Internal strain (stress) in an SiCAl particle-reinforced composite: an X-ray diffraction study

Materials Science and Engineering, 89 ( 1987 ) 53-61

53

Internal Strain (Stress) in an SiC-A1 Particle-reinforced Composite: an X-ray Diffraction Study H. M. LEDBETTER and M. W. AUSTIN

Fracture and Deformation Division, Institute for Materials Science and Engineering, National Bureau of Standards, Boulder, CO 80303 (U.S.A.) (Received September 6, 1985; in revised form July 16, 1986)

ABSTRACT

SiC and aluminum alloy 6061 possess very different thermal expansion coefficients: 3.3 X 10 -6 K -1 and 22.5 X 10 -8 K -~ respectively. Thus, we expect large internal strains and stresses in these composites because the two constituents form interracial bonds at high temperatures and are cooled to ambient temperatures. From a simple elastic model, we expect a hydrostatic tensile stress in the aluminum matrix and a hydrostatic compressive stress in the SiC particles. Using conventional back-reflection diffraction geometry with Cu Kol radiation, we studied three surfaces o f a plate specimen. For both phases, we determined the unit-cell dimensions for two situations: unmixed and mixed in the final composite. The SiC particles showed a compressive stress and the alum i n u m matrix a tensile stress, equal to 75% o f the yield strength. Measurements and theory s h o w that both stress tensors are approximately hydrostatic.

1. INTRODUCTION

In this study, we consider the internal strain and accompanying stress in a two-phase material: an aluminum alloy matrix reinforced with SiC particles. The stress arises thermally from three conditions: (1) a strongly b o n d e d particle-matrix interface, (2) composite consolidation at a high temperature and (3) dissimilar particle-matrix thermal expansion coefficients ~. A small temperature change generates a large stress. For example, the temperature decrease A T required to cause a hydrostatic tensile stress equal to the aluminum alloy yield strength oy is only 0025-5416/87/$3.50

AT =

0"y

3B m (oLra -- oP)

= 64 K

(1)

where B denotes the bulk modulus, the superscript m the matrix and the superscript p the particle. Here, we use the ambient temperature physical properties: B m = 74.9 GPa, ~P = 3.3 X 10 -6 K -1 and ~ra = 22.5 X 10 -6 K -i . The alloy's yield strength equals 274 MPa. We obtain eqn. (1) by assuming hydrostatic stress and equating the mechanical and thermal strains. Previous studies on internal stress in composites include those by Barrett and Predecki [1], Tsai et al. [2] and Arsenault and Taya [3]. Barrett and Predecki studied stress in non-crystalline polymers and in polymer matrix composites by inserting (during material manufacture) fine powders of diffracting materials such as aluminum. Tsai et al. studied a graphite-fiber-reinforced aluminum. In the aluminum, they detected a tensile residual stress ranging from 33 to 228 MPa. Arsenault and Taya studied an SiC-fiberreinforced aluminum and reported compressive stress in the aluminum. The present study differs from the previous studies principally in that a particle-reinforced composite is studied and internal strains for both the occluded phase and the matrix phase are obtained.

2. AN ELASTIC MODEL

For the present study, a simple model provides much essential information. This includes symmetry of the macroscopic strain and stress tensors, relative values of particle and matrix © Elsevier Sequoia/Printed in The Netherlands

54

stresses, relationship of macroscopic stress to local stress, and stress magnitude. We recognize the inadequacy of a purely elastic model when plastic deformation occurs in the matrix. However, an elastic model contains some utility. It describes the situation preceding plastic deformation. It establishes the key parameters. It provides a basis for proceeding to a plastic deformation model. Mura and Taya [4] gave a plasticity model based on Eshelby's ellipsoidal inclusion model and Somigliana dislocations. For our elastic model, we adopt one originated b y Bitter [5] and improved and elaborated by Eshelby [6]. We consider the SiC particles as elastic spheres in a finite aluminum matrix as schematized in Fig. 1. In this diagram, R represents the radius of the totalvolume sphere, r2 the radius of the embedded particle, r0 the radius of the non-constrained particle and rz the radius of the hole in the non-constrained matrix. At some higher temperature, r2 = rz = r0. Upon cooling, the matrix hole tries to contract to rz ; the particle, with a lower thermal expansion, tries to contract only to r0. Thus, at ambient temperature, we have R >~ r0 > r2 > rz, with ro ~ r l ~ r2 • Many researchers give equations for the elastic state of this simple, but widely useful,

model. We start with those given by Teodosiu [7] in the natural spherical r-O-~) coordinates for the problem. Within the inclusion, the stress a is pure hydrostatic: O'rrP = O'ooP = 0 " ~ p

= --

4GmC

(2)

ro 3

where G denotes the shear modulus, p the particle, m the matrix and C the center-ofdilatation strength: ~iv c =

4r(1 +

(3)

4Gra/3BP)

where ~v denotes the difference in volume between the non-constrained matrix hole and the non-constrained particle: ~tv = } ~r(ro 3 _ rz s)

(4)

In the matrix the stress characterizes the presence of a dilatation center: 4GmC

ore_

~1 - - r3

(5)

aO 0 m _-- (700 m

2GmC ( -

-

rS

2rS~

(6)

1 + Ra ]

Figure 2 gives a graphical representation of eqns. (5) and (6). Both art m and aoo ~ are relatively short ranged, decaying effectively to zero at approximately r = 3r0. Because we make strain measurements in a cartesian coordinate system, we need to

1.0]

t

I

I 1.5

I

I

2.0

2.5

~----0.5

-1.or 1.0

3.0

r/r 0

Fig. 1. Spherical elastic i n c l u s i o n o f n o n - c o n s t r a i n e d radius r 0 forced into a hole o f radius r 1 and reaching a radius r 2 in a spherical matrix o f radius R.

Fig. 2. In the matrix the variation in art, 000 and a¢¢ w i t h distance r.

55

transform from spherical to cartesian coordinates in the usual way: (7)

(1iJ = ~ik °qz(1kl

o

sin 20 cos24 Orr + (cos20 cos24 + sin24) (1¢0

(8)

+ (cos20 sin24 + cos24) (1¢¢ (133

cos20 (1= + sin20 qo0

(112 :

sin20 cos 4 sin 4 (1=

o

Oll r 2 sin 0 dr dO d e V

(15)

ro

where 2~ R

sin20 sin24 (1=

(122 :

~r 27r R

o =fff

This transformation leads to 011

position. Thus, we seek the averaged value 5/j m . For (111, for example,

(9)

y=ff fr2sinOdrdOd4 0 0

(10)

ro

= ~Ir(R 8 -- ro a)

+ (cos20 cos 4 sin 4 -- sin 4 cos 4) (1~

(11)

(16)

Combining eqns. (8), (15) and (16) gives 4GmC R3

(17)

4GmC Ra

(18)

~11 m

013 = sin 0 cos 0 cos 4 a= -- cos 0 sin 0 cos 4 0oo (Y23

----

(12)

sin 0 cos 0 sin 4 (1=

5 22 m -

-- sin 0 cos 0 sin 4 (100

(13)

In cartesian coordinates, for the inclusion, the stress remains pure hydrostatic and is (111 p :

(122 p = (133 p

= ~ trace{a(r, 0, 4)} 4GmC -

Similarly,

(14)

a

4GmC -

033

m

_

and all other 0/j m equal zero. Thus, in cartesian coordinates, the stress tensor 0u m is a pure hydrostatic stress, independent of r. Equations (14) and (17)-(19) lead to the result

ro with all other (1up equal to zero. In cartesian coordinates, for the matrix, we need to average over r - 0 - 4 because, as shown in Fig. 3, a~jm depends on the angular

I

X

(19)

R 3

Oil m = - - Oil

p[ro? ~kR)

(no sum)

(20)

Thus a compressive stress in the inclusion means a tensile stress in the matrix, and lower by the ratio (to/R) 3. Below, in eqn. (45), we estimate that, for the composite under study, (ro/R ) 3 equals 0.405.

Surface

3. X - R A Y D I F F R A C T I O N A N A L Y S I S

Fig. 3. Diagram to relate cartesian and spherical polar coordinate systems. The diagram shows also that diffracting atomic planes (horizontal, parallel to surface) are extended in some regions and compressed in others. This occurs because the stress source possesses spherical symmetry.

We determined stress components by t w o methods: (1) comparing unit-cell dimensions and (2) sin2~ analysis in the modern version to account correctly for the ai8 stresses. Concerning the sin2~ method, in 1976, Evenschor and Hauk [8] (see also ref. 9) gave a general expression for the lattice strain: eaa' - d ~ -- do do = 8 1 (011 "~- (122 "~ 033)

(21)

56

+~

eqn. (21) becomes

( a n cos2¢ + a12 sin 2¢

$2

ess' = S l ( a n + a22 + ass) +--~- ass

+ a22 sinZ~b) sin 2 4

2 $2 - - e s s ) sin 4 + - ~ a l s

+~(an

2 ass c°s2~

sin 24 (27)

+-S-2(al s cos ¢ + a2s sin ¢) sin 24

If we define u = sin~4

Here d¢~ denotes the interplanar spacing for the direction ¢ ~ , d o denotes the spacing for the non-stressed case, and $1 and $ 2 / 2 denote the X-ray elastic constants. It should be noted that eqn. (21) contains nine terms; older models [10] contain only six terms because t h e y omit all the a~s stress components. For the usual Bragg-Brentano parafocusing X-ray diffraction geometry, ¢ = $ = 0 and eqn. (21) becomes

where

$2 ess = S l ( a ~ + a22 + ass) + -~- ass

c = 2els

(22)

then eqn. (27) becomes simply ess ,

S1

$12

=

=

_ - -

a + bu + c(u

u2)1/2

(29) (30)

b = e n -- ess

(31)

and

()e33'

(sin 2 @)

(23)

E

(32)

= b + c cot 2~ = e n - - e s s + 2els cot 24

(33)

If no shear strains exist, then a graph o f ess ' vs. sin2@ appears linear with a slope

-

Sn

--

Sn ~6ss'

a (sin 2 @)

= 2S44

2 (Gll

(24)

E where S~j denote the Voigt notation elastic compliances, E denotes Young's modulus and v denotes Poisson's ratio. Thus, 1 ess = ~ {ass -- v(o'n + a22)}

(25)

If the stress is hydrostatic ( a n = a22 = ass), then 1 -- 2v -

-

-

-

E

033

B = 30ss

(26)

where B denotes the bulk modulus. For a sin2@-type analysis [9]~ if we choose scanning in the x l - x s plane so that ¢ = 0,

= e l 1 - - eS s

$2

l+p -

633

--

Thus the slope of eqn. (27) equals

and

S2 2

=

a = ess

On the assumption of elastic isotropy,

P

(28)

0"33)

(34)

Whether or not shear strains exist, the intercept of this curve is given by eqn. (22). Because cot 2@ vanishes at @ = 45 °, we can obtain a n -- O3s directly from the slope at sin24 = 0.5. Because cot 2@ becomes infinite at 4 = 0, the low @ region of the ess'-sin24 curve provides little useful quantitative information. If the stress is hydrostatic, a n = ass. If, also, the elastic constants are isotropic, then e n = ess, and the slope in eqn. (34) equals zero. We can obtain the shear strains e u by considering eqn. (27) for both +@ and --@ at = 45 °. We obtain a

ess

i =

ess'(+~) - e~

=

2613

t

(-~) (35)

57

from whence E a l 3 --

l+v

(36)

C13

assume that Cu K s X-rays penetrate approximately 50/~m, or approximately 10r 0 . Thus, considering measurement inaccuracy, we can neglect the decay of at3 near the surface.

4. N E G L E C T OF S U R F A C E E F F E C T S

5. E X P E R I M E N T A L D E T A I L S

Mechanical equilibrium requires that, at a surface with a normal x3, all ei3stresses vanish:

5.1. Material We obtained materials from a commercial supplier in the form of 1 cm plates. The supplier started with commercially available aluminum alloy 6061 and SiC particles. These powders were blended, compacted and sintered to produce billets measuring 25 cm × 30 cm × 4 cm. Billets were h o t rolled at 7 0 0 - 7 8 3 K with a reduction per pass ranging from 10% to 50%. Rolled plates were subjected to a standard T6 heat treatment: solution treat at 800 K (980 °F) for 2 h, water quench and age at 436 K (325 °F) for 18 h. The plate contained nominally 30 vol.% SiC. We chose the coordinate system of x3 as the plate normal, xl the rolling direction and x2 the inplate perpendicular to the rolling direction. Figure 5 is a photomicrograph of the composite.

o13 = e23 = 033 = 0

(37)

Most previous X-ray diffraction stress analyses assumed that eqn. (37) held throughout the near-surface diffracting volume [10]. Recent studies [ 11-13 ] dispute this assumption and recognize that the at3 stresses show sharp gradients near the surface. The X-rays sample this gradient and reflect an effective stress value. In the present study the near-surface stress gradients are ignored. A basis for this assumption exists in a mathematical study by Mindlin and Cheng [14]. They considered an elastic inclusion e m b e d d e d near the surface of an elastic body. From their study, Fig. 4 shows the variation in 033 with depth r/r o below the surface. Within approximately t w o radii, the stress reaches 90% of its maximum value. We

1.0

I

5.2. M e a s u r e m e n t s

The measuring apparatus consisted of a commercial horizontal Bragg-Brentano focusing computer-automated diffractometer with a 22 cm diffractometer radius. The diffractometer contained a scintillation

0.8-

0.6 N

I~

0.4

0.2 0 1.0

I 2.0

3.0

r/r o Fig. 4. M a g n i t u d e o f stress component azz at a p o i n t r - - r 0 as a f u n c t i o n o f d e p t h r o f inclusion b e l o w the surface. OzZ d e n o t e s the magnitude for an infinite b o d y ( w i t h no surface effects).

Fig. 5. P h o t o m i c r o g r a p h o f an SiC-particle-reinforced

aluminum alloy matrix composite. The particle volume f r a c t i o n equals 30%. The particle size equals a p p r o x i mately 5 pm.

58 counter and a m o n o c h r o m a t o r , a curved graphite (Johann) type, located between the specimen and the counter. We used Cu K s radiation at an exciting potential of 45 kV and 40 mA. We prepared fiat specimen surfaces by chemically polishing the composite with a mixture of phosphoric acid (6:9), sulphuric acid (2.25:9) and nitric acid (1:9) for 15 min at the boiling point. Typically 0.1 mm of surface was removed from the studied surface. Aluminum lines 111 to 422 and discernible SiC lines (10i (i = 1, 2, 4, 7, 9), 208, 209, 213, 1 0 . 1 5 , 2 1 9 , 20.15) were scanned in 0.01 ° steps for 60 s. We obtained peak positions by fitting to a Pearson type VII function:

4.06

{1 + (o -_ 2W 2

)

I

•L-'•'•••B 4os 4.04 0

u Ik

I

I

1.0

2.0

Composite

3.0

C068 cot8

Fig. 6. For the aluminum matrix the variation in unitcell dimension with cos 0 cot 0 where 0 denotes the Bragg diffraction angle.

3.085

3.084 15.2

--rrI

y =

I

3.082

(38)

where/max denotes m a x i m u m intensity, 0 the diffraction angle and W the full peak width at half-maximum intensity. We chose m = 2, making y(0 ) a lorentzian. Both Cu K~I and Cu K~2 lines were measured using a deconvolution procedure, and the measured lattice spacings were least squares fitted vs. the cos 0 cot 0 extrapolation function. In both the incident beam and the diffracted beam, we used 1 m m horizontal Soller slit packs with 1.5 ° vertical divergence slits and 0.2 m m receiving slits. The uncontrolled temperature in the diffractometer chamber varied from 299 to 302 K. For aluminum, a variation of +1.5 K produces an error Aa of +0.00014 A, too large for careful study, but n o t fatal to the present study and its conclusions. In the present study, Aa values were f o u n d ranging up to 0.0044, 30 times larger than the temperature-induced error.

6. RESULTS Figure 6 shows the variation in the matrix alloy cubic unit-cell dimension with the geometrical function cot 0 cos 0 . 0 represents the Bragg diffraction angle. When this is extrapolated to 0 = 90 °, we obtain a = 4.0496 A. Figure 7 shows a similar diagram for the SiC particles, which have a hexagonal unit

o3

o~

15.1

==

o

3.08G

3.078

3.076

0!5

110

1!5

20

25

5.0,4.,

cos 8 cot 8

Fig. 7. For the SiC inclusion the variation in unitcell dimension with cos 0 cot 0 where 0 denotes the Bragg diffraction angle. We take the axial ratio 7 = c/a to be equal to 4.9079.

cell. We achieved a single line by using the axial ratio 7 = 4.9079, which does not depend on 0. When this is extrapolated to 0 = 90 °, we obtain c = 15.1243 )k and a = 3.0816 A. Both Fig. 6 and Fig. 7 represent diffraction results from a constituent powder. (The composite was made from powders of aluminum and SIC.) Table 1 contains the study's principal results. For both the f.c.c, aluminum matrix and the hexagonal inclusion, Table 1 shows unit-cell dimensions a and c, the associated strains e = A a / a , the volume strain, and the stress estimated from AV o = B -V

(39)

where B p = 223.4 GPa and B m --- 74.9 GPa. Figure 8 shows a A d / d o vs. sin2$ diagram for one of the six principal diffraction geometries. Table 2 contains numerical results for Ou - - ass, for the differences in principal stress

59 TABLE 1 For both matrix and particles, the unit-cell dimensions, strains and calculated stresses

Face

Matrix

Particle

a

e

AV/V

(XlO-3)

(.~)

(XlO-3)

oii

a

c

(MPa)

(A)

(A)

ea

(XlO -3)

ec

AV/V

(XlO -3)

(XlO -3)

oii

(MPa)

xI x1

4.0532 4.0536

0.896 0.990

2.692 2.974

202 223

3.0805 3.0808

15.1171 15.1180

-0.357 --0,282

-0.479 --0.418

-1.193 --0.982

-266 --219

x2 x2

4.0532 4.0540

0.894 1.077

2,684 3,233

201 242

3.0804 3.0806

15.1195 15.1181

--0.409 --0.334

--0.321 --0.413

--1.139 --1.082

--254 --242

x3 x3 xa

4.0498 4.0495 4.0510

0.047 --0.035 0.343

0.141 --0.104 1.030

11 --8 77

3.0809 3.0807 3.0806

15.1198 15.1217 15.1208

--0.247 --0.308 --0.334

--0.298 --0.176 --0.237

--0.791 --0.792 --0.905

--177 --177 --202

Reference

4.0496

3.0816

15.1243

c o m p o n e n t s and f o r t h e shear stresses o u (i =/=j ). We o b t a i n e d these f r o m t h e results in Fig. 8 a n d its c o m p a n i o n figures, w h i c h we o m i t here. We e s t i m a t e t h e e x p e r i m e n t a l e r r o r in a , as +20 MPa. F r o m t h e results in Tables 1 and 2, we cons t r u c t t h e f o l l o w i n g stress tensors, r e p r e s e n t e d as t h r e e - b y - t h r e e matrices, f o r t h e m a t r i x and t h e particle:

0il ra

=

I22i

0.4

I0 v ¢o ~¢0 0.2

O. 1

0

0.3

0.4

0.5

sin2~/

Fig. 8. Lattice strain e33' vs. sin2~ for the 420 reflection from the aluminum matrix. This diagram represents diffraction from the xl surface scanned in the Xl-X 2 plane. Thus, it corresponds to the s t r e s s a 2 2 - -

- -

207 --3

O.E

18 2

(711.

206 -+

0

206

0

0

(40) 206

TABLE 2 For the matrix phase the results of sin2~ analysis

and

--248 0

--185

---219

-*

0 0

--219 0

°

1

ij

e i i - ej] (X10 -3)

O i i - g]J (MPa)

eli (× 10 -3)

OiJ (MPa)

31 21 23

--0.827 --0.403 --0.459

--44 --21 25

--0.060 --0.063 --0.051

--3.2 --3.4 --2.7

(41)

--219

T h e matrices o n t h e first lines o f these equat i o n s r e p r e s e n t t h e m e a s u r e m e n t s . T h e matrices o n t h e s e c o n d lines o f these e q u a t i o n s represent the imposed constraint of hydrostatic stress. In t h e particle case, we did n o t m e a s u r e t h e shear strains e u (i C j ) or stresses ou(i - ¢ j ) . In t h e m a t r i x case, we n e g l e c t e d the

x3 face results in Table 1; these results disagree d r a m a t i c a l l y with o u r o t h e r measurem e n t s ; t h r o u g h f u r t h e r s t u d y , we h o p e t o resolve this dilemma. We e m p h a s i z e t h a t eqns. (40) and (41) refer t o a cartesian (measurement} c o o r d i n a t e s y s t e m and r e p r e s e n t the average over a v o l u m e c o n t a i n i n g very m a n y particles.

60 Our measurement analysis neglects possible, and probable, textures both in the aluminum matrix and among the SiC particles. We have just begun a neutron diffraction pole figure analysis of this composite, an analysis which we plan to report later. Also, in future studies, we hope to represent the SiC particles as prolate ellipsoids rather than as spheres.

First, we note that 0U m is approximately a hydrostatic tensile stress. Approximately, (Tum = o6 u where a denotes a scalar stress and 8 u the Kronecker ~ function. The elastic sphere-in-hole model described above predicts such a stress. Although such stress states occur infrequently in composites, they do occur. Garmong [15] remarked that "sintered p o w d e r alloys m a y be the best examples of such structures". Second, we caution that Oil~ represents a macroscopic average of a volume large relative to particle size. Furthermore, 0ij ~ represents an average over all directions. Locally, we should describe the stress in spherical r - 0 - ¢ coordinates. Taking r = Xl, 0 = x2 and ¢ = x3 this gives

0~o m =

0 0

0

0

Oo0 ~ 0

0 0¢¢ m

] (42)

(Here the bar over o means an average value between r o and R .) We can use several approaches to obtain numerical values for 0rrm and 0oo m = ~ m. Here, we consider two.

First, from eqn. (5), we can calculate the average stress

R

R

a. =fo, dr/fd ro

I"o

- ~R

2ro

2ro 2

Similarly, O0 0 m = OCpdpm

_4GmC( R3

=21/2 f=0.405

(45)

where f denotes the particle volume fraction, equal to 0.3. This gives ro/R = 0.74 and {in megapascals)

7. DISCUSSION

[ orrrn

and Ooom from eqns. (17), (40), (43) and (44). We estimate the upper b o u n d on ro/R (and probably a good estimate of ro/R) by assuming a close-packed arrangement of spheres. Then,

R

R2) 1 + 4to + 4r02

Thus, if we k n o w

R/ro,

(44)

we can calculate On.m

0~ m=

o o0J

0

369

0

0

(46)

369

We recognize that this approach to estimating ro/R is weakened by a non-homogeneous particle distribution, as shown in Fig. 5. In a second approach, we require mechanical equilibrium at the particle-matrix interface for the radial stress component:

0rr m = ~rr p = --219 MPa

(47)

At the particle-matrix interface, this gives (in megapascals)

Or0~m =

0

419

0

0

(48) 419

where we require that trace a = trace O, which can be proved for the elastic case. Equations (46) and (48), derived by different physical assumptions, show surprising compatability. Third, we remark that the stress components in eqns. (46) and (48) exceed the matrix yield strength of 274 MPa. These stress components represent the local condition near the particle, and not the macroscopic average detected by X-rays. These large stresses suggest severe local work hardening in the matrix, especially near the particle-matrix interface. The argument for work hardening becomes even more compelling if we consider the von Mises yielding criterion [16]. By this criterion, yielding occurs when the effective stress ae exceeds the yield strength. The effective stress is given by 1 Oe = ~ - ' ~ ( {O'rr -- 0"00) 2 -}- {0"00 -- O'¢~(p)2

+ (0"~ b -- art)2) l/2 = 0"00 -- O'rr

(49)

61 which equals 490 MPa, 80% higher than the non-work-hardened yield strength. F o u r t h , and finally, we observe t h a t our measurements do n o t satisfy the mechanical equilibrium condition for average stresses in a two-phase material. Mura [17] gives t he relationship f p + (I -- f)m = 0

(50)

where f denotes the volume fraction of particle. The ratio of average stress components equals <%>m _

p

f

1 --f

onal stress com ponent s. Second, shear strains are negligibly small. (5) Transforming the stress tensor into local spherical polar coordinates reveals stress c o m p o n e n t s (~00 TM = ~ ¢ m ) t hat exceed the matrix yield strength.

(51)

which equals --0.43 for the present case where f-- 0.3. From eqns. (40) and (41), we find --0.94 for the macroscopicaveragecartesian coordinate hydrostatic stress. The reasons for not satisfyingmechanical equilibrium remain speculative. They include (1) only a surface sampling by the X-rays and a different sampling of the matrix and particles, (2) texture in either phase, especially among the particles and (3) a non-linearrelationship between stress and particle concentration combinedwith a homogeneousparticle distribution.

8. CONCLUSIONS F r o m this study, five conclusions emerged. (1) A simple Bitter-Eshelby sphere-inhole elastic mo d el predicts hydrostatic compression in the inclusion and hydr os t a t i c tension in the matrix. (2) Conventional Bragg-Brentano focusing X-ray diffraction shows t ha t the matrix exists in tension ( a p p r o x i m a t e l y hydrostatic), on average with a stress of 206 MPa, 75% of the matrix yield strength. (3) Similar diffraction measurements show t h a t the particles exist in compression, on average with a stress of 219 MPa. These two results fail to satisfy mechanical equilibrium. (4) Measurements by the sin2~ m e t h o d provide two results t hat s uppor t an approximately hydrostatic stress in the matrix. First, only small differences exist bet w e e n the diag-

ACKNOWLEDGMENT Ming Lei, guest scientist from the Institute for Metals Research, Shenyang, China, contributed several calculations and a critical reading.

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