Time integration scheme for elastoplastic models based on anisotropic strain-rate potentials

Transcription

1 Time integration scheme for elastolastic models based on anisotroic strain-rate otentials Meziane Rabahallah, Tudor Balan, Salima Bouvier, Cristian Teodosiu To cite this version: Meziane Rabahallah, Tudor Balan, Salima Bouvier, Cristian Teodosiu. Time integration scheme for elastolastic models based on anisotroic strain-rate otentials. International Journal for Numerical Methods in Engineering, Wiley, 2009, 80 (3), <0.002/nme.2640>. <hal-09277> HAL Id: hal htts://hal.archives-ouvertes.fr/hal Submitted on 3 Se 205 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/0985/9908 To cite this version : Meziane RABAHALLAH, Tudor BALAN, Salima BOUVIER, Cristian TEODOSIU - Time integration scheme for elastolastic models based on anisotroic strain-rate otentials - International Journal for Numerical Methods in Engineering - Vol. 80, n 3, Any corresondence concerning this service should be sent to the reository Administrator : archiveouverte@ensam.eu

3 Time integration scheme for elastolastic models based on anisotroic strain-rate otentials MEZIANE RABAHALLAH,2, TUDOR BALAN,*,, SALIMA BOUVIER 2, CRISTIAN TEODOSIU 2 LPMM-CNRS, Arts et Métiers ParisTech Metz Camus, 4 rue A. Fresnel, Metz Cedex 03, France 2 LPMTM-CNRS, Université Paris 3, Institut Galilée, 99 rue J.-B. Clément, Villetaneuse, France SUMMARY Modelling of lastic anisotroy requires the definition of stress otentials (coinciding with the yield criteria in case of the associated flow rules) or, alternatively, lastic strain-rate otentials. The latter aroach has several advantages whenever material arameters are determined by means of texture measurements and crystal lasticity simulations. This aer deals with a henomenological descrition of anisotroy in elastolastic rate-insensitive models, by using strain-rate otentials. A fully imlicit time integration algorithm is develoed in this framework and imlemented in a static-imlicit finite element code. Algorithmic details are discussed, including the derivation of the consistent (algorithmic) tangent modulus and the numerical treatment of the yield condition. Tyical sheet-forming alications are simulated with the roosed imlementation, using the recent non-quadratic strain-rate otential Sr Numerical simulations are carried out for materials that exhibit strong lastic anisotroy. The numerical results confirm that the resented algorithm exhibits the same generality, robustness, accuracy, and time-efficiency as state-of-the-art yield-criterion-based algorithms. KEY WORDS: stress udate algorithm; backward Euler; lastic strain-rate otential; elastolasticity; sheet metal forming simulation. * Corresondence to: Tudor Balan, Arts et Métiers ParisTech Metz Camus, 4 rue A. Fresnel, Metz Cedex 03, France tudor.balan@ensam.fr.

4 INTRODUCTION The descrition of lastic anisotroy is recognized as a key factor for the accuracy of finite element simulations of sheet metal forming rocesses. This is articularly true when final art roerties, such as sringback or forming limits, are to be investigated. The initial lastic anisotroy of metal sheets is classically modelled either by micromechanics or henomenological yield criteria, the latter being referred whenever numerical simulations are erformed. In order to henomenologically reresent the rate-insensitive, incomressible lastic behaviour of materials, classical constitutive models use a yield function (for a yield surface descrition), the associated flow (or normality) rule, and a hardening law. The first two exress anisotroic relationshis between the stress and lastic strain rate comonents at a given material oint. Ziegler [] and Hill [2] have shown that, based on the lastic work equivalence rincile, a meaningful strain rate otential can be associated with any convex stress otential (or yield surface). Therefore, an alternative aroach to describing lastic anisotroy is to rovide a strain rate otential, which is exressed as a function of the lastic strain rate tensor, while its gradient gives the direction of the stress deviator. Arminjon et al. [3, 4] and Van Houtte et al. [5] roosed fourth-order and sixth-order strain rate functions, resectively. Barlat and Chung [6], Barlat et al. [7], Chung et al. [8], and Kim et al. [9] introduced strain-rate otentials that were seudo-conjugates of yield functions ublished earlier. Formally, the stress or strain-rate otential aroaches are identical. A strong driving force for the develoment of strain-rate otentials has been their convenient arameter identification by means of micro-mechanical calculations based on Taylor-tye models [3, 5]. For some alications, such as rigid-lastic finite element (FE) simulations [0-2], minimum lasticwork ath calculations [7], inverse one-ste analysis [3], and analytical calculations in material forming, the strain-rate otential aroach can be comutationally more convenient. Nevertheless, numerical imlementation of strain-rate otentials has been also tackled in the framework of elastolasticity. In their finite element imlementation, Bacroix and Gilormini [4] roosed a solution to overcome the lack of an exlicit yield condition in this modelling framework. They used a finite-difference tangent modulus and membrane elements to simulate a simlified cu drawing rocess. A method used to derive the algorithmic tangent modulus for strain-rate-otential-based elastolastic models has been roosed by Szabo and Jonas [5]. Van Houtte et al. [6], Hoferlin et al. [7], and Li et al. [8, 9] develoed imlicit FE imlementations and erformed sheet forming simulations with their sixth-order lastic otential [5]; Zhou et al. [20, 2] imlemented similar fourth-order [4] and sixth-order [22] otentials; Kim et al. [23] recently roosed a general lane-stress finite element imlementation for use with shell elements. One can note, however, that most of the former comuter imlementations have been designed with feasibility in mind, rather than generality. They were develoed for secific strain-rate otentials, e.g., those involving comlex numerical treatments to deal with convexity issues; 2

5 hardening models were either oversimlified [4, 20, 2] or very secific [9], with microstructural and textural relevance being the main objectives. These develoments have been conducted in the context of texture-based arameter identification, and hence the materials science background has been taken into account throughout the develoment of the numerical scheme. A comlete minimization technique has been develoed and used to determine the intersection of the trial stress increment with the yield surface, which is not really a requirement for the return maing algorithm. On the other hand, strain-rate otentials have been originally roosed for secific alications, whereas yield criteria were considered better suited for the elastolastic FE imlementation. Today, however, flexible strain-rate otentials have been roosed that exhibit excellent redictive abilities for general alication [24]. Thus, the availability of strain-rate otentials in finite element codes would bring a real added value to the sheet metal forming community, both in research work and industrial alications. The aim of this aer is to roose a generic imlicit time integration algorithm for anisotroic elastolastic, time-indeendent constitutive models using the strain-rate otential aroach. Comared to the state-of-the-art integration schemes develoed for yield criteria, the roosed algorithm is exected to exhibit the same generality, robustness, accuracy, and time-efficiency. Existing rigid-lastic FE imlementations revent the use of strain-rate otentials for sheet forming simulations including unloading and sringback. The structure of the aer is as follows. The strain-rate otential-based elastolastic modelling framework is briefly resented in Section 2. Section 3 is devoted to the develoment of the imlicit time integration algorithm, which is resented in detail and ket as general and comlete as ossible. In Section 4, the quadratic Hill otential, as well as the recent, nonquadratic otential Sr are taken as articular cases of strain-rate otentials, together with a non-linear isotroic-kinematic hardening model. The resulting algorithm is imlemented into the software Abaqus/Standard. Validation and comarison with seudo-conjugate yield criteria are erformed through numerical simulations of tyical sheet metal forming rocesses. 2 CONSTITUTIVE MODEL The henomenological elastolastic modelling adoted here is rate indeendent (without viscous effects) and restricted to cold deformation. Classical rate-indeendent models utilize a hyoelastic law defining the stress rate with resect to the elastic strain rate, a yield function delimiting the elastic zone, a lastic flow rule, and a set of internal state variable evolution laws defining the work hardening during lastic deformation. In the current setting, the yield function is relaced by a lastic strain-rate otential and the flow rule is modified accordingly. The frame objectivity issue, which arises when large deformations are intended, is solved by writing the constitutive equations in an aroriate rotating orthogonal frame. Vector and tensor variables are denoted by bold-face symbols. Comonents, whenever used, are referred to a Cartesian orthogonal frame. The summation convention over reeated indices of 3

6 such comonents is used throughout the aer. Let A, B denote second-order tensors and C a fourth-order tensor. The double-contracted tensor roducts between such tensors are defined as: A : B = A B, ( C : A) = C A, A : C : B = A C A () ij ij ij ijkl kl ij ijkl kl The norm of A is defined as A = A : A, and its direction, if A is non-zero, as A A. The norm of C is defined by = CijklCijkl C. Finally, ( ) = Aij Bkl A B. Note that all second- and fourth-order tensors that enter the modelling described hereafter are suosed fully symmetric. ijkl 2. Rotation-comensated tensor quantities and equations The sheet undergoes generally large deformations in metal forming and its elastolastic behaviour is described by rate constitutive equations. In order to achieve material objectivity, objective rates must be used. A very convenient aroach used to ensure material objectivity, while keeing the constitutive equation simle in form, consists of reformulating these equations in terms of rotation-comensated variables. More recisely, if A and C designate second- and fourth-order tensors, resectively, the corresonding rotation-comensated tensors (labelled by a suerosed hat) re defined by Aˆ = R R A, Cˆ = R R R R C, (2) ij ki lj kl ijkl i qj rk sl qrs where R is an orthogonal rotation matrix, generated by a skew-symmetric sin tensor Ω using T ɺR R =Ω, where the suerosed dot on R denotes time differentiation and the suerscrit T indicates the transose of R. The main interest of this aroach is that objective derivatives (labelled here by a suerosed circle) are simly related to the material time derivatives of their rotation-comensated counterarts via equations similar to Eq. (2), i.e., ˆɺ A = R R A, ˆɺ kl C = R R R R C ij ki lj ijkl i qj rk sl qrs (3) Clearly, R should satisfy, in turn, the objectivity condition under suerimosed rigid-body motions. For examle, the Jaumann derivative is obtained by setting Ω = W, while using R = R leads to the Green-Naghdi derivative. Here W denotes the total sin, while R is the orthogonal tensor in the olar decomosition of the deformation gradient. It should be noted that, following Mandel s ioneering work on the average lastic sin of olycrystals [25], other henomenological models have been roosed to describe the evolution of the lastic sin (see, e.g., [26-28]), and their comuter imlementation has also been studied [29, 30]. In the following, we assume throughout that all tensor variables turn with the sin W (i.e., Jaumann rates are considered), and that they are rotation-comensated. Consequently, simle 4

7 time derivatives are involved in the constitutive equations, making them identical in form to a small-strain formulation. For simlicity, the suerosed hat (^) is omitted thereafter. 2.2 Modelling framework The total strain rate tensor D is decomosed into an elastic art e D and a lastic art D : e D = D + D (4) and, therefore, the linear, hyoelastic resonse of the material is described by ( ) e σɺ = C : D D, (5) e where σɺ is the rate of the Cauchy stress tensor σ and C is the fourth-order elasticity tensor. In e s the case of isotroic linear elasticity, C = 2 G I 4 + K I I, with K and G being the bulk and shear moduli, resectively. Finally, I is the unit second-order tensor, whose comonents are the s Kronecker s deltas, i.e. I kl = δ kl, while I 4 is the fourth-order symmetric deviatoric unit tensor, s whose comonents are I 4ijkl = ( 2)(δikδ jl + δδ il jk ) - ( 3)δδ ij kl. In associated rate-indeendent incomressible lasticity, the lastic strain rate tensor suosed roortional to the gradient of a yield function Φ, defined as ( σ X) ( σ X) D is Φ,τ, = σ τ = 0. (6) Here, the scalar variable τ is a measure of the size of the elastic domain (and describes the isotroic hardening), X locates the centre of this elastic domain in the stress sace (and introduces the kinematic hardening), and σ is the equivalent stress defining its shae. The flow rule reads D P = ( σ,τ, X) Φ ɺλ, (7) σ where λ ɺ is the lastic multilier, which is suosed strictly ositive for lastic loading, and equal to zero for neutral lastic loading, for unloading, and in elastic state. In the current work, the dual otential Ψ of the yield function Φ is used instead: Ψ ( D ) = ɺ λ. (8) The flow rule becomes 5

8 Ψ T = τ, D (9) where T = σ X is the effective stress tensor and σ denotes the deviator of the stress tensor σ. In this work, we restrict ourselves to functions Ψ and Φ that are first-order homogeneous with resect to ositive scalar multiliers. Although the existence of dual otentials is theoretically demonstrated, the analytical exression of a strain-rate otential is seldom calculated as the dual of a given yield function. The two hardening variables τ and X evolve with the lastic strain. Their evolution equations are sought in the generic form X ɺ = h ɺ λ, τ ɺ = h ɺ λ. (0) x Note again that objective rates must be used in Eq. (5) and (0), should the model be written in a fixed frame rather than the articular rotating frame adoted here. By using the consistency condition for lastic loading, the following linear tangent relation can be derived between the stress rate and the strain rate tensors where the analytical tangent modulus τ ana σɺ = C : D, () ana C takes the form [3] C ana e e ( C : N) ( N : C ) e = C α, (2) e N : C : N + N : h + Ψ( N) Ψ( N) ( h ) X τ with α = for lastic loading and α = 0 otherwise; N = D / D denotes the lastic strain rate direction. When the elasticity is linear and isotroic, this relation reduces further to 2 ana s 4G N N C = K I I + 2 G I 4 α. (3) 2 G + N : h + Ψ( N) Ψ( N) ( h ) X τ 3 TIME INTEGRATION ALGORITHM The elastolastic model introduced in Section 2 has been imlemented in the static imlicit code ABAQUS/Standard. At each equilibrium iteration, a dislacement increment is redicted in each node of the mesh. From this, the kinematic equations are emloyed to calculate the strain increment at each integration oint of the finite elements. These stes are carried out by the FE code, so only the udate of the state variables needs to be erformed in order to verify the 6

9 equilibrium at the end of the loading increment. In this section, we develo the state udate methodology and derive the consistent tangent modulus necessary to iteratively reach the equilibrium at the end of each loading increment. The total and lastic strain increments ε and ε are defined as tn+ tn+ ε = D dt, ε = D dt (4) tn tn and must be further aroximated since the values of D and D are only available at the two ends of the time increment t = tn+ tn. Several aroximations for the strain increment have been roosed in the literature and they are already imlemented in the finite element codes, as the strain increment is an inut variable for the constitutive algorithm. For the lastic strain increment, the backward Euler time integration scheme assumes that D is constant over the increment and equal to its value at t, n + i.e. n+ t. ε D (5) As a consequence, due to the first-order homogeneity of the lastic otential with resect to scalar multiliers, one can write the incremental form of Eq. (8) in any of the following equivalent forms: ( ) ( ) ( ) ( ) λ = Ψ( ε ) = t Ψ( D ) = t D Ψ N = ε Ψ N, (6) n+ n+ n+ n+ where the direction of the lastic strain rate at the end of the increment can be written as N D ε n+ n+ = = Dn+ ε (7) The derivatives of the lastic otential can also take different equivalent forms, which can rove to be useful in articular contexts: ( n ) ( ) Dn ( ε ) ( ) Ψ D Ψ ε Ψ N ξ = = = N + n+ + n+. (8) Equations (6) and (8) show that the effects of the norm ε and the direction N n+ of the lastic strain increment can be searated. This observation may be used at the algorithmic level to locally reduce the number of unknowns. 3. Discrete equations of the constitutive model 7

10 In the revious section, Eqs. (5) and (8)-(0) were shown to comletely define the constitutive model. The FE imlementation of such a model requires the numerical integration of these equations over a time increment, from a known state at time t n to the unknown state at t n+, given the total strain increment ε. The most widely used method is the fully imlicit, backward Euler integration scheme (see, e.g., [32-34]), which is also emloyed in this work. The incremental form of the hyoelastic Hooke s law is written as e n n : ( ) σ = σ + C ε ε. (9) + The imlicit time integration schemes of rate-indeendent lasticity models include an elastic try e trial σ = σn + C : ε followed, when necessary, by a lastic correction. This two-ste rocedure is illustrated in Figure, together with some of the notation used in this section. The decision about whether the trial stress corresonds to an elastic state or an elastolastic one requires a articular attention here since, in the framework of strain-rate otentials, no exlicit yield criterion is available. This issue will be clarified in Section 3.2. try σ σ n+ σ n udated yield locus initial yield locus Figure. Elastic rediction and lastic correction during a tyical elastolastic increment; grahical illustration in the deviatoric stress sace. 8

11 try The elastic trial stress σ can be comuted exlicitly since the total strain increment is known at the beginning of the time ste, together with the sherical (hydrostatic) art of the final stress ( 3) tr ( ) ( 3) tr ( ) K tr ( ) σ = σ I = σ I + ε I (20) sh n+ n+ n since it also deends only on known quantities. However, the deviatoric art of the final stress deends on the yet unknown increment of lastic strain, since e n+ n C : ( ) σ = σ + ε ε. (2) An alternative way of comuting the deviatoric stress at the end of the increment is rovided by the incremental form of Eq. (9): Ψ σ n+ = Xn+ + τ n+. ( ε ) Combining Eqs. (2) and (22) yields the following nonlinear system of algebraic equations: (22) Ψ e Xn+ + τ n+ C : ( ε ε ) σ n = 0. (23) ( ε ) Consequently, the time integration roblem of the constitutive model is reduced to solving this system, the rincial unknown being ε. Then, the udated stress is comuted as σ = σ + σ. (24) sh n+ n+ n+ For the Newton-Rahson solution of Eq. (23), one defines the residual function ρ( ε ) as Ψ( ε ) e ρ( ε ) = Xn+ + τ n+ C : ( ε ε ) σ n = 0. (25) ( ε ) An initial value ε (0) for the lastic strain increment is calculated and then corrected at each iteration k with the correction term ( ( k ) ) ρ ε δ ε = : ρ ε ε ( k ) ( k + ) ( ) The Jacobian ρ ( ε ) is calculated by differentiation of Eq. (25): (26) 2 ρ( ε ) X Ψ( ε ) τ Ψ( ε ) e = + + τ + C : I 2 ( ε ) ( ε ) ( ε ) ( ε ) ( ε ) s 4. (27) It is imlicitly assumed here that each of the state variables τ n + and n+ X can be exlicitly written in terms of ε. This assumtion is easily verified for the combined isotroic-kinematic model used in section 4.; it has also been shown [9, 35] to be true for the much more comlex model of Teodosiu and Hu [36]. 9

12 The solution of Eq. (25) requires the calculation of the lastic otential and of its first- and second-order derivatives: 2 Ψ( ) Ψ( ) λ ε ε =Ψ( ε ), ξ =, ζ =, 2 ( ε ) ( ε ) (28) as well as the values of the internal variables τ n+ and X n+ and of their first-order derivatives τ ( ε ) and X ( ε ). These terms are the only ones secific to the articular forms of the anisotroy and hardening models; some examles are given in Section 4. The rest of the rocedure is general and can be used with any other model. 3.2 Yield condition In classical rate-indeendent lasticity, the elastic trial stress is used to evaluate the yield function. If the yield condition is not verified (nor violated), then the increment is elastic and try σ = n+ σ ; otherwise, the lastic correction should be alied. In the resent case, no exlicit yield condition is available. To overcome this difficulty, Bacroix and Gilormini [4, 37] have develoed a strain-rate-otential-based yield condition, using the following function of the lastic strain rate direction: ( ) ( ) g N = τψ N T : N. (29) Based on the maximum work rincile, the authors have shown that for a given stress state, a yield condition can be written as: < 0 if σ lays outside the yield surface, Min τψ ( N) T : N = 0 if σ lays on the yield surface, N > 0 if σ lays inside the yield surface. (30) Four indeendent angles θ, θ 2, θ 3, and θ 4 are used to define the comonents of the unit-length tensor N, their collection being denoted by θ. This comact notation is described in more detail in the Aendix. The minimization of g with resect to θ is associated to the solution of the following equation: where g( θ) = 0, (3) θ ( ) Ψ ( ) g θ N T N = τ : θ N τ θ. (32) 0

13 The BFGS minimization algorithm is used in order to avoid analytical calculation of the Hessian 2 2 g θ θ, which is instead aroximated numerically. The BFGS algorithm has also the ( ) advantage of roviding a better convergence whenever the initial guess is far from the solution or the function g is not roven to be convex. In Eq. (32), the term Ψ ( ) N N deends on the chosen otential, while N θ deends only on the definition (56) and is comuted once for all as s s2s3 s4 c c2s3s4 c s2c3s4 c s2s3c 4 c s2s3 s4 s c2s3 s4 s s2c3 s4 s s2s3c4 N = 0 s2s3s4 c2c3 s4 c2s3c 4, (33) θ 0 0 s c 3s4 3c s4 where c = cosθ and s = sin θ ; i =,4. i i i i As comared to the classical elastolastic models, the verification of the yield condition seems more exensive here, since a minimization roblem has to be solved. In ractice, this extra cost can be avoided in most cases. As underlined by Hoferlin [38], the trial stress is surely elastic and try no check needs to be made whenever the deviatoric effective trial stress σ X is much smaller than τ n, e.g., try σ X τ 0. (34) n When the initial stress σ n lies on the yield surface, the following simle condition guarantees that the trial stress lies outside the yield surface [32], and hence that the increment is elastolastic: try ( n ) : n 0 σ σ N, (35) where N n is the normal to the initial yield surface, which can be stored at each increment for future use. The use of Eq. (35) renders the minimization unnecessary in most situations. Finally, in the remaining cases, when the minimization must be erformed, it can be stoed as soon as a tensor N is found so that g(n)<0. Indeed, the minimum is guaranteed to be negative in this case, so the increment is elastolastic. Note that the lastic strain rate direction N that minimizes g N in Eq. (30) has no relevance with resect to the solution of Eq. (25). Consequently, there ( ) is no need to calculate its exact value. In ractice, several simle initializations for N already fulfil this condition in most cases. Thus, the minimization rocedure seldom needs more than

14 one iteration, which is equivalent to the classical yield condition in terms of comutational cost. These conjectures will be further substantiated in Section Consistent tangent modulus The user material routine in a finite element code must udate the stress (and other state variables) over a strain increment, and it must also rovide the modulus defining the tangent relation between the stress increment and the strain increment. This so-called algorithmic (or alg consistent) tangent modulus C is required for the finite element equilibrium iterations. In the framework of strain-rate otentials, algorithmic tangent moduli have been derived by Bird and Martin [39] in the case of elastic-erfectly lastic materials, by Szabó and Jonas [5] for isotroic hardening, and by Hoferlin [38] for combined nonlinear isotroic-kinematic hardening models. The last modulus can be alied for the resent model; its exression reads e s ( ) ( ) D σ n+ = + 4 : : D( ) C I C ε, (36) with C = h ξ + h τ ξ ξ + τ + ζ being the tangent oerator linearly relating the deviatoric stress X n increment to the lastic strain increment, Dσ = C : D( ε ). n+ Calculation of the algorithmic modulus with Eq. (36) involves two matrix inversions. This numerical inconveniency can be avoided if the tangent modulus is derived in a slightly different manner, starting from Eq. (25). By differentiation of Eq. (25) and convenient rearrangement of terms, one obtains: X n+ τn+ e s e + ξ + τ n : 4 : D( ) : D( ) + ζ + C I = ( ) ( ) ε C ε. (37) ε ε K Therefore, the following relationshi can be written: e D ε = K : C : D ε (38) where K is a matrix related to C, yet different. The major advantage of this formula is that K ρ( ε ) ( ε ), as one can easily see from Eq. (27). Consequently, K and its inverse have already been comuted during the calculation for ε. Finally, the incremental form of Hooke s law is differentiated in its slit form (20)-(2) and then combined using Eq. (38) to obtain: D σ = C ε C = C C K C. (39) + alg : with alg e e : : e n D 2

15 An interesting characteristic of this new formulation, in addition to its simlicity, is that no extra matrix inversions are needed, excet for the Jacobian of Eq. (25), which is already available in triangular form. Note that in the case of isotroic linear elasticity, the algorithmic tangent modulus further simlifies as alg s 2 C = KI I + 2GI 4 4GK. (40) 3.4 Sub-steing rocedure In order to ensure a quadratic convergence of the Newton-Rahson resolution of Eq. (25), a consistent tangent modulus has been derived from the discrete equations. However, the solution of the nonlinear Eq. (25) can fail to converge for large strain increments, esecially when the lastic otential exhibits strong variations of curvature. For examle, this aeared to be the case for the Yld yield criterion [40], when small values of the exonent b are considered, as shown by Yoon et al. [4]. The strain-rate otential Sr2004-8, roosed by Barlat and Chung [42] and Kim et al. [9], has also shown such difficulties when the validations that will be shown in Section 4 were erformed. Similar numerical difficulties have been reorted in the literature, when using highly flexible anisotroic yield criteria. A sub-steing rocedure has been adoted to solve this roblem, insired from revious yield-surface based works erformed in the classical lasticity framework [43-47]. When the initial solution ε (0) induces a too large value of the residual, which revents convergence, this value is used to generate a user-defined number m of constant vectors, with the following rule: ( ) m i ρi = ρ ε (0), i =, m. (4) m Then, the following series of equations are solved sequentially, using the solution ε of ( i) equation i as an initial guess for equation i+ : ( ) ρ ε( i) ρ i = 0, i =, m. (42) At the end of this rocedure, the solution of the initial equation is obtained, since according to Eq. (4), ρm = 0. Note that this sub-steing rocedure is activated only at the time stes and integration oints where the direct solution of Eq. (25) fails. Consequently, the imact on the overall comutation time is reduced. The user-defined number of sub-stes m can be increased automatically by the code in case of divergence. The initialization of the lastic strain rate increment has an imortant imact on the convergence. In articular, the initial value for the lastic strain increment cannot be zero, since it enters the very definition of the lastic otential. The most satisfactory initialization was 3

16 found to be ε(0) = λ(0) N (0), where N (0) is the direction normal to a von Mises yield surface assing through the elastic trial stress. Note that, just for avoiding similar cases where the nonlinearity of the lastic otential would revent the convergence of the minimization roblem (3), a similar sub-steing algorithm has been imlemented that allows for a robust solution of both equations. 3.5 Overall time integration algorithm The numerical imlementation has been erformed in Abaqus/Standard via a UMAT routine. For clarity, the numerical algorithm is summarized hereafter. It has been used in the next section for several validations and alications, in order to address its robustness and usefulness.. Inut data: σ, X, τ, ε (strain increment, initial stress and internal variables) n n n sh 2. Comute σ, sh n ε, σ n, ε (sherical and deviatoric arts) try e 3. Elastic rediction: σ = σ + C : ε try 4. Plastic yield condition: n ( ) ( n ) n { N σ X N } sgn min τψ :? N 5. If elastic increment: try sh σ = σ + σ + Ktr( ε) I X n+ = X, τ = τ n+ n n+ n n alg e C = C 6. Otherwise (elastolastic increment): Initialize ε Reeat: Calculate λ, ξ( ε ), ζ( ε ) X n+ τn+ Calculate Xn+, τ n+,, ( ε ) ( ε ) 7. Return n+ (secific to chosen otential) (secific to hardening model) ρ( ε ) Calculate ρ( ε ), ; udate ε (BFGS algorithm) ( ε ) Until convergence Udate stress and state variables; calculate the consistent tangent modulus alg σ, state variables, and C to check equilibrium. alg C 4 ALGORITHM VALIDATION AND APPLICATIONS The constitutive algorithm develoed in Section 3 has been imlemented in the finite element code Abaqus/Standard and alied to tyical sheet metal forming roblems. One aim of this 4

17 section is to validate the state udate algorithm with resect to equivalent models available in Abaqus. Secondly, an advanced anisotroic strain-rate otential is imlemented in the same framework and its redictions are comared to a yield function similar in form and shown to behave as its quasi-dual [9]. 4. Plastic otentials and hardening model The constitutive model resented in Section 2 must be comleted by the mathematical exressions of the lastic otential Ψ( D ) and of the internal variables τ and X. Two examles of lastic otentials are considered hereafter. One is the quadratic strain-rate otential dual to the classical Hill yield criterion, mainly used for validation uroses. Next, the anisotroic strain-rate otential Sr [9, 42] has been imlemented. As for the hardening, a non-linear isotroic-kinematic model is imlemented, which is also available in Abaqus. Indeed, hardening is not an issue in this work validation is the main issue here, and more comlex hardening models can be adoted within the resent framework. Quadratic lastic otential The yield criterion roosed in 948 by Hill [48] is a quadratic exression with six material arameters (F, G, H, L, M, and N). In the material orthotroic frame, this criterion can be written as: ( ) ( ) ( ) ( ) Φ σ, X, τ = F T T + G T T + H T T + 2LT + 2MT + 2NT τ (43) where the three axes, 2, and 3 are the rolling, transverse and normal direction, resectively, in the case of a rolled, orthotroic metal sheet, and T ij are the comonents of the effective stress tensor. Its dual strain-rate otential can be rigorously derived [4] and takes the form: 4F 2 4G 2 4H Ψ ( D ) = ( D ) + ( D22 ) + ( D33 ) + ( D23 ) + ( D3 ) + ( D2 ) L M N (44) where = FH + FG + HG. For an isotroic material, F = G = H = 2 and L = M = N = Sr lastic otential An extension of the Sr93 strain-rate otential [6] has been recently roosed by Barlat and Chung [42] and Kim et al. [9]. This otential, named Sr2004-8, has consistently roven to have suerior flexibility and ability to describe the anisotroy of sheet metal for a large range of materials [24, 49]. Its mathematical exression involves 8 material arameters and makes use of two linear transformations of the lastic strain-rate tensor D : 5

18 ( ) ( ) b b b b b b b Ψ D = Eɶ + Eɶ + Eɶ + Eɶ + Eɶ + Eɶ + Eɶ + Eɶ + Eɶ 2 b 2 + 2, (45) where E ɶ and,,3 i E ɶ i i = are the rincial values of two linear transformations hereafter: D ɶ and D ɶ, defined resectively by the Dɶ = A I D, (46) s 4 Dɶ = A I D. (47) s 4 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: 0 a a a0 a a 0 a a 0 a a a a a A = and A = (48) a a a a a a 9 8 In order to use these comact notations, the D -like tensors are written as 6-comonent vectors; D = [ D D D D D D ] T, with comonents in the frame of material i.e., symmetry. The isotroic case is obtained for a = a2 =... = a8 = and b = 4/3 or 3/2 for bcc or fcc materials, resectively. The Sr93 otential can be recovered by enforcing A I = A I = A. (49) s s 4 4 Recently, an extension of Sr93 and Sr has been roosed by Rabahallah et al. [50], involving an arbitrary number of linear transformations. The first-order derivatives of the exression (45) are rovided in [9]. The imlicit time integration rocedure also requires the calculation of the second order derivatives. Hardening model Modelling of hardening has been a very active research field in the last decades, esecially for sheet metal forming alications, due to an increased interest in the accurate descrition of strain-ath changes in finite element simulations (e.g., strain reversal, orthogonal loading etc.; 6

19 an overview can be found in Haddadi et al. [5]). Several advanced hardening models have been roosed in the last two decades [36, 52-55]. In order to be imlemented directly in the resent algorithm, the hardening model must take the form (0). This may seem a restrictive condition; however, it has been demonstrated that even hardening models as comlex as the Teodosiu-Hu model [36, 5, 56, 57] can be cast in such a simle form, without any alteration [9, 38, 58, 59]. For the sake of comarison, the non-linear isotroic-kinematic hardening model imlemented in the current algorithm is the one already available in Abaqus/Standard [60]. This model involves two internal variables (R, X). The scalar variable R describes the isotroic hardening, and the second-order tensor X describes the kinematic hardening. With the notation of Section 2, the differential equations and the initial conditions describing the evolution of the hardening variables are: τ = τ0 + R, (50) Rɺ = CR ( Rsat R) ɺ λ, R(0) = 0, (5) Xɺ = Cx ( X sat ξ X) λ, X( 0) = 0, (52) where τ 0, CR, Rsat, Cx and X sat are material arameters. The backward Euler scheme is used for the time integration of these rate equations, leading to the udate equations: Rn + CRRsat λ τn+ = τ 0 +, + C λ X n+ X = R + C X ξ λ. + C λ n x sat n+ x (53) The following derivatives are also required for the calculation of the algorithmic tangent modulus: 2 ( λ λ ) X C x X satζ Cx + + C n x X satξ ξ C + xxn ξ = 2 ( ε ) + ( CRRsat RnCR ) 2 ( C λ) τn+ = ( ε ) + R ξ ( C λ ) x (54) These articular exressions of the lastic otential and hardening laws, as well as their derivatives, simly feed into the general algorithm without any other modification. Any other model that fits the requirements of Section 2 can be imlemented with this algorithm in the same way. 7

20 4.2 Algorithm validation: bulge test and S-shae rail forming These alications are meant to validate the numerical imlementation of the constitutive algorithm with resect to reference results, and to address its comutational efficiency with resect to the more classical yield-function-based algorithms. To do so, two tyical sheet metal forming tests have been selected: a bulge test and an S-shae rail forming. The main difference between the two is the highly nonlinear contact evolution for the S-rail, which may cause convergence roblems that further interact with the overall equilibrium convergence sequence. In both cases, Hill s quadratic otential is used. Since the quadratic Hill yield criterion imlemented in Abaqus/Standard is the exact dual of the quadratic otential used here, an identical resonse should be exected. The material arameters, corresonding to an -mm thick AA582 aluminium sheet, are given in Table (from Haddadi et al. [5]). The incomatible-modes enriched, hybrid dislacementressure element C3D8IH is used throughout this section. Table. Material arameters used for the numerical simulations. Elasticity Hill 48 anisotroy Hardening E [MPa] ν F G H L M N τ 0 [MPa] C R R sat [MPa] C X X sat [MPa] The results for the bulge test simulations and the S-rail simulations are shown in Figures 2 and 3, resectively. The two figures clearly show that the results obtained with the Abaqus built-in model and algorithm, and with the current imlementation via UMAT, do coincide with each other. This erfect corresondence has been also noticed in terms of individual comonents of stress, strain, and internal variables. Thus the numerical imlementation of the constitutive algorithm seems accurate and error-free, at least in such tyical alications. Table 2 summarizes the comuting time required by two simulations, using the Abaqus built-in Hill quadratic criterion, as well as its dual otential via UMAT. The calculations have been run on a PC comuter with a Pentium.8 GHz dual core rocessor. The two dual quadratic otentials (and the corresonding comuter imlementations) give not only identical redictions, but the comuting time is almost the same (less than 5% larger in the strain-rate otential case). 8

21 Figure 2. Finite element simulation results for the bulge test. Left: using the Abaqus/Standard built-in model and algorithm; Right: using the current model and algorithm imlemented in Abaqus/Standard via UMAT. Isovalues of equivalent stress. Figure 3. Finite element simulation results for the S-rail. Left: using the Abaqus/Standard builtin model and algorithm; Right: using the current model and algorithm imlemented in Abaqus/Standard via UMAT. Isovalues of equivalent stress. Table 2. CPU time and number of increments for the numerical simulations of bulge test and S- shae rail. Forming roblem Code CPU time Number of increments Bulge test Abaqus 32 min 73 UMAT 33 min 73 S-shae rail Abaqus 2 min 240 UMAT 29 min 240 9

22 4.3 Cu drawing simulation with Sr The aim of this alication is twofold. First, the currently develoed algorithm is alied to one of the most comlex strain-rate otentials available in the literature, in order to address its robustness. Second, its finite element redictions are comared to those of its seudoconjugate yield function, Yld2004-8, for a first investigation of their relative equivalence. The Yld criterion has been used for the cu drawing simulation of an AA2090 aluminium alloy by Yoon et al. [4]. Kim et al. [23] have recently used lane-stress versions of both Yld and Sr for this alication. The arameters of several strain-rate otentials, including Sr2004-8, have been identified for the same material by Rabahallah et al. [24]. Therefore, the same cu drawing simulation was erformed with the algorithms develoed in this aer, using the anisotroy coefficients of the AA2090 material. The hardening arameters are those used in [4]. Table 3 summarizes the material arameters used in the simulation. The geometry of the test is given in Figure 4. The sheet thickness was.6 mm, and the blank holder force was 5500 N. The Coulomb friction coefficient between the sheet and the tools was 0.. One layer of linear hybrid (dislacement-ressure) C3D8IH solid elements has been used to mesh one quarter of the sheet metal, as arametric numerical studies have shown that the number of solid element layers does not influence the earring rofile [59, 6]. Table 3. Material arameters used for the cu drawing simulation. (a) Elasticity and hardening arameters Elasticity Hardening E [MPa] ν τ 0 [MPa] C R R sat [MPa] C X X sat [MPa] (b) Anisotroy arameters a a 2 a 3 a 4 a 5 a 6 a 7 a 8 a a 0 a a 2 a 3 a 4 a 5 a 6 a 7 a b=

23 0.48 Die Samle Blank holder Punch Figure 4. Test geometry for the cu drawing simulation. Dimensions are in mm. Figure 5 shows the results in terms of cu height rofiles. The results obtained with the current imlementation are comared to those available in [4] and [23]. One may notice that, although the four redictions are slightly different, they resent similar characteristics: six ears are redicted, at the same location, and the errors with resect to the exeriments are of the same order and similarly distributed. It is noteworthy that very few other stress or strain-rate otentials allow for the rediction of six ears in cu drawing. Thus, this simulation underlines the mathematical flexibility of the Sr otential and its ability to describe comlex anisotroic behaviour within a henomenological framework. Both Sr redictions underestimate the ears at 0 and 80, as comared to Yld Also, the differences between three-dimensional and lane stress formulations are as large as those between the stress and strain-rate otentials, resectively. A more detailed interretation of the differences existing between the four redictions is difficult for several reasons. The arameter identification method used for our model is different from the three other. Also, two different finite element codes have been used (namely, MSC.Marc for the 3D Yld model and Abaqus for the other models), with different elements. Thus, the comarison in Figure 5 cannot be refined any further, unless all the simulations are erformed with the same finite element code, mesh, and arameters identified in the same manner. Such a comarison would allow a more detailed investigation of the quasi-duality of the two functions from a ractical oint of view. The recent results in [23] suggest that the redictions of the in-lane variation of tensile yield stress and anisotroy coefficient may be slightly different between Yld and Sr2004-8, thus exlaining art of the variability in Figure 5. 2

24 Yoon et al. [4] Current simulation 50 Cu height [mm] Exeriments, Yoon et al. [4] Yld2004-3D, Yoon et al. [4] Yld lane stress, Kim et al. [23] Sr lane stress, Kim et al. [23] Sr2004-3D, current simulation Angle from rolling direction [ ] Figure 5. Results of the numerical simulation of cu drawing with Sr2004-8; cu height rofiles for the current simulation and the reference simulations in [4] and [23], comared to exeriments. 5 CONCLUSIONS A fully imlicit state udate algorithm for strain-rate-otential-based time-indeendent, anisotroic, large strain elastolasticity models has been develoed and described in detail. This aer rovides a generic framework for the numerical imlementation of various models that fall into this category. The numerical imlementation of this algorithm in the commercial FE code Abaqus/Standard is accurate and robust enough to simulate tyical sheet forming oerations. 22

25 The minimization roblem that overcomes the absence of an exlicit yield condition has been shown to induce virtually no additional cost. Therefore, yield-function-based and strain-rateotential-based constitutive algorithms can be considered equivalent in terms of numerical efficiency. The recent Sr has been imlemented in this framework and has shown redictive caabilities similar to the Yld yield criterion, esecially the ability to redict six-ear cu drawing rofiles. ACKNOWLEDGEMENTS The authors are grateful to Frédéric Barlat for roviding material data and simulation results, and to Kwansoo Chung and Brigitte Bacroix for fruitful discussions. The three-year financial suort from the Région Lorraine and from the Centre National de la Recherche Scientifique (CNRS) for the first author is gratefully acknowledged. APPENDIX COMPACT NOTATION FOR DEVIATORIC AND UNIT-LENGTH TENSORS Most tensor quantities involved in the lasticity equations (e.g., N) are symmetric and deviatoric, hence a five-comonent notation can be adoted to reduce the number of indeendent unknowns in the calculations: N = N N 2 ( ) 22 3 N = N + N 2 N N N ( ) 2 22 = 2N 3 23 = 2N 4 3 = 2N 5 2 (55) Moreover, the minimization in Eq. (30) is erformed with resect to the unit-length deviatoric symmetric tensor N which has only four indeendent comonents. In order to reduce the size of the roblem and to avoid minimization under constraints, four angles are used to define N, as follows: 23

26 N N N N N = cosθ sinθ sinθ sinθ = sinθ sinθ sinθ sinθ = cosθ sinθ sinθ = cosθ sinθ = cosθ 5 4 (56) where 0 θ 2π and 0 θ i π, for i between 2 and 4. This contracted notation has several useful roerties. Thus, it is easy to verify that the scalar roducts of two tensors A and B verify the following equalities A : B = A B = A B ; I =,5 ; i and j =,3 (57) ij ij I I and, as a consequence, the norm of the tensor A verifies A = A A = A A. (58) ij ij I I REFERENCES. Ziegler H. An introduction to thermomechanics. North-Holland: Amsterdam, Hill R. Constitutive dual otentials in classical lasticity. Journal of Mechanics and Physics of Solids 987; 35: Arminjon M, Bacroix B. On lastic otentials for anisotroic metals and their derivation from the texture function. Acta Mechanica 99; 88(3-4): Arminjon M, Bacroix B, Imbault D, Rahanel JL. A fourth-order lastic otential for anisotroic metals and its analytical calculation from the texture function. Acta Mechanica 994; 07(-4): Van Houtte P, Mols K, Van Bael A, Aernoudt E. Alication of yield loci calculated from texture data. Textures Microstruct 989; : Barlat F, Chung K. Anisotroic otentials for lastically deforming metals. Modelling and Simulation in Materials Science and Engineering 993; (4): Barlat F, Chung K, Richmond O. Strain rate otential for metals and its alication to minimum lastic work ath calculations. International Journal of Plasticity 993; 9():

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