Since texture causes anisotropy, it is important to quantitatively predict the deformation texture evolution and the corresponding mechanical response of materials under plastic deformation. The main challenge of texture modeling lies in how to treat the stress and strain conditions at grain boundaries. In 1928 and 1938, Sachs and Taylor firstly proposed the models with the upper and lower boundary conditions, respectively, i.e. the
‘non constraint’ Sachs model (NC-Sachs) [Sachs1928] and the ‘full constraint’ Taylor (FC-Taylor) model [Taylor1938]. Both models are based on the assumption that deformation distributes homogeneously within a grain and crystallographic slip is the only cause for lattice rotation. The major difference between these two models is that, the Taylor model assumes a continuity of plastic strain ε throughout the polycrystal, i.e. the strain tensor of each grain is equal to that of the entire polycrystal
!ij Sample
=!ij
Crystal (also written as !ij S=!ij
C) (2.2.1)
whereas the Sachs model assumes a homogenous stress σ throughout the polycrystal. It can be seen that the Taylor model causes stress incompatibilities at the grain boundaries, whereas the Sachs model causes strain incompatibilities at the grain boundaries.
A condensed comparison of these two models made by Hirsch and Van Houtte [Hirsch1988c, Van Houtte2002] pointed out that, even though the FC-Taylor model violated the stress continuity at the grain boundaries, it remained a better approach than the Sachs model. Therefore, most of the current deformation texture models were developed basing on the Taylor’s assumption, such as the Visco-Plastic Self-Consistent (VPSC) model [Lebensohn1997/2004, Tomé1991/2002, Proust2007/2009], the Crystal Plasticity Finite Element Model (CPFEM) [Kalidindi1992/1998/2001/2009], the advanced-Lamel (A-Lamel) model [Van Houtte2002/2005] and the Grain Interaction (GIA) model [Wagner1994, F-Bühner1998, B-Hill1951, Crumbach2004/2006a/2006b,
21 Mu2010/2011a, Engler2005, Raabe1998]. For materials deformed by crystallographic slip, such as the face centered cubic (FCC) and body centered cubic (BCC) materials with high or medium stacking fault energies (SFEs), the FC-Taylor model can make a qualitative approximation and show the main evolution of texture components during rolling process. However, in terms of quantitative prediction, the final orientations can have a deviation of up to 10°, and the intensities and volume fractions of the most important texture components can be overestimated by a factor of up to 2 [Van Houtte2002]. It was also pointed out that this problem most likely lay in the stress incompatibilities at the grain boundaries. A better assumption was therefore desired, with a compromise of both strain and stress compatibilities at the grain boundaries.
An improvement to the FC-Taylor model was achieved by abandoning the global homogeneity of strain throughout the polycrystal, but allowing for certain shear components to be relaxed, i.e. deviations from the prescribed strain state. This is called the Relaxed-Constraint (RC-Taylor) model [Kocks1982, Van Houtte1982/2005, Honneff1978]. In such a case the shear stress corresponding to the relaxed shear is set to be zero automatically, and thereby the compatibility of stress is accomplished in those directions. Based on the geometry of the rolling process, there are several variations of the RC-Taylor model, such as the ‘pancake’ model in which both of the !13 and !23
strain components can be relaxed (1, 2 and 3 represent for the rolling, transverse and normal directions of sample, respectively). In the ‘lath’ model only the !13 component is relaxed. The shear component !13 is also called Copper-shear, since it promotes the Copper component in FCC type texture. The shear !23 is called S-shear for an equivalent reason. The Brass component in FCC type texture is promoted by the in-plane !12 shear, which would create large strain incompatibilities between the ‘pancake’ grains, therefore
!12 is not reasonable to be relaxed in a common RC model. As a conclusion, both FC- and RC-models cannot predict strong Brass component in FCC metals. However, better predictions of the texture evolutions during the rolling process can be obtained with the RC-Taylor models [Hirsch1988b].
However, in the RC-Taylor models, the choice of the relaxed shear components is quite artificial, and the one relaxation that is justified in one certain strain path may be meaningless for most other deformation paths. For example, the ‘pancake’ model with
22
relaxations in both !13 and !23 components is a quite good choice for rolling process modeling, in which most of the grains have a flat, elongated shape. However, its application for other arbitrary strain paths would make no sense. The most successful attempt to make a non-artificial choice of relaxed strains was to take the interaction between neighboring grains into account. For example, in the VPSC model each grain is treated as a visco plastic ellipsoidal inclusion embedded in a homogeneous equivalent medium with uniform properties. While each grain is representative of all grains with the same orientation, the homogeneous equivalent medium represents the average neighbourhood of those grains. With such a structure, the VPSC model can treat the grain interaction in an average and ‘long range’ manner. On the contrary, in the CPFEM model each crystallite consists of at least one finite element mesh, the interactions among all grains are taken into account at once. Therefore, the CPFEM model achieves both stress and strain compatibility simultaneously at grain boundaries in a ‘short range’ manner.
However, it demands high computing power or CPU time. Both models mentioned above achieved substantial improvements in prediction of the texture evolution by taking the grain interaction into account. However, a compromise solution that provides a precise description of the grain interaction with reasonable CPU time was still desirable.
In the present work, one of the most advanced cluster-type deformation texture models, the Grain-Interaction (GIA) model is introduced and applied. Compared to the VPSC model, the GIA model is advantageous in its treatment of the grain interaction. This cluster type Taylor model has a grain structure consisting of 8-grain aggregates embedded in a homogeneous matrix. During deformation, the whole aggregates are deformed under a fully prescribed strain state of the polycrystal, like in the FC-Taylor model, which avoids strain incompatibilities between aggregates. But each grain within the aggregates is allowed to be relaxed in all three spatial directions. Roughly speaking, by adding a proper number of geometrically necessary dislocations (GNDs) at the boundaries, the model accommodates the misfit and gap caused by relaxations of individual grains and describes the grain interaction more precisely than the VPSC model.
Meanwhile, the GIA model does not demand as much computing power as the CPFEM model. The GIA model has been successfully applied to predict deformation texture evolution of cubic materials, especially of Al and its alloys, in which crystallographic slip is the only cause for lattice rotation and texture evolution. In the following, the basics of
23 the FC- and RC-Taylor models as well as the main features of the GIA model are summarized (also see [Wagner1994, F-Bühner1998, B-Hill1951, Raabe1998, Crumbach2004/2006a/2006b, Mu2010, Engler2005,]).