Precipitation hardening by misfitting particles and its comparison with experiments

Scripta

METALLURGICA

V o l . 13, pp. 8 9 5 - 8 9 8 , 1979 Printed in t h e U . S . A .

Pergamon Press Ltd. All rights reserved.

PRECIPITATION HARDENING BY MISFITTING P A R T I E S

AND ITS

COMPARISON WITH EXPERIMENTS

v. Gerold and H.-M. Pham Max-Planck-Institut fuer Metallforschung Institut fuer Werkstoffwissenschaften D-7000 Stuttgart I, W.-Germany

(Received

June

28,

1979)

Introduction In the literature there exist three estimates (1,2,3) on the amount of precipitation hardening if the interaction between dislocation and precipitate is assumed to occur via the coherency strain field surrounding the particle and if the maximum interaction forces K are small compared with m twice the line tension T. All estimates for the increase of the critical resolved shear stress (CRSS) AT result in the equation o AY = ~(GE) 3/2 / (-~--~/2~ (I) o and differ only by the value of the numerical factor ~ which is 3.0, 4.2 and 3.7, respectively. The remaining parameters are the (isotropic) shear modulus G of the matrix, the modulus e of the misfit parameter, the volume fraction f, the length of the Burgers vector b, and the radius R of spherical particles. The strength of the interaction can be characterized by the following parameter A: A = (2R E/b) It will be shown that the estimate, Eq.

(Gb2/2T).

(2)

(1), is valid for A values below 0.2.

The three estimates differ by the way they average the maximum interaction forces K in the different slip planes at different distances from the particle center. The first estimate by Gerold and Haberkorn (i) uses a linear averaging procedure. The limits of the range of slip plane positions are chosen in such a way that the average gives a maximum value for AT . However, the authors made a numerical mistake in their derivation of the average obstacle distance along the dislocation. Compared to the well-known Friedel approximation (4) a correction factor / - ~ has to be applied which changes the value for e from 3.0 to 3.7 (5). Thus, this estimate leads to the same result as the one derived by Jansson and Melander (3). The latter authors used a more accurate averaging technique developed by Hanson and Morris (6) which is termed circle rolling procedure. This procedure applies statistical branching theory and again maximizes the applied shear stress. The third estimate, by Brown and Ham (2), uses a parabolic averaging technique which gives the slightly larger value of 4.2 for e. In sun~nary, all these estimates lead to comparable results. The purpose of the present paper is to extend the estimate to larger A values where most of the experimental results are obtained. This estimate will be compared with such results on precipitation hardened Cu-Co (7,8) and Cu3Au-Co (9,10) single crystals. Extension to Larger Interaction Forces The extension of the estimate to the case of stronger interaction is based on calculations of the maximum interaction forces K as a function of the relative position z/R of the slip plane and of m the strength parameter A, as performed by Wiedersich (11). The dislocation is approximated by an elastic string which is bent by external and internal forces. This leads to a non-linear differential equation which is solved by computer methods. As for Eq. (1), the calculation is made for an edge dislocation because it shows the maximum interaction force K . The results can be exm pressed by E

m

= 4 GERb kS(z/R, A).

895 0036-9748/79/090895-04502.00/0 Copyright (c) 1 9 7 9 P e r g a m o n Press

(3)

Ltd.

896

PRECIPITATION HARDENING BY MISFITTING PARTICLES

Vol.

15, No. 9

In this equation, ~ is a function of both parameters z/R and A (Fig. I). For small values of A (0 < A < 0.2) ~ is nearly independent of it (rigid line approximation). For this range, the estimate, Eq. (I), for the CRSS is valid. For larger A values the interaction force is no longer proportional to R and levels off to K = 2T (for example, for A = I on Fig. 1). This indicates m the transition from the cutting mechanlsm to the by-passlng or Orowan mechanism. In order to estimate the CRSS, the same averaging procedure as in (I) was applied. Since the result is proportional to /--~, the dimensionless quantity S

=

AT

o

/G

/-~

(4)

was chosen which is plotted versus the particle radius R/b in a log-log plot (Fig. 2) for variou~ misfit parameters E. The dashed lines (case a) were calculated with the assumption T = Gb-/2 whereas for the solid lines (case b), a variable line tension T

=

Gb2(I-2 ~ + 3 ~ sin20)/2(1-u)

(5)

with H = I/3 was chosen where O is the an~le between the local and the original direction of the dislocation. For 0 ~ O °, T becomes Gb-/4 which is half the value of case (a). For weak obstacles, the curves follow the approximation, Eq. (I), and differ by a factor /--~.

SO0 0,~

i

i

-I#

i

i

i

i

-o~

i

,

,

,

,

~ ,

o t e l Position of SlipPlane z / R

,

, ;

,

,

,~,

2

I0

20

50

lO0

200

tel. Porticle Rodius R / b

FIG. i ~(z/R)

5

FIG. 2

as a function of parameter A

Normalized CRSS as a function of R/b and 6.

On the right-hand side of the diagram all curves run into a straight line with negative slope - I. This line is determined by the Orowan mechanism. It has shown by Ashby (12) that this line must be shifted to lower S values because the two branches of the dislocation which b y - ~ s s the particle attract each other. This effect can be described by a lower effective value of the line tension. The reduction of S increases with decreasing particle radius. Comparison with Experimental Results To compare this estimate with experiments, the systems Cu-Co and CuRAu-Co were chosen since a number of careful experiments have been made with these alloys (7 t5 10). The negative misfit parameter e was calculated from strain-free lattice constant differences ~ = Aa/a using Eshelby's equation (13) = 6/(1 + 4 Gin/3~ )

(6)

where G is the shear modulus of the matrix and ~ is the bulk modulus of the precipitate. In order t~ get AT from the experimental CRSS, the contribution T of the depleted matrix has to be substracted.°This has been done in two different ways, namel~ AT O = T o - Tm,

(7a)

AT 2 = T 2 - T 2. o o m

(7b)

Vol.

13, No. 9

PRECIPITATION HARDENING BY MISFITTING PARTICLES

The quantity S determined from Eqs. that S 1 is more realistic.

(7a) and

897

(3) is called SI, the other one S 2. It is assumed TABLE 1

Values of e, f and T

Matrix

m

used in the calculation

Co at.%

-£ %

f %

Ref.

Cu

1.4

1.49

1.26

4

7

Cu

2.0

1.49

1.60

4

8

Cu3Au disordered

1.5

4.28

1.27

55

9

Cu3Au disordered

1.6

4.28

1.08

55

10

Cu3Au ordered

1.5

4.14

1.27

28

9

Cu3Au ordered

1.6

4.14

1.19

28

10

T

M~a

Table 1 summarizes the values of e, T and f used in the present analysis. The calculated S. values are plotted versus R/b in Fig.m3, the solid lines reproduce the theoretical curves f~om Fig. 2. The difference between S I u n d S_ is remarkable only for the ternary case where T reaches m relatively large values. As predicted from the theory, the S. values for the ternary alloys are 1 shifted to the left-hand top corner of the diagram as compared with the binary alloys. The experimental points follow the theoretical curves reasonably well. However, there are two marked differences: (i) the experimental points do not reach the Orowan line, (ii) the theoretical values of £ do not agree with the experimental data from Table I.

' ~ "/,

~

O~ CujAu-Co ~i~*de,'ed --------.

50O

Cu - Co

FIG. 3

=. 20e

Experimental results and theoretical curves.

~= 10C

50

0,5/~. I~ 2

5

10

20

50

I00

200

re/ P a r t l c l e Radius R / b

Case (i) has been explained already at the end of the last section. The experimental points at the right end agree very well with the lowering of the line tension (12). The disagreement with the e values is more serious. One of the reasons may be the anisotropy of the shear modulus of the matrix. In the derivation of Eq. (I), G is the shear modulus which determines the shear stress component of the coherency strain field which interacts with the dislocation. From the elastic constants, the shear modulus acting in the slip system is found to be G = 30,500 MPa for Cu, compared with G = 42,000 MPa as an average value. The latter one is use~ to express the line tension T, and has also been used to normalize the quantity S, Eq. (3). Since G

s

enters Eq.

(I) with the power 3/2, like e, one can correct for this difference by

898

PRECIPITATION

HARDENING

BY

MISFITTING

PARTICLES

Vol.

13,

No.

9

changing the ~ e o r e t i c a l e values to ec r = Ge/Gs = 1.38C .If one chooses the lower theoretical curves (T = Gb-/2) as the correct ones ~ e experimental points for the binary alloy would give C "~ 0.85 % instead of 1.49 %, and for the ternary alloys ~ ~ 2.3 % instead of 4.2 %. The corr. . corr remalnlng dlscrepancy can be explained within the scope of the present theory elther by a much higher line tension for the cutting mechanism or by a coherency strain field whose maximum strain close to the particle interface (which mostly determines Km) can no more be described by linear elasticity. Acknowledgements The authors would like to thank Prof. E. KrOner for stimulating discussions on anisotropy effects and Dr. R. Bauer for supplying the necessary data to calculate the coherency strain field for the anisotropic case. References I. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

V. Gerold and H. Haberkorn, phys. stat. sol. 16, 675 (1966). L. M. Brown and R. K. Ham, Strengthening Methods in Crystals, p. 9. Applied Science Publ., London (1971). B. Jansson and A. Melander, Scripta Met. 12, 497 (1978). J. Friedel, Les Dislocations, Gauthier-Villars (1956); Dislocations, Pergamon Press (1964). V. Gerold, Precipitation Hardening. Discolations in Solids, ed. F. R. N. Nabarro, Vol. 4: Dislocations in Metallurgy, North Holland Publishing Comp., in press. K. Hanson and J. W. Morris Jr., J. Appl. Phys. 4-6, 983 and 2378 (1975). M. Witt and V. Gerold, Scripta Met. ~, 371 (1969). K. E. Amin, V. Gerold and G. Kralik, J. Mater. Sci. 10, 1519 (1975). I. A. Ibrahim and A. J. Ardell, Acta Met. 25, 1231 (1977). V. Gerold and H.-M. Pham, Z. Metallkunde, in press. H. Wiedersich, Trans. Japan Inst. Met. 9 Suppl., 34 (1968). M. F. Ashby, Proc. Second Bolton Landing Conf. on Oxide Dispersion Strengthening. Gordon and Breach, p. 229 (1968). J. D. Eshelby, Solid State Phys. ~, 79 (1956). R. K. Leutz and R. Bauer, Computer Physics Comm. 11, 339 (1976).

*) The coherency stress field can be calculated on the basis of anisotropic elasticity theory (14). It has been checked that this leads to comparable results for the shear stress component in the slip system.