Elastic constants of polycrystals

Inr.J. Engng Sci. Vol. 8, pp. 49-6 1.

ELASTIC

Pergamon Press 1970.

CONSTANTS

Printed in Great Britain

OF POLYCRYSTALS

PETER R. MORRIS Armco Steel Corporation, Middletown, Ohio, U.S.A. Abstract-Kneer’s method for obtaining polycrystal elastic constants from single crystal constants and orientation distribution is extended to orthotropic physical symmetry trolled sheet) and to cubic crystal symmetry. The o~entation dist~bution is expanded in a series of generalized spherical harmonics. The limiting cases of randomly oriented and completely oriented (sin&e crystal) crystals are used to test the theory. Dirac-deita functions are used to derive the coefficients of the distribution function for those settings of a cubic crystal which possess orthotropic physical symmetry. Numerical examples employing data obtained on samples of copper plate and low-carbon aluminum killed steel are used to illustrate the method. 1. INTRODUCTION THEORETICAL work of Eshelby [ 1,2] has been extended by Kroner[3] and Kneer {4,5] to the calculation of the elastic constants of an oriented assembly of spherically shaped, elastically anisotropic crystallites. Kroner and Kneer have assumed that the nonhomogeneous environment of a crystallite can be replaced in a statistical sense by a homogeneous environment having the macroscopic elastic properties of the polycrystal. In contrast to previous treatments, stress equilibrium and compatibility equations are satisfied on the boundary between matrix and inclusion in the solutions of Eshelby, Kroner and Kneer. Developments due to these authors which are relevant to the present extension of Kneer’s theory will first be reviewed.

THE

2. PRIOR

Following performed.

Eshelby,

Kroner

2.1 The transformation

problem

THEORETICAL

and Kneer

DEVELOPMENT

a series of hypothetical

experiments

is

From a homogeneous stress-free matrix having elastic constants, ClqtS, a spherical region (the inclusion) is removed and subjected to a homogeneous stress-free deformation (tr~sformation), eL. The inclusion is elastically restored to its original shape by applying surface tractions -u$dS; to each element dS; of the surface, where $* = C,&. The inclusion is replaced in the cavity from which it was removed, and rejoined with the matrix. The layer of body force is removed by application of an equal and opposite distribution -to$dSI over S’. At this point, the total deformation of the inclusion is ezW,the sum of the stress-free deformation, e&, and the elastic deformation, (eC?WJ -
The stress-free and total deformations

are proportional,

(2) If an additional elastic deformation,

&&, is superimposed 49

on the body, the stress in the

P. R. MORRIS

50

inclusion becomes

If the components of wtvawcan be determined, equations (2) and (3) provide a relation between the stresses and strains in a region which has undergone a deformation, which in the absence of the matrix would have been stress-free.

2.2 The single crystal inclusion The inclusion is again removed, and replaced by a single crystal which had originally the same shape as the undeformed inclusion and which has been subjected to an eiastic deformation, (e&,+ Z,&. The single crystal is then joined with the matrix. The elastic constants, cfmnw,of the single crystal are chosen so that the stress,

czm= CZmnw (eZw + GIW> 9

is equal to that present prior to the substitution. (4) may thus be equated,

(4)

The right sides of equations (3) and

(5) Equation (5) may be solved for eG,

(6) where C * * w denotes the double scalar product C~~~w~~. Introducing (7) and substituting ec from equation (6) in equation (4) gives (8)

Premultiplying

both sides of equation (8) by S, and replacing &Jby S * - C yields

By employing the definition of u noted in equation (7), equation (8) may be written

Following Eshelby, Kroner and Kneer, the single crystal constants shall now be assumed to be specified and the polycrystal constants to be determined. If the components of w can be determined, equations (7), (9) and (10) provide relations between the strain in the single crystal and stress in the matrix and between the stress in the single crystal and strain in the matrix. The transformation and single crystal problems

Elastic constants of polycrystals

51

were first solved by Eshelby [ 1,2] for an isotropic inclusion and matrix. The solutions were extended to an anisotropic inclusion by Kroner[3] and to anisotropic inclusion and matrix by Kneer [4,5]. 2.3 The polycrystal constants Assuming that the nonhomogeneous environment of a grain in a polycrystal can be represented in a statistical sense by a homogeneous matrix having the macroscopic elastic properties of the aggregate, equations (9) and (10) may be written e=

(S+t)

cr= (C+r)

*.F,and

(11)

**P.

(12)

Averaging equations (11) and (12) over all grain shapes and orientations, to frequency of occurrence yields

according

f= 0,and

(13)

P= 0.

(14)

Kroner [3] called r the ‘elastic susceptibility’ and r - - 2 the ‘elastic polarization’ per unit volume. Equations ( 13) and (14) form the basis of Kroner and Kneer’s method. Kneer has expressed the result in a theorem: “Within a macroscopically homogeneous polycrystal the average of the elastic polarization over all orientations and grain shapes vanishes.” From equations (9) through ( I2), r and t may be expressed r=C*-(I-w)*.u,and

(15)

t=u*-S.

(16)

In the case of a spherical grain shape w is independent of the orientation of the crystal axes, and from equations (15) and (16) it is apparent that equations (13) and (14) have a common solution, D = 0.

(17)

If the expression for u in equation (7) is averaged over all crystal orientations according to the frequency of occurrence, the polycrystal constants may be obtained by an iterative procedure which minimizes this average. In order to carry out the average, w must first be found. 2.4 The components of w In order to obtain the components of the tensor w, it is first necessary to introduce the function U/n,,(r-r’), which gives the displacement, u,(r), at r due to a unit point force F,(r’) acting at r’. By expressing the strains as partial derivatives of the displacements, and the stresses in terms of the strains, the stress equilibrium equation becomes

CZTIMZP Unq,pdr-r’) +h&(r-r’hnpp)

= 0,

(18)

P. R. MORRIS

52

where the comma denotes partial differentiation, (np) denotes the part which is invariant with respect to interchange of n with p, a,, is the Kronecker delta, and 6(r - r’) is the Dirac delta function. Equation (18) defines U&r-r’), called the ‘fundamental integral of anisotropic elastic theory’ by Kroner [6]. Denoting the Fourier transform of U&r - r’) by o,,(k)

y

Unp(r-r’)

C?,,(k) eik’(‘-“) dVkV,,

=&

where k * (r-r’) = /&(rs-r~), and dvk = &,dk.&,. equation ( 19) with respect to x, and xI gives Unp,pr(r-r’)tnPj

= -&

The Fourier transform of 6 (r - r’ ) is &$

(19)

I “k

1

Taking

partial

derivatives

dk;“‘-“’ dF/k(,,,.

k,k,~,,(k)

b’k

I Vk

of

(20)

eik”r-r” d r/,. The Fourier transform of equa-

tion (18) is thus -- 1 C,,,,k,k,~,,,(k) 87r3 I Bk

e”““”

dV,+&

1

&,,9e’k’~‘-“’dVkCnp,= 0.

(21)

V

From equation (2 1) C mwbk~~w(W,n,,

= &n,

(22)

= ClmnplCpkl.

(23)

Let &z,(k) Equation (22) may then be written D,,(k)&,(k) Premultiplying

(24)

= &,.

both sides of equation (24) by D;k( k) , the inverse of D,, (k) gives o,,(k)

= D;,(k),

a result obtained by Liebfried [7]. By substituting ( 19), Kneer obtained U&r-r’)

The displacement, given by

u,(r),

u,(r)

= &

I “k

(25)

l?,,(k)

D;:(k)

eik”r-r’) dV,.

at r due to the distribution = CW & I s, U&r-r’)

from equation (25) in equation

(26)

of surface forces +o& dS’ is dS’.

(27)

E-la&c constiults

of polycrystis

53

Applying Gauss’ divergence theorem to equation (27) gives un(r) = CW eg I v, Unq,lp(r-r’) dV’.

(28)

The strains e%,(r) may be obtained by differentiating equation (28), &Jr)

= %z&

f

~, Ua~&=--r~) dV&,, .

t.291

According to Eshelby [I], the strains within an ellipsoidal region, subject to the surface tractions noted are uniform, so that e&=-

1 &J(r) dV. V.I

(30)

Pram equations f2), (29) and (30), 1 -I =- V W,,rv

IV

dV

I V’

C IWU ~&-r’)

dVI,,,.

(31)

Equation (3 I) is due to Kneer[4]. It provides a general form of the solution to the problem of determining the elastic field in an ellipsoidal region which has undergone a deformation which would have been stress-free in the absence of the matrix. 2.5 Evaluating the integrals In order to evaluate the integrals in equation (3 l), m = w-l and E = m - 6S we introduced, where S are the elastic compliances of the matrix. Postmultiplying both sides of equation (3 1) by Sn,.8yields

where (nw) (rs) denates the part which is invariant with respect to interchange of n with w, of r with S, and of the (nw) pair with the (KS)pair. Substituting Unr(r -r’) from equation (26) in equation (32) gives 133) By introducing k = lkj and dV* = k2 sin 8 de d+ dk, equation (33) may be written (34) The integral e-ik.r

dv’

e ““dV

(35)

P. R. MORRIS

54

is the same for all crystal systems. Following Kneer, the integral over V for fixed k is calculated by introducing a spherical coordinate system, r, 4, 8, where the z axis coincides with k.

Iv e”“dV=

J; Pdr s-+: eiRrcosed (cos 0) c” d+ = $(sin

ak - ak cos ak)

.

(34)

‘J.‘he Sv,e-*.“dV’ yields the same result, and from equation (35), A =-$ =- 2 S-V

(sinak-akcosak)2dk k4

sin2 ak -I-2ak sin ak cos ak- a2kzcos2 ak - 2a2k2sin2 ak w 3k3

sin ak;os ak dk).

0

(37)

The limit of the first fraction approaches 0 as k + 03. By repeated application of L’Hospital’s rule the limit of this fraction is found to approach 0 as k 3 0. Hence, with V = 4ma3/3, A reduces to A=$

“sinakyakdk=&. f0

(38)

For a spherical inclusion equations (34), (35) and (38) yield

By introducing 101, the determinant of D, and

where D,*,(k) are the elements of the adjoint of D and setting 6 = cos 8, equation (39) may be written (41) Since D* and IDI are of orders 4 and 6 respectively in the components of k, e/lDl in equation (41) is of order zero in the components of k. The substitutions k, = sin 6 cos 4, $ = sin 9 sin (b and kS = cos 8 may therefore be made in the integration of equation (41). This equation was derived by Kneer[4]. Equations (7), (17), (23) and (41) permit dete~ination of the polycrystal constants, providing a satisfactory represen~tion for the distribution of crystal orientations can be found. Kneer suggested expansion of the orientation distribution in a series of generalized spherical harmonics. Since no method

55

Elastic constants of polycrystals

was then known by which the coefficients of the expansion could be determined for the general case, Kneer restricted his solution to hexagonal crystal and fiber physical symmetries. 3. EXTENSION

TO ORTHOTROPIC

PHYSICAL

SYMMETRY

3.1 The integrand of E,,:,, The D,,(k) defined in equation (23) have been given for orthotropic Kroner [8]. They are c1px2+ c=y2 + CSJZ2 D=

(Cl2

+

(Cl2 &x2

w-v

(CM + C5s)xz

+

(Cl,

w-v

c22y2 + C@Z2

+ tc23

+

C44)YZ

+

symmetry

Ghz

G4)YZ CJJX” + c4y2 + c&z2 cc23

+

by

19

(42)

where the Voigt two-index notation for CUkl has been used, and x = kI, y = k2 and z = ks. With the substitutions k, = sin 8 cos $J,, k2 = sin 19sin 4 and ka = cos 8, the following quantities are obtained from equation (40):

IDI = twl+a2~2++~4+a7xs)ul+ ta~~+w7+aer,)ut+ (agxs+alox9)u3+a3u4, e 1111

--

(allxl+aler4+a12x6)ul+

(a15x3+alBX3)u2+a13X6u3,

e 2222=

(a23x4+a2er2+a2d6)ul+

(a15x7+a27x3)u2+a25X9u3,

e 3333=

(a32x6+a33X7+a28X3)u2+

(a2ti6+a14x9)h+a34u4,

e 2233 =

(a36x7+a2~3)u2+a31x9u3,

e 3311 =

(a29x6++21x3)u2+a22xsu3,

e 1122 -

(a17x4+al~6)ul+a19x3u2,

e 2323=

{[a32~4+a33x2+a26&l%+ +

e1212 =

[a2,x6+

i [tall +

+

a26+

[2(al6

=

+

cll~22c33 +

c13c23c66

-

c22G3

[%x6+

(a24+a14+2a39)x7+2(a26+a29)~31~2

(a16+a~+2a3,)x91u3++26u4}/4,

2a17b4

&9)x3

where a, = C,,C,Cm, a4

(43)

+

a12x2 +

+

(al4

a27%1uP

+ +

64

+

[al3xS

2a16)x6 +

+

%6x61

a23x11

ul

u31/4T

a2 = C22CMCm, a3 = C&&s,

4&&~66 +

-

+

+&6x7

+

c13c44c66

cll~

-

2 (~11~~3~~ +

~23c66~66

+ -

cl2cl3c44 c22c13c63

c&2,

a5

=

GlC22G6

+

GlG4G6

-

G&T2

-

2G2G6G6,

a6

=

c&3c66

+

cllc44&

-

&cf3

-

2cl3&&,

a7 = CllC2,C,

+ C&Jm

- C&f2 - 2C12CJ&,

a8 = C2&Cm

+ C&&

- C,C& - ZC&~C,,

+ -

c12c23c66 ~&23~44

+ -

~12~4&6 c33c12c66)

P. R. MORRIS

56 Q9 =

G&G4

+

G&&?6

-

c**e3

-

2G,C**G,,

a 10 =

C22G3G5

+

c33G4G6

-

GIG3

-

2~23G4G5,

u 11 =

G5G6,

al5

C33G

=

%2 +

=

c22c44,

C44G5,

%3

ai6

=

=

C33Gt

C22C33

-

a14 G3

-

=

G2G5+

G&6:,,

2C23C44,

a 17=-C55(C12+C66),alR=-C44(C12+Cs6), a 19=

(C,3+C36)(C23+C~)-C33(C12+Css),a2o=-C6s(C13+C55),

azl

=

(G2+

a 23 =

Cl&r

a 26 =

c&44

+

&&jr

a 28 =

CllC22

-

C%

a29=

(C,2+C~~6)(C13+C~5)-C11(C23+C~~.

Gd

a24

a 30 =

-GdG3+C44),

as

C2&,

=

fC23-t

=

C44)

C44&+

=

=

GJ,

a22

=

-C44(G3+

Gd,

c33&

cd33

-

CT3

-

2c13c55,

2C12C86,

a3] a34 =

C22(C13+

a25

a27 -

-

=-C5J(C23+C44),

C44C55, x1 = cos6#,

a32 =

x2 =

Cl&,

sin6#,

~~=cos~~sin~~,x~=cos~~sin~~,x,=cos~~,x~=cos~~sin~~, x7= sin4+,x,= u3 = c4( l-4’))

cos2ct,,x9=

sin*4,v,

= (1-~2)3,u2=~2(1-[2)2,

u4 = f6, andg = cos 8.

Using E = m - - S, where

m

=

w-l,

equation (7) may be written

where fi symboiicalilly represents the orientation of a single crystal specified by three Eulerian angles (+, 8,+). The c(G) are related to the principal elastic constants c by (45) where the al, are the elements of the rotation matrix

ffim =

(cos $I cos e cos # -sin* sin#) (sin~cosecos~ +cosJI sin#) -sine cos# -

(- cos + cos e sin cp cos $ sin e -sin$coscb) (-sin~cosesin~ sin~sine , +cost@cos~) sin e sin # cos e 1

(446)

and the set of Eulerian angles corresponds to that used by Roe [ lo]. The average, II, indicated in equation (17) is taken over all crystal orientations according to frequency of occurrence. For this purpose the orientation distribution,

Elastic constants of polycrystak

g(G) is expanded in a series of generalized

57

spherical harmonics (47)

where Z,,,([) are the Jacobi polynomials augmented by the square root of the weight function, and 5 = cos 8. These polynomials have been tabulated for orthotropic physical and cubic crystal symmetries through 16th order by Morris and Heckler[9]. A method for determining the coefficients of the series from conventional pole figure data was developed by Roe [ 10,111. Equation (45) contains expressions of at most 4th degree in trigonomet~c functions of & 8 and #. The generalized spherical harmonic expansion of c~~~(~), the expression in rectangular brackets in equation (44) and its inverse therefore contain no terms with I > 4. Because of the orthogonality of these functions, for calculation of the polycrystal elastic constants the orientation distribution is represented exactly by setting A = 4 in equation (47). For orthotropic physical and cubic crystal symmetries only three linearly independent coefficients, Wdoo, Wazo and WMo of the expansion are required. Wow = l/4&2, and W404, W424 and Wbd4 are obtained from W4m4= 5 W,,,fd70. If in equation (44) the inverse of the quantity in rectangular brackets is denoted by h, the double scalar product can be formed by expressing both h and E-l as 6 X 6 matrices using Voigt two-index notation. The relations between E& and E;: are the same as those between Ciikl and C,,, and those between hurl and h,, are the same as those between SziKland S,,. The elements along the principal diagonals of the C, E-l, h-l and h matrices are generally larger than the off-diagonal elements, so that a change in Cue, generally has a greater effect on &kl than on any other component. This assures the convergence of an iterative process which minimizes the average of u over all crystal orientations according to frequency of occurrence. In this process the (n + 1)th estimate, Q;z’), is obtained from the nth estimate by

where f is a factor &ecting the convergence. If f is too small, convergence is slow, if too large, the process will diverge. Typical values range from 10 to 30. The arithmetic mean Hill[l2] of the Voigt[13] average ctikl and the inverse of the Reuss]l4] average ( s)$ is used as the first estimate, G&. The problem of averaging fourth rank tensors with weight functions was considered in a previous paper [ 151 for orthotropic physical and orthorhombic crystal symmetries, and the procedure for extending the results to tetragonal, hexagonal and cubic crystal symmetries was indicated. The weighted average of c in Voigt two-index notation may be written E&ii]=

@fjkll

[ckllt

(49)

where EUand cRIare column vectors. For orthotropic physical symmetry the subscripts ij are restricted to 11, 22, 33, 23, 31, 12, 44, 55 and 66. For cubic symmetry kl are restricted to 11, 12 and 44. For these symmetries the FUkl are the elements of a 9 X 3 matrix, given by

P. R. MORRIS

58

where W,,, are coefficients of the orientation distribution, and the values of A, depend on the indices ijkl. The F tiklare given in Table 1 for cubic crystal symmetry, where I&,= l/30, B4 = 2lY840, B5 = Y2/210 and B, = (35)‘j2/420. Table 1 contains fifteen TUKI.The remaining twelve are given by relations of the form F,144= 2F,,,,, %*a = 2%312, %a, = f2;231x and %,a = &,,2 + ?;m Table 1. TLfkL for cubic symmetry Aooo

IF 1111 1112 2211 2212 3311 3312 1211 1212 1311 1312 2311 2312 4412 5512 6612

400

Box 18 12 18 12 18 12 6 24 6 24 6 2; -6 -6

B,X 72 -72 72 -72 192 -192 24 -24 -% % -% % E -24

420 B,X -24 24 24 -24

24 -24 -24 24 24 -24

440 B,X 24 -24 24 -24 -24 24

24

3,3 Calculation procedure Calculation of the polycrystal constants proceeds in the following manner. 3.3.1 Determine weighted Voigt and Reuss averages from Table 1 and take asfirst approximation GiLl = ( cGkt+ S&) /2. 3.3.2 Calculate ERwtinurxrs) from equations (41) and (43). For the symmetry con-

sidered here, it is sufficient to carry out the integration over the range 0 =S6 s 1 and and 0 G $I s rr/2, where 8 = cos 8. Express in Voigt two-index notation, where E,, = E illlr El2 =.L2 and E;bl= 4G3, and invert the resultant 6 X6 matrix to obtain E& .

3,3.3 Average u(Q) given in equation (44) on a JI,5, # grid. At each grid point the CQ are determined from equation (46). The quantity inside the rectangular brackets (denoted h-l by Kneer) of equation (44) is then evahrated, and expressed in Voigt twoindex notation. The hU are next obtained by inverting the resultant 6 X 6 matrix. The h Ornnare determined from hIllI = hII, h,,, = h12 and h2323= hJ4. The product hmnEGkKifnltn is formed, i.e. hU,.E;l&,KkO = (hu,&;;,‘,k, + hktnznE&,)/2, and &jkl = (&&+ &&)/2 is subtracted. The result is multiplied by g(sZ) determined from equation (47), and by the incremental angular volume associated with the grid point and the resultant product is summed over ah grid points to obtain i&& 3.3.4 Obtain next estimate, CfR+lf,of C from C@,$? = C$& -fiil3#r. 3.3.5 Test for satisfactory solution. If ] (cl(ffnl) - ~~~~~1~~~ 1 < c print C$$lr as the sohttion set, if 3 l perform next iteration. In the present work Ewas set equalto 10-a.

Elastic constants of polycrystals 4. TEST

59

OF THEORY

A satisfactory theory must yield proper values for the polycrystal constants in the limiting cases of completely oriented crystals. The (001) - [loo], (100) - [Ol 11,(110) [OOl] and (110) - [liO] settings of a cubic single crystal have orthotropic physical symmetry. The coefficients, W,,,,,,, of the generalized spherical harmonic expansion in equation (47) may be obtained for a single crystal by replacing the continuous orientation distribution g (CI) by a Dirac delta function which has non-zero values only for the 24 equivalent (Ml) - [uuw] settings. The W,,, calculated for the single crystal settings Table 2. IV,,, for some single and polycrystals Sample

W 420

W400 +0.03134464 +0*03134464 -0M783616 -0.007lU616 -0*01166825 -0XIO125081 0

(am)-[1W (loo)-[all]

(1W-[oo_11

(llO)-[llO] a-Fe OFHC-Cu Random

WUO +0*01873201 -0~01873201 +0~01404900 +0*01404900 +0~00155135 -0*00014738 0

0 0 -0.02478012 +0*02478012 -0*alO64850 -o*ooo13986 0

Table 3. Polycrystal constants Sample

cu Rand

a-

Fe

OFHC cu

Av.

CI1

C,,

Css

CB

C,,

C,,

C,,

Cw

Cw

K K*t

21.06 19.11 20.09 20.20 20.20

21.06 19.11 20.09 20.20

21.06 19.11 20.09 20.20

10.12 11.09 1060 10.52 10.55

10.12 11.09 10.60 10.52

10.12 11.09 10.60 10.52

5.47 4.01 4.74 4.87

5.47 4.01 4.74 4.87 4.83

5.47 4.01 4.74 4.87

VRH K

1690 16.99

16.90 1699

16.90 17.00

12.19 12.18

12.19 12.18

12.19 12.18

7.55 7.60

7.55 7.60

7.55 760

VRH K

22.10 22.15

22.10 22.15

16.90 17.00

12.19 12.17

12.19 12.17

7.00 7.01

7.55 760

7.55 7.60

2.36 2.42

VRH K

1690 17.02

22.10 22.07

22.10 22.07

12.19 12.19

12.19 12.19

2.36 240

7.55 7.55

7.55 7.55

VRH K

22.10 22.09

16.90 17.04

22.10 22.08

12.19 12.22

7.00 7.06

12.19 12.22

7.55 7.55

2.36 2.39

7.55 7.55

V R H K

29.53 2766 28.59 28.68

29.71 27.88 28.79 28.88

31.17 29.67 30.42 30.55

10.28 llwl 10.64 10.53

10.46 11.22 10.84 10.73

11.91 13.02 12.47 12.39

7.78 6.46 7.12 7.18

7*% 6.59 3:g

9.41 797 8.69 8.84

V R H K

21.21 19.28 20.24 20.28

21.24 19.30 20.27 20.31

21.30 19.37 20.34 20.38

10.14 11.11 10.63 10.42

10.17 11.14 10.65 10.45

10.24 11.20 10.72 10.52

5.35 3.92 4.63 4.76

5.38 3.95 4.67 4.79

5.45 4.01 4.73 4.87

V R H

3::

tK* Values due to Kneer[4]. All values are x 101ldyn/cm*.

60

P. R. MORRIS

above and obtained from experimental data for a sample of OFHCet copper plate and of low-carbon aluminum-killed steel sheet [9] are given in Table 2. Weighted Voigt (V), inverse Reuss (R), Hill (H) averages, and the polycrystal constants calculated according to the present extension of Kneer’s method (K) are shown in Table 3 for the samples noted above. The single crystal constants used for copper, cl, = 16905, cl2 = 12193 and cti = 7.550 X 10” dynlcm2, are attributed by Kneer[4] to G. Bradfield. For the steel sample, the single crystal constants, cl1 = 23.7, C 12 -- 14.1 and ca4 = 11.6 X 10f* dyn/cm2, are those reported by Mason[l6] for iron. In the case of the randomly oriented copper sample the results of the present calculation may be compared to values obtained by Kneer (K*). A 64 x 64 net over the range 0 s [ s 1 and 0 6 Q, d 7r/2 was used in the numerical integration of equation (41).A32~32~32netovertherangeO~$~~~/2,Oc4~~ landO~#~7r/2was used in the numerical integration of equation (44). Each iteration required 4 hr on an IBN-360 computer. In most cases a satisfactory solution was obtained after three iterations. 5. DISCUSSION

OF RESULTS

Examination of Table 3 reveals that for the case of randomly oriented copper the disagreement between the results of the present calculation and Kneer’s results for C,l, C,, and Cqq is in each case less than 1 per cent of the quantity calculated. In the case of the various single crystal settings considered the disagreement between the present results and the Voigt-Reuss-Hill averages (which are in this case exact) is typically O-3 per cent for C,, and C,, constants and 0.9 per cent for Ch4 type constants. The differences noted may be attributed to propagation of errors in the rather lengthy computer calculations. It is perhaps worth noting that in the case of the two polycrystal samples examined the differences between the values of the constants obtained by the present extension of Kneer’s method in all cases differed by less than 3 per cent from the more easily obtainable Hill averages. Acknowledgrnenfs-The author wishes to thank Professor J. D. Eshelby and Dr. G. Kneer for correspondence related to their papers and dissertation respectively, Professor E. Kroner for supplying a copy of his dissertation, W. Granzow for making the experimental measurements, J. W. Flowers and J. A. Peterson for reviewing the manuscript, and the Armco Steel Corporation for permission to publish. The OFHCa copper sample was graciously supplied by the Anaconda American Brass Company. REFERENCES [l] J. D. ESHELBY, Proc. R. Sec. AM, 376 ( 19.57). 121 J. D. ESHELBY,froc. R. Sot. AZS2,561 (1959). ]31 E. KRONER, 2. Phys. 151,504 (1958). [4] G. KNEER, Dissertation, Bergakademie Clausthal, Technische Hochschule [.5] G. KNEER, Phys. StatusSolidi9,825 (1965). [6] E. KRONER, Z. Phys. 136,402 (1953). [7] G. LEIBFRIED,Z. Phys. 13523 (1953). [8] E. KRONER, Dissertation, Technischen Hochschule Stuttgart (1953). [9] P. R. MORRIS and A. J. HECKLER,Adu. X-ray Analysis 11,454 (1968). [lo] R. J. ROE,J. appl. Phys. 36 2024 (1965). [ 111 R. J. ROE, f. appt. Phys. 37,2069 (1966). 1121R. HILL, Proc. phys. Sot. Lend. A65349 (1952). t@ Trademark of American Metal Climax, Reg. U.S. Pat. Office.

(1964).

Efastic constants of polycrystals

61

[ 131 W. VOIGT, Lerbuch der Kristuffphysik p. 962. Taubner, Leipzig (I 928). 1141 A. REUSS. Z. anpew Math. Mech. 9.49 (19291. ilSj P. R. MORRIS,J:appf. Phys. 40,44; (1969). ’ [16] W. P. MASON, Piezoefectric Crystals and Their Application to Ultrasonics, p. 417. Van Nostrand (1950). (Received 30 June 1969)

R&nrme- L’auteur &end au cas de la symetrie physique orthotrope (toe laminee) et B la symitrie cubique des cristaux la methode proposde par Kneer pour determiner les constantes Clastiques dun poiycristal a partir des constantes et de la repartition de I’orientation dun monocristal. La repartition de I’orientation est developpee en une serie d’harmoniques spheriques generalises. I1 verifie ensuite la validite de sa theorie pour les deux cas limites des cristaux a orientation aleatoire et des cristaux complitement orient& (monocristaux). L’emploi de fonctions delta de Dirac lui permet de calculer les coefficients de la fonction de repartition pour les orientations dun &stat cubique qui prdsentent une symetrie physique orthotrope. t’auteur illustre I’emptoi de sa methode de calcul par des exemples numeriques utilisant des resultats obtenus sur des ~chantillons de plaques de cuivre et d’acier a faible teneur en carbone et calme avec ~~uminium. ZnsammenfassungKneer’s Methode zum Erhalten elastischer Konstanten fur Vielkristalle von Einzelkristalikonstanten und Orientierungsverteihtng wird auf orthotropische Kiirpersymmetrie (Walzblech) und auf kubische KristaRsymmetrie ausgedehnt. Die Orientierungsverteilung wird in einer Reihe verallgemeinerter spherischer Harmonischen entwickelt. Die Grenzftile zufillig orientierter und vollsttidig orientierter (Einzelkristall) Kristalle werden beniitzt urn die Theorie zu priifen. Dirac Deltafunktionen werden verwendet urn die Koeffizienten der Verteilungsfunktion fur jene Stellungen eines kubischen Kristalles abzuleiten, die orthotropische Korpersymmetrie besitzen. Zahlenbeispiele, die Daten verwenden die an Proben von Kupferplatten und Aiuminium beruhigten nied~gekohienstoffenhaltigen Stahl erhalten wurden, werden zur Veranschaulichung der Methode verwendet. Smnmario-H metodo di Kneer per ottenere costanti elastiche pohcristalhne da costanti monocristalhne e distribuzione dell’orientamento e esteso alla simmetria fisica ortotropica (foglio avvolto) e aIia simmetria cristahina cubica. La distribuzione dell’orientamento & portata a una serie di atmoniche sferiche generahzzate. Per provare la teoria si usano i casi limitativi di cristalli orientati a case e completamente orientati (monocristallo). Si adoperano le funzioni di Dirac-delta per derivare i coefficienti della funzione di distribuzione per le impostazioni di un cristallo cubic0 avente simmetria fisica ortotropica; per ihustrare il metodo si danno esempi numerici the fanno uso di dati ottenuti su campioni di lastra di rame e acciaio calmato all’alluminio con basso tenore di carbonio. A6crpain--Meron Hnpa nonyqeHna KoHfTaHT ynpyroc~w nnx nonnKp~cTannoB Ha ~HO~H~K KoHCTaHT msi onmiornbtx tcpucrannoa, a rakxce no3tyvenria pacnpeneneriun opnenrauw~ pacrsfusaercn au i$H3~YSKyfO OPTOTPOtEfyIO CRMMCTpUK) (KaTaHb& JIHCTOBOi%MaTepHan) H tiB K5’6H’ieCKYM CHMMeTpHK) KpHCTannOB. Pacnpeneneme opuetiraun8 pa3naraerca B pan o6o6meunbtx C~pEWCKHX TapMOHHK. ~PaHHYHbleCny'iaH6eCLiOpJl~O'IHOOpHeHTH~BaHHblX HCTpOrO OpHeHTHpOBaHHblX (OllWHO'IHbleKPHCTaJlJlbl)KpHCTaJUlOB HCIlOnb3yK)TCR LIJIRIIpOBepKH TeOpUU. yllOTpe‘%lWTC5l &'HKURH &tpaK-AeJlbTa NlR fsbmefleHHx Ko3+$mHemoe @YHKUHH pacnpeJreness* mr raxnx ycranoeok Ky6uqeciroro xpncranna, y KOTOpblX HMeeTCR O,,TOTpOilHaSl~H3FiWCKa~ CHMhleTpHII. i+2ilOIIb3yKWCR PEICfleHHble tIpW+lepbIHa OCHOBaHHH nannbtx, nonyrieunbtx Ha o6pasuax ~oncTonucroaoi+i Mew w ~C~OKO~HHO# antobfmHsfeM H~3uoyrnepo~~~ToU crann, ilns unn~T~u1~~

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