REVIEW OF DETERMINISTIC METHODS OF CALCULATION OF POLYCRYSTAL ELASTIC CONSTANTS

The paper is a critical review of non-stochastic (referred here as deterministic) methods of calculation of effective elastic constants of single phase statistically homogeneous polycrystals. The methods are briefly described. They are compared and grouped according to their basic assumptions. Their degree of generality is discussed. The analysis of their compliance with formal requirements is provided. Moreover, the possible but unknown completions and generalizations of some methods are considered. For most methods, example calculations are carded out for a special case of quasi-isotropic aggregate with cubic crystal symmetry. To make the paper self-contained, well-known essentials of the problem of effective constants are included.


INTRODUCTION
The essence of the problem of the effective elastic properties is to calculate the elastic constants of a composite basing on the known properties of its components and on the structure of the material. For polycrystal the special case of the composite the constituents are crystallites with elastic properties determined by their orientations and single crystal data.
Diverse methods of solving the problem have been proposed. They can be roughly divided into stochastic methods and other, non-stochastic or deterministic ones. The deterministic methods are characterized by simple assumptions, relatively simple calculations and unique results. The involved information about the structure of the material is reduced to orientation distribution. At the same time, they lack strict theoretical basis. Such basis is a feature of the stochastic methods which use more sophisticated statistical information (many-point probability functions) about composition of material. Moreover, for real polycrystal, the available information cannot be exhaustive, thus, instead of a unique result, only bounds for effective constants can be obtained. In fact, the problem is, to f'md optimal (i.e., narrowest) bounds for a given portion of statistical information.
One can argue that according to the above distinction, also the self-consistent method (Hershey, 1954;Marutake, 1956;KrOner, 1958) or the singular approximation method (e.g., Fokin, 1972;Kuzmenko, Komeev and Shermergor, 1983) have deterministic character. Because of their close relation to the stochastic methods they will not be discussed here. First, for completeness, some standard assumptions will be repeated. The existing deterministic methods deal only with local theory. The static case, without body forces will be considered here. Therefore, divergence of the stress tensor t is assumed to vanish and t is assumed to be symmetric, i.e., in an established cartesian coordinate system OjOij 0 (Jij i (1.1) (The set of second rank symmetric tensors will be denoted by IF; hence, Furthermore, only small deformations will be taken into account. They will be described by the strain tensor e 1F defined as symmetrized gradient of the displacement vector field u % 0tiuj) (1.2) For a given (twice differentiable) tensor field eelF, the existence of a displacement u satisfying (1.2) is guaranteed, if the compatibility conditions are fulfilled (1.3) The e0k denotes the symbol of Levi-Civita.
Although there exist some deterministic solutions for the effective constants of the third order, only those concerning the linear theory (second order elastic constants) will be analyzed here. Hence, the elastic energy U is assumed to be a quadratic function of the components of strain tensor 2U e c e (1.4) and the stress and strain tensors are related through Hooke's law o c e, (c e)0 C0klGa (1.5) Following definition (1.4), tensor c satisfies the symmetry conditions and co qi cijn (1.6) c0k Cklij (1.7) The set of fourth rank tensors with symmetry given by both, (1.6) and (1.7), conditions will be denoted by 1P; hence, ce 114. Moreover, let Tr be the algebra of tensors satisfying at least (1.6) with the multiplication rule (tt/)ij tOm t2mnkl t,tZTr (1.8) and the unit tensor I defined by iijld ='l (ikj + iljk (1.9) It is shown in the Appendix A that vectors and symrn,etric matrices, respectively, can be assigned to tensors of l and 1P. Because the elastic energy is positive for any non-zero deformation, the matrix corresponding to is positive definite. Hence, it is not singular and, subsequently, there exists the inverse matrix and the corresponding tensor s II 4 cs sc I (1.10) The tensor s can be used to write down the energy (U=tst/2) and Hooke's law (e=so). Below, to express the positive definiteness of te 1P the symbol t>0 will be used.

Volume Average and Statistical Homogeneity
The average of a quantity q over a volume V (large in comparison with the inhomogeneities grains) is defined by <q>:= -q(x)d3x (1.11) v The notion of statistical homogeneity can be precisely def'med using the ensemble average. However, the idea is intuitively obvious and can be also clearly explained basing on the volume average: Let the sample be large compared to V. The sample is considered to be statistically homogeneous (regarding a given quantity q) if the average <q> does not depend on the location of V. This means that q is stationary on the sample and its value fluctuates around that constant average value.

Orientation Average
Further, it is assumed that the material properties (and hence the tensor field q (=e or s)) at a given point x V are determined by the orientation geSO(3) of the crystallite containing this point. That is, in fact, the field is determined on the orientation space and indirectly at points of the sample. Let O:m-g be the mapping, which to the point x, assigns the orientation g=O(x) of the erystallite containing the point. Hence, writing in a compact, although not completely precise way, q(x)=q(O(x)). Moreover, q(x) can be given in the form (1.12) SO(3) The volume average can be replaced by the orientation average A. MORAWIEC where f:SO (3) .> / given by f(g):= -1 1/5(O(x)'g)d3x (1.14) v denotes the texture function (Morawiec, Pospiech, 1992) and M is the operator of orientation averaging with f as weight function.

Definition of Effective Elastic Constants
From the macroscopic viewpoint, the statistically homogeneous material appears as homogeneous. Thus, one expects the existence of a constant effective tensor * which describes the overall elastic properties. Due to the assumed linearity, the elastic energy should be expressible as the quadratic function of a macro-strain and, moreover, the macro-strain should be related to a macro-stress through macroscopic Hooke's law. In both cases, tensor c* should supply appropriate coefficients. The question is, how to express the macro-variables through local variables. In the considered case, it can be done relatively easily. Because of statistical homogeneity, the macro-variables are constant, provided the boundary conditions are homogeneous. Therefore, the standard procedure of defining the effective tensors uses such boundary conditions. Let they be given by  (1.15-16), one has to use again the Gauss theorem and the conditions of local equilibrium (Bishop & Hill, 1951

Reciprocity Condition
This is the demand to obtain the same result from a procedure of effective constants calculation, regardless of the fact which of equivalent equations are the basis of calculation. For simple tensor averaging, it means that a method satisfies the-condition, if, when applied to mutually reciprocal tensors, it gives the mutually reciprocal results; shortly, if the operation of averaging commutes with the matrix inversion.

VOIGT AND REUSS AVERAGES
The methods proposed by Voigt (1889Voigt ( , 1928 and Reuss (1929) were chronologically the first ways to determine the effective elastic constants. Moreover, for other reasons explained below, the averages are important for the whole problem. The Voigt method consists in recognizing the average of the stiffness tensor as the tensor describing effective properties c* <c> =:cv. (2.1) The above average is related to the assumption that any deformation is homogeneous throughout the material. From e(x)--const=<e> and c*<e>=<ee> (or <e>c*<e>=<eee>) one has c*<e>=<c><e> (<e>c*<e>=<e><c><e>; because <e> is arbitrary, then c*=cv. The Reuss average is dual to the Voigt average: the inverse of the effective stiffness tensor is taken to be the average of c-l(=:s), i.e., c $ <C-I>-I=: di or equivalently s* <s>. (2. 2) The existence of mutually dual relations causes some analogies; the corresponding formulae can be obtained by replacing symbols c*, c and e by s*, s and o, respectively. the simplified version (homogeneous boundary conditions) of the well known classical variational principles of elasticity will be shortly considered.

Variational Principles
According to the first variational principle for the first boundary problem, among all differentiable displacements which fulfill the given boundary conditions, those satisfying the equilibrium equations lead to minimum of potential energy. More precisely: Let the differentiable displacements u and u* be given on V. Moreover, let eij..--O0uj), E*ij-----O(iU*j) and let both, u and u*, satisfy the homogeneous boundary conditions u v-<e>jxj=uivv. If ce satisfies the equilibrium relations (i.e., Oj(Cijk)----O), then <ece> < <e*ee*>. (2.3).
On the other hand, if o and o* satisfy the equilibrium relations and, moreover, so satisfies the compatibility conditions, then < 20<e.> OSO > > < 20*<e> O'SO* >. (2.4) Similarly, for the second boundary problem, let e, e* satisfy the compatibility The relations (2.5-6) based on the second boundary problem give the same inequalities.
Summarizing, Voigt and Reuss averages are upper and lower bounds for the tensor of effective constants (Hill, 1952).
Orientation Average of c and s Let the properties (i.e., the stiffness tensor c(x)) at a given point of the polycrystal be determined by the orientation of the crystallite containing this point. Let c ijid be components of the stiffness tensor in a coordinate system related to the crystal lattice of the crystallite and gij be components of the orthogonal matrix corresponding to orientation g. The components of c(g) in the laboratory (sample) coordinate system are given by (c(g))ij gsigtjggwc ,,,,. (2.10) The position of subscripts follows the convention used in texture analysis. According to Eqs (1.12-14), the volume average can be expressed as the orientation average with texture function as a weight function Analogously, for Reuss average, it occurs In general, the problem of the determination of Voigt and Reuss averages, due to Eqs (2.10-12), is reduced to the calculation of the integrals (Ganster & Geiss, 1985) for the comprehensive description of a method allowing to get those coefficients.
The first modulus is the same in both relations. This is a special case of a more general feature: for cubic symmetry the bulk modulus is invariant of a very general averaging procedure (given by *=<A> with <A>=I, see Appendix D).

Geometrical Sense of lsotropic Voigt and Reuss Averages
The Voigt and Reuss averages have an interesting geometric property which will be shortly explained. Let 'o' denote the scalar product in ]I 4 tot2:= tr2(tt2), t,t2e114. (2.21) The norm Iol and the distance d can be defined in the standard way Itl:=/(tot), tE114, d(tl, t2):= Itl-t2l tl,t2114 (2.22) (Rychlewski, 1984;Krause, Kuska & Wedell, 1989). From the definition, the isotropic tensors preserve their form when rotated, i.e., tiju=gsigtjgugwts for isotropic t. Thus, the linear space of isotropic tensors is the subspace of 1I4, which is invariant under rotations, and 1I 4 is the direct sum of the subspace of isotropic tensors and its orthogonal complement. In other words, any t ]I 4 can be uniquely decomposed into t=ti+ta, where t is the isotropic part of t and t satisfies the orthogonality relation tiot'=0. Moreover, the distance d(t,xi) as the function of isotropic tensor xi 1I 4 takes the absolute minimum for xi=ti.
The clue of the above remarks is that for the uniform orientation distribution, the average M(t) is the isotropic part of t, i.e, ti=M(t). Moreover, M(t) is the isotropic tensor closest to tensor t in the sense of the distance d. One can check it applying the standard procedure of calculation of extrema for the function given by d(c,iso(,),x)) (with two variables ,/x).
.Going back to effective properties, Voigt average M(c) is equal to the isotropic part of the stiffness tensor c. Analogously, Reuss average M(s) is the isotropic part of tensor s.

HILL METHOD
These results are the basis for defining Hill average. To satisfy the above inequalities it is usually taken to be the common arithmetic or geometric mean of bounds for the bulk and shear moduli, i.e., or * (R + v)/2 (3 + yv)/2 (=:) (3.4) , (v)r2 (3/v) r (3.5) (Hill, 1952). Along with (2.18-19) these are the most commonly used formulae for polycrystal elastic constants calculation. The example results are shown in Figure 1.  i.e., the harmonic average of bounds. The formulae (3.6) and (3.7) give different results, thus the reciprocity condition is not satisfied. To achieve this objective, let's consider a new solution in the form of the series where c:=c v and e0:---c R. (3.9) When the series are convergent to the same limit, one can accept this limit to be the effective tensor lim c lim c c*. (3.10) The Hill average modified in this way satisfies the reciprocity condition.

OTHER METHODS CONSTRUCTED TO SATISFY THE RECIPROCITY CONDITION Aleksandrov Method
As was mentioned before, to tensor teTr there corresponds a unique 6x6 matrix. The det(t) will be considered to be the determinant of this matrix. Aleksandrov (1965) and Aleksandrov & Aizenberg (1966) proposed the method of the moduli determination based on the assumption that the determinants corresponding to the effective stiffness tensor and to the stiffness tensor of the single crystal are equal det(c*) det (c). (4.1) The bulk modulus of polycrystals with cubic crystal symmetry is given by *=:, and only the shear modulus is to be determined. The single equation (  Peresada's Suggestion Peresada (1971) (independently) proposed a scheme analogous to Aleksandrov procedure but for general symmetries. Because, in such a case, an additional relation is necessary, Peresada assumed that :*=(:vra)u2; hence T* (det(c)) 1/5 / (cVk'R) m The formulae for particular symmetries can be found in (Peresada, 1971). To define the generalized Aleksandrov method, one has to determine two mappings of 1I 4 into 1I 4 Let the exponential mapping be given by tn exp: a t= >- ;-,, (l--t)n ]I 4 (4.7) n n=l satisfying the condition (exPlog<v))-l=log is correctly defined. The neighborhood / is described by /={telP: II l-t I1<1 }. For tl,t2 / such that tit2 E / and tlt2=t2tl, there occurs log(tt2)= log(t)+log(t2).

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A. MORAWIEC Let us define the average of a tensor te / by exp<log(t)>. Such operation is homogeneous (i.e., exp<log(0t)> o exp<iog(t)>, e /) and, moreover, it commutes with the operator of tensor inversion exp<log(t-1)> (exp<log(t)>) -1 (4.8) Assuming that the effective tensor c* is given by the average of c c* exp<log(c)>, (4.9) the reciprocity condition is satisfied automatically (Morawiec, 1989). The condition that the argument of the log mapping has to belong to / is not a strong limitation; due to the homogeneity of the operation one can reduce the stiffness tensor to / by multiplying it by a non-negative number. In the general case, determination of / is rather complicated. In the special case of cubic symmetry, it is described by the inequalities 0<0r, tip, :2. Thus the average (4.9) can be calculated for any positive definite tensor of cubic symmetry.
Replacing the volume average by the orientation average, the expression for the effective tensor takes the form c* exp(Mf(iog(c))). (4.10) In practice, to apply this formula, it is necessary to use definitions (4.6) and (4.7), and take the same average as in Voigt method but of log(c) instead of c.
To the invariants trp (p=l,2) of the M operation there correspond the invariants Trp(t):= exp(trp(log(t)) of the operation determined by Eq (4.10), i.e., for te V, there occurs Trp(t)=Trp(exp(Mf(log(t)))). One can show that Tr2(t)=det(t). It means that the described average is a generalization of Aleksandrov procedure.
There should occur y R<y geom<, v or in full 2p+33' (4.14) For positive p and , both conditions can be reduced to the same form (2x+3y)5-55x2y (32x3+304x2y+1296xy2+243y3)(x-y) > 0, where (x----y=p) corresponds to the first, and (x=p, Y---T ) to the second of the inequalities (4.14). It is enough to notice that Eq (4.15) is true for all non-negative Imhof Method Looking for a method satisfying the reciprocity condition, Irnhof (1989) proposed an iterative averaging procedure. It is based on two series  (i.e., the effective tensor) is outside Voigt-Reuss bounds (see Figure 3) and, moreover, it always has isotropic form except the case of monocrystal (f(g)= (g0)) when c*=e.
Thus, in spite of the author's claims (Imhof, 1989), applicability of the method to the textured material is doubtful.

BRUGGEMAN METHOD
The method given by Bruggeman (1930,1934) has specific features characteristic for more contemporary concepts about the effective properties. Bruggeman admitted not only texture but also grain shapes to influence these properties.
It is assumed that grains have lamella-like shapes and thus some components of strain as well as some components of stress tensors are constant throughout the material. More precisely, for the direction normal to the lamellae being parallel to 'x 3' axis, the components el, 122' 133' 23' 31' t12 are assumed to be constant. To explain the main idea of the method, let each of these components be denoted by with proper subscripts. Using Hooke's law, the elastic energy density U can be expressed as u ci%e x(e)i ;i ; (5.1) where X is a (not-tensor) function of Cijkr On the other hand, using Hill's definition, one has <2U> c*iju<e>ij<e>l=X(c*)iju ij kl" (5.2) Averaging (5.1) over all orientations and comparing to the last formula, one obtains the expression for X(c*) X(c*) f X(e(g)) f (g) dg, (5. 3) so (3) where e(g) is determined by the crystal properties and crystallite's orientation. It remains to solve the above equation with respect to the components of the tensor e*. But it involves only 12 from among the 21 components of c*; for the axial symmetry of c* 4 out of 5 independent constants. To avoid this problem, instead of U, Bruggeman (1930) used a slightly different function tp given by 2tp (-1) 5 (i)3Cijkltij__,kl. (5.4) The calculations were carried for the cubic and hexagonal crystal symmetries and orthorhombic and axial sample symmetries. The dual equations can be obtained by replacing c and c* in the starting equations (5.1-4) by s and s*, respectively. In the original work, such mutually dual relations are used by turns, according to which is more convenient. Moreover, in case of this method, one can proceed in another way, not noticed by Bruggeman: Replacing c and c* by s and s* in final equations, one gets the result correspbnding to the exchanged roles of t and e, with the components (11' (22' 133' t23' 131 and O;12 being constant.
The appropriate material is composed of needles parallel to the 'x 3' direction.
Besides the (primary) aggregate described above, Bruggeman considered a 'secondary POLYCRYSTAL ELASTIC CONSTANTS aggregate' being a random composition of blocks of primary aggregates. To obtain the secondary effective constants, the Voigt procedure was applied to the primary ones. The curve in Figure 4 was calculated in that way.

Huber-Schmid and Boas Methods
The method of Huber and Schmid (1934) concerns only quasi-isotropic material. The effective Young (E*) and shear (G*) moduli are calculated by averaging (s3333(g))and (2(s2323(g)+s1313(g))) -(6.1) respectively, over all orientations g. One should mention here, that calculation of two moduli in the quasi-isotropic case of cubic crystal symmetry is in contradiction to the fact that one of the two moduli describing the effective properties is established 0*=), and only one remains to be determined.
The procedure led to relatively complicated integrals. Huber and Schmid (1934) gave the solutions for cubic and hexagonal symmetries, whereas Boas (1934) solved the problem for tetragonal symmetry. Furthermore, Boas (1935) changed the expression for shear modulus, slightly correcting the results. The second formula of (6.1) was replaced by 2[S3323(g)+s3313(g)] )-1 2[$2323 (g)-I-Sl313(g)]-$3333(g) (6.2) Again, the averaging gave complex integrals but their form was close to those used by Bruggeman (1930), and this allowed to apply the already existing solutions. The final results of Huber-Schmid and Boas methods (as well as the Bruggeman method) are relatively intricate and, therefore, the example will be omitted here. The curves for shear modulus are given in Figure 4. Only their displayed parts lie between Voigt-Reuss bounds. Shirakawa and Numakura (1958)

BUNGE METHOD
The method was originally presented in Fourier series formalism of texture analysis (Bunge, 1974). Here, it will be described in notation used for other methods. Let (prime) denote a deviation from the average. Because <e'>=0=<e'>, the elastic energy density U=<ece>/2=<(<e>+e')(<c>+e')(<e>+e')>/2 can be written as Assuming that <e'c'e'> is small in comparison to other components and omitting it, one finds that the variation of U with regard to e' leads to the following stationarity condition <c>e' + c'<e> 0.
Short calculation shows that the result (7.5) is above Reuss limit, provided P-7 <0. On the other hand, (7.7) is below Voigt limit, only if P-7 >0. Therefore, for materials with anisotropy coefficient A:----7/p less than 1, Eq (7.7) would be the proper formula, whereas for A greater than 1, relation (7.5) would be correct.

METHODS BASED ON AVERAGING OF THE VELOCITY OF ELASTIC WAVES
Assuming a given one-to-one relation between elastic constants and the velocity of elastic waves, one can calculate the effective properties according to the following scheme: crystal elastic constants --> velocity in crystal --> velocity in polycrystal ---> effective elastic constants. The constants considered here are adiabatic, contrary to the previous methods concerning the isothermic constants. The velocities v of the longitudinal (i=l) and two transverse (i=2,3) modes of elastic monochromatic planar wave can be obtained from the roots of Chdstoffel equation det(cijanjn qv2ik) 0, where q denotes density and the integration is over all possible directions of n. Methods considered below assume analogous relation to occur for polycrystalline aggregate det(c*ijanjn q2i.) O. Hence, one gets effective moduli, provided the velocities in the aggregate (i) are known.

Method of Ledbetter and Naimon
The method of Ledbetter and Naimon (1974) is based on the assumption that Debye temperatures of the polyand the single crystal are equal. Analogously to the crystal case, the Debye temperature 0 of the polycrystal is considered to be proportional to the average phase velocity {v} which is given by V (V (II))-3d2n fn i-- where n is the unit vector with the direction of the wave propagation vector and the integration is performed over all directions of n. After substituting i (i=1,2,3) from (8.6), the integral can be easily calculated because i does not depend on n. As the result one gets 3 3/3 (:* + 2) -3t2 + 4/2 ()-3 (8.9) With the left hand-side known (the same as for the single crystal) and *=1 (cubic symmetry) one obtains the equation for the effective shear modulus /*.

Markham Method
It is worth mentioning that the Ledbetter and Naimon assumption of the equality of Debye temperatures is secondary in relation to the assumption of equality of the average phase velocities {v} for single-and polycrystal. In fact, a similar idea is the base of a method proposed by Markham (1962) and later by Middya, Paul and Basu (1985).
The average velocities for longitudinal and transverse modes are assumed to be given The following arguments led the authors to the formulae: In a polycrystalline material, where velocity in a given direction is , the time of passing the distance L is equal to the sum of intervals necessary for passing subsequent grains on the wave's path.
I.e., L/rc=,plp/v v, where v and vP are the distance and the velocity in p-th grain, respectively. Assuming uniform orientation distribution and such a grain shape that no crystallographic direction is preferred, one gets as harmonic average of the velocities in all possible crystallographic directions.
Having Eqs (8.10-11) and using Eq (8.6) one can calculate the effective shear and bulk moduli.

Gold Method
Gold (1950), while estimating the elastic constants of beryllium, used an analogous method for calculation of the effective constants of the quasi-isotropic polycrystal. The assumption there is that the squares of longitudinal and transverse wave velocities in the polycrystal ( are equal to arithmetic averages (over all crystallographic directions) of squares of velocities in the crystal ((vi(n))2). Formally, this corresponds to the modified relations (8.10-11).
In the original paper Gold assumed weak anisotropy and equality of the mean velocities r 2 and re of transverse modes. To obtain the curve shown in Figure 7 the harmonic average of V and r was used in Eq (8.6).

CONCLUDING REMARKS
The degree of generality of the particular methods is different. The original Aleksandrov procedure or the Ledbetter and Naimon method can be applied only to quasi-isotropic materials with cubic crystal symmetry. Besides in those, quasi-isotropy is the immanent assumption of the methods by Huber-Schmid-Boas, Shirakawa-Numakura, Verma-Aggarwal and Imhof. The Markham and Gold methods, originally proposed for the quasi-isotropic case, can be generalized to include non-trivial textures or grain shapes but, to the author's knowledge, this has not been attempted up to now. The group of procedures considered to be general without restrictions, i.e., applicable to all symmetries and arbitrary textures, is not numerous. It contains the Voigt and in the main body of the paper. As to the exact solution for bulk modulus of cubic materials, only early papers on the subject (e.g., Huber & Schmid 1934) took no account of it. The situation is specific for the procedures based on averaging of the velocity of elastic waves. In fact, the adiabatic constants are considered there, but the authors assume them to be equal to the isothermic ones. The equality of singleand polycrystal bulk moduli does not occur for those methods. Procedures using the definition of the effective stiffness tensor are based on the following scheme. First, the strain tensor is expressed as e=A<e>. Substituting it in the standard definition, due to arbitrariness of <>, one gets e*=<eA>. Analogously, Hill's definition gives c*=<ATeA>, with 'T' denoting matrix transposition. Among the considered methods only three (Voigt, Reuss and Bunge procedures) can be easily reduced to the scheme. In case of Voigt method A=I and both definitions give the same effective tensor. For Reuss average, A=se* and the formula c*=<s> follows from Hill's definition, whereas the standard definition reduces to identity. The example of A for Bunge method can be easily read from Eq (7.3) (A=l--<e>-e'). As was stated before, in this case, the standard and Hill's definitions give different results.
The Bruggeman method has to be treated separately. It is an early attempt to take into account the texture as well as the shape of grains. This is close to the essence of statistical methods. On the other hand, it deals with a material of a very special internal structure. A generalization to other structures and symmetries, although possible, would require extensive transformations. Moreover, some arbitrary assumptions are included (e.g., there is no firm justification of the form of the potential {p). Generally the procedure is very cumbersome; calculations are carded out not in the compact tensor form, but using directly separate tensor components. It is difficult to check if the formal requirements listed above are satisfied. In particular, it concerns obeying the Voigt-Reuss bounds. As to the effective bulk modulus of the cubic material, it is equal to the single crystal bulk modulus due to the extra condition imposed in this case.
The generalizations of Hill method and methods described in part 4 are constructed ad hoc, to satisfy some of the formal requirements for effective constants. Besides those conditions, they are arbitrary and physically groundless. Similarly, there is no firm justification of averaging separately Young or shear moduli.
The methods based on velocity averaging use assumptions of a somehow external nature. There are no arguments in support of the Gold procedure. The assumption of Ledbetter and Naimon (equality of Debye temperatures) is doubtful. It would be more reasonable, to go in opposite direction, i.e., to consider the differences between Debye temperatures basing on the known effective properties as was done by, e.g., Anderson (1965) or Ledbetter himself (1973). Among those methods, the most clear is Markham's justification. One has to remember, however, that it is an extreme simplification of the phenomenon of wave propagation in the heterogeneous medium of the polycrystal. In other words, simplistic assumptions of the static problem are replaced by a trivial dynamic model. The approximation used in the elegant Bunge method appears to be too crude and, in effect, the result falls outside Voigt-Reuss bounds.
Summarizing, none of the reviewed methods can be considered as firmly justified and, as giving satisfactory results, at the same time. Each of them includes some arbitrary assumptions. In fact, this is the way to evade the lack of complete statistical information about the composition of the material. Unfortunately, none of such substitutes is satisfactory and application of more complete statistical information seems to be inevitable, especially for crystals with strong anisotropy. APPENDIX A: Matrices corresponding to tensors of if4 Due to the symmetries (1.6) calculations involving tensors of Tr can be carried out equivalently using (6x6) matrices. It is not convenient to use traditional Voigt assignment because it is not the same for all tensors. The other ascription used, e.g., by Wooster (1949) requires a special kind of 'matrix' multiplication (Morawiec, 1989). The best way to describe the correspondence is the following one (see, e.g., Gubernatis & Krumhansl, 1975): a) pairs of tensor subscripts 11, 22, 33, 23, 13 and 12 correspond traditionally to matrix subscripts 1, 2, 3, 4, 5 and 6, respectively, b) if one of the matrix subscripts, either of column or of row, is greater than 3, then to obtain the matrix element, one has to multiply the corresponding tensor element by /2. c) if subscripts of both, column and row, are greater than 3, then to obtain the matrix element, the corresponding tensor element should be multiplied by 2.

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A. MORAWIEC Elements of the set 1P correspond to symmetric matrices. Using rules a) and b) elements of 1 can be prescribed to column matrices 6xl (vectors) and the Hooke's law will also have the form of matrix multiplication.

APPENDIX B: Inequalities
Let for te 1I 4 the symbol t>0 denote that the matrix corresponding to t is positive def'mite, i.e., if t>0, then for any 0 (el) t>0.
It is well known from linear algebra that symmetric positive definite (semi-definite) matrices can be diagonalized with positive (non-negative) eigenvalues on the diagonal.
The main aim here is to prove that for t,t>0, from the inequality t>t follows tz->ti -. Having this proven, one can easily show that t>t and t>t7 are equivalent.
The same is true for tl_>t 2 and t t . Now, let's prove t>t2=>t>t{ .B ecause ti>0 (i=l,2), then there exists the orthogonal matrix uiW, such that ti=uiti7 , where t is diagonal matrix with positive diagonal elements. Let be the diagonal matrix with elements being square roots of diagonal elements of ti a, i.e., yi-ti , i>0. Defining yi:=uiY? one has yiy=ti along with yi>0. Let z:=y{y2y_y{ , thus z>0. Inequality t-t>0 is equivalent to y(t-t2)y=I-z>0. On the other hand, t-t>0 corresponds to z-l-I>0. It is enough to show that I-z>0 leads to z--I>0. Because z>0, there exists an orthogonal matrix u z, such that z a given by za=-uZUz is diagonal. I-z>0 leads to l=#=u(l=z)u,>0, i.e., each of the diagonal elements of # is greater than zero and less than one. Thus (#)--I>0 and z--I= Uz[(#)--I]u->0 which ends the proof of the statement.

APPENDIX C: Cubic and isotropic tensors of ]V
The tensors of 1P corresponding to the material with cubic crystal symmetry (T,Th,Td,O,Oh) are described by three independent parameters. Let for t these parameters Therefore, there occurs the simple multiplication rule eub(Kl,P,) eub(K2,O2,T2) eub(KiK,PlP2,%/2).
Let's consider the effective properties. If e=A<e> for arbitrary e, then <A>=I, and from the definition c*<e>=<ce>, the formula for the effective tensor is c*=<cA>. This is the form of exact stochastic solutions of the problem.