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A NEW APPROACH TO CRYSTAL SPIN CALCULATION DURING DEFORMATION TEXTURE DEVELOPMENT

 

R.E. Bolmaro, A. Fourty, A. Roatta, M.A. Bertinetti and J. Signorelli

 

Instituto de Física Rosario - Fac. de Ciencias Exactas, Ingeniería y Agrimensura. Consejo Nacional de Investigaciones Científicas y Técnicas, Bv. 27 de febrero 210 bis, 2000, Rosario, Argentina.
bolmaro@ifir.ifir.edu.ar

 

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Abstract:

The paper presents a simple simulation model for intra-grain misorientations and texture development during plastic deformation. The model is presented as an improvement over viscoplastic self-consistent models and its main features are assessed by comparison with experiments and FEM results. The effectiveness of the model to simulate textures is quite unexpected due to its simplicity. However, a detailed analysis of its microscopic consequences shows that the model is able to empirically capture the main characteristics of grain fragmentation during plastic deformation. A brief discussion of different levels of heterogeneity, as an introduction to the model, is used to ground the approach and to settle its capabilities.

 

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Introduction:

The development of strain heterogeneities in polycrystalline materials can span over different length scales. Those heterogeneities are the source for residual strains when the strain heterogeneity we are dealing with is of elastic nature. When the heterogeneities are established  in the plastic regime one of the main effects is over the texture development. Independently of the scale of heterogeneity the common aftereffect is the smoothing of textures or general decrement of texture intensities.

1.- The scale comprising many grains, reaching a level comparable to the sample size, is usually called the macroscopic heterogeneity. That level of heterogeneity has been successfully approached by FEM and it will be referred from now on, in analogy with residual strains nomenclature, as 1^st kind heterogeneity [1-2].

2.- The second known level of heterogeneity is established between neighbor grains due mainly to crystal anisotropy. Two different ways of dealing with the problem have carried both understanding and limitations to the field of texture interpretations: Self-Consistent micromecanical models [3-4] and FEM by the scheme of one element-one crystal [5-6]. It will be denominated 2^nd kind heterogeneity.

3.- The third level of heterogeneity, called 3^rd kind heterogeneity, is developed inside each grain and is mainly due to dislocation arrays of different origin. Recently, this level of heterogeneity has been studied by FEM in the scheme one grain-many elements [7-8]. N-sites self-consistent models have also been developed with that purpose [9-10].

The main goal of texture development simulation is the reliable understanding and prediction of textures. So far the simulations have been able to explain the presence of different components but they have not been so successful in the understanding and/or prediction of textures severities or component intensities. Slight differences in strain rates, grain size and or alloying compositions, through stacking fault energy changes, can influence the rate of development and the final severity of textures. Nevertheless the main limitation of simulations with respect to experiments is the evident lack of severity matching by, sometimes, an order of magnitude at high deformations. Component texture prediction and severity matching is quite good at medium strains but the simulations do not do so well at very low and high deformations. There is general agreement about the main reason for that mismatch: All three levels of heterogeneity smooth the textures contributing in a different degree to severity decrement. By the sake of what is called the 3^rd kind heterogeneity effect, each grain does not follow exactly the Taylor assumption neither follows some of the Self-Consistent model calculations. They actually deform very heterogeneously, achieving strain compatibility and equilibrium by developing different dislocation structures in each grain. Those dislocation structures are grain orientation dependent and interrelated, at different deformations, in a not completely well known way. Notwithstanding they bear the common feature of been able to accumulate dislocations of the kind that are activated by the general Taylor rules. Minimum energy principles for the accumulated dislocations are the master rules to decide which dislocations are indeed active. The experiments conducted in this field are quite profuse [11-13]. It is by now pretty clear that which were in principle assumed to be the main dislocation sink areas of a polycrystal, the grain boundaries, are actually not. The dislocations are accumulated in regions internal to the grains and, there, they develop quite complex interactions and consequently carry the local deformations and crystal lattice spin (rotation rate). The experiments also show quite large misorientations among the many fragments developed in a grain. Some dislocation arrangements, produced by Geometrically Necessary Boundaries (GNB), introduce larger misorientations than others, called Incidental Dislocation Boundaries (IDB) or statistical accumulated dislocations. Grain orientation is also responsible for a different apportion of misorientations to those main two dislocation arrangements. In any case, the development of misorientation regions away from the original grain boundaries is a well-established experimental fact either at medium or very high deformations. Deformation Banding (DB) has been also referred as one of the main features of highly deformed materials. Optical microscopy was this time the technique used to characterize the DB features held responsible for grain subdivision. By that technique the number of DB appearing for each grain, in the direction of rolling deformation in highly deformed samples, has been determined to be close to 2. DBs are usually rather parallel to the rolling plane while the dislocation features known as GNB, comprising Dense Dislocations Walls (DDW) and Micro Bands (MB), keep a measurable angle, usually not less than 30^o, with the rolling plane. Duggan and Lee [14] have shown a common origin for those medium (GNB) and large deformation (DB) features.

Regarding the misorientations between the different regions undergoing different, but compatible, deformation paths Hansen et al. [13] have found by SEM and TEM techniques that the frequency of High Angle Boundaries ( >20^o ) (HAB) is much larger than the estimated frequency of the original grain boundaries. The growing number of HAB shows that the deformation process is not creating a short-range misorientation order by texture development but just a long-range statistical misorientation order among grains. As a consequence it is pretty clear that the spin of the different areas is concentrated not in the grain boundaries but in the newly created DBs, GNBs or IDBs. The dislocation sink regions are mainly located in the interior of the original grains where they get trapped creating spin accumulation regions.

The model: How to deal with lattice spin in a fragmenting (Balkanized) world?

A schematic representation of the medium and high deformation arrays of GNBs is shown in Fig. 1 as it seems to be coming from the specialized literature. IDBs are not shown in the scheme and are not considered in the current model. Currently it is assumed that the dislocation structures formed inside each grain are responsible for both strain rate and spin. Moreover two companion grains will be allowed to freely deform following the rules coming from Self Consistent models but only spinning together like a unique entity. Statistically each grain will represent only half of a grain, which other part is assigned as a companion to other randomly chosen grain. The two parts forming the same grains are initially oriented in the same direction but will spin and deform following the closest influence of the first neighbor. From the statistical viewpoint it is not even necessary to have both fractions present. The many randomly chosen grains, representative of the starting texture, are capable of representing the textures in a statistical sense.

A question about the mechanism by which the grain boundaries are allowed to keep the same misorientation with the closest neighbor should be answered. Some authors have suggested that very close to the grain boundary a HAB can be developed and the actual GB could remain unchanged. Otherwise the dislocations should go all the way through the grain (half-grain in the current approach) and change the relative misorientation when they reach the interface. Some other phenomena like climbing and recombination have been also suggested as a way of keeping the misorientation stable. The grain boundary characteristics obtained by solidification, for instance, could be very stable and difficult to change by plastic deformation.

The crystal arrangement designed for the current simulations is appropriate for computing the misorientation evolution between two parts of a grain (intra-grain misorientation) and among all grains (statistical  or texture dependent misorientation). The misorientation between the two closest neighbors not belonging to the same original crystal will be kept constant assuming that the grain boundaries are orientationally stable. Texture is achieved by creating intra-grain misorientation regions that fragment the grains and the orientational order is a long-range order created at expenses of short-range orientational disorder. First neighbor misorientation is kept constant for the original grain boundaries and the misorientation angle between parts of the same grain becomes larger due to the strong interaction with the closest companion.

Results and discussion

Hughes et al. [12] have found some experimental results that should be reproducible by any numerical modeling of texture development attempting to take in account misorientation effects. The scaling phenomena of the probability  misorientation angle (in an angle-axes scheme) reveals a scale invariance, with varying equivalent deformation, by using the average misorientation angle as scaling parameter. The scaling phenomenon seems to be quite evident for IDBs but it is just approximate for GNBs. Nevertheless, in such approximate way, the model should show some correlation between experiments and computed misorientations. Another experimental result that should be reproduced by modeling is the variation of the average misorientation in function of the equivalent deformation. The slope in a log-log plot is approximately equal to 1/2 for IDBs and to 2/3 for GNBs. The model is assumed to be capable of simulating the fragmentation of grains due to GNBs not reproducing the statistically stored dislocations in any way. The same results have been simulated with fairly good agreement by FEM techniques by Mika and Dawson [7].

We simulated a channel die deformation (idealizing rolling deformation) by imposing homogeneous pure shear for an FCC material until a Von Mises equivalent strain of 2.0 is reached. One thousand grains were randomly pare with identical grains in such a way that each pare of identical grains form a couple with different grains and eventually spin differently. The evolving misorientation between both grains is taken as the intra-grain misorientation to be compared with experimental results. The usual 12 (111) <110> slip systems are considered with a stepwise linear hardening simulating literature values for copper. Texture and misorientations were the output of the simulations for 0.025, 0.05, 0.10, 0.15, 0.20, 0.25 and every 0.25 Von Mises strain from there on. 

Fig. 2 shows the MacKenzie plot of misorientation angles for the starting misorientation of the randomly oriented grains. Those misorientations do not evolve along the whole deformation process because closest  
companions are constrained to spin together. Mika and Dawson have shown that the evolution of the original grain boundaries misorietation is quite slow [7]. Moreover, experiments show that the original grain boundaries remain identifiable as HAB along large deformations. Some not conclusive experimental results, taken by EBSD technique, in Ag-Ni two-phase materials show that the evolution of inter-phase misorientations during rolling deformation is rather absent [15]. Fig. 3 shows the probability density functions of the misorientation angles, normalized  by the average misorientations, and scaled by the average misorientations at each deformation step. The scale invariance is rather clear from the plots. The probabilities were fitted by following the same ansatz used by Hughes et al. [12]:

             (1)

By looking for the best matching for a=2.5 the misorientations were found to have a bias to larger misorientation angles of 0.08 the average misorientation (between 0.5⁰ and 2.0⁰ by increasing the Von Mises equivalent strain from 0.25 to 2.0). The ansatz suggested by Hughes et al. [12] has so far no clear physical meaning. The process of intra-grain dislocation accumulation is dependent on a complex interaction of probabilistic phenomena like grain orientation, grain-to-grain misorientation, stochastic generation of dislocations, dislocation trapping by pre-existent GNBs, etc.. All those phenomena contribute to P(w) by the product of the probabilities of every single phenomena, considering them as mutually independent:

 

                                                   (2)

 

That equation can be written as:

 

                                (3)

 

If the individual probabilities are closely independent and satisfy the weak condition of having a finite variance, the Central Limit Theorem can be used and ln P(w) has a normal distribution of the kind [16]:

 

                                                                              (4)

 

Fig. 4 shows the curve for all misorientation angles for every calculated deformation together with a fit by eq. (4). The fit is quite perfect as shown by the c² test. However, despite the agreement between different fits and simulations, the simulated average misorientation angles are larger than the experimental values by a few degrees: 3⁰ and 9⁰ for 0.25 and 1.0 Von Mises equivalent deformations, respectively. Fig. 4 shows the power law relationship between the average misorientation angles and the equivalent strain. For large deformations the slope is 0.68, which is quite close to the experimental value of 2/3, but the values tend to be overestimated. The values simulated by Mika and Dawson [7] are also shown in Fig. 5, which tend to be underestimated in comparison with the experimental values. In our simulation the bias is probably coming from two effects not considered in the model: a) The original grain boundaries do evolve in real materials to allow the grains to reorient and b) IDBs are also misorientation boundaries permitting texture development. Another topic that deserves discussion is the fact that the model allows a single GNB to be formed inside each grain, which is representative of small size grain materials. From 5 to 27 GNBs (for rolling and torsion respectively) can be found in deformed materials [13]. The agreement between the current model and the experiments is clearly better at low deformations when the average number of GNBs per grain is close to two. At larger strains the GNBs of the model develop misorientations larger than the experimental ones. Analyzing the scaling invariance from more fundamental viewpoints it is expectable that the grains developing intra-grain misorientations will develop a new GNB when the misorientation between the first two parts of a grain reaches the largest possible value for that configuration. From there on the GNB will behave as a common grain boundary, inducing new fragmentation at more or less half-way from the next GNB and keeping the average misorientation angle lower. As far as the model does not contain any dimensionally related information, different deformation states can be considered, with regards to misorientation quantities, as a new starting state statistically equivalent to the non-deformed one. Model improvement can be envisaged by allowing further fragmentation whenever the misorientation between two half-grains reaches a certain pre-established limit. However, considering the high complexity of the phenomena we are trying to simulate, the agreement between simulated and experimental textures and misorientations is quite perfect. The (111) pole figure shown in Fig. 6 presents components and intensities quite close to the experimental ones. The severity is obtained without any Gaussian smoothing other than the one coming from the 5⁰ x 5⁰ imposed grid. The model is able to capture the main characteristics of grain fragmentation with very low computer power requirements. Whichever the other localization phenomena responsible for texture development, its relative importance with respect to the GNBs (or DBs in Duggan's nomenclature [14]) is much lower. Regarding previous attempts to simulate grains fragmentation it is worthy to mention the models of Lee et al. [17] and van Houtte et al. [18], which have been quite successful in modeling rolling textures of FCC and BCC materials. In our understanding, at least part of the success of the models is mainly due to a somehow hidden way of averaging spins between the components of  the unitary lamel composite, simulating stacked co-deforming grains.

 

 Acknowledgements: This work has been performed with fundings provided CONICET, UNR, Argentina. The authors kindly recognize profitable discussions with Dr. A. Ceccatto about the physics of Lognormal distributions.

 

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