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September 8th, 2015

It has already been mentioned that, to model processes such as texture evolution and its influence on deformation-induced anisotropy, models based on crystal plasticity have been employed in numerous studies. In these studies it has been assumed that the textures employed are representative of the entire volume. However, most forming processes do not produce materials with uniform spatial distributions of texture. Indeed, without sufficient care in the forming process, significant texture gradients develop (e. g., surface-to-midplane texture gradients in rolled materials, surface-to-core gradients in wires). Thus from a theoretical or practical point of view, it is important to investigate these texture gradient effects on plastic deformation properties.

In this section, a thin, ortho tropic sheet specimen submitted to uniaxial tension is modelled (Fig. 2 where 40 x 56 elements are employed) under the assumption of plane strain conditions. The analyses assume no initial geometric imperfection. Localized deformation occurs as a result of the so-called “clamped” boundary conditions applied at the ends (x1= ±L0). With the tensile axis aligned in the x1 direction, and x3 being the direction normal to the sheet, the boundary conditions are

u3 = 0 along x1 = ± L0

Uj = V (applied velocity) along x1 = L0 (6)

iij = – V (applied velocity) along x1 = – L0

Figure 2. Finite element mesh used in the simulations |

A set of discretized orientations of approximately 400 grains, measured at 7 different locations through the thickness of the aluminium alloy AA5754 is employed in the simulations (Figs. 3a-g). It can be seen that the initial textures become sharper towards the centre of the sheet. Thus, the degree of anisotropy increases towards the centre of the sheet. The values of the material properties used in the simulations are

r0= 95 MPa, h0/ r0 = 1.2, ts/t0= 1.16, hjr0 = 0, q = 1.0 (7)

The slip system reference plastic shearing rate f0 and the slip rate sensitivity parameter m are taken asf0 = 0.001s-1, and m=0.002, respectively with the crystal elastic constants taken as C11 =206 GPa, C12 =118 GPa and C44 =54 GPa.

Figures 3a-g. Initial textures of the aluminum alloy AA5754 represented by {111} stereographic pole

figures from the surface towards the centre of the sheet

A quantitative representation of shear band development is presented in Fig. 4 where contours of true strain (in the rolling direction) are plotted versus normalized elongation. It can be seen that at an elongation of U/L0 = 0.065, a shear band passing through the centre of the specimen has already developed (Fig. 4a). With further stretching (U/L0 = 0.07), even though strain has began to concentrate in this well defined shear band, a second shear band has developed perpendicular to the first one (Fig. 4b). The fully developed shear bands at U/L0 = 0.15 are presented in Fig. 4c. Note that, although there are two fully developed shear bands intersecting at the centre of the specimen, the primary (first formed) shear band is sharper and wider than the secondary shear band. This pattern is due to the existing through-thickness texture gradients. Previous studies (Inal et al. (2002b, 2002c)) have indicated that when a single layer of texture was employed in the simulations (no texture gradients), multiple shear bands occurred simultaneously with the same intensities. It should also be mentioned that simulations of plane strain tension, where only a single layer of the initial textures (Figs. 3 a-g) was employed (no texture gradients), always predicted a single shear band.

A recent study by Inal et al. (2002b) has shown that when texture evolution is excluded from the analyses, localized deformation in the form of shear bands was not predicted during plane strain tension. To investigate the effect of through-thickness texture gradients on the predicted localisation modes (necking and/or shear banding), the simulation described above was performed once more, but with texture evolution excluded from the polycrystal model. Thus the stretching and rotation of the lattice vectors were excluded in the numerical analysis. Simulations have shown that, when through

thickness texture gradients are considered, even without texture evolution, localised deformation in the form of shear bands were predicted during plane strain tension (Fig. 5).

EPS

0. 26

0. 2!

0. 1 S 0.11 0.05

The Effects of Strain Paths on Localized Deformation in Drawing Quality (DQ) Sheet Steel

Many industrial processes require sheet metal to be subjected to several complex strain paths before the final product is manufactured. One such process is tube hydroforming, where tubed material is

typically bent into a desired shape, placed within a die, then hyrdoformed to alter the tube crosssection. The strain path is complex in that the sheet material is first subjected to a near-plane strain path in the axial direction (i. e., bending), followed by a near-plane strain path in the circumferential direction (i. e., hydroforming). To simulate this process, sheet specimens were first pre-strained in the rolling direction (RD), then were rotated 90° clockwise and pulled along a second (orthogonal) path, such that the transverse direction (TD) was then aligned along the tensile axis (Fig. 6).

A so-called finite element (FE)/grain model together with a unit cell approach was employed to simulate the strain paths described above. In this model, each element of the finite element mesh represents a single crystal, and the constitutive response at a material point is that given by the single crystal constitutive model. A unit cell is defined as a globally small region of the sheet that contains all the essential micro-structural and textural features that characterize the sheet (e. g., Inal et al. (2005)). The sheet itself is subject to plane stress conditions (i. e., &33 = 0). Orientations within the measured texture data are randomly assigned in the mesh/unit cell. In other words, each element of the mesh represents an orientation from the measured texture. The loading imposed on the edges of the unit cell is assumed to be constant (Fig. 7), such that

where s22 and <6^ are the (principal) logarithmic strain rates. The initial texture for the DQ steel represented by 400 grains/orientations is shown in Fig. 8. In this figure X! and X2 correspond to the rolling and transverse direction of the sheet respectively. The values for the material parameters in the crystal plasticity analysis are y0 = 0.001s-1, m=0.05, h0 /r0 = 28, r0 = 54.5 MPa, n= 0.18 and q=1.

Figure 7. Schematic representation of a unit cell |

Figure 8. Initial texture represented in terms of {111} pole figure. |

The strain paths observed in the experiments (Fig. 6) were imposed at the edges of the unit cell (Equation 8) and numerical simulations were performed with 1600 elements. Thus each grain in the initial texture is represented four times in our simulations. Figs. 9-10 present the measured and predicted textures after the strain paths defined in Fig. 6. It can be seen that the simulated texture is in good agreement with the measured texture. However, the simulated texture is slightly sharper than the measured one. This is probably due to the major drawback of the FE/grain model; i. e., the inevitable inhomogeneous spatial orientation distribution introduced numerically (since the initial texture is assigned randomly to the finite elements). To reduce the effect of spatial distribution of texture components in numerical simulations, the measured crystal orientations are randomly assigned N times to elements in the mesh. In our simulations N, was taken as 4. Usually the higher the N, the lower the effect of the inhomogeneous spatial orientation distribution introduced numerically. Thus, employing higher values on N will improve the overall predicted macroscopic and microscopic responses; predictions were slightly improved when N was taken as 8. However, employing electron backscattering diffraction (EBSD) data as input is the most efficient technique to minimize the effects of the spatial orientation distribution dependency.

Figure 10. Simulated texture by FE/grain model |

In this paper, the implementation of crystal plasticity constitutive relations in the numerical modelling of large strain phenomena was discussed. In particular, the effects of through-thickness texture gradient.

The common metals of industrial practice are polycrystalline aggregates which consist of single crystals or individual grains with lattice structures. As has already been discussed, the mechanical properties of a polycrystalline metal depend on many attributes of its microstructure. Thus, accurate modelling of large strain phenomena should include the effect of the initial microstructure and its evolution. Although phenomenological models are acceptable for many applications, they cannot explicitly include the basic physics of plastic deformation. However, large strain phenomena can be modelled more accurately based on crystal plasticity theories where the initial texture and its evolution, as well as the anisotropy induced by the evolution of microstructure and microscopic properties, are accounted for. Furthermore, with proper parallelization techniques and supercomputers, realistic applications with crystal plasticity theory are now feasible.

Numerical simulations have shown that both the Taylor model and the FE/grain model can be employed to simulate large strain plasticity phenomena in poly crystalline metals. It should be mentioned that the FE/grain model accounts for the grain morphologies (e. g., grain shapes and sizes), which cannot be modelled with the Taylor approach. However, the major drawback of the FE/grain model is the inevitable inhomogeneous spatial orientation distribution introduced numerically (since the initial texture is assigned randomly to the finite elements). By contrast, the spatial orientation distribution does not present this type of problem in simulations with the Taylor model since this model employs average values obtained from individual grains that do not depend on their relative locations.

This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). K. W. Neale holds the Canada Research Chair in Advanced Engineered Material Systems, and the support of this program is gratefully acknowledged.

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