A model for rolling deformation with grain subdivision. Part II: The subsequent stage

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Abstract

Part I of the present work dealt with the initial stage of plastic deformation with grain subdivision into two band families. It covered the situation with different average strains in the individual grains (the non-Taylor case) and the situation with identical average strains in the individual grains (the Taylor case). For the latter situation part I included solutions with 5 plus 3 and with 4 plus 4 active slip systems in the two band families, respectively. In part II we deal with the subsequent stage of plastic deformation (finite strains) for the Taylor case with four active slip systems in each band family. In the subsequent stage the cooperation between the two band families leads to an energetic advantage for grain subdivision, and it leads to a simulated texture with decreased sharpness in accordance with experimental observations. Finally, the physics behind grain subdivision are discussed on the basis of the results in Parts I and II and various general considerations.

Introduction

In part I (Leffers, 2001) the general experimental background for grain subdivision into bands or cell blocks was outlined with reference to aluminium. Two different sets of equations were formulated for (rate-insensitive) plastic deformation of fcc materials by rolling (plane strain) with grain subdivision into two families of bands: one set for the “non-Taylor case” and one set for the “Taylor case”. The geometry of the band structure was described in part I, and this description is not to be repeated here. The possible types of solutions were investigated for the initial stage (the first deformation step) where the lattice orientations in the two band families were identical.

In part II we shall take the deformation to the subsequent stage where different lattice orientations develop in the two band families. In order to limit the number of possibilities we shall only consider the Taylor case. The non-Taylor case will be dealt with later. For the Taylor case two types of solutions were described in part I: 4+4 solutions with four active slip systems in each band family and 5+3 solutions with five active slip systems in one band family and three in the other. In order to achieve a further concentration in the scope we shall only consider one type of solution, 4+4. The specific choice of 4+4 will be discussed later.

Section snippets

Mathematics in the subsequent stage

The final equations to be solved are , , , , , , , [Eqs. (20)–(27) from part I]:Σdγin1id1i=K1dEΣdγjn1jd1j=K1dEΣdγin2id2i=K2dEΣdγjn2jd2j=K2dEΣdγin1id2i+n2id1i=2K3dEΣdγjn1jd2j+n2jd1j=2K3dEΣdγin1id3i+n3id1i+Σdγjn1jd3j+n3jd1j=4K4dEΣdγin2id3i+n3id2i+Σdγjn2jd3j+n3jd2j=4K5dE

In these equations n1, n2, n3 and d1, d2, d3 are the coordinates in the band coordinate system (with x3 perpendicular to the plane of the bands) of slip plane normal and slip direction, respectively, dγ is the corresponding shear

Specific problems in the subsequent stage

In part I we only pointed out the possible solutions for the slip patterns in the two band families. We did not actually perform any deformation. Therefore, it was not necessary to define a solution to the “ambiguity problem” (which appears for rate-insensitive Taylor-type models), and the lattice rotations were not relevant. In part II we deal with the subsequent stage where we do perform deformation, and hence we must have a solution to the ambiguity problem, and we must know the rules for

Results

In part I we only found the physical solutions for the shears dγ on the different slip systems without performing the lattice rotations which would split up the grains into two band families with different lattice orientations. Therefore, we could quote detailed information about the solutions.

In part II we actually perform (a computer) plastic deformation in many steps, and we let the grains split up. This means that the number of parameters is too large for detailed investigations like those

Discussion

It has been demonstrated, for the Taylor case with 4+4 solutions, how the principles for the initial stage with grain subdivision outlined in part I may be extrapolated into the subsequent stage where the two band families have different lattice orientations. There is a very significant change from the initial stage: once the two band families have developed different lattice orientations, there is an energetic advantage associated with the difference in slip pattern between the two band

Conclusions

It has been demonstrated how the Taylor version of the model for plastic deformation with grain subdivision described in part I (Leffers, 2001) may be extrapolated into the subsequent stage with significant strains and, as a consequence, with different lattice orientations in the two band families. The model leads to an fcc rolling texture which is closer to the experimental copper-type texture than that derived from the simple Taylor model. However, there are two very important (and probably

Acknowledgements

The author wants to acknowledge helpful discussions with H. Christoffersen, N. Hansen, G. Saada and G. Winther. The work has earlier been supported by the Danish Materials Technology Development Programme. Now it is part of the activities within the Engineering Science Centre for Structural Charaterization and Modelling of Materials.

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