The effect of particle shape on dispersion hardening

Scripta

METALLURGICA

Vol. 6, pp. 6 4 7 - 6 5 6 , 1972 P r i n t e d in the U n i t e d S t a t e s

Pergamon

Press,

THE EFFECT OF PARTICLE SHAPE ON DISPERSION HARDENING P.M.

Kelly

Australian Atomic Energy Commission Research Establishment, N.S.W., (Received

Lucas Heights

Australia

May

9,

1972)

INTRODUCTION With very few e~ceptions the various refined versions of the 0rowan dispersion hardening mechanism are based on the simple model of spherical particles in a metal matrix (I-3).

Elongated plates or rods are however

very common in two phase alloys and recent work has indicated that particle shape may have a marked influence on dispersion strengthening (4).

The

object of the present paper is to examine the effect of particle shape on strength.

Two types of particle are considered - rod shaped particles

and an alisned array of plate shaped particles.

Results for each case are

compared with the strengthening produced by the same volume density (Nv) and volume fraction (f) of spherical particles.

In both cases two factors

are important - the effect of particle shape on the number of particles intersecting the slip plane (NA) and any possible influence of the shape on the ease with which dislocations can bypass the particles. ROD SHAPED PARTICLES A dispersion of thin rod shaped particles of length ~and diameter 6 (~) ~=

will intersect a slip plane as a set of ellipses of mean area %62/2 (minor axis

6, average major axis 26)! 5)

The ~ccentricity

of this ellipse is only 2) so it can be approximated by a circle of diameter ~26 and its shape in the slip plane neglected.

The rod shape

does however affect the number of particles intersecting the slip plane. This is given by:

....

647

(1)

Inc

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PARTICLE

SHAPE

AND DISPERSION

HARDENING

Vol.

For an equivalent number of spherical particles of diameter d having the same volume as the rod:

,,,,,a

,5

....

(2)

....

(3)

t,~-~s W

where A = ~6. Combining (i) and (3) gives:

•

-~

--

,v,~

c,b.~,) . . . .

(4)

so that NA (rods) is greater than hA(spheres) in the range of A for which e
The change in shape from

spheres to elongated rods therefore causes an increase in strength since the strength is related to ~ ½ . The currently accepted version of the Orowan equation for spherical particles is(3):

%.-o-/(*D~

'

k ~

[z.J 7

d

....

(sa)

where r ° is the dislocation core radius jf2~ ~ J d is the effective sphere diameter in the slip plane, B is L is [N A

(spheres)] -½

0.85 ~b

2~(l-~)'~

for an isotropic material and

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PARTICLE

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649

In the case of rods the corresponding equation is:

.... (6)

....

"g~0a,) = I l l . - - - - - - - - - - -

Comparing equations

~"

(6a)

L

(5a) and (6a) gives:

/"C~t,~ )

JU3 s 6,,

with

equation r

o

(3)

= b = 2.5

(a)

8 = i00

(b)

0 = I000

(c)

0 = I ~m

(d)

6 --->

to express ~ are

infinity

0 in

shown i n

-

i.e.

terms

Figure

neglecting

(7)

....

(7a)

___

ie

using

....

of

d.

1 for

the

The v a l u e s four

log

different

of

this

ratio

cases:

terms.

This figure shows that for typical values of rod size and shape,

(6 : i00 to

i000 ~ and A--~e25) rod shaped particles result in 1.75 times more strengthening than the equivalent number o] spherical particles of the same volume. This shape enhancement factor increases with both rod diameter and aspect ratio. ALIGNED PLATE SHAPED PARTICLES The above discussion on rod shaped particles is valid for both aligned and randomly arranged rods.

The case of plates represents a more difficult

problem since the plates intersect the slip plane to give a much more

6S0

PARTICLE

SHAPE

AND

elongated profile than rods.

DISPERSION

HARDENING

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The mean area s of the intersection of a

thin plate of diameter D and thickness t (D~ t) with a slip plane is given by(5):

= Dt

.... (8)

The shape of this intersected area is appro>imately rectangular with _=

width 2 t and length %-2D. This elongated obstacle shape can affect the ease with which dislocations bypass the particles merely because the effective interparticle spacing is altered.

A calculation for randomly

oriented line obstacles has not yet been published,

but the simpler case of

parallel line obstacles has been treated by Foreman et al (6).

A reason-

ably good approximation to this computer calculation can be obtained by using either the square array model adopted by Bush and Kelly (4) and calculating the averase end to end spacing (not the minimum end to end spacing as in Ref. (4)), or the triangular array model shown in Figure 2. Both approximations give the variation of the average, effective spacing as a function of P/L where P is the length of the line obstacle and L is the centre to centre spacing (i.e. L = (NA)-½).

The results of the

square and triangular array models are shown in Figure 3 along with the computed values of Foreman et al.(6).

All these results for aligned particles

can be adequately represented by the appro:imationj

....

(9)

Equation (9) applies to aligned, line obstacles only, and to take account of the finite thickness of plate shaped particles the model shown in Figure 4 can be used.

The average intersection of a plate with the

2D t slip plane is taken to be an elongated strip of length ~-, width ~-~t and rounded ends of radius -~-. This is assumed to be equivalent to a 2D - 2 ) and the spacing ~ line obstacle of length F = (~this basis.

calculated on

~t The finite thickness is then allowed for by subtracting --~

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PARTICLE

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651

from this value of ~' to give the effective interparticle spacing~j

i.e. A - A ' -

where L is

INA

--

(plates)l-½ , .

J+P/£

The appropriate value of N A (plates) is D~f(5)

Substituting this into equation (i0) an ~ putting D = A, gives the appropriate version of the 0rowan equation for alisned plate shaped particles

.... (ll)

where B is

and C is

I-// \ |I/[.1~J I--

For an equivalent number of spherical particles of diameter d having the same volume as the plates

D 2~A,)~ =

d

.... (12)

•

Substituting this value into equation (II) gives

_

.... (lla)

This could now be compared directly with equation (5a) to give ~(plates)/~spheres), e,-uations differ.

except for the fact that the log terms in the two

__t The term in t[~-,, "2D

which is based on the particle

length rather than the width, probably represents an overestimate of the log term. in I~-o i

As a result it would appear justifiable to replace

equation (lla) by the slightly smaller term in

particularly as this considerably simplifies the comparison of equations (5a) and (lla).

Using this altered log term the strengthening enhance-

ment due to aligned plates can be expressed in the form

zr J

C

.... (13)

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PARTICLE

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DISPERSION

HARDENING

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Results of this expression for different values of A and f are given in Figure 5, which shows that for typical values of A (A~25), plate shaped particles are twice as effective as equivalent spherical particles, when the volume fraction is 5%.

This strength enhancement factor increases with both

volume fraction and aspect ratio. Very little experimental evidence on elongated plate (or rod) shaped particles is available in the literature.

An enhancement of strengthening

by elongated particles in bainitic steels has been reported (4), and the results appear to be in reasonable agreement with equation (II). best be demonstrated as follows.

This can

In the experimental work on bainite (4'7),

a number of specimens were transformed to give aligned elongated particles and then tempered to spherodise the carbides without altering their number. In the case of the as-transformed material the end to end spacing between the elongated carbides was measured.

~t This corresponds to ~' - ~- in Figure 4.

After the tempering treatment the interparticle spacing for the spherical carbides (i.e. d ~ l - ¢ ~ ,

l was also measured.

Provided that the tempering

treatment does not alter the number of carbides per unit volume then the value of

~'

-

~t ~- in the elongated carbide case should equal the corresponding

spherical carbide spacing ~ f ~ - ~ 3 ~

d divided by~(plates)/ ~(spheres).

This means that the effective interparticle spacing for elongated carbides in as-transformed material can be calculated from the spacings measured on tempered material with spherical carbides and then compared with experimental data from as-transformed specimens. shown in Figure 6.

This comparison is

The correlation is remarkably good, particularly in

the case of the higher carbon alloy. CONCLUSIONS Theoretical equations for the dispersion hardeningJdue to rod shaped particles or to an aligned array of plate shaped particles indicate that in both cases the hardening is greater than that produced by an equivalent

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PARTICLE

SHAPE

AND

DISPERSION

number of spherical particles with the same volume. generally give greater enhancement

of strengthening

particles and for both types the enhancement particle aspect ratio.

HARDENING

The aligned plates tha~ the rod shaped

factor increases with the

The limited evidence currently available

aligned plates indicates

653

on

that the estimates of the shape effect are

reasonably accurate. ACKNOWLEDGEMENTS The author is indebted to Professor P. B. Hirsch (Oxford University) for providing him with an advance copy of the computer calculations parallel

line obstactles

for the experimental aligned plates. (A.A.E.C.)

on

(Ref. 6) and to Dr. M. E. Bush (I°R.D., Newcastle)

data used in assessing

the theoretical

expression

for

The author would also like to thank Mr. J. G. Napier

for his assistance

in cDmputing

the values o f ~ ( p l a t e s / ~ ( s p h e r e s ) .

REFERENCES i)

E. Orowan - Symposium on Internal Stresses Inst. of Metals,

2)

M.F.

p.451

in Metals and Alloys.

(1948)

Ashby - Physics of Strength and Plasticity. M.I.T. Press, p.l13

Ed. A.S. Argon,

(1969)

3)

P.B.

Hirsch and F. J. Humphreys - ibid~ p.189

4)

M.E.

Bush and P. M. Kelly - Acta Met. 19, 136~ (1971)

5)

R.L.

Fullman - J. Metals ~, 447 (1953)

6)

A.J.E.

Foreman,

of dislocation

p. B. Hirsch and F. J. Humphreys

"Fundamental

theory", Nat. Bureau of Standards

317 Vol. II, p.i083 (1970)

Aspects

spec. publication

654

PARTICLE

SHAPE

AND

DISPERSION

HARDENING

Vol.

!3 ,/_i o

~o

~o

~o

~o

~o

ASPECT RATIO OF ROD (A = t/6 )

Figure I )

Values of t(rods)/ t(spheres) calculated from equation (7a) for different values of the rod diameter 6.

The dotted

extrapolation to A = I represents the region where the rods can no longer be regarded as thin and equation (i) is not strictly valid.

////

x

x:~ ~O-o~sL?+ p~-2-,sp, cos.o]ao Figure 2)

Triangular array of parallel line obstactles used to calculate the influence of particle length P on the effective interparticle spacing)%'.

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PARTICLE

S H A PE A N D

DISPERSION

v

HARDENING

i

A = -'~'~'~ ~ /.,~"

0

A ~ A Y . PRESENT PAPER

v • COMPUTER CALCULATION

REF~6)

C 0

Figure 3)

0"4

1.2

The variation of the effective particle spacing (expressed as

--~e) with obstacle length P for the case ot parallel line

obstacles.

lit 2

I

I I I

2D

/L /

I I l Figure 4)

Model used to allow for finite plate thickness in the case of aligned plate shaped particles.

655

656

PARTICLE

SHAPE AND DISPERSION

HARDENING

Vol.

4

• o,os

• o-o ~

~.c-~ ~_ ~.o.O~ ~.C~002

I

I 0

40

20

60

BO

100

ASPECT RATIO OF PLATE (A-=/T) ,

Fisure 5 )

Values of [(plates)/~(spheres)

calculated ~from equation (13),

plotted against aspect ratio A for various volume fractions of alisned plate shaped particles. The dotted extrapolation to A = i represents the region where the plates are no f to be longer thin enough for the relation N A (plates) = ~-[ valid.

1500

/ ~ " KXX~

/

/ / /o

/

/

/

h I CORRELATIOI~

/

0

/

/

e/ /o

/o / /

u, ~C

/

/

/

o O.SS~.

C]

, o.a9 % c

IREE C7~

/ / / 0 0

s6o MEASURED

Fisure 6)

~

,soo

SPACING (~)

The effective end-to-end spacings ( ~

- ~t/2)

for aligned

carbides measured in as-transformed bainite compared with calculated values obtained from the corresponding material.

tempered

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8