Abstract
The aim of this paper is to investigate the impact of the microscopic yield surface (i.e., at the single crystal scale) on the forming limit diagrams (FLDs) of face centred cubic (FCC) materials. To predict these FLDs, the bifurcation approach is used within the framework of rateindependent crystal plasticity theory. For this purpose, two micromechanical models are developed and implemented. The first one uses the classical Schmid law, which results in the formation of vertices (or corners) at the yield surface, while the second is based on regularization of the Schmid law, which induces rounded corners at the yield surface. In both cases, the overall macroscopic behavior is derived from the behavior of the microscopic constituents (the single crystals) by using two different scaletransition schemes: the selfconsistent approach and the Taylor model. The simulation results show that the use of the classical Schmid law allows predicting localized necking at realistic strain levels for the whole range of strain paths that span the FLD. However, the application of a regularized Schmid law results in much higher limit strains in the range of negative strain paths. Moreover, rounding the yield surface vertices through regularization of the Schmid law leads to unrealistically high limit strains in the range of positive strain paths.
Keywords:
crystal plasticity; rateindependent theory; selfconsistent model; Taylor model; bifurcation approach
1 INTRODUCTION
During forming processes, metal sheet instabilities may occur in several ways. Though the origins of these instabilities are diverse and varied, they can be categorized into two main classes. The first class includes all instabilities that are basically related to the shape of the metal part and one refers to them as geometric instabilities. Buckling and wrinkling are typical examples of such geometric instabilities. The second class pertains to the instabilities that are intrinsically related to the material behavior and one refers to them as material instabilities. The localization of deformation into narrow shear bands, as usually observed in threedimensional solids, and localized necking in thin sheet metals are typical examples of material instabilities. In the literature, many investigations have been devoted to predicting the occurrence of localized necking, since this phenomenon is considered as limiting to material formability. In this field, we can quote the pioneering works of (Keeler and Backofen, 1963Keeler, S.P., Backofen, W.A. (1963). Plastic instability and fracture in sheets stretched over rigid punches. Trans. ASM 56:2548.) and (Goodwin, 1968Goodwin, G.M. (1968). Application of strain analysis to sheet metal forming problems in press shop. Metallurgia Italiana 60:767744.), who introduced the concept of forming limit diagram (FLD) to characterize the occurrence of localized necking in metal sheets. For each strainpath parameter ρ ranging from uniaxial tension (ρ =  0,5) to equibiaxial tension (ρ = 1), the inplane principal strains, corresponding to the deformation state at the onset of localized necking, are reported on a curve designated as FLD. Accordingly, the deformation states below the FLD are expected to produce flawless parts, while the deformation states above the FLD may lead to potential defects. The FLDs can also be obtained experimentally; however, their experimental determination turns out to be complex and expensive. An interesting alternative to this experimental procedure is the numerical modeling approach. To set up this latter approach, two main ingredients are required: the modeling of the mechanical behavior of the studied sheet and a localization criterion, which is then coupled with the constitutive equations. In the related literature, several phenomenological and micromechanical models have been coupled to various localization criteria in order to predict FLDs for sheet metals. Among these diffuse or localized necking criteria, we can cite the maximum force principle initiated by (Considère, 1885Considère, A. (1885). Mémoire sur l'emploi du fer et de l'acier dans les constructions. Annales des Ponts et Chaussées 9:574775.), the imperfection approach developed by (Marciniak and Kuczynski, 1967Marciniak, Z., Kuczynski, K. (1967). Limit strains in processes of stretchforming sheet metal. International Journal of Mechanical Sciences 9:609620.), the bifurcation theory (Drucker, 1950Drucker, D.C. (1950). Some implications of work hardening and ideal plasticity. Quarterly of Applied Mathematics 7:411418.; Drucker, 1956Drucker, D.C. (1956). On the uniqueness in the theory of plasticity. Quarterly of Applied Mathematics 16:3542.; Hill, 1958Hill, R. (1958). A general theory of uniqueness and stability in elasticplastic solids. Journal of the Mechanics and Physics of Solids 6:239249.; Stören and Rice; 1975Stören, S., Rice, J.R. (1975). Localized necking in thin sheets. Journal of the Mechanics and Physics of Solids 23:421441.; Rice, 1976Rice, J.R. (1976). The localization of plastic deformation. In: 14th International Congress of Theoretical and Applied Mechanics 207220.) and the perturbation method (Molinari and Clifton, 1987Molinari, A., Clifton, R. (1987). Analytical characterization of shear localization in thermoviscoplastic solids. Journal of Applied Mechanics 54:806812.; Dudzinski and Molinari, 1991Dudzinski, D., Molinari, A. (1991). Perturbation analysis of thermoviscoplastic instabilities in biaxial loading. International Journal of Solids and Structures 27:601628.). A more detailed discussion along with a comparative analysis of the various plastic instability criteria can be found in (AbedMeraim et al., 2014AbedMeraim, F., Balan, T., Altmeyer, G. (2014). Investigation and comparative analysis of plastic instability criteria: application to forming limit diagrams. International Journal of Advanced Manufacturing Technology 71:12471262.). In the present work, attention will be restricted to the bifurcation approach developed by (Stören and Rice, 1975Stören, S., Rice, J.R. (1975). Localized necking in thin sheets. Journal of the Mechanics and Physics of Solids 23:421441.), which is adopted here as localization criterion. This approach is characterized by the formation of narrow localization bands corresponding to jumps in some constitutive ratevariables (e.g., strain rate tensor) across interfaces. Such material instability has also been shown to correspond to the loss of ellipticity of the partial differential equations governing the associated boundary value problem. The condition of strain localization is reached when the first zero eigenvalue is seen in the acoustic tensor (Rice, 1976Rice, J.R. (1976). The localization of plastic deformation. In: 14th International Congress of Theoretical and Applied Mechanics 207220.).
From a numerical point of view, the bifurcation criterion is rather easy to implement compared to other localization criteria, such as the imperfection approach. However, without introducing some key destabilizing effects within the material constitutive modeling, the application of this theory unalterably predicts unrealistically high strain levels in the positive inplane biaxial stretching range (see, e.g., Rice, 1976Rice, J.R. (1976). The localization of plastic deformation. In: 14th International Congress of Theoretical and Applied Mechanics 207220.). In the case of phenomenological constitutive modeling, the destabilizing effects may be accounted for by introducing either damageinduced softening effects (Haddag et al., 2009Haddag, B., AbedMeraim, F., Balan, T. (2009). Strain localization analysis using a large deformation anisotropic elasticplastic model coupled with damage. International Journal of Plasticity 25:19701996.; Mansouri et al., 2014Mansouri, L.Z., Chalal, H., AbedMeraim, F. (2014). Ductility limit prediction using a GTN damage model coupled with localization bifurcation analysis. Mechanics of Materials 76:6492.) or by considering deviation from normality in the plastic flow rule (see, e.g., Rudnicki and Rice, 1975; Kuroda and Tvergaard, 2001). In the case of micromechanical modeling, the destabilizing mechanism is a natural outcome of crystal plasticity multislip and the associated yield surface vertex effects, which is accounted for by using the classical Schmid law (Schmid and Boas, 1935Schmid, E., Boas, W. (1935). Plasticity of Crystals. Chapman and Hall, London.). The bifurcation approach has been successfully coupled in (Franz et al., 2009aFranz, G., AbedMeraim, F., Lorrain, J.P., Ben Zineb, T., Lemoine, X., Berveiller, M. (2009a). Ellipticity loss analysis for tangent moduli deduced from a large strain elasticplastic selfconsistent model. International Journal of Plasticity 25:205238., ^{2009b}Franz, G., AbedMeraim, F., Ben Zineb, T., Lemoine, X., Berveiller, M. (2009b). Strain localization analysis using a multiscale model. Computational Materials Science 45:768773.) with a polycrystalline model based on the selfconsistent scaletransition scheme within a fully threedimensional framework. More recently, the bifurcation approach, under the planestress assumption, has been applied in (Yoshida and Kuroda, 2012Yoshida, K., Kuroda, M. (2012). Comparison of bifurcation and imperfection analyses of localized necking in rateindependent polycrystalline sheets. International Journal of Solids and Structures 49:20732084.) to crystal plasticity based on classical Schmid's law and the fullconstraint Taylor model. Furthermore, micromechanical models have the advantage of accounting for physical phenomena relevant at the single crystal scale, such as texture evolution, distribution of dislocation densities, morphological texture, residual stresses, in contrast to phenomenological models, for which some associated parameters do not have a direct physical meaning. However, the use of the classical Schmid law in micromechanical modeling is known to exhibit a major difficulty. Indeed, the application of classical Schmid's law leads in many cases to the wellknown issue of ambiguity in the determination of the set of active slip systems and their corresponding slip rates. In the literature, this ambiguity problem has been tackled by a number of authors, and the most widely used remedy to eliminate the associated nonuniqueness in the selection of active slip systems is through the consideration of a ratedependent approach. But, again, the use of a ratedependent approach has also several shortcomings. On the one hand, the choice of ratedependent modeling is not very realistic when the viscous effects are really limited. This is typically the case in cold forming processes. On the other hand, the use of elastoviscoplastic behavior models in conjunction with the bifurcation approach leads to unrealistic limit strains for the whole range of strain paths (ranging from uniaxial tension to equibiaxial tension). To avoid at once the ambiguity in the determination of the set of active slip systems and the influence of undesirable viscous effects, the regularized Schmid law has been proposed as an interesting alternative to the classical Schmid law for the modeling of the mechanical behavior at the single crystal scale.
As discussed above, the application of the bifurcation approach leads to unrealistically high limit strains in the righthand side of the FLD, when the flow theory with associative plasticity and smooth yield surface is used (Rice, 1976Rice, J.R. (1976). The localization of plastic deformation. In: 14th International Congress of Theoretical and Applied Mechanics 207220.; Ben Bettaieb and AbedMeraim, 2015Ben Bettaieb, M., AbedMeraim, F. (2015). Investigation of localized necking in substratesupported metal layers: Comparison of bifurcation and imperfection analyses. International Journal of Plasticity 65:168190.). In the framework of micromechanical modeling, based on crystal plasticity, one can wonder how realistic the predicted FLDs would be if a rounded yield surface is used instead of the sharper yield surface associated with the classical Schmid law. One of the main objectives of the current work is to address this question. To this end, a regularized Schmid law, quite similar to that proposed by (Gambin, 1992Gambin, W. (1992). Refined analysis of elasticplastic crystals. International Journal of Solids and Structures 29:20132021.), is adopted as plastic flow rule to model the single crystal response. The main feature of this regular form of the Schmid law is that its associated yield surface is practically the same as the classical Schmid yield surface, except that the corners are rounded. Both the Taylor model and the selfconsistent scheme are used to derive the overall constitutive response of the polycrystal starting from the behavior of its microscopic components (single crystals). A similar behavior model has been proposed by (Yoshida et al., 2009Yoshida, K., Brenner, R., Bacroix, B., Bouvier, S. (2009). Effect of Regularization of the Schmid Law on selfconsistent estimates for rateindependent plasticity of polycrystals. European Journal of Mechanics A/Solids 28:905915.) to evaluate the response of an FCC rigidplastic material under small strains.
The remainder of the paper is organized as follows:

The second section will be devoted to the modeling of the single crystal behavior. Thereafter, a comparison will be performed between the classical Schmid law and the regularized Schmid law.

In the third section, the theoretical framework for both homogenization models (namely, the Taylor model and the selfconsistent approach) will be presented.

In the fourth section, the bifurcation approach will be developed in the general framework, as well as its specialization to the planestress conditions.

The fifth section details the numerical results at the macroscale and includes the corresponding comparisons and discussions.
Notations and conventions
CSL and RSL will be used hereafter as abbreviations for classical Schmid law and regularized Schmid law, respectively. As to the selfconsistent model, it will be referred to as SC.
Vectorial and tensorial fields are designated by italic bold letters and symbols
Scalar variables and parameters are represented by italic thin letters and symbols
Einstein's convention of summation over repeated indices is adopted. The range of the free (resp. dummy) index is given before (resp. after) the corresponding equation
time derivative of •
•^{1} inverse of •
•^{T} transpose of •
• ⋅ • inner product
• : • double contraction product
• ⊗ • tensor product
det(•) determinant of tensor •
sgn(•) sign of •
2 MODELING OF THE SINGLE CRYSTAL BEHAVIOR
As a starting point, the total deformation gradient f is taken to be multiplicatively decomposed into an elastic part f^{e} and a plastic part f^{p}
The elastic part f^{e} may itself be multiplicatively decomposed into a stretching tensor ν^{e} and a rotation tensor r
Rotation r defines the orientation of the coordinate system related to the intermediate configuration relative to its counterpart in the current one.
The Eulerian velocity gradient g, expressed in the current configuration, is therefore given by the following relation:
As is the case for most metallic materials, the elastic deformation is often assumed to be very small compared to unity. Accordingly, the stretching tensor ν^{e} is very close to the secondorder identity tensor
Combining Eqs. (3) and (4), we obtain
The symmetric and skewsymmetric parts of g, denoted as d and w, respectively, are defined by
where the elastic and plastic parts, d^{e} and d^{p} , of the strain rate tensor d, as well as the elastic and plastic parts, w^{e} and w^{p} , of the rotation rate tensor w are given by
In the current work, the plastic strain is assumed to be solely due to the slip on the crystallographic slip systems. Each slip system β is defined, in the deformed configuration, by two vectors β), are denoted by R^{β} and S^{β} , respectively. In the intermediate configuration related to the crystallographic lattice, the counterparts of and and as well as between and (which is known as the Schmid orientation tensor for slip system ⊗ representing the slip direction and the normal to the slip plane, respectively. The symmetric and skewsymmetric parts of the tensor product remain fixed throughout the deformation. They are denoted by and , respectively. Similarly, the tensors corresponding to Rβ and Sβ in the intermediate configuration are denoted by Rβ0 and Sβ0, respectively. The relations between and are given by the following expressions:
The detailed numbering of Appendix A. for FCC single crystals is given in and
As the plastic deformation is solely due to the slip on the crystallographic systems, tensors d^{p} and w^{p} can be expressed as
where:

β. The algebraic values of the slip rates have the same sign as their respective resolved shear stresses, denoted τ^{β} . is the absolute value of the slip rate on the slip system

N_{s} is the total number of slip systems (equal to 12 for FCC single crystals).
In order to satisfy the objectivity principle, the lattice corotational derivative σ^{∇} of the Cauchy stress tensor s is related to the elastic strain rate d^{e} by the following hypoelastic law:
where C^{e} is the fourthorder elasticity tensor.
The resolved shear stress τ^{α} is defined as the projection of the Cauchy stress tensor on the Schmid orientation tensor
The rate form of Eq. (11) allows us to obtain
Combining Eqs. (12), (10), and (9)(1), can be expressed as
where R ^{α}: C^{e} . is equal to
The evolution of the critical shear stress is given by the following hardening law:
where H is a hardening matrix, whose analytical expression will be given in details in Appendix A.
The constitutive law of the single crystal relates the nominal stress rate g, using the microscopic tangent modulus l to the velocity gradient
To derive the analytical expression of l, one has to express the slip rates in terms of the velocity gradient
where γ ^{α} is a secondorder symmetric tensor, corresponding to slip system α, whose expression will be given later, as it depends on the plastic flow rule considered (i.e., classical Schmid law or regularized one).
The analytical expression for the microscopic tangent modulus reads
where ^{1}_{σ}L and ^{2}_{σ}L are fourthorder tensors that contain convective terms of Cauchy stress components
Depending on the plastic flow rule considered, which may involve CSL or RSL, different analytical expressions for γ ^{α} can be obtained. These expressions will be derived in Sections 2.1 and 2.2.
2.1 Plastic Flow Rule Associated with the Classical Schmid Law
In this section, the slip rates are derived from the classical Schmid law (Schmid and Boas, 1935Schmid, E., Boas, W. (1935). Plasticity of Crystals. Chapman and Hall, London.). This law states that slip may occur on a given slip system α only if the absolute value of its resolved shear stress τ^{α} reaches its critical value τ^{α} _{c}
Accordingly, the CSL corresponds to a multisurface yield criterion, where each slip system α is defined by a yield function χ ^{α}
A global yield function can therefore be defined as
It is noteworthy that the CSL satisfies the normality rule. Indeed, one can easily verify that Eq. (9)_{(1)} is equivalent to
In order to compute the slip rates, the consistency condition is used
where A is the set of active slip systems (α ∈ A ⇔ >0).
One obtains eventually
where M is the inverse of matrix P defined by the following index form:
γα being the secondorder symmetric tensor used in Eqs. (16) and (17).
When the CSL is used, the analytical expression of γα is given as follows:
2.2 Plastic Flow Rule Associated with the Regularized Schmid Law
The regular form of the Schmid law (RSL) is drawn from the work of (Gambin, 1992Gambin, W. (1992). Refined analysis of elasticplastic crystals. International Journal of Solids and Structures 29:20132021.). The corresponding yield function reads
where n is a positive number acting as a regularization parameter. As will be shown in Section 2.3, this parameter accounts for the sharpness of the yield surface.
Contrarily to the CSL, for which the computation of the slip rates requires matrix inversion, the slip rates have the following generic form in the present case:
where such that is a positive scalar, which is derived from the normality rule. Indeed, the normality rule states that there exists a positive value
By taking , one obtains relation (9)(1). In order to derive the analytical expression of the plastic multiplier , the consistency condition is used
Furthermore, by using Eqs. (13), (14), and (28), one obtains the following relation:
Combining Eqs. (30) and (31), the analytical expression of is eventually obtained
where ℱ is a secondorder symmetric tensor defined as
and k _{0} is a scalar defined as
Once the plastic multiplier Eqs. (32)(34)), the slip rate on each slip system can be determined using Eq. (28) is calculated (through
Recall that tensor γ ^{α} is used in Eq. (17) for the determination of the microscopic tangent modulus. In the framework of RSL, the analytical expression of γ ^{α} reads
The generic form of Eq. (28) is quite similar to that used for viscoplastic materials. We briefly recall the analytical expression of the slip rates in the framework of ratedependent theory (Asaro and Needleman, 1985Asaro, R.J., Needleman, A. (1985). Texture development and strain hardening in rate dependent polycrystals. Acta Metallurgica 33:923953.)
where m the ratesensitivity parameter. is a reference shear rate and
Although similar in form, a major difference between the ratedependent formulation (see Eq. (37)) and the RSL (see Eq. (28)) is that the latter preserves its rateindependent character and, hence, it is still applicable to bifurcation analysis, while the former is not. It must also be noted that, from a numerical point of view, forms (28) and (37) generally lead to very stiff equations, especially when exponents 2n1 and 1/m take high values. Consequently, the CPU time required to integrate the constitutive equations based on the regularized form of the Schmid law is generally high. This CPU time is quite comparable to that required for the integration of viscoplastic single crystal models, when the value of exponent 2n1 (in the regularized form (28)) becomes very close to the magnitude of exponent 1/m in the viscoplastic evolution law (37).
2.3 Comparison Between the CSL and the RSL
As pointed out by several authors (Gambin and Barlat, 1997Gambin, W., Barlat, F. (1997). Modeling of deformation texture development based on rate independent crystal plasticity. International Journal of Plasticity 13:7585.; Imbault and Arminjon, 1998Imbault, D., Arminjon, M. (1998). Deformation textures of fcc materials predicted with a regular form of the Schmid law. Materials Science Forum 273:371376.), the value of the regularization parameter n determines the degree of sharpness of the yield surface. The higher the value, the sharper the corresponding yield surface. From a certain value of n, the obtained yield surfaces are very close to that determined by CSL. This feature is illustrated in Figure 1, where the RSLbased yield surfaces for an FCC single crystal, corresponding to two different values of n (namely, n=5 and n=20), are compared with the CSLbased yield surface. The three Euler angles that define the initial crystallographic orientation of the single crystal (Bunge, 1968Bunge, H.J. (1968). The orientation distribution function of the crystallites in cold rolled and annealed lowcarbon steel sheets. Physica Status Solidi (b) 26:167172.) are randomly chosen and are respectively equal to 293°, 124°, and 305°. The Young modulus is equal to 210 GPa and the Poisson ratio is equal to 0.3. The hardening parameters τ_{0,} h _{0} ,n _{1} (see Appendix A for more details) are equal to 40 MPa, 390 MPa, and 0.35, respectively.
Comparison between the RSL yield surfaces, corresponding to two different values for the regularization parameter, and the CSL yield surface: (a) RSL (n=5); (b) RSL (n=20); (c) CSL.
Figure 2 (a) depicts the stressstrain uniaxial tensile response as obtained with both flow rules (CSL and RSL). For n=5, the RSLbased response is significantly different from its counterpart yielded by CSL. Nevertheless, for n = 500, the RSLbased curve matches the CSLbased one for a certain range of strain (0 up to 0.2). Beyond 0.2, the deviation between the two tensile responses is a consequence of slip system activity, which is not the same in both models. Figure 2 (b) shows the evolution of some components of the microscopic tangent modulus as a function of the tensile strain. We can observe that for the RSL, the evolution of the tangent modulus components is independent of the regularization parameter n. Compared to CSL, we can see that there is only a difference in the shearing components, where those predicted by CSL are reduced beyond the elastic range, whereas those determined by RSL keep their elastic values. These qualitative comparisons are consistent with those reported by (Yoshida et al., 2009Yoshida, K., Brenner, R., Bacroix, B., Bouvier, S. (2009). Effect of Regularization of the Schmid Law on selfconsistent estimates for rateindependent plasticity of polycrystals. European Journal of Mechanics A/Solids 28:905915.) in the framework of small strains.
Uniaxial tensile test for an FCC single crystal: (a) Stressstrain responses; (b) Components of the single crystal tangent modulus.
3 MODELING OF THE POLYCRYSTAL BEHAVIOR
In this section, we briefly present the incremental selfconsistent scheme used to derive the macroscopic tangent modulus ℒ. The latter relates the macroscopic nominal stress rate tensor G to the macroscopic velocity gradient
This scheme has been initially developed by (Hill, 1965aHill, R. (1965a). Continuum micromechanics of elastoplastic polycrystals. Journal of the Mechanics and Physics of Solids 13:89101., ^{1965b}Hill, R. (1965b). A self consistent mechanics of composite materials. Journal of the Mechanics and Physics of Solids 13:213221.), in the framework of small elastoplastic strain, and extended by Lipinski (Lipinski and Berveiller, 1989Lipinski, P., Berveiller, M. (1989). Elastoplasticity of microinhomogeneous metals at large strains. International Journal of Plasticity 5:149172.; Lipinski et al., 1995Lipinski, P., Berveiller, M., Reubrez, E., Morreale, J. (1995). Transition theories of elasticplastic deformation of metallic polycrystals. Archive of Applied Mechanics 65:291311.) to large deformation. Using this scaletransition scheme, the macroscopic tangent modulus is obtained by the following relation:
where f^{I} represents the volume fraction of grain I, l^{I} its microscopic tangent modulus, while A^{I} denotes the concentration tensor that relates the microscopic velocity gradient g^{I} to its macroscopic counterpart G. When the Taylor model is considered, Eq. (39) is still valid provided that the concentration tensor A^{I} of each grain I be replaced by the fourthorder identity tensor I.
4 THEORETICAL FRAMEWORK FOR THE PLANESTRESS BIFURCATION APPROACH
Figure 3 illustrates the localization of deformation in a metal sheet, which is represented by a localization band defined by its unit normal vector .
In the current study, the threedimensional formulation of the bifurcation approach implemented in (Franz et al., 2009aFranz, G., AbedMeraim, F., Lorrain, J.P., Ben Zineb, T., Lemoine, X., Berveiller, M. (2009a). Ellipticity loss analysis for tangent moduli deduced from a large strain elasticplastic selfconsistent model. International Journal of Plasticity 25:205238., ^{2013}Franz, G., AbedMeraim, F., Berveiller, M. (2013). Strain localization analysis for single crystals and polycrystals: Towards microstructureductility linkage. International Journal of Plasticity 48:133.) is adapted to the planestress framework. With the planestress condition, the nominal stress rate tensor and the velocity gradient satisfy the following mixed boundary conditions:
where ρ is the strainpath parameter, and symbol ? designates the unknown components. Though in the present work the adopted tangent modulus is different from that developed in (Franz et al., 2009aFranz, G., AbedMeraim, F., Lorrain, J.P., Ben Zineb, T., Lemoine, X., Berveiller, M. (2009a). Ellipticity loss analysis for tangent moduli deduced from a large strain elasticplastic selfconsistent model. International Journal of Plasticity 25:205238., ^{2013}Franz, G., AbedMeraim, F., Berveiller, M. (2013). Strain localization analysis for single crystals and polycrystals: Towards microstructureductility linkage. International Journal of Plasticity 48:133.), the bifurcation criterion has the same form
where ℒ^{PS} is the 2D tangent modulus that relates the inplane components of the nominal stress rate tensor to the inplane components of the velocity gradient. The analytical expression for the 2D tangent modulus ℒ^{PS} is derived from the general expression of the 3D tangent modulus ℒ by the classical relation
5 NUMERICAL RESULTS AT THE MACROSCALE
In this section, we investigate the influence of both flow rules (namely, the CSL and the RSL models) on the overall stressstrain responses as well as on the predicted FLDs at the polycrystal scale. As a starting point, an FCC polycrystalline aggregate composed of 1000 grains is considered. The hardening parameters as well as the elastic constants are the same as those used in Section 2.3. Figure 4 shows the random initial texture taken for this polycrystal.
Figure 5 compares the stressstrain responses, obtained with both flow rules, for three different inplane strain paths ρ: uniaxial tensile test (ρ = 0.5), planestrain tension (ρ = 0), and positive biaxial stretching (ρ = 0.5). For the RSL, the regularization parameter n that measures the sharpness of the corresponding yield surface is set to 20. This value is motivated by a reasonable compromise between the computation time and the stressstrain curve matching of both flow rules. Indeed, the computation time increases significantly with the value of n, because the time step requires to be divided into several subincrements in order to accurately satisfy the yield criterion. For all the computations related to the RSL presented hereafter, parameter n is fixed to 20. In Figure 5 (a), the Taylor model is used as homogenization scheme. From this figure, one can observe that at large strains, the gap between the RSL and the CSLbased responses increases as the strainpath parameter r increases. However, in the small strain range, it is found that both flow rules yield the same response, as previously mentioned by (Yoshida et al., 2009Yoshida, K., Brenner, R., Bacroix, B., Bouvier, S. (2009). Effect of Regularization of the Schmid Law on selfconsistent estimates for rateindependent plasticity of polycrystals. European Journal of Mechanics A/Solids 28:905915.). In Figure 5 (b), the SC model is used to assess the macroscopic behavior of the material. In this latter case, except for the uniaxial tensile test, the deviations between the RSL and CSL occur at much earlier stage of deformation, with a softer response yielded by the CSL. On the other hand, comparisons between the Taylor model and the SC scheme (see, e.g., Figures 5 (a) and 5(b)) reveal that the use of the RSL leads to the same overall stressstrain response, irrespective of the homogenization model, whereas the use of CSL leads to a softer response with the SC approach. These observations are fully consistent with those made by (Yoshida et al., 2009Yoshida, K., Brenner, R., Bacroix, B., Bouvier, S. (2009). Effect of Regularization of the Schmid Law on selfconsistent estimates for rateindependent plasticity of polycrystals. European Journal of Mechanics A/Solids 28:905915.).
Overall stressstrain responses for three different inplane loading paths: (a) Taylor model; (b) Selfconsistent model.
Figure 6 depicts the evolution of some components of the tangent modulus as function of the major strain. It is wellknown that the incremental elastoplastic tangent modulus plays a major role in the framework of bifurcation theory. In this regard, the evolution of the components ℒ_{1111}, ℒ_{1122}, and ℒ_{1212}, which are representative of the other components, is investigated. It is worth noting that, in the case of RSL, the predicted components keep their high elastic values in the course of loading. Quite the opposite trend is observed with the CSL, where the shearing components of the tangent modulus, represented by ℒ_{1212}, are significantly reduced beyond the elastic range of deformation. In spite of this significant reduction in the shear components, notice that the stressstrain responses yielded by both flow rules are practically the same with the Taylor model (see, e.g., Figure (5)).
Evolution of the components of the tangent modulus for different strain paths: (a) Taylor model (ρ= 0.5); (b) SC model (ρ= 0.5); (c) Taylor model (ρ= 0); (d) SC model (ρ= 0); (e) Taylor model (ρ= 0.5); (f) SC model (ρ= 0.5).
Figures 7 and 8 show the evolution of the minimum of the determinant of the acoustic tensor, over all possible band orientations, as a function of the major strain, for different strain paths. In Figure 7, the Taylor model is used as scaletransition scheme. Recall that strain localization is marked by the singularity of the acoustic tensor (i.e., as soon as its determinant reaches zero). In the range of negative strain paths (see, e.g., ρ=0.5 and ρ= 0), we observe a quasicontinuous drop in the determinant of the acoustic tensor, for both flow rules, which regularly decreases and eventually reaches zero. However, for positive strain paths (ρ> 0), the trend is completely different for the RSL framework, in which the determinant decreases very slightly for ρ= 0.1 and keeps practically a constant value for ρ= 1 or ρ= 0.5. As discussed in the literature, most often within phenomenological constitutive frameworks (see, e.g., Rice, 1976Rice, J.R. (1976). The localization of plastic deformation. In: 14th International Congress of Theoretical and Applied Mechanics 207220.; Hutchinson and Neale, 1978Hutchinson, J.W., Neale, K.W. (1978). Sheet Necking  II. Timeindependent behavior. In: D.P. Koistinen, N.M. Wang (Eds.), Mechanics of sheet metal forming. New York: Plenum Press, 127153.; Ben Bettaieb and AbedMeraim, 2015Ben Bettaieb, M., AbedMeraim, F. (2015). Investigation of localized necking in substratesupported metal layers: Comparison of bifurcation and imperfection analyses. International Journal of Plasticity 65:168190.), it is found that when the shearing components of the tangent modulus keep their high elastic values, the use of bifurcation theory as localization criterion yields unrealistically high localization strains in the positive biaxial stretching range. Here, the localization predictions suggest values for the critical strains far beyond 2. In Figure 8, the same analysis as that conducted in Figure 7 is followed, but with the SC model as homogenization scheme. As can be observed, the SC approach (see Figure 8) reveals trends quite similar to those reflected by the Taylor model. Besides, for the CSL, the predicted localization strains given by the SC model are much lower in the whole than those obtained with the Taylor model. For the RSL, however, there are no significant differences between the Taylor and the SC localization predictions.
Evolution of the minimal determinant of the acoustic tensor for different strain paths: (a) RSL (n=20) coupled with the Taylor model; (b) CSL coupled with the Taylor model.
Evolution of the minimal determinant of the acoustic tensor for different strain paths: (a) RSL (n=20) coupled with the SC model; (b) CSL coupled with the SC model.
Figure 9 displays the localized necking strains predicted for all of the strain paths ranging from uniaxial tension (ρ= 0.5) to equibiaxial expansion (ρ=1). The two homogenization models (namely, the Taylor and SC scaletransition schemes) are coupled with both plastic flow rules (i.e., CSL and RSL). As previously mentioned, the limit strains predicted by the RSL are extremely high in the range of positive inplane biaxial stretching. This trend is consistent with the observations made in Figure 6, where it was shown that the shearing components of the tangent modulus keep their elastic values when the RSL is used, which leads to unrealistically high limit strains in the positive biaxial stretching range. In the negative strainpath range, however, the RSL necking strains are predicted at realistic levels, although they are much higher than those obtained with the CSL (see Figure 9).
FLDs predicted by the two different homogenization models for both plastic flow rules: (a) Taylor model; (b) SC model.
Figure 10 shows the evolution of the localization band orientation, corresponding to the predicted FLDs, as function of the strainpath parameter ρ. With the Taylor model, the predicted band orientations are almost the same for both plastic flow rules in the range of negative strain paths (ρ < 0) (see Figure 10 (a)). For the SC model, however, the differences in terms of predicted band orientations are more noticeable in the range of negative strain paths (see Figure 10 (b)). Also, with the SC approach, and for the studied polycrystalline aggregate (see its initial texture shown in Figure 4), the localization bands predicted by the CSL are found to remain always normal to the major strain direction in the range of positive strain paths (ρ > 0).
6 CONCLUDING REMARKS
The effect of the plastic flow rule, as modeled at the single crystal scale, on the forming limit diagrams of polycrystalline sheet metals has been investigated using the bifurcation approach as localization criterion. To this end, two different plastic flow rules have been considered at the single crystal scale. The first one is based on the classical Schmid law, which results in a yield surface with sharp vertices, while the other uses a regular form of the Schmid law, which leads to a smoother yield surface (i.e., with rounded vertices). In both cases, the Taylor model as well as the selfconsistent scaletransition scheme are used to relate the micro and macro scales. From the numerical results, it is found that, in the range of negative strain paths, the FLDs predicted by the regularized Schmid law are realistic, although much higher than those predicted by the classical Schmid law, and this conclusion is valid irrespective of the scaletransition scheme considered (Taylor model or incremental selfconsistent approach). However, in the range of positive strain paths, the predicted limit strains given by the regularized Schmid law are unrealistically high, in contrast to those obtained by the classical Schmid law, which remain quite realistic. It is also found that when the regularized Schmid law is used, the overall stressstrain response of the material as well as the predicted necking strains are almost insensitive to the scaletransition scheme, at variance with the use of the classical Schmid law.
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Appendix A
The set of FCC slip systems is given in the table below:
The hardening matrix used in this study is of powerlaw type, which is given by the following relations:
where τ_{0} is the initial critical shear stress, h _{0} is the initial hardening rate and n _{1} is the powerlaw hardening exponent.
Publication Dates

Publication in this collection
Dec 2016
History

Received
09 Sept 2015 
Reviewed
08 Jan 2016 
Accepted
01 Mar 2016