be comuted as = W ɺ ( D ) W ɺ ( D )f (g)dg, (6) P P P g g g with f(g) denoting the orientation distribution function which describes the texture of the material, and the integration is erformed over the whole orientation sace. The reresentative volume element is described using 06 crystallograhic orientations in the Euler sace. Since the lastic ower is homogeneous of degree one with resect to D, it is sufficient to work with the direction N = D / D of D. The normalized stress tensor S/τ c is also considered. 0

13 3. Princile of the identification rocedure Given a lastic strain rate direction N, the corresonding average lastic ower ɺ P ( N) P the normalized exression Π ( N) c W and S = : N can be comuted according to the revious τ section. Eq.(6) is used in order to calculate of the same quantities by using the lastic otential. Moreover, the different otentials used in this work are described by homogeneous function of degree one. Therefore, Eq.(6) can be rewritten as ( N) ψ = Wɺ P ( N) τ. (7) In other words, for any strain rate direction N i, the reviously defined two functions P ( i ) Π N and ( ) ψ N corresond to the lastic ower associated to a unit-norm strain rate i tensor and normalized by τ c. The coefficients of the lastic otential ψ can then be identified by minimizing the objective function: ( N ) ψ ( N ) P wi Π i i i= Tex(material arameters) = P Π ( i ) N i= F, (8) with resect to the coefficients of the chosen otential. The sum is erformed over a number of selected strain rate directions. In order to swee the 5D strain rate sace uniformly, 80,000 directions are selected in this sace. The rocedure for generating these directions is given in (Arminjon and Bacroix, 99). Then, the values ψ(n) are comuted for all these directions. This is a lengthy task, but it has to be erformed only once for each material. When the exression of the otential is a linear function of the material coefficients; e.g. Eqs.Erreur! Source du renvoi introuvable. and (7); the identification corresonds to a linear least-squares roblem that may be solved in one iteration (Van Houtte et al., 989; Arminjon and Bacroix, 99). In order to extend the identification to any strain rate otential, the non-linear least-squares-roblem is solved using a Levenberg-Marquardt minimization algorithm (Denis and Schnabel, 983). This algorithm requires the calculation of the objective function and its first order derivatives with resect to the arameters subject to identification. It is worth noting that the terms of Eq. (8) are weighted by w i, where

14 q + cos π ( N3,i ) + ( N3,i ) w i = + β, (9) while β and q are two real constants with values between 0 and (although mathematically seaking, q may take any ositive value). This weight function allows to increase the imortance of the in-lane loading (i.e. the sheet lane) with regard to the out-of-lane loading. 4. Identification rocedure based on mechanical data 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. While the algorithm to determine the coefficients can be general for 3-D deformations, higher weights are given to the sheet in-lane data in the articular case of sheet forming alications. Also, while a variety of measurements can be considered, the combination of in-lane uniaxial tensile strength and r values along various directions, as well as the strength σ b( = σ xx = σ yy ) and strain rate ratio εɺ = ε yy r b( ) ɺ xx under the balanced biaxial stress condition are considered here. Out-of-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 states could be considered for the out-of lane roerties. When all the inut data are selected, the coefficients are obtained in this work by minimizing the following objective function (see Kim et al., 006): m m m m m µ ψ ε µ ψ ε33 σ µ ψ ε µ ψ ε33 Mech = m + m m σ σ σ F w w µ ψ εxx µ ψ εzz σ µ ψ ε b xx µ ψ εyy + wr + wr σ σ σ n n µ ψ εij τij + wn n σ σ. (0) 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.

15 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. The weight can be used to differentiate longitudinal, transverse or other stresses. However, in this work, these weights are identical for the in-lane roerties of the sheet. Moreover, because some of the inut data are not known but aroximated under the isotroic assumtion, the weights corresonding to these inut data are made lower than the weight of exerimental data, which are more reliable. Tyically, in this work, weights for the in-lane and out-of-lane roerties were of the order of.00 and 0.0, resectively. 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 (Chung and Richmond, 993). 5. Alication to steel and aluminium sheets For the investigation of the anisotroic lastic otentials, several BCC and FCC sheet materials have been selected. These materials rovide a certain diversity of anisotroic behaviors with different initial crystallograhic textures, leading to different initial anisotroy in terms of uniaxial yield stresses, r-values, and yield surface shaes. Either crystallograhic texture measurements, or mechanical tests, or both have been erformed on the selected materials. Therefore, an interesting basis is rovided for the investigation of the ability of different strain rate otentials to describe lastic anisotroy. The resent work has been carried out on a large range of industrial materials. However, for conciseness, only a selected set of characteristic examles are commented hereafter. 3

16 5. Crystallograhic texture-based identification Figure and Sr004-8 Sr93 Hill,948 Exerimental data Sr004-8 Sr93 Hill,948 Exerimental data σ /τ c 3.9 σ /τ c α ( ) (a) α ( ) (b) σ /τ c Sr004-8 Sr93 Hill,948 Exerimental data α ( ) (c) Figure show an examle of the anisotroy of the r-value and yield stresses (normalized by the critical resolved shear stress τ c ) redicted when the micromechanical-based adjustment rocedure is used on different materials. The exerimental data obtained using off-axis uniaxial tensile tests, are added in the figures for comarison. With regard to yield loci (normalized with the critical shear stress τ c, Figure 3), the redictions of the quadratic otentials are clearly imroved by the Sr004-8 and otentials. However, discreancies are observed in the rediction of the r-value secifically when the anisotroic behaviour is more ronounced (as for DC06 and AA60-T43). It is noteworthy that with the 4

17 texture-based aroach, the r-values are not used for the arameter identification and thus can be used for validation uroses. 3.5 Sr004-8 Sr93 Hill,948 Exerimental data. Sr004-8 Sr93 Hill,948 Exerimental data r(α).5 r(α) α( ) (a) α( ) (b) r(α).. Sr004-8 Sr93 Hill,948 Exerimental data α( ) Figure r-value redictions for several otentials when crystallograhic texture-based identification is adoted. (a) IF mild steel DC06, (b) Aluminium alloy AA60-T43, (c) Dual Phase DP600 steel. (c) 5

18 3. 3. Sr004-8 Sr93 Hill,948 Exerimental data Sr004-8 Sr93 Hill,948 Exerimental data σ /τ c 3.9 σ /τ c α ( ) (a) α ( ) (b) σ /τ c Sr004-8 Sr93 Hill,948 Exerimental data α ( ) (c) Figure Yield stresses redictions for several otentials when crystallograhic texture-based identification is adoted. (a) IF mild steel DC06, (b) Aluminium alloy AA60-T43, (c) Dual Phase DP600 steel. 6

19 σ /τ c 0 - Sr004-8 Sr93 Hill, 948 σ /τ c 0 - Sr004-8 Sr93 Hill, σ /τ c σ /τ c (a) (b) 4 3 σ /τ c 0 - Sr004-8 Sr93 Hill, σ /τ c (c) Figure 3 Yield surface redictions for several otentials when crystallograhic texture-based identification is adoted (comutation carried out with { 0} and { } sli families for bcc materials and { } 0 sli family for fcc materials). (a) IF mild steel DC06, (b) Aluminium alloy AA60-T43, (c) Dual Phase DP600 steel. 7

20 Mechanical data-based Crystallograhic data-based Potentials AA60- AA090- AA008- AA60- DP600 T43 T3 T4 T43 DC06 DP600 b b b b b b b b b b b b b b b b b b µ c c c c c c µ α α α α α α α α α α α α α α α α α α α α α α F G H L M N Sr004-8 Sr93 Hill 48 Table Material arameters of the different lastic otentials for the materials under investigation, as identified with the two methods. Yield stresses normalized by σ 0. 8

21 The sets of material arameters obtained in the current investigation are summarized in Table. Figure 4 shows the comarison of the error function values, (Eq.(8)), obtained after the identification of the investigated otentials for the different BCC and FCC materials. The conclusions drawn in (Bacroix et al., 003) are again verified for the materials tested in this work: Sr93 rovides better redictions than for aluminium alloys, while does better for all steels. In contrast, the recent Sr004-8 otential rovides a better fit than both Sr93 and in all cases. The main interest of the texture-based identification rocedure is that it rovides such consistent comarison between different otentials. The above conclusions have been found for more than ten different aluminium alloys and ten different steel sheets. Thus one may consider that, among the otentials considered in this work, Sr004-8 has great chances to rovide the best fit for virtually any metal sheet in these categories. 0-3 Sr004-8 Sr93 Hill Error function AA60-T4 AL58 AA606-T4 DC06 DP600 HSLA-340/ HSLA340/ Figure 4 Error function for several BCC (IF mild steel DC06, Dual Phase steel DP600 and micro-alloyed steels HSLA with different thicknesses) and FCC materials (Al-Mg AA58-O, Al-Mg-Si AA60-T4 and AA606- T4) using different otentials and texture-based identification. 9

22 The sensitivity of the redicted results to some numerical asects (initial guess for the arameters, objective function weights, the number of iterations ) has been carefully investigated in this work. Although the adoted algorithm is known to be robust and efficient, it is sensitive to the initial arameters guess, like most of the minimization algorithms seeking for the closest local minimum. Figure 5 and 6 resent the sensitivity of the material arameters identification to the initial introduced solution for fixed β and q arameters. As one can see, the imact of the initial set of arameters on the shae of the yield surface is negligible (Figure 6). However, when the r-value or the yield stress anisotroies are checked, some differences are found. This suggests that these two imortant measures of anisotroy could also be included in the objective function for a better controlled identification. Nevertheless, it should be noted that the r-values redicted by the Taylor model are less reliable than the yield surface itself. In the examle from Figure 5, the imact of the initial guess is smaller than the ga between the micromechanical and the exerimental r-values.. 3. r(α) Solution Solution Solution 3 Solution 4 Exerimental data σ/τ c Solution Solution Solution 3 Solution 4 Exerimental data α( ) α( ) Figure 5 Aluminium alloy AA60-T43: Sensitivity analysis to the initial guess of arameters for crystallograhic texture-based identification carried out on Sr004-8 strain rate otential with β = and q =. 0

23 4 3 σ /τ c 0 - Solution Solution Solution 3 Solution σ /τ c Figure 6 Aluminium alloy AA60-T43: Sensitivity analysis to the initial guess of arameters for crystallograhic texture-based identification carried out on Sr004-8 strain rate otential. When crystallograhic-data-based identification is adoted, the cost function uses 80,000 exerimental values on the yield surface, Eq. (8). A secific weighting factor w i, Eq. (9) for every strain rate direction can be imosed. This factor is intended to give a smaller weight to the through-thickness shear modes (w i = when the comonents of the lastic strain rate tensor lie in the lane of the sheet and w i = - β for urely through-thickness shear modes). Thus β = allows for a comlete discrimination of the later deformation modes. When β 0, the q arameter intensifies the effect of β. With regard to the numerical results in Figure 7 and Figure 8, the same conclusions can be drawn as for the sensitivity to the initial arameter guess: the imact on the yield surface is subtle, while there is a significant sensitivity of the Hill coefficients of anisotroy and uniaxial tensile yield stress to the weights adoted in the cost function. In our comarative studies, both the coefficients β and q have been ket equal to one for consistent comarisons. No need to say here that such sensitivity asects were already reorted in ublished researches relative to the otimization algorithm. However, the general trends reorted later were obtained taking under consideration such sensitivity studies.

24 r(α) q = 0 q = 0.5 q = q = q = 0 Exerimental data σ/τ c q = 0 q = 0.5 q = q = q = 0 Exerimental data α( ) α( ) Figure 7 Aluminium alloy AA60-T43: Sensitivity analysis to the value of q, for crystallograhic texture-based identification carried out on Sr004-8 strain rate otential with β =.. β = r(α) β = 0 β = 0.3 β = 0.5 β = 0.8 β = 3 Exerimental data.95 σ /τ c β = 0 β = 0.3 β = 0.5 β = 0.6 β = 0.8 β = Exerimental data α( ) α( ) Figure 8 Aluminium alloy AA60-T43: Sensitivity analysis to the value of β, for crystallograhic texturebased identification carried out on Sr004-8 strain rate otential with q =. 5. Mechanical data-based identification Figure 9 and Figure 0 deict an examle of the best fit redictions obtained with the five otentials when mechanical data-based adjustments are used. The redictions of the quadratic otentials are clearly imroved by the three other otentials (Sr93, Sr004-8 and ). Moreover, Sr004-8 systematically imroves the redictions of Sr93 (Figure

25 ). When mechanical data-adjustment aroach is adoted, Sr004-8 seems to exhibit more flexibility in accurately describing the yielding behaviour of the examined aluminium alloys (e.g. Figure 9, Figure and Figure 3). Consequently, the comarison of the different otentials flexibility leads to similar conclusions as when the texture-based identification is used. However, aears here to erform better than Sr93 even for aluminium alloys, in terms of objective function. In fact, a more careful examination of the results reveals that for aluminium alloys, the best-fit set of arameters for often leads to a non-convex yield surface, as illustrated in Figure 4. The major difficulty for this otential aears to be the simultaneous fit of both the uniaxial tensile exeriments and the biaxial one. Reducing the weight of the biaxial oint in the cost function leads to accetable solutions (convex yield surface). This is how the results for have been obtained in the figures above. Yet, the corresonding cost functions cannot be comared anymore to the other ones since the considered weights are different. It is noteworthy that with the texture-based identification, no loss of convexity has been detected for. This is attributed to the use of numerous, evenly distributed, equally weighted reference oints for the identification, which revent any unrealistic distortion of the yield surface...04 r(α) Sr004-8 Sr93 Hill48 Exerimental data σ /σ b α( ) 0.96 Sr004-8 Sr Hill48 Exerimental data α( ) Figure 9 Aluminium alloy AA60-T43: Yield stress and r-value redictions for several otentials when mechanical data-based identification is adoted. 3

26 σ /σ b Sr004-8 Sr93 Hill, 948 Exerimental data σ /σ b Figure 0 Aluminium alloy AA60-T43: Yield surface redictions for several otentials when mechanical data -based identification is adoted. 0 0 Sr004-8 Sr93 Hill Von mises AA090-T3 AA008-T4 AA60-T43 DP600 Figure Error function for several materials using different otentials and mechanical data-based identification. 4

27 Sr004-8 Sr93 Hill,948 Exerimental data r(α) Sr Sr93 Hill, Exerimental data α( ) σ /σ α ( ) Figure Al-Li aluminium alloy AA090-T3: Yield stress and r-value redictions for several otentials when mechanical data-based identification is adoted. r(α) Sr004-8 Sr93 Hill,948 Exerimental data σ /σ α( ) 0.9 Sr Sr93 Hill, Exerimental data α ( ) Figure 3 Aluminium alloy AA008-T4: Exerimental yield stress and r-value and redictions based on several otentials. Mechanical data-based identification. 5

28 . 0.8 σ /σ (w r = 5) (w r = ) Exerimental data σ /σ 0 Figure 4 Predictions of the otential for aluminium alloy AA090-T3: best fit results (thin line) and accetable solution that does not violate convexity (thick line). The arameter w r corresonds to the weight affected to the équibiaxiale yield stress as defined in Eq. (0). When the material exhibits weak in-lane anisotroy (e.g. the dual hase steel DP600 in this work), the identified yield surfaces obtained with the different otentials are very close to the yield surface. Nonetheless, the redicted Hill coefficients of anisotroy can be very different as indicated by Figure (c) for DP600 steel. For this material, when the mechanical data-based aroach is adoted, a significant imrovement in the redicted Hill coefficients of anisotroy is achieved (Figure 5) with general yield loci close to the surface (Figure 6). A careful insection of the redicted yield locus using the two identification strategies clearly indicates that the normal to the yield surfaces can be very different (Figure 7, a). These differences can be even larger when the in-lane anisotroy become more ronounced as for the AA60-T43 material (Figure 7, b). 6

29 Sr004-8 Sr93 Hill,948 Exerimental data Sr004-8 Sr93 Hill,948 Exerimental data r(α) 0.9 σ /σ α( ) α ( ) Figure 5 Dual Phase DP600 steel: Exerimental yield stress and r-value and redictions based on several otentials. Mechanical data-based identification..5 σ /σ Sr004-8 Sr93 Hill, 948 Exerimental data σ /σ 0 Figure 6 Dual Phase DP600 steel: Yield surface redictions for several otentials when mechanical databased identification is adoted. 7

30 .5 Mechanical data-based Crystallograhic texture data-based.5 Mechanical data-based Crystallograhic texture data-based σ /σ 0 0 σ /σ (a) DP600 (b) AA60-T σ /σ 0 σ /σ 0 Figure 7 Determined Sr004-8 yield surface using different strategies of material arameters identification. Deending on the rocedure used to identify the arameters, the resulting redictions of lastic anisotroy are slightly different. It can be observed that when mechanical test data is used for the identification, the redictions of the r-value and yield stress anisotroy with all the non-quadratic otentials are closer to each other while the scatter in the rediction of the yield surface is more significant (Figure 0). On the contrary, a larger scatter in r-values and yield stress anisotroy redictions is obtained with the texture-based identification (Figure -b 8

31 and Sr004-8 Sr93 Hill,948 Exerimental data Sr004-8 Sr93 Hill,948 Exerimental data σ /τ c 3.9 σ /τ c α ( ) (a) α ( ) (b) σ /τ c Sr004-8 Sr93 Hill,948 Exerimental data α ( ) (c) Figure -b). Figure 6 and Figure 8 rovide very tyical examles, where the rediction of the lane strain tension stresses is very different from a otential to the other, while the mechanical-test based anisotroy is redicted almost identically. The origin of this behaviour is again the discreancy in the number, tye and location of the reference data used for the arameter fit in the two identification rocedures. Figure 9 shows the resective reference (exerimental) data used in both cases. The texture-based identification makes use of an extensive, evenly-distributed set of reference oints (lastic strain rate directions), while the mechanical tests rovide a more reduced set of secific data, 9

32 including both stress values and lastic strain rate directions. More secifically, the lane strain tension area is oorly reresented in the commonly used set of exeriments which may exlain e.g. the scatter in Figure 8 corresonding to this area. However, diversifying the mechanical tests is an extremely challenging task σ /σ Sr004-8 Sr93 Exerimental data σ /σ 0 Figure 8 Yield surface redictions for several otentials when mechanical data -based identification is adoted: Aluminium alloy AA008-T4. 30

33 σ /σ Texture oints Uniaxiale tensile tests Shear tests σ /σ 0 Figure 9 Tri-comonent (σ - σ - σ ) normalised yield surface reresentation and location of the reference oints for the two identification rocedures. A fictitious isotroic, -like material is illustrated here. 3

34 One can easily dulicate the available tests by means of the micromechanical model, so that similar data and the same rocedure can be used for the identification. Figure and Sr004-8 Sr93 Hill,948 Exerimental data Sr004-8 Sr93 Hill,948 Exerimental data σ /τ c 3.9 σ /τ c α ( ) (a) α ( ) (b) σ /τ c Sr004-8 Sr93 Hill,948 Exerimental data α ( ) (c) Figure show the r-value and yield stress anisotroy as measured and as redicted by the Taylor model, for three materials. It is obvious that for some materials, the Taylor model leads to oor redictions of these anisotroy indicators. Consequently, in such situations the texture-based identification method does not rovide a good accuracy, unless a more accurate micromechanical descrition of the lastic anisotroy is obtained. For this urose, different micromechanical models (e.g. self-consistent models, finite element olycrystalline models etc.) could be used in the framework of the current aroach, without any limitation. 3

35 The texture-based identification rocedure can also be enalized in terms of final accuracy if an equal weight is considered for all the reference oints in the 5D deviatoric lastic strain rate sace. Indeed, while this guarantees a consistent fit of the whole sace of ossible straining modes, ractical sheet forming alications exhibit near lane-stress loading situations. Thus, one should rescribe lower weights to the out-of-lane shear terms, e.g. with the weight function (Eq.(9)). 5.3 Combined strategy for material arameters identification When both mechanical test data and crystallograhic texture are available, one can combine the advantages of the two arameter identification methods by combining the two error functions in a single one: ( ) F = β F + β F, () Tex Mech where F Tex is defined by Eq. (9) and F Mech by Eq.(0). This aroach has been alied here for the AA606-T4 aluminium alloy, by using the Sr93 otential. The Taylor-redicted data is combined with exerimentally determined shear yield stresses. As shown in Figure 0, when only the texture-based function is used (e.g. β =) the anisotroy of the exerimental shear stresses is very oorly reroduced by the identified arameters. Decreasing the value of β allows for a rogressive imrovement of the fit, until the results of the mechanical-testbased identification rocedure are comletely recovered. In such cases, when the Taylor model does not accurately describe the r-values and/or uniaxial yield stresses, one should use the mechanical test data to identify as many arameters as ossible. Nevertheless, several arameters cannot be identified with the exerimental data alone; instead of arbitrarily keeing their values to the isotroic case, using the Taylor model contributes to imrove the final identification result. 33