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

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Abstract

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.

Introduction

There is a need to incorporate deformation texture models in finite element (FE) software for the simulation of forming processes. Several FE software packages already take texture-based anisotropy into account. Some of them try to simulate the texture evolution during the deformation. But deformation texture prediction software that would be quantitatively reliable for arbitrary strain paths does not really exist yet. Most efforts to develop reliable deformation texture models have so far been focused on the deformation of rolling textures in f.c.c. or b.c.c. metals deformed at room temperature. The strain history is not varied then, but the initial texture can be varied; the models should perform well for all possible initial textures. The results obtained can be briefly summarised as follows:

•All models use a microscopic model which describes the plastic deformation of an elementary volume within a crystallite. It provides a relationship between, on one hand, the strain rate and the rate of rigid body rotation (both combined in the velocity gradient tensor), and on the other hand, the slip rates, lattice rotation rates and deviatoric stress. The relationship is based on the laws of crystal plasticity in one form or another, such as the generalised Schmid law or the visco-plastic model for crystal plasticity (see for example the overview by Aernoudt et al., 1993).

•In the full-constraints (Taylor, 1938) model (FC model), it is assumed that the velocity gradient tensor of the elementary volume is equal to the one of the entire polycrystal. This model implies homogeneity of plastic strain throughout the polycrystal, but it neglects stress equilibrium at the grain boundaries. For cubic metals, the predictions of this model are first-order approximations of deformation textures for all possible strain histories and all possible initial textures, though they do not perform well for low stacking-fault energy f.c.c. metals (brass, silver etc.). “First-order approximation” means that the predictions are good from a qualitative point of view (and they show that Taylor's ideas truly form the basis of understanding of deformation texture prediction), but not from a quantitative point of view, since the locations of the final orientations may be up to 10° off, and intensities and volume fractions may be up to a factor 2 wrong. In the opinion of the present authors, any more advanced model is to be regarded as an improvement of the Taylor model, adding corrections to its predictions (“2nd order approximations”). The most successful attempts to improve the Taylor model have been those which have focused on the problem of the stress equilibrium at the grain boundaries (see for example the overview by Aernoudt et al., 1993):

•The relaxed constraints models (RC models) abandon Taylor's strict requirement of strain homogeneity, but not for all components of the velocity gradient tensor. Most components lij of the velocity gradient still need to be equal to the Lij of the polycrystal velocity gradient. However, for some components, this condition is not maintained. The best known example is the pancake model, which is advocated for materials with flat, elongated grains (Fig. 1). Here, l13 and l23 are “relaxed” — the figure shows only relaxation l13. They represent shears that would not cause any rotation of the top or bottom surfaces of the flat, elongated grains. It is believed (by the advocates of this model) that the relaxation of such shears would cause fewer problems of strain misfits with neighbouring grains that other kinds of relaxations would. Indeed, it should not be forgotten that in such model, l13 and l23 are calculated without taking the interaction with other grains into account. As a result, l13 and l23 will take a different value in each crystallite, depending on the lattice orientation. Two methods exist to calculate l13 and l23, but both imply either explicitly or implicitly that the shear stresses, which correspond to the shears that are relaxed, are set to zero (Van Houtte, 1982). In the case of the pancake model, this means that σ13 and σ23 are supposed to be zero. So the microscopic model for elementary volumes within crystallites is augmented with this condition, which allows for the calculation of the strain rates d13 and d23. Since d13 must be the average of l13 and l31, the latter still being zero (because L31 is zero in the case of rolling), l13 can be obtained. This method can also be used for l23.

Relaxed constraints models of this type have led to some improvements of the prediction of the location in Euler space of certain deformation texture components, but they have not led to more precise predictions of the ODF intensity of the deformation textures.

•The choice of relaxations in a RC model seems somewhat artificial. The choice made for the pancake model can be justified in case of rolling, but this justification is meaningless for most other deformation modes. So the pancake model cannot be used for an arbitrary strain history. A model that would make an automatic choice of the strains to be relaxed could be an answer to these remarks. The generalised relaxed constraints method (GRC) is such a method (Van Houtte and Rabet, 1997). It has many features in common with the visco-plastic self-consistent (VPSC) method proposed by Molinari et al. (1987). Both methods have been used for predictions of deformation textures of cubic materials (Wagner et al., 1991, Van Houtte et al., 1999). Although in some cases better results have been obtained than with the FC or RC models, a truly quantitative agreement between predicted and measured deformation textures has not been achieved. It is perhaps possible to improve that agreement by adjusting certain parameters within these models, but that needs further investigation.

FC and RC models are “1-point” models, which only take one crystallite at a time into account. The GRC and VPSC models look at one crystallite but also take the macroscopic properties of the entire polycrystal into account. Recently, several models have been proposed which look at two grains simultaneously (“2-point models”) or even more (“n-point models”). The purpose of such models is to capture some specific effects due to the interaction of neighbouring grains, which apparently have not been captured by any of the previous models. One such model is the “Lamel model” (Van Houtte et al., 1999), which will be extensively dealt with below.

With the aid of finite-element (FE) software, models can be constructed which take the interaction between many grains into account, thereby simultaneously achieving stress equilibrium and strain compatibility (see for example Kalidindi et al., 1992). In such a model, each crystallite consists of at least one element of the FE mesh. The constitutive equations of the microscopic model (crystal plasticity model) are implemented as a material model. Note that regarding microscopic strain heterogeneity, such model is in principle superior to methods such as the FC, RC or Lamel models. However, the required calculation time is at least 1 order of magnitude larger, which makes the method unsuited for large-scale FE simulations of a forming process of a fine-grained material. It is nevertheless interesting to confront the texture predictions of the FC, RC or Lamel models not only with experimental results, but also with predictions of a FE-based polycrystal model. This is the purpose of the work described in the present paper.

Section snippets

The Lamel model

The Lamel model (Van Houtte et al., 1999) is a “2-point” model which looks at 2 grains at the same time. It was originally developed for rolling. It is assumed that after some cold reduction, the grains tend to become flattened and elongated. This assumption is also used in the RC-pancake model. The smallest “element” considered by the model is a set of 2 such grains which (before deformation) have the same shape and size and which lie exactly on top of each other, the grain boundary which

Prediction of an IF-steel cold-rolling texture

The purpose of this section is to illustrate how accounting for grain interactions allows for better simulations of the evolution of the global crystallographic texture during plastic deformation. The reference material is an IF-steel that has been industrially cold-rolled to a reduction of 70%. The experimental ODFs before and after deformation have been derived from 4 pole figures using an harmonic method adapted for incomplete pole figures (Van Houtte, 1984) and applying a ghost correction (

Comparison of the grain interaction models with the other models

Visual inspection of Fig. 5 shows that the 3 grain interaction models (FEM, original Lamel, Lamel+Type III) lead to predicted ODFs which agree much better with the experimental result than the two 1-point models (Taylor FC, Taylor RC pancake). It does not allow detecting significant differences between the performance of the 3 grain interaction models.

The quantitative diagrams Fig. 6, Fig. 7 show, that the predictions of the 1-point models (Taylor FC, Taylor RC pancake) are really much worse

Conclusions

• For cold rolling of b.c.c. metals at thickness reductions which cause significant flattening of the grains, much better texture predictions have been obtained by models which take grain interaction into account (FE-model, Lamel models), as compared to models which do not (FC Taylor model, RC Taylor model).

• In principle, the best results should have been obtained by the FE-model; however, the two Lamel models performed slightly better. It is suspected that this problem can be solved by using

Acknowledgements

Financial support for this work has been obtained from the Federal Government of Belgium (DWTC) through the contract IUAP P4/33, and from the FOM/NIMR project 99.1098 (The Netherlands). L. D. acknowledges his grant as ‘aspirant’ of the Flemish FWO, and S. R. K. his fellowhip from the Katholieke Universiteit Leuven.

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