Laurent Delannay

Comparison of two grain interaction models for polycrystal plasticity and deformation texture prediction

The reader is briefly reminded that there are no models, yet available, capable of truly quantitative deformation texture predictions for arbitrary strain paths, although such models are clearly needed for accurate finite element (FE) simulations of metal forming processes. It is shown that for cold rolling of steel, the classical models (full-constraints and relaxed constraints Taylor, self-consistent) are clearly outperformed by new 2-point or n-point models, which take grain-to-first-neighbour interactions into account. Three models have been used: the 2-point “Lamel”-models (two variants) and the micromechanical finite element-model developed by Kalidindi et al. (J. Mech. Phys. Sol. 40 (1992) 537). Extensive comparisons of the results of the Lamel-model with experimental data has been published before (by Delannay et al. (J. Phys. IV France 9 (1999) 43) and van Houtte et al. (Textures and Microstuctures 31 (1999) 109). Emphasis of the present paper is a confrontation of the Lamel model with the micromechanical finite element-model. It was found that for the case study at hand, the solutions of each model can be regarded as approximations of the solutions of the other. It is, however, believed that the FE-model would really be able to produce reference results (macro and micro deformation textures) if more elaborate meshes are used that describe the microstructure more closely.

Quantitative Prediction of Cold Rolling Textures in Low-Carbon Steel by Means of the Lamel Model

Rolling textures of low-carbon steel predicted by full constraints and relaxed constraints Taylor models, as well by a self-consistent model, are quantitatively compared to experimental results. It appears that none of these models really performs well, the best results being obtained by the Pancake model. Anew model (“Lamel model”) is then proposed as a further development of the Pancake model. It treats a stack of two lamella-shaped grains at a time. The new model is described in detail, after which the results obtained for rolling of low-carbon steel are discussed. The prediction of the overall texture now is quantitatively correct. However, the γ-fibre components are better predicted than the α-fibre ones. Finally it is concluded that further work is necessary, as the same kind of success is not guaranteed for other cases, such as rolling of f.c.c, materials.

Quantitative analysis of grain subdivision in cold rolled aluminium

A procedure is proposed for a statistical characterisation of deformed microstructures from the data collected using Orientation Imaging Microscopy (OIM). This procedure has been applied for a characterisation of 352 grains in commercial purity aluminium cold-rolled to a reduction of 40%. The results demonstrate the different behaviour of grains with different orientations: (i) grains having orientations near the {001} and {025} components develop orientation gradients over distances of 10–20 μm; (ii) grains with orientations close to the {205} component form fragments with relatively large misorientations, and (iii) grains within the β-fibre form fragments with relatively small misorientations. The experimental results are compared to previous observations by transmission electron microscopy (TEM) and good agreement is found. Finally, possible applications of the present observations for advanced texture modelling are discussed.

Quantitative prediction of textures in aluminium cold rolled to moderate strains

Two aluminium plates with different hot band textures were cold rolled to moderate thickness reductions (55 and 60%). The formation of a rolling texture was analyzed quantitatively by focusing on crystallographic fibers in the orientation distribution function. These measurements served to test several polycrystal plasticity theories that rely on different assumptions on how individual grains perceive the macroscopically exerted load. It was found that N-site models in which each grain interacts with a specific neighborhood, yield improved predictions of rolling textures compared to ‘full constraints’ and ‘relaxed constraints’ Taylor models. The β-fiber representing stable lattice orientations was reproduced satisfactorily by the N-site models but none of the models captured the behavior of grains initially oriented near {0 0 1}〈1 0 0〉. The latter is perhaps attributable to the fact that the continuum theories evaluated do not account for the fragmentation mechanisms active in such grains.

Adaptive mesh refinement and automatic remeshing in crystal plasticity finite element simulations

In finite element simulations dedicated to the modelling of microstructure evolution, the mesh has to be fine enough to: (i) accurately describe the geometry of the constituents; (ii) capture local strain gradients stemming from the heterogeneity in material properties. In this paper, 3D polycrystalline aggregates are discretized into unstructured meshes and a level set framework is used to represent the grain boundaries. The crystal plasticity finite element method is used to simulate the plastic deformation of these aggregates. A mesh sensitivity analysis based on the deformation energy distribution shows that the predictions are, on average, more sensitive near grain boundaries. An anisotropic mesh refinement strategy based on the level set description is introduced and it is shown that it offers a good compromise between accuracy requirements on the one hand and computation time on the other hand. As the aggregates deform, mesh distortion inevitably occurs and ultimately causes the breakdown of the simulations. An automatic remeshing tool is used to periodically reconstruct the mesh and appropriate transfer of state variables is performed. It is shown that the diffusion related to data transfer is not significant. Finally, remeshing is performed repeatedly in a highly resolved 500 grains polycrystal subjected to about 90% thickness reduction in rolling. The predicted texture is compared with the experimental data and with the predictions of a standard Taylor model.

Modelling the initial stage of grain subdivision with the help of a coupled substructure and texture evolution algorithm

The substructure development in f.c.c. monocrystals and polycrystal grains under cold rolling is modelled with the help of evolution equations for the densities of redundant cell wall and non-redundant fragment boundary dislocations as well as of mobile and immobile disclinations in six cell wall and fragment boundary families. The slip rates for the 12 f.c.c. slip systems are calculated by a full constraints Taylor algorithm. The critical resolved shear stresses are derived from the dislocation and disclination densities. Substructure and crystal orientation are updated alternately in each integration step. The model is able to predict cell wall and fragment boundary spacings as well as misorientations in reasonable agreement with experimental data obtained in TEM studies on aluminium. The subset of preferably splitting grains in a polycrystal is found in reasonable agreement with OIM results.

Prediction of intergranular strains in cubic metals using a multisite elastic-plastic model

A novel approach is adopted for determining the elastic and plastic strains of individual grains within a deformed polycrystalline aggregate. In this approach, termed “multisite modeling”, the deformation of a grain does not merely depend on the grain lattice orientation. It is also significantly influenced by the interaction with one or several of the surrounding grains. The elastic-plastic constitutive law is integrated by identifying iteratively which dislocation slip systems are activated within the grains, and the local stress tensor is shown to be the solution of a linear equation set. Several micro–macro averaging schemes are considered for the distribution of the macroscopic load over the polycrystalline aggregate. These averaging schemes are tested by simulating the development of intergranular strains during uniaxial tension of MONEL-400 as well as commercial purity aluminium. Neutron diffraction measurements of the elastic lattice strains are used as a reference in order to discriminate between the various predictions. The results demonstrate the relevance of “multisite” grain interactions in f.c.c. polycrystals.

Simulation of cup-drawing based on crystal plasticity applied to reduced grain samplings

This paper presents a means of reducing the computational cost of finite element (FE) simulations coupled to polycrystal plasticity theory. One typically assumes that a polycrystal with a large number of grains underlies every integration point of the FE mesh. Instead, it is suggested here using reduced samplings of grains which differ from one integration point to another. On average, every set of 5 to 25 finite elements contains a variety of lattice orientations that is representative of the macroscopic texture. The model is applied to deep-drawing of a cylindrical cup made of steel. In a first set of simulations, grains are assigned orientations representative of a cold rolling texture and the “earing” profile is compared to experiment. In a second set of simulations, lattice orientations are random and an isotropic deep-drawing result is expected. It is demonstrated that using a minimum of 20 grains per integration point allows properly predicting the final shape of the cup and the texture development.

A disclination-based model for grain subdivision

Grain subdivision under cold rolling of fcc polycrystals is modelled by formulating evolution equations for the statistically stored dislocation densities (cell structure), the propagating partial disclination dipole densities, and the immobile partial disclination densities (cell block structure) at the slip system level. The development of the mean cell and cell block sizes and of the mean misorientations across the cell block boundaries (cbb) is predicted for several grain orientations. Two types of grain subdivision were distinguished, cube, rolling direction (RD)- and transverse direction (TD)-rotated cube oriented grains showed a finer subdivision and nearly no spread of misorientation between the cbb families, while Goss, brass and S-oriented grains showed a coarser subdivision and a significant spread of the misorientations. Copper oriented grains exhibit an intermediate behaviour. The predictions of the model for the evolution of the cell and cell block sizes are in reasonable agreement with experimental results.

Application of the Lamel model for simulating cold rolling texture in molybdenum sheet

The cold rolling texture in a molybdenum sheet was simulated using different plasticity models. The Lamel model made a prediction closer to the experimentally determined texture than the “full constraints” (FC) Taylor model and the “relaxed constraints” (RC) Pancake model. However, the Lamel model did not perform as well as in previously investigated low-carbon steels. Having observed that the experimental fibre-profiles appear intermediary between the predictions of the Lamel and the Pancake model, two revised compromise models are developed, by which the simulated results can be further improved. The applicability of the various models is physically motivated.