SURFACE REPRESENTATION OF POLYCRYSTAL PHYSICAL PROPERTIES: ALL CRYSTAL CLASSES, SIMPLE AVERAGE APPROXIMATION

Algorithms for polycrystal physical properties estimation are presented. Bunge’s spherical harmonics treatment of surface representations, under simple average approximation, is applied. Specific formulae for so-called longitudinal magnitudes are given. Physical properties associated to tensors of second-, third- and fourth-rank are considered. All crystal and sample symmetries are covered.


INTRODUCTION
The calculation of physical properties in polycrystalline materials, starting from the corresponding properties of their constituent crystallites, is one of the important objectives in texture investigations. A general theoretical background for this problem has been given by Bunge (1982), who also wrote a pioneering work (1969) on ODF prediction of polycrystal mechanical properties. Recent developments in this field are associated to the use of orientation correlation and misorientation distribution functions, to self-consistent methods and presently to Matthies' (1994) geometrical mean approximation. The papers by Bunge (1989), Humbert (1991) and Mainprice (1994) may be representative of important moments in this working area. A large number of specific studies have been performed in metallic (highly symmetric) systems. Special interest has been conferred to mechanical and magnetic properties. For the above mentioned cases, suitable algorithms and computer codes, ranging from simple averages to more sophisticated methods, have been established. However, in the case of low-symmetry materials, even for simple average estimations, we have not found a systematic presentation of such procedures.
Our purpose, with this work, is to contribute somewhat to a systematic treatment of polycrystal tensor properties. Specific algorithms for surface representation of properties associated to tensors of rank r (2 < r < 4) are presented. Simple average approximation and Bunge's spherical harmonics ODF expansions are used. All crystal and sample symmetries are considered. THEORETICAL SUMMARY (for further details see Bunge, 1982) Any macroscopic single-crystal tensor property E Y/X may be represented by one or more scalar functions E(h) of the crystallographic direction h. This gives place to the so-called surface representation (Nye, 1957). X and Y are measured quantities.
The mentioned functions may be described by expansions in series of surface spherical harmonics of the crystal symmetry K/(h) The coefficients e/2 characterize the property, r is the tensor's rank and M(l) the number of linearly independent functions K/(h).
The coefficients e/ may be calculated by means of the integral Let us consider a macroscopically homogeneous polycrystalline sample with a texture described by the orientation distribution function (ODF) 1=0 /2=1 v=l The ODF, or likewise the coefficients Cl/2v, characterize the volume fraction of crystallites in the neighborhood dg of the orientation g. Tl/2v(g) are symmetry-adapted three-dimensional spherical harmonics. If the crystallites of the considered sample show the property E(h), then the polycrystal simple average property 1 will be a function of the sample direction y and may be calculated, in the so-called Voigt's approximation, by the general formula N(l) l(y) /=0 v=l N(1) is the number of linearly independent functions K/V(y) of the sample symmetry. The acceptance of the Voigt approximation is equivalent to the assumption of a constant value for the variable X over the entire polycrystal. The coefficients l summarise the effects of both single crystal property and crystallite orientation distribution, characterized by el and CI/2v respectively. They are calculated by M0 .2 Cl/2 el" The symmetric functions in (1), (3) and (4) used in this work are those real functions described by Raymond et. al.. If the orientation distribution is rotationally symmetric about the Z-axis of the sample coordinate system, then the expansion (3) coincides with that of the fibre axis inverse pole figure R(h).  (9) where hi are the direction cosines of the crystal direction h.
With respect to such properties all crystals have at least orthorhombic symmetry. Equation (9) may be expressed, according to (1), by means of orthorhombic spherical surface harmonics. The coefficients e/ and the functions K/(h)are summarized in Table 1. Table 2 shows the coefficients l and the functions KlV(y) corresponding to equations (4) and (5) for a polycrystal, with all common sample symmetries taken into account. Subindex identifies triclinic coefficients and functions For fibre textures, according to (7) and (8) Table 3. So-called reduced (matrix) notation of the independent components d 0 is used. The non-centrosymmetric nature of the considered property implies that only odd-/ coefficients are non-null. Table 4 gives el for other crystal systems, with reference to those reported in Table 3. K?(h) functions follow the same relationship    For polycrystal average piezoelectricity one obtains, if sample symmetry is triclinic: d/,,(y) I/(y) + /(y) + 313(y) + /(y) + /(y) + 33/33(y) + (y) + 3/ (y) + 3/(y) + 73/37(y) with coefficients calculated from (5). For example, if the crystal symmetry is Cs or D3, then these coefficients are given in Table 5. The functions K/v (y) in (12) are the same as those reported in Table 3, with v =/z. For other sample symmetries the average value of the property may be obtained from (12), using Table 6. In these cases, the number of linearly independent functions depends on symmetry requirements (see Raymond et al.).

TENSOR PROPERTIES OF FOURTH RANK
Elasticity is the characteristic property associated to fourth rank tensors. It may also be described by longitudinal surfaces according to the transformation law: S[111 .a li alja,, an Sijkl (1 6) ij,k,l If S i#.l is the compliance tensor, then the surface Su gives the inverse of Yo.ung's modulus or eventually elasticity modulus. Expressions for e coefficients and K (h) functions are shown in Tables 7 and 8 for all crystal symmetries, excluding the cubic case. The reduced notation Si.i is used. Due to centrosymmetric nature of the considered property, all odd-/ coefficients in (1) and (4) are null. [S16 q-S26 + --(S36 -1-S45)]  [3(Sll " S22) q-8(S33 S13 S22) 4(S44 + S + S66 + S12)]  For a polycrystal, it is possible to express the average elasticity in the form (12). For sample triclinic symmetry, then the expansion of function S( (y) will have one term for 0, five for 2 and nine for 4. Other sample symmetries are solved by the same method as that leading to Table 6, taking into account Tables 7 and 8 and using equations (4)-(5). For example, if crystal-sample symmetry is D4h--D2h one obtains: i',(Y) g,(Y) + g2K2t(y) + gI,(y) + g,KS4,(y)+ g4k74t(y)  (20) are the same as those reported in Table 7 for v /.