Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 54, 4, pp. 1147-1156, Warsaw 2016 DOI: 10.15632/jtam-pl.54.4.1147 IDENTIFICATION STRATEGY OF ANISOTROPIC BEHAVIOR LAWS: APPLICATION TO THIN SHEETS OF ALUMINIUM A5 Amna Znaidi, Olfa Daghfas National School of Engineers of Tunis, Departement of Mechanical Engineering, Tunis, Tunisie e-mail: amna.znaidi@Laposte.net; daghfasolfa@yahoo.fr Amen Gahbiche National School of Engineers of Monastir, Mechanical Engineering Laboratory, Monastir, Tunisie e-mail: amen gahbiche@yahoo.fr Rachid Nasri National School of Engineers of Tunis, Departement of Mechanical Engineering, Tunis, Tunisie e-mail: rachid.nasri@enit.rnu.tn Numerical simulation provides a valuable assistance in the controlling of forming processes. The elasto-plastic orthotropic constitutive law is based on the choice of an equivalent stress, a hardening law and a plastic potential. An identification of the model parameters from an experimental database is developed. This database consists in hardening curves and Lank- ford coefficients of specimens subjected to off-axis tensile tests. The proposed identification strategy is applied to aluminumsheets.Thebehavior of thismaterial is studiedunder several solicitations. The anisotropic behavior of the aluminum plate is modeled using the Barlat criterionand the hardening law. The obtained Lankford coefficients are compared to those which are identified by a different strategy. Keywords: anisotropy, strategy, identification, behavior law, off-axis testing 1. Introduction Except for certain processes such as molding, a large majority of metal parts is obtained by forming processes during which the material is plastically deformed. They are optimized to reduce the cost, which requires manufacturers to increasingly use numerical simulation and, therefore, need to describe the material behavior. These simulations are often flawed by a simplified description of the plastic behavior of the material; particularly the anisotropy of rolled sheets (Ghouati and Gelin, 2001; Znaidi, 2004; Haddadi et al., 2006; Kim et al., 2007; Barlat and Lian, 1989). Therefore, it is important to accurately model the plastic behavior of metals in large deformation in order to better predict their behavior of the part during the forming processes (Boubakar and Boisse, 1998; Manget and Perre, 1999; Dogui, 1989). To describe the plastic behavior of the material, it is necessary to clarify the concept of the load surface related to a plasticity criterion (Barlat and Brem, 1991) that indicates conditions of plastic flow. The experimental determination of these areas through various mechanical testing andma- thematical modeling has been the subject of many inquisitive efforts such as those using the VonMises criterion because of its implementation inmost commercial finite element codes. This criterion is called the energy criterion in which the elastic deformation energy of the material must not exceed a limit value to remain within the elastic range. 1148 A. Znaidi et al. In the case of ametal sheet, thematerial is considered as having orthotropic plasticity where it reserves three preferred directions as it is used in the Hill criterion (Hill, 1948). Also, to describe the asymmetric behavior in tension and compression such as the anisotropy of a compact hexagonal structure of a metal sheet, Cazacu et al. (2008, 2009) proposed a new orthotropic criterion (Barlat and Lian, 1989; Gronostajsk, 2000; Plunkett et al., 2006). Theobjective of thiswork is toprovide amodel for numerical simulation of formingprocesses by plastic deformation of thin metal sheets. Hence, the importance of developing a general framework for elasto-plastic orthotropic models (initial orthotropic and isotropic hardening) based on the choice of an equivalent stress andusing theBarlat criterion especially for aluminum alloys (Barlat and Brem, 1991), a hardening law and a plastic potential (Znaidi et al., 2009; Baganna et al., 2010). An identification of themodel parameters from an experimental database is developed. This database consists of many hardening curves from various tests interpreted as homoge- neous (Znaidi, 2004) and their Lankford coefficients (Lankford et al., 1950). Those plates are obtained from a hot-rolling process.We use in our work amethod called the Simplex algorithm for computer programming to identify the constitutive parameters of thematerial behavior.This is an important step.Anew identification strategywith its validationwill be presented, followed by a comparative study using the Hill criterion. 2. Anisotropic elasto-plastic constitutive laws The formulation of the anisotropic elasto-plastic behavior in large deformations is well under- stood now (Znaidi, 2004; Haddadi et al., 2006; Boubakar and Boisse, 1998; Manget and Perre, 1999): using the formalism of the rotating frame ensures the objectivity of the behavior law regardless of the natural (isotropic, anisotropic) constitutive model functions. In this work, we focus on the plastic hardening behavior. The materials are considered as incompressible with negligible elastic deformations. The plastic hardening constitutive laws that we have to study fall within the following framework (with the stress tensor) f(σ,α)¬ 0 α =Q[α] (2.1) with Q being the transformation tensor of Lagrange state to Eulerien state. α represents the internal hardening variable D= λh(σ,α) α̇ = λl(σ,α) (2.2) with λ as the plastic multiplier that can be determined from the consistency condition ḟ = 0 andD is the plastic deformation rate tensor. 3. Orthotropic plasticity model Models are formulated for standard generalized materials with isotropic hardening which are described by an internal hardening variable, a law of evolution and an equivalent deformation. 3.1. Internal variable hardening This work is limited to plastic orthotropic behavior. In fact, the material is initially ortho- tropic and remains orthotropic. The isotropic hardening is assumed to be captured by a single Identification strategy of anisotropic behavior laws... 1149 scalar internal hardening variable denoted by α. In particular, we will assume that the elastic range evolves homothetically, the yield criterion is then written as follows f(σD,α)= σc(σ D)−σs(α) (3.1) whereσD is the deviator of the Cauchy stress tensor (incompressible plasticity). The function σc(σ D) satisfies the following condition σc(aσ D)= aσc(σ D) for all a > 0 (3.2) This property implies that the normal nc = ∂σc/∂σ D is homogeneous of the zero degree with respect toσD nc(aσ D)=nc(σ D) for all a > 0 σ D :nc = σc (3.3) 3.2. Evolution law We assume the existence of a plastic potential g(σD,σs(α)) as follows g(σD,σs(α)) = σp(σ D)−σs(α) (3.4) The hardening function σs(α) plays the role of the thermodynamic function associated with the internal hardening variable α. The function σp(σ D) is assumed to beorthotropic positively homogeneous in the first degree with respect to σD. The evolution law can be written as follows D= λnp np = ∂σp ∂σD α̇ =−λ ∂g ∂σs = λ (3.5) The hypothesis of associated plasticity implies that the plastic potential g is identically σc. The behavior model will be defined by data from two equivalent stresses, σc and σp, which are unique and similar.We consider the case of non-associated normality. 3.3. Equivalent deformation The equivalent deformation is obtained according to one of the following definitions: —Equivalent deformation under the criterion εc ε̇c = 1 σc (σD :D)= 1 σc (σD :D)= λσp σc (3.6) —Equivalent deformation according to the potential εp ε̇p = 1 σp (σD :D)= 1 σp (σD :D)= λ (3.7) 4. Orthotropic equivalent stresses Any orthotropic function σc(σ D) with respect to σD is written in a general way according to the following invariant: — isotropy: det(σD), |σD| — transverse isotropy: σD :m3, (σ D)2 :m3 — orthotropy: σD : (m1−m2), √ (σD)2 : (m1−m2) 1150 A. Znaidi et al. wheremi = ~mi ⊗ ~mi (without summation), ~mi being the orthonormal orthotropy landmark. The application of this configuration to the stress deviator gives us x1 = √ 3 2 (σD11+σ D 22) x2 = 1√ 2 (σD11−σD22 x3 = √ 2σD12 x4 = √ 2σD23 x5 = √ 2σD13 (4.1) Using the special setup of the space deviators, we can write the stress deviator as follows |σD|= √ x21+x 2 2 det(σ D)= x1√ 6 ( x22− x21 3 ) x1 = √ 3 2 (σD1 +σ D 2 ) x2 = 1√ 2 (σD1 −σD2 ) (4.2) We introduce the angle θ which defines the orientation ofσD in the deviatory plane x1 = |σD|cosθ x2 = |σD|sinθ (4.3) We define the off-axis angle ψ (angle which defines the orientation of the loading directions with respect to the preferred direction of the material) as in the following x1 = x1 x2 = x2cos2ψ x3 = x2 sin2ψ (4.4) This allows us to write them in terms of the two angles θ and ψ x1 = |σD|cosθ x2 = |σD|sinθcos2ψ x3 = |σD|sinθ sin2ψ (4.5) Using the special setup of the space deviators, the general form of the equivalent orthotropic plane constraint, is thus σc(σ D)= σc(x1,x2, |x3|)= |σD| f(θ,2ψ) (4.6) Any type of criterion (4.6) can be written in the form f(θ,2ψ)= |σD| σs(α) (4.7) where θ is the angle that defines the test andψ the off-axis angle (Baganna et al., 2010; Lankford et al., 1950). Table 1.Values of θ relative to various tests Test Expansions Simple Large Simple equibiaxes traction traction shear θ 0 π/3 π/6 π/2 5. Identification procedures 5.1. Basic assumption In this Section, we focus on the phenomenology of plastic behavior; especially modeling plasticity and hardening based on experimental data represented as families of hardening cu- rves, and Lankford coefficient data. In order to simplify our identification process, the following assumptions are adopted: Identification strategy of anisotropic behavior laws... 1151 • Hypothesis 1 – Identification through “small perturbations” process. • Hypothesis 2 – The tests used are considered as homogeneous tests. • Hypothesis 3 – We neglect the elastic deformation; the behavior is considered as rigid plastic incompressible. • Hypothesis 4 – The plasticity surface evolves homothetically (isotropic hardening). • Hypothesis 5 – All tests are performed in the plane of the sheet resulting in a plane stress condition. 5.2. Limitation of the model The identified model is defined by: • An equivalent stress σc(A :σD), σc is an isotropic function. It is assumed that the shape is defined by coefficients of the form mi. A – 4th order orthotropic tensor defined by anisotropy coefficients ai. • Apotential equivalent stress σp(Ap :σD), σp is defined by coefficients of the form m′i and anisotropic coefficients a′i. • Hardening curve σs(α) The tests used for the identification of this model are “radial” and “monotonous” tests σ= σa (5.1) with σ > 0 and increasing, and a is constant. And the deformation tensor is ε= εb (5.2) Knowing that σ(ε) is determined from experimental tests as r(ψ). According to the yield criterion (3.1) σc(q)−σs(α)¬ 0 (5.3) where q=A(ai) :σ, thus q= σA(ai) : a. So the equivalent stress (positively homogeneous of one degree) σc(σA : a)= σσc(A :a)= σae (5.4) The equivalent deformation is determined from the duality relation ε̇c = σε̇ σc (5.5) Similar to the equivalent deformation relative to the potential, we write εp = εbe (5.6) When the potential identifies the criterion, we have be =1/ae. Under these conditions, our hardening curve may be written as in the following σae = σs(εbe)⇒ σ = σs(εbe) ae (5.7) 1152 A. Znaidi et al. First comment If our hardening curve is shown by an analytical law such as the following law σs = k(α0+α) n ⇒ σ = k(α0+εbe) n ae = ktest(αtest +ε) n ktest = kbne ae αtest = α0 be (5.8) when n is the same for all tests. A second analytical law σs = σ0+kα n ⇒ σ = σ0+k(εbe) n ae σs = σ0test +ktestε n with σ0test = σ0 ae ktest = kbe ae (5.9) In conclusion, we can say that regardless of the analytical law representing the hardening curve and whatever the test, the value of n does not change. Second comment We can also begin our identification procedures using the Lankford coefficient as determined from the tensile test by r = ε̇2 ε̇3 =− 1 1+ ε̇1/ε̇2 (5.10) We can notice that the Lankford coefficient is independent of ε. This coefficient is equal to one in the case of isotropy, and remains constant in the case of transverse isotropy. However, in the case of orthotropy, it varies depending on the off-axis angle ψ. This coefficient completely characterizes the anisotropy of the sheet when loaded in its plane ε̇ = λnp(σa) aTψ = ε̇1 ε̇2 = [np(a)]11 [np(a)]22 r(ψ)=− 1 1+aTψ (5.11) So identifying this coefficient, means determining aTψ relative to a well chosen model. 6. Results and discussion This identification strategy requires: (a) an experimental database, (b) criterion for anisotropic plasticity and (c) validation strategy. In the particular case of aluminum sheets, where anisotropy is present, the identification of this constitutive law requires the identification of: • The hardening coefficients k and n, • The anisotropy coefficients f, g, h, and n notes that ai has the form factor of m, • The Lankford coefficient r(ψ). The test specimens are cut in different directions relative to the rolling direction of the sheet, according to the geometry defined in Fig. 1. Identification strategy of anisotropic behavior laws... 1153 Fig. 1. Test piece in tension. Definition of the systems axis and current dimensions of the active area (b0 =12.5mm) 6.1. Identification of k and n Using the Hollomon law σs = kε n (6.1) And using the Barlat criterion (Barlat and Brem, 1991) σmc = |q1−q2|m + |q2−q3|m + |q1−q3|m (6.2) where q1, q2 and q3 are the eigenvalues of the tensor q defined by Eq. (5.3). Using the plastic Barlatmodel and respecting the assumptions, the identification of the thin aluminum sheet is equivalent to choosing the coefficients of the model while minimizing the squared difference between the theoretical and experimental results. In Table 2, we present the values of k and n for different tractions tests. Table 2. Identification of the constants of the hardening law ψ 0◦ 45◦ 90◦ K [daN/mm2] 15.1117 17.2835 13.8759 n 0.253 0.2316 0.2485 Knowing that the coefficient n is the same for all tests as demonstrated at the beginning of this work. By convention we choose n for traction in the direction ψ = 0◦ as a reference. For n =0.253, we present different values of k (Table 3). Table 3. Identification of the constant hardening law for the fixed n ψ 0◦ 45◦ 90◦ K [daN/mm2] 15.1132 18.0828 14.002 In Figs. 2a and 2b, the experimental hardening curves (exp) and the curves identified from the model (ident1) using an average value of n and the curves identified by our model (ident2) are represented. For tensile tests in ψ =45◦, the twomodels (ident1) and (ident2) give a clear fit between the theoretical and experimental results. However, for tests in ψ =0◦, identifying these results provides better validation for a good agreement between the experimental and theoretical results for the model (ident2) presented in this work. 6.2. Identification of anisotropic coefficients ai and shape coefficient m Our second identification determines the coefficients of anisotropy (f, g, h, n) and the shape coefficient m (Table 4), considering the Barlat criterion (6.2). 1154 A. Znaidi et al. Fig. 2. Identification of the hardening curve: (a) ψ =0◦, (b) ψ =45◦ Table 4. Identification of anisotropic coefficients and the shape coefficient m F g h n m 0.2854 0.2064 0.3335 0.8921 6.9584 6.3. Evolution of the anisotropic and the Lankford coefficient Using the identified anisotropic coefficients, we represent in Fig. 3a the developments of the Lankford coefficient and the evolution of the anisotropy based on off-axis angles (Fig. 3b). Fig. 3. (a) Evolution of the Lankford coefficient and (b) evolution of the anisotropy based ψ Wefind a good agreement between the experimental results and those from themodel using the Barlat criterion.We notice an important anisotropy at ψ =45◦ (see Fig. 3b). However, the identification using the Hill criterion is not validated by the Lankford coefficient. 6.4. Evolution of the load surface The model (ident2) allows us to study the load surface on each test. We note that this material is resistant to simple shear muchmore than to simple traction. In contrast, the plastic flow in wide traction (i.e. the specimen length is comparable to its width) is quickly reached. 7. Validation Weuse tensile tests atψ =90◦ in order to validate ourmodel.The results showagoodagreement between the theoretical results on themodel (using anisotropic coefficients, shape coefficient m) and experimental data (Fig. 5). Identification strategy of anisotropic behavior laws... 1155 Fig. 4. Evolution of the load surface in the deviatory plane (x2,x3) Fig. 5. Validation of the hardening curve at ψ =90◦ 8. Conclusion In this work, we show that the identification strategy results can be extracted. This identifica- tion has focused both on plastic material parameters of the constitutive law and the Lankford coefficients. Thus, the plastic behavior model: the Hollomon law and the Barlat criterion with 5 parameters are identified. Validation by comparing the models with the experiment data base has been performed. The model using the Barlat criterion is in good agreement with the experimental results relating to the Lankford coefficient. It is better than the Hill model. Following this strategy, we observe very pronounced anisotropy of Aluminum A5 and the load surface for different tests at the end of this identification. Reference 1. Baganna M., Znaidi A., Kharroubi H., Nasri R., 2010, Identification of Anisotropic Plastic Behavior Laws for Aluminum2024 afterHeatTreatmentMaterials fromOff-AxisTesting, Toulouse 2. Barlat F., Brem D.L.J., 19991, A six components yield function for anisotropic materials, International Joural of Plasticity, 7, 693-712 1156 A. Znaidi et al. 3. 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Lankford W.I., Snyder S.C., Bausher J.A., 1950, New criteria for predicting the press per- formance of deep- drawing sheets,Transactions of American Society for Metals, 42, 1196-1232 14. Manget B., Perre P., 1999, A large displacement formulation for anisotropic constitutive laws, European journal of Mecanique, 18, 5 15. Plunkett B., Cazacu O., Barlat F., 2006, Orthotropic yield criterion for hexagonal closed packedmetals, International Joural of Plasticity, 22, 1171-1194 16. Znaidi A., 2004,Plasticity orthotrope in large deformation».PhDthesis at theFaculty of Sciences of Tunis 17. Znaidi A., Soula M., Guellouz S., Nasri R., 2009, Constitutive Equations Identification Strategies Elastoplastic Anisotropics Heet Material Using Associated and Unassociated Laws of Plasticity, Marseille Manuscript received March 5, 2015; accepted for print February 8, 2016