Texture and anisotropy engineering · Crystal plasticity

Minimum earing in aluminium: an inverse texture design computed with the texture component crystal plasticity finite element method

Z. Zhao (MIT), W. Mao (UST Beijing), F. Roters, D. Raabe (Max Planck Institute for Iron Research)

Acta Materialia 52 (2004) 1003–1012 · doi:10.1016/j.actamat.2003.03.001 · received 12 December 2002, accepted 6 March 2003

Deep drawing of a textured aluminium sheet leaves a rim with ears. The study reported here, published in Acta Materialia in 2004, asks a different question from the usual one. Instead of computing the ears of a given texture, it searches for the texture that gives the smallest ears, and it does so quantitatively, with the texture evolution during the forming step included in the calculation. The method is a texture component crystal plasticity finite element method in which the orientation distribution is represented by a small number of spherical Gaussian components, each with a center orientation, a volume fraction, and a full width at half maximum. The components are mapped onto a finite element mesh, the cup is drawn in simulation, and the rim profile is read off. The result is a concrete recipe: a mixture of 62.75 vol% S component and 37.25 vol% Cube component, with an initial spherical scatter of 15 degrees at half maximum, produces the flattest rim among the combinations tested. The ratio 1.67 to 1, rather than the intuitive 1 to 1, is one of the more informative numbers in the paper.

The problem and the practical route

Earing in deep drawn sheet is a direct consequence of plastic anisotropy, and in sheet with more than 10 000 grains that anisotropy is set by the crystallographic texture inherited from rolling and annealing. Beyond the irregular rim, the effect is an uneven distribution of wall thickness and of the local mechanical properties, because volume conservation forces the strain rate to vary around the circumference. The trivial fix is a random starting texture, which carries no shape anisotropy at all. In practice this route is closed: a random distribution is difficult to produce in the first place, and it does not remain random once the sheet is deformed. Most metallurgical and mechanical processing sharpens orientation distributions rather than randomizing them, which is true in particular for face centered cubic metals without bulk phase transformation, such as aluminium [1,2].

The workable route is compensation. Rolling textures of aluminium carry the S, Brass, and Copper components, while annealing adds the Cube and Goss components. The two families behave oppositely in the drawing test: Cube and Goss grow their highest ears at 0 and 90 degrees to the rolling direction, whereas S, Brass, and Copper place theirs near 45 and 135 degrees. A sheet in which the two families are mixed in the right proportion should therefore show a flatter rim than any of its constituents. The question the study answers is what that right proportion is, and how the scatter of the components enters the answer [3].

The texture component crystal plasticity finite element method

The texture component method, introduced by Lücke and co workers and improved by Helming, approximates the orientation distribution function as a sum of spherical Gaussian central functions [3]. Each component is fixed by three parameters with direct physical meaning: the center orientation given as an Euler triple, the volume fraction of the component, and the full width at half maximum, which measures the angular spread around the center. A unit intensity random background completes the sum, and the sum of all volume fractions is one. The Gaussian shape is chosen because it is a normalized central function whose pole figures can be written in closed form with generalized Bessel functions, which makes the least squares fit to experimental pole figures fast and stable. Few components are enough to describe an orientation distribution, whatever the grain count of the real material, and that compactness is exactly what a finite element model needs, since every integration point must carry a discrete set of orientations that can be updated during loading.

The mapping onto the mesh is done in two steps, and it is the core of the method. In the first step the center orientation of a component is assigned to every integration point, so that locally the specimen is a single crystal. In the second step these orientations are systematically reoriented, by a weighted Monte Carlo sampling in Euler space, until the resulting distribution reproduces the prescribed Gaussian function. The scatter, which was defined only in orientation space, is thereby converted into a spatial distribution of orientations across the mesh. Where several components are present at a location, the local Taylor assumption, a local strain rate homogenization, allows more than one orientation with its own volume fraction at the same integration point. Once this representation is built, the components themselves are dropped from the simulation. During the calculation every orientation follows its own slip driven reorientation path under the local boundary conditions, and the texture update is the direct result of these individual paths.

The constitutive description is the fully implicit crystal plasticity model of Kalidindi and co workers, whose finite element implementation goes back to Pierce, Asaro, and Needleman [4,5]. The slip rate on each system follows a rate sensitive power law with rate sensitivity exponent m, the critical shear stress is updated with a latent hardening matrix that distinguishes self and latent hardening through the ratio q, and the stress is obtained from the elastic deformation gradient of the polar decomposition. For aluminium, twelve {111}<110> slip systems are used, and the hardening is fitted to pure aluminium at room temperature.

Simulation set up

The cup drawing is computed in ABAQUS with a user material subroutine, using the quarter of the blank allowed by the orthotropic symmetry of the sheet. The mesh contains 432 C3D8 and 80 C3D6 elements, with the angular element density increasing from the blank center toward the rim so that the final border of the cup is well resolved. The blank holder is clamped in two steps with an exponential soft contact law, and the punch then travels 80 mm. The friction coefficient is set to zero. Earlier simulations with friction between 0 and 0.2 showed only a small effect on the relative ear height, and the reason is physical rather than numerical: nonuniform deformation and strain localization arise naturally from crystallographic slip in a polycrystal, which is different from classical J2 continuum models [6,7].

Five ideal components are used as building blocks, with the Euler triples listed in Table 1. Because of the orthorhombic symmetry of the sheet, each single orientation has to be balanced by its symmetry equivalent variants: four variants for the S and Brass components, two for Copper, and one for Cube and Goss, each variant carrying the corresponding share of the volume fraction. The full width at half maximum of the components is varied from 0 to 15 degrees in the two simulation series, and earing is quantified by the ear height and by the ear area shown in Fig. 4.

Results

Single components. The five components were first drawn as ideal single crystals, with zero scatter about the main orientation before loading. The Cube component gives maximum ear height at the rolling and transverse directions with ideal orthotropic symmetry. The Goss component shows the same 0 / 90 type of pattern, with its maxima shifted about 15 degrees toward the transverse direction. The S, Brass, and Copper components place their peaks at about 45 degrees plus or minus 6, 60, and 30 degrees to the rolling direction, respectively. The predictions agree with the ear shapes measured in earlier work [1,2]. The five components thus split into two groups with opposite earing behavior, the 0 / 90 group (Cube, Goss) and the 45 / 135 group (S, Brass, Copper), and the two groups are the natural candidates for a compensating mixture.

Mixing S and Cube. The second series combines the two components that produce opposite ears. Both the ear height and the ear area pass through a minimum at 62.75 vol% S and 37.25 vol% Cube, a ratio of 1.67 to 1, and the same composition is optimal for both measures. The deviation from a 1 to 1 mixture is the important finding. The opposite ear shapes of the two components would suggest equal volumes, and the calculation shows that the interaction between texture components is strongly nonlinear: the Cube orientation has a disproportionately large effect on the anisotropy of deep drawing, so that a clearly larger share of the S component is needed for compensation. The earing curves are also steep around the minimum. A small change of the volume ratio, which corresponds in practice to a modest change of the pole figures, moves the ear profile strongly toward optimization or toward degradation. The texture has to be held within a narrow window, and the steepness of the response is something a process engineer has to know before the recipe is used [1].

Scatter width. The initial spherical scatter width acts as a second optimization variable. For the Cube component alone, increasing the full width at half maximum from 0 to 15 degrees lowers the ear height from 19 to 14 mm. For the optimum S and Cube mixture, an increase of the scatter from 0 to 15 degrees smooths the rim substantially, and the drawn cup in Fig. 9 shows a rim that is visibly calmer than the single crystal cases. As a rule, the earing decreases with increasing initial scatter width of the texture components. One qualification matters here: the zero scatter starting components evolve during the simulation into an array of orientations that reorient individually according to the local boundary conditions, so the specimens do not remain single crystalline. It is the local, orientation by orientation resolution of these reorientation and strain paths that the method provides, and the optimum ratio of 1.67 to 1 is the result of that resolution, not of a volume average [1].

Conclusions of the paper. The ear peaks of the Cube and Goss components appear at the rolling and transverse directions, while the ears of the Copper, S, and Brass components occur near 45 and 135 degrees to the rolling direction. Minimum earing is obtained by mixing the Cube and the S components at a volume ratio of 1.67 to 1. The optimization depends on both the crystallographic orientation and the scatter width of the components, and the earing decreases as the spherical scatter width increases.

Significance

The study changed the direction of texture engineering in a concrete way. In the classical practice of the period, texture was an inherited side effect of processing, known as such and accepted as it came. The paper inverts this logic: the desired anisotropy defines the texture, and the process has to be tuned to deliver it. This inverse procedure, the physical tailoring of a texture for a prescribed product property under prescribed processing conditions, is the program that crystal plasticity finite element methods have served since, and the texture component approach used here is a compact and robust way to feed experimental textures into such simulations without enumerating individual grains. The method is part of the development path toward the open source framework DAMASK, which solves the same class of problems, texture evolution, local strain, and formability, at the grain scale of real materials [8].

The practical content is equally clear. For high stacking fault energy face centered cubic sheets such as aluminium, a low earing texture is a designed mixture of a 45 / 135 type component and a 0 / 90 type component: S and Cube at a volume ratio of about 1.67 to 1, with a moderate initial scatter of about 15 degrees at half maximum. Both the ratio and the scatter width are quantities that can be influenced in rolling and annealing practice, and the steep dependence of the ear profile on composition, which the simulations quantify, tells the process engineer that the texture has to be held within a narrow band once the recipe is fixed [1].

Key references

1. Z. Zhao, W. Mao, F. Roters, D. Raabe, A texture optimization study for minimum earing in aluminium by use of a texture component crystal plasticity finite element method, Acta Materialia 52 (2004) 1003–1012, doi:10.1016/j.actamat.2003.03.001.
2. A.D. Rollett, G.R. Canova, U.F. Kocks, in: Metal Formability and Microstructure, TMS-AIME, Warrendale (1986) p. 147; and the earing and r value literature cited therein, e.g. J. Morris, Scr. Metall. Mater. 31 (1994) 387; O. Daaland, E. Nes, Acta Mater. 44 (1996) 1389; J. Savoie et al., Acta Metall. 44 (1996) 587; J.W. Yoon, F. Barlat et al., Int. J. Plast. 16 (2000) 1075.
3. K. Lucke, J. Pospiech, K.H. Virnich, J. Jura, Acta Metall. 29 (1980) 167; K. Helming et al., Z. Metallkd. 85 (1994) 545.
4. D.G. Pierce, R.J. Asaro, A. Needleman, Acta Metall. 30 (1982) 1087; A. Needleman, R.J. Asaro, Acta Metall. 33 (1985) 923.
5. S.R. Kalidindi, C.A. Bronkhorst, L. Anand, J. Mech. Phys. Solids 40 (1992) 537.
6. D. Raabe, M. Sachtleber, Z. Zhao, F. Roters, S. Zaefferer, Acta Mater. 49 (2001) 3433; M. Sachtleber, Z. Zhao, D. Raabe, Mater. Sci. Eng. A 336 (2002) 81.
7. H. Dève et al., Acta Metall. 36 (1988) 341; S.V. Harren, R.J. Asaro, J. Mech. Phys. Solids 37 (1989) 191.
8. DAMASK, open source crystal plasticity finite element framework, https://damask-multiphysics.org/ (and the references therein).
9. D. Raabe, homepage and publications, https://www.dierk-raabe.com/.