FUNCTIONS DESCRIBING ORIENTATION CORRELATIONS IN POLYCRYSTALLINE MATERIALS

Statistical functions are presented, which can be applied as characteristics of the polycrystalline 
microstructure when the parameters are the orientations at separate points of the sample. Basing on 
the formal definition of the ODF the corresponding definition of the two-point coherence function 
describing the orientation correlation is given. This immediately gives relations involving this function. 
Also a discretized form of such a two-point function is presented. Moreover, the relations of this 
function to other functions describing orientation correlation are considered.


INTRODUCTION
It is a well-known fact that the existence of correlations between orientations of spatially separated points in the polycrystalline sample influence the properties of a material.For example the determination of the effective properties of a polycrystal as heterogeneous continuum on the basis of statistical continuum theory requires the knowledge of many-point correlation functions of appropriate quantities (see e.g.Beran, 1971, Kr/Sner, 1986).The coherence functions, which we are interested in, are the tools for the description of such correlations.
These functions are also useful from the view-point of description of microstr- ucture and they play an essential role in analysis and modeling of processing polycrystalline materials (e.g.recrystalisation, deformation).
The growing interest in coherence functions follows the enormous progress of the experimental techniques for measuring individual grain orientations, which enables determination of such functions.
Throughout the paper statistical homogeneity of the sample and its infinite dimensions are assumed.These assumptions are necessary to the averaging procedure (over a suitably large area of the sample) used here.

OCF EXPRESSED BY DISTRIBUTIONS
We shall present a new kind of formal definitions of the ODF and of the recently discussed (see e.g.Adams, Morris, Wang, Willden and Wright, 1987 or Adams,  68   A. MORAWIEC AND J. POSPIECH Wang and Morris, 1988) two-point orientation coherence function (OCF) which complete their previous descriptive definitions.This will first be done for the intuitively clear case of the ODF determined on a discretized orientation space.
The ODF as a density function appears in expressions of the type f dgf(g)A(g) with A being an orientation dependent quantity.As we shall see, the function f can be written in such a form that the averaging process takes place in the specimen area.
Let G indicate the function which assigns the orientation of the crystallite to every point of the (homogeneous in large scale) area V cut out of infinite specimen G:3 V rg ff. (1) The definition given above requires previous indication of the symmetrically equivalent area ff in the orientation space; it is assumed here that this area is equal to the whole space of proper rotations.Let us define V as the volume of area V V fv d3r and f dg l. (2) One can divide the orientation space into cells numbered by e K t and having the volume Ag.Thus, we have the mapping d:f2 9 g-i e.K. (3) Let us define the function :V x Kas 1 ri0(r, i) g/6i where j dt(G(r)), j e K. (4) By integrating it over the specimen area we get where AV(i) is the volume of all crystallites having the orientations belonging to dill(i).Therefore f in formula ( 5) is the discrete form of ODF.
Moreover, the following formulae hold: The above given sequence of relations can be replaced by its compact form.For this puose we shall assume that the volume Ag := maxiK {Agi} proceeds to zero and we shall replace i]]Agi by the distribution 6(g', g) with g' G(r) and the summation over the set K by integration over the orientation space , [ dg(.).
DISCRETIZED FORM OF THE TWO-POINT COHERENCE FUNCTION In a common experiment individual orientations are measured at points arranged in uniform grids.Thus, the specimen area is treated as a discrete set.On the other hand, because of computational difficulties, the orientation space often has also to be discretized, i.e. one assumes that the functions over the rotation space have constant values in defined cells.
Therefore, let the orientation space be divided in the same way as in the first paragraph and moreover, let the specimen be represented by a discrete structure (grid).
If the ODF is available in continuous form one can get the components of the texture vector by assuming that the value of its ith component is equal to the mean value of the texture function over the ith cell 1 fd f(g) dg. (23) Let the points of the grid (numbered by Greek indices) form the set M containing M elements.The function G is re-defined now as G'M--K.Analogously to (3) one can write do" V 9 r/ e M (24) Moreover, when du(g) and j dt(g') we shall write du(gg')= i.j.Similarly, when/ dr(r), v dv(r') we shall write dv(r + r') =/ + v.
The formulae from the first paragraph can be written in the discrete form.In particular we have Defining fi fi Agi, e c Agi Agj (27) one can re-write (26) in the simpler form M Z. C ij =/" (28) Because the coherence matrices are non-negative we conclude that if vanishes, the ith row and ith column of arbitrary coherence matrix vanish, too.
We shall assume below that 0 for all K.
One can define stochastic matrices s ii c i/f corresponding to the conditional probability that the point situated at the distance r from the arbitrary point having the orientation g has the orientation g': c(g'lg, r):= c(g, g'lr)/f(g).In the simple one-dimensional case, assuming that the orientation at point a correlates with the orientation at point only through point g () one can write += s.
(29) k It can be shown (using the Perron-Frobenius theorem), that lim (s v).to, where toi =f (30) and from the definition of the s matrix one gets the particular case of the statement that c(g, g'lr)=f(g)f(g') for large *--++ _...-:--+ j lim ei ]itoij f'l'" (31) The last result expresses the fact that the orientations of remote points are not correlated.0 1,0 2.0 3.0 x,r Fibre l The 7 distribution (a) and corresponding q(rn 1) distribution (b) for te 1.5 and fl-0.5.
where and fl depend on n and Fy indicates incomplete Euler F-function i.e.
(38) Figure 1 shows an example for the 7 distribution and the corresponding q(rn I1) distribution.
Formula (35) can be written in the discrete form as ,h , e;S.,q(v I1) with rhi defined analogously to i.
FINAL REMARKS Among the functions describing correlations between orientations the n-point (in particular two-point) coherence functions are most universal.But the usage of them is influenced by calculation and experimental restrictions.From the view-point of economy of data storage the best way of dealing with two-point coherence function is to calculate the Fourier coefficients (21 + 1)(2l' + 1) T,m,,(G(v))TT,m,n,(G(v +/,)).( 40) OTfl'm'nn'(l) M For a quantitative analysis of various aspects of the polycrystalline microstruc- ture some other functions describing orientation correlations are of interest.Those of them, which are discussed in the present paper, contain less information than the two-point coherence function.