EffEffect of Sing ect of Single Cry le Crystal Orientation on F stal Orientation on Formingorming

AAbstrbstractact.. Among processes involving plastic deformation, sheet metal forming requires a most accurate description of plastic anisotropy. One of the main sources of mechanical anisotropy is the intrinsic anisotropy of the constituent crystals. In this paper, we present the single-crystal yield criterion recently developed by Cazacu et al. [1] and its application to the prediction of anisotropy in uniaxial tension of strongly textured polycrystalline sheets. Namely, it is shown that using this single crystal yield criterion the Lankford coefficients exist and have finite values for all loading orientations. Moreover, the variation of both the yield stress and Lankford coefficients with the crystallographic direction can be expressed analytically. An application of this criterion to forming a cylindrical cup from a single crystal of (100) orientation is presented. Finally, we show that using this single-crystal model, one can describe well the effect of the spread around an ideal texture component on the anisotropy in uniaxial tensile properties of a polycrystal.


Intr Introduction oduction
Since the publication of Lankford landmark paper [2] in which it was shown that the material's initial anisotropy may have a beneficial influence on forming performance, extensive experimental and theoretical efforts have been undertaken towards development of appropriate anisotropic yield criteria. For polycrystalline metallic materials, versatile orthotropic yield criteria have been developed (for example see [3], [4]). In comparison very few yield criteria have been proposed in the literature for the description of the yielding anisotropy of single crystals (e.g. see [5]). Recently, in [1] were demonstrated the benefits of using a single crystal yield criterion that is defined for any 3-D loadings. Specifically, it was shown that this criterion involves the correct number of independent anisotropy parameters it captures the differences in yield stress anisotropy between different crystals, e.g. the differences in the anisotropy in the tensile uniaxial yield stresses with the orientation between the loading direction and the crystallographic directions (e.g. see the examples provided in [1] for aluminum and copper single crystals). Moreover, it is possible to obtain analytically the expression of the Lankford coefficient and tensile uniaxial yield stress along any crystallographic orientation (see [6]). Another advantage of using this criterion is the fact that it is C 2 differentiable. This in turn facilitates its implementation into finite element (FE) codes. Furthermore, using for the description of the individual grain behavior this single-crystal model, one can describe well the anisotropy of polycrystalline materials (e.g. see [7]).
In this paper, we present an application of this single crystal yield criterion to forming a cylindrical cup from a singlecrystal sheet of (100) orientation. Moreover, we show that using this single-crystal model, one can describe well the effect of the spread of orientations around (100) on the anisotropy in uniaxial tensile properties.  2 Single-cry le-crystal yield crit  stal yield criterion erion The single-crystal yield criterion is defined for any stress-state. It is written in terms of cubic invariants that were deduced using rigorous theorems of representation of tensor functions. Therefore, the properties of invariance of the yield function with respect to the intrinsic symmetries associated to crystals belonging to the cubic system are automatically satisfied. Moreover, by using these representation theorems, it is ensured that the criterion involves the correct number of anisotropy coefficients such as to satisfy the crystal symmetries and the condition of yielding insensitivity to the hydrostatic pressure (for full mathematical proofs and further details, see [6] and Cazacu et al. [1]).
Let us define as Oxyz the Cartesian coordinate system associated with the crystal axes (i.e., the <100> crystal directions). In this coordinate system, the expression of the effective stress associated with the single-crystal yield criterion [1] is given by: In Eq. (1), σσ denotes the Cauchy stress deviator, m1, m2, n1, n3, n4 are anisotropy coefficients while the parameter c describes the relative importance of the second-order and third-order cubic stress-invariants on yielding of the crystal. For example, all the parameters of the criterion can be determined using the experimental tensile uniaxial yield stresses in four crystallographic directions. A schematic view of the cup-drawing set-up is shown in Fig. 1. The tool and blank dimensions are given in Table 1. As a result of material and geometrical symmetries of the problem, only one-quarter of the cup needs to be considered in the F.E. analysis. The blank is meshed with 4200 Abaqus C3D8H elements (8 node brick elements with constant pressure, see ABAQUS [8]). The circular blank has an initial thickness of 0.813 mm and an initial diameter of 79 mm. A blank force of 1000 N is used.  where K0, ε0 and n are parameters, and ε̅ p is the equivalent plastic strain associated with the effective stress given by   It is well known that even in the case of strongly textured polycrystalline sheets for which it can be considered that the texture has a single component, more a spread is generally observed around this component. Therefore, it is of great interest to estimate the effect of this spread on the plastic anisotropy in the uniaxial properties at the polycrystal level.
In the next section, using the single crystal yield criterion [1] for describing the plastic behavior of the constituent crystals, we describe the anisotropy in uniaxial tensile properties for sheets containing only the (100) component and being characterized by various scatter widths from this component (see Fig.3).
The effective stress of the polycrsytal, σ̅ poly(σ σ) corresponding to the applied stress tensor, σ σ, is expressed in the loading frame is considered to be of the form: with N being the number of crystals considered in the polycrystal, σ̅ grain j denotes the effective stress of the crystal j  with the effective stress σ̅ given by Eq. (1). It can be shown that irrespective of the values of the parameters of the criterion, r(θ)=r(90°-θ), and in particular, r(0°)=r(90°)=1 (for full proof, see [5]). In Fig. 4, the evolution of Lankford coefficients r(θ) and yield stress ratios σ(θ)/σ(0) with the loading direction θ for the sheets with textures given in Fig.   3, were calculated with the same set of values for the parameters describing the crystal level yielding behavior , namely m1=1.0, m2 =0.38, n1=0.98, n3=0.04, n4=0.08, c=2.3. Lankf Lankfor ord coefficients d coefficients r r( (θ θ) in the plane of the cube t ) in the plane of the cube te extur xtured pol ed poly ycry crystalline sheets. The t stalline sheets. The te extur xtures f es for diff or differ erent scatt ent scatter er width ar width are sho e shown in Fig.3. The r wn in Fig.3. The results f esults for the ideal t or the ideal te extur xture e ω ω0 0=0 w =0 wer ere obtained with the anal e obtained with the analytical f ytical formulas (E ormulas (Eq.(5)-q.(5)-E Eq.(6)), r q.(6)), respecti espectiv vel ely y. .
Note that even for the ideal cube texture both the predicted yield stresses and Lankford coefficients vary smoothly with the loading orientation (Fig.4), for the values of the parameters considered, there are extrema only at 0º , 45º, and 90º to the [100] direction in the (100) plane. It is worth recalling that for an ideal cube texture, the yield stress variation with the loading orientation according to the Taylor-Bishop-Hill model displays two cusps and infinite values are predicted for the Lankford coefficients along the 0º (i.e. RD) and 90º (i.e. TD) tensile loadings (see [9]). Additionally, the TBH model predicts a smooth variation only if the texture is characterized by a very large spread (see simulation results presented in [9]).