Observations of the effect of an applied stress on the morphology of discontinuous precipitation in a CuCd alloy

Acta metall, mater. Vol. 41, No. 4, pp. 1183-1188, 1993 Printed in Great Britain. All rightsreserved

0956-7151/93 $6.00+ 0.00 Copyright © 1993Pergamon Press Ltd

OBSERVATIONS OF THE EFFECT OF AN APPLIED STRESS ON THE MORPHOLOGY OF DISCONTINUOUS PRECIPITATION IN A Cu-Cd ALLOY WENXU

GUO 1,

J. R. D R Y D E N 2 and G. R. P U R D Y t

tDepartment of Materials Science and Engineering, McMaster University, Hamilton, Ontario and 2Department of Materials Engineering, The University of Western Ontario, London, Ontario, Canada (Received 13 August 1992)

Alwlract--The influence of an applied tensile stress on the morphology and growth rate of the discontinuous precipitation product in a Cu-3.8 wt% CA alloy was first reported by Sulonen. In this contribution, we investigate and further quantify a number of factors associated with the role of applied stress in this alloy, including absolute and relative growth rates of interfaces with normals parallel to and perpendicular to the tensile axis, and the morphological stability of discontinuous precipitation front as a function of the applied stress. Once morphological instability has occurred, the amplitude and wavelength of the protuberences formed depends on the value of the applied stress, and on the angle between the average interface normal and the tensile axis. It is suggested that the process is best viewed as one of transformation-assisted viscoelastic deformation.

INTRODUCTION

EXPERIMENTAL

In 1964, Sulonen presented the results of a series of experiments which demonstrated the profound and surprising effect of an applied tensile stress on the growth rates of discontinuous precipitates in a Cu-3.81 wt% Cd alloy [1]. He found that, for grain boundaries with normals parallel to the applied stress, the growth rates of discontinuous precipitate colonies originating at these boundaries were decreased. In contrast, the rates were increased for colonies originating at boundaries with normals which were perpendicular to the applied stress. We will term these interfaces parallel and perpendicular respectively. Qualitatively similar effects were found for other alloy systems [2], but the effect was most pronounced and best quantified in the Cu--Cd alloy system. Interpretations of Sulonen's results have been put forward by Sulonen [1, 2], Hillert [3] and Dryden and Purdy [4]. Sulonen and Hillert concentrated on the elastic interaction of a misfitting solute field (presumed to exist in advance of the discontinuous precipitation front) with the applied stress. Dryden and Purdy proposed a model based on the inelastic interaction between the applied stress and the misfit (or transformation) strain of the transformed regions. In this contribution, we report on investigations of the morphology of the transformation product, with emphasis on the tendency (also apparent in Sulonen's micrographs [1]) for certain transformation fronts to become morphologically unstable. It appears that the applied stress acts not only to accelerate the growth of perpendicular interfaces, but to destabilize them.

Pre-weighed charges of Cu and Cd were sealed in evacuated quartz tubes, and slowly heated to 1150°C in a programmable furnace, then held at that temperature for 3 h, and quenched in water. The alloys were next homogenized at 720°C for 1 week, then subjected to hot isostatic pressing at 820°C and a pressure of 150 MPa for 5 h. This procedure resulted in fully dense alloy specimens which were cut into strips 2 x 2 × 10ram, and solution treated at 630°C for 3 h before quenching into water. The specimens were then treated at 400°C under uniaxial tensile stress in a dead-loading jig for varying times. This treatment is similar to that employed by Sulonen, in his studies of the same alloy system. Plastic strains were estimated using E = ( i o - !1)/1o, where 10 and lm are fidicial lengths measured before and after the test. Reductions in area were similarly determined using ( Ao - A l )/ Ao . In a few cases, specimens were first transformed stress-free, then under an applied stress to see what effect the stress might have on a pre-existing planar front. The influence of prior plastic deformation was also investigated by abrading or otherwise deforming one surface of the solution-treated specimens. All specimens were sectioned, then polished and etched in FeCl3-ethanol. Optical and scanning electron microscopy were employed, and qantitative data determined using standard image analysis techniques. RESULTS

When the Cu-Cd supersaturated solid solution is transformed stress-free at 400°C, discontinuous

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et al.:

EFFECT OF STRESS ON MORPHOLOGY

precipitation colonies develop equally and apparently randomly from the grain boundaries, as shown by Sulonen [1]. The precipitate colonies often initiate at grain corners, then spread over the boundary areas. Both single and double seam morphologies are present. The fl phase is present in the discontinuous product in the form of rods, Fig. l(b). When the alloys are transformed under stress, one finds the characteristic behaviour summarized in the sequence of Fig. 2, which represents a series of longitudinal sections from specimens transformed isothermally and at constant stress varying from 1 to 16 kg/mm 2. (We have chosen to represent stress in

these units for easy comparison with Sulonen's original paper.) The thickness of precipitate colonies is stress and orientation dependent; growth at parallel interfaces is inhibited, and that at perpendicular interfaces accelerated with increasing stress, up to about l0 kg/mm 2, as shown in Table 1. Perpendicular growth fronts become morphologically unstable for stresses greater than about 6kg/mm:; parallel interfaces remain stable. The "wavelength" is rather well defined along each straight boundary segment, and decreases with increasing tensile stress, as shown in Table 2. These data were obtained from perpendicular boundaries, and for early transformation times in order to minimize the effects of any possible "ripening" (see below). For nonperpendicular or curved boundary segments, we find a correlation between the wavelengh and the angle 0 (defined as the angle between the grain boundary normal and the stress axis). This behaviour is summarized in Table 3, and suggested by the composite micrograph of Fig. 3. Once instability has occurred, it seems that the spikes that develop from the initial planar front become platelike, with their broad faces perpendicular to the applied stress. This is suggested by the cross section of Fig. 4. It also appears that the growth rate of a plate increases with the plate length; the longer plates grow more quickly, so that shorter plates eventually lose the competition, Fig. 5. From the two-stage transformed specimens, it was found that the instability could be induced at previously-formed stable interfaces by the application of stress (Fig. 6); conversely, the removal of stress at temperature resulted in a reversion to a stable front. Concerning the influence of prior deformation, it was determined that discontinuous precipitation originated preferentially from those surfaces which had been abraded prior to transformation heat treatment, stress-free, at 400°C. Discontinuous precipitation fronts also were seen to develop from the free surfaces of uniaxially stressed specimens; for the more highly stressed samples, the wavelengths of instabilities were of the same order as for grain-boundary initiated colonies. The macroscopic plastic strain data are summarized in Fig. 7, which shows that this quantity becomes small and immeasurable (using the present methods) for applied stresses less than about 4 kg/mm 2. The time-dependence of the strain at higher stresses is suggested by Fig. 8, which shows a near-linear strain time plot for a fixed stress of 12 kg/mm 2. DISCUSSION

Fig. 1. Mierostructures of Cu-3.8 wt% Cd alloys, solution treated then aged, stress free, for various times at 400°C; (a) 2 min, (b) 5 rain, (c) 10 min.

In this section, we will focus almost entirely on the instability and its consequences. We begin by making two very general observations: First, the observed unstable transformation behaviour occurs within a range of stresses for which the alloy undergoes

WENXU GUO et al.:

EFFECT OF STRESS ON MORPHOLOGY

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Fig. 2. Some representative microstructures from longitudinal sections of specimens reacted at 400°C under various loads. The stress axis is vertical in each case. (a) 10 min, 4 kg/mm2; (b) 10 min, 6 kg/mm2; (c) 10 min, 8 kg/mm2; (d) 2 min, 12 kg/mm2; (e) 10 min, 12 kg/mm2; (f) 16 rain, 16 kg/mm 2. measurable plastic d e f o r m a t i o n c o n c u r r e n t with the phase t r a n s f o r m a t i o n ; secondly, u n d e r the deadloading conditions of these experiments, the macroscopic result of the phase t r a n s f o r m a t i o n is the increased rate of reduction o f the potential energy (or free energy) of the loading system. W e conclude t h a t it is very likely t h a t the m a i n coupling between the applied stress a n d the t r a n s f o r m a t i o n front is t h r o u g h some form of plastic d e f o r m a t i o n .

It is difficult to carry this a r g u m e n t m u c h farther with precision, in part because we presently have n o way o f k n o w i n g with certainty the state o f strain in the t r a n s f o r m e d regions. W i t h D r y d e n a n d P u r d y [4], we suppose t h a t the t r a n s f o r m a t i o n strain is capable of partitioning in such a m a n n e r t h a t the applied stress is m o s t effectively a c c o m m o d a t e d , a n d the c o r r e s p o n d i n g plastic strain maximized. W e are left to speculate o n the details o f this effect, since the

Table 1. Estimated average growth rates for interfaces with normals perpendicular and parallel to the tensile stress (y) axis; Cu-3.8 wt% Cd transformed for 10 min at 400°C Applied stress Perpendicular Parallel o'~ (kg/mm2) v (,um/s) v (/~m/s) 0 4.0 4.0 2 4.4 3.3 4 5.1 2.9 6 6.3 2.6 8 7.0 2.5 10 7.5 2.4 12 6.8 1.6 14 6.0 1.3

Table 2. Estimated wavelengths as a function of applied stress; Cu3.8 wt% Cd, transformed 10 min at 400°C; perpendicular interfaces Applied stress Wavelength a~ (kg/mm2) 2 (urn) 4 6.5 6 4.3 8 3.4 10 3.0 12 2.5 14 2.2

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WENXU GUO et al.: EFFECT OF STRESS ON MORPHOLOGY Table 3. The wavelength 2 as a functionof the angle0, between the stress axis and the grain boundarynormal; try= 14kg/mm2 0 (degrees) 2, (/~m) 30 5.8 47 4.1 58 2.8 74 2.3 90 2.2

degree of partitioning of the transformation strain would necessarily be a function (unknown) of the applied stress. It is also of interest to consider the extent to which elastic stresses might play a part in determining the morphology of the precipitate colonies. An estimate of the relaxation time for the specimen can be obtained from a knowledge of the "viscosity", r/and the elastic modulus, E, of the material. We find z = r//E ,~ 10 s

for the present conditions. Thus for a front velocity of 10-7111/8, we would expect an unrelaxed region approximately 1 0 - t m in thickness immediately behind the transformation front. In this case, the material is expected to behave in a viscoelastic manner, the freshly transformed regions exerting an elastically accommodated stress upon the matrix. Given that a shape instability is required as a precursor to the development of the platelike protuberances, we next investigate the stability against perturbation of a virtual planar transformation interface. The surface free energy, 7, will act to stabilize against perturbations of high wavenumber, cc The opposing force, driving the instability, will be taken as the virtual mechanical force, derived from the vertical (y) component of the applied stress. For a planar interface of unit area in the y - z plane, the plastic work accomplished when the interface moves through fix is the product of the stress try, the y component of the transformation strain e*, and tSx.

3 0 pm

Fig. 3. A composite scanning electron micrograph of a specimen reacted at 400°C for 2 min under an applied stress of 14 kg/mm2. The tensile axis is vertical. The information given in Table 3 was obtained from this specimen.

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!

Fig. 4. Morphology of the discontinuous precipitate, as seen on a section taken perpendicular to the axis of applied stress. 10min at 400cC under an applied stress of 10 kg/mm 2. Thus we obtain to first order a normal force of magnitude U~ = aye*. If the interface is now perturbed by a corrugation of amplitude a and wavenumber ct (i.e. a cos =y), then the misfit stress in the matrix at the interface will also develop a sinusoidal component. The important stress for the present purpose is tr* due to the y component of misfit, and this is given (see Appendix) by try* = {2(C - A + B/ct) + 2#C}~t cos ~ty where 2 and # are Lain6 constants, and A , B / ~ and C are combinations of elastic constants and transformation strains as defined in the Appendix. The stress in the matrix at the interface is now the sum of the applied stress, try, and the misfit stress try*; the virtual force in the x direction will be t~x

= O ya e y* "-I- try* e y*

Fig. 6. The microstructure of a specimen first annealed stress-free for 5 min at 400°C, then annealed under an applied stress of 12 kg/mm 2 for a further 5 min; stress axis vertical. The total force, including a contribution (-V~t2a cos cry) from the effect of surface free energy, is a * tr yey + (Ee .2 --Vet)eta cos cry. The second term is responsible for the instability. Setting the term in parenthesis to zero yields an order of magnitude for the wavelength of 10 -5 m, in reasonable agreement with the present observations. Thus we believe that the instability is caused by the stress variation along the length of the perturbed interface, and the interfacial response to that stress variation. Note also, that according to this analysis, the instability will be favoured if the transformation 12 Error

bar

I

//

400°C, l0 min ~-"

8

/ /

--

h.,

!

4-

.d / i

Fig. 5. Cu-3.8 wt% Cd, reacted at 400°C for 20 min under an applied stress of 12 kg/mm 2. The stress axis is horizontal in this micrograph. Two shorter spikes (arrowed) are in the process of losing a growth competition to their longer neighbours.

0

5

i

i*

l0

15

Applied stress (kg/mm 2) Fig. 7. The plastic strain as a function of applied tensile stress, Cu-3.8 wt% Cd, transformed at 400°C for 10 min.

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WENXU GUO et al.:

EFFECT OF STRESS ON MORPHOLOGY

15 --

APPENDIX 4 0 0 " C , 12 k g / m m 2

.~

S t r e s s at a C o r r u g a t e d G r o w t h F r o n t

Consider first a planar interface in the y z plane, moving in the x direction. The origin (x = 0) is set on the moving interface. For small perturbations, a cos ~y, the effect of a transformation strain can be modelled by the superimposition on the growth front of a thin region of misfit;

10 --

¢1 tj

e* a cos oty6(x )

gh

e~ a cos,,y6 (x)

0

[ 5

I 10

I 15

I 20

I 25

Time (rain) Fig. 8. Plastic strain as a function of time; 400°C, 12 kg/mm2.

e* a cos oty6(x)

where the e~"s are the components of the transformation strain, and di(x) is the delta function. We observe that u = f ( x ) cos ~y, and v = g ( x ) sin ~y,

strain is partitioned in the direction of applied stress, and thereby promoted by higher applied stresses. We have no simple suggestion for the experimental verification of this effect; it invites further study. Once instability has occurred, it is believed that the growth of the proturberances is further enhanced by the stress-concentrating effects of the partially unloaded transformed regions, i.e. that the interaction between the applied stress and the precipitate tip will increase the value of ay in the matrix at the tip. Growth then becomes in a sense autocatalytic.

(3. + #)ae/dx + #V2u = (2#ex* + 2e*)a cos 0cy6'(x) (2 + #)de/dy + #V2v = -(2#ey* + 2e*)ao sin ay6(x)

(3)

where 2 and # are Lain6 constants, e = Ou/Ox + dv/Oy, e * -- e x* + e y * + e z* ,

and 6'(x) is a dipole. Using (2) and (3), we obtain two coupled ordinary differential equations, which can be solved forf(x) and g(x) f ( x ) = e-~lXl{A(2H - 1) + B x }

g(x) = e-¢~{C + Blxl}

(4)

where h is a Heaviside function and ( A - C ) ~ ( 2 + # ) = B(2 + 3#). Two more expressions relating A, B and C to the misfit are found from (3). 2A(2 + 2#) -- a(2ge~* + 2e*) 2~ {A (2 + #) + (7# } - 2#B = a~(2#ey* + 2e*).

(5)

Solving for B and C B = (a~/2)[(2 + #)/(3. + 2#)]{e.**- er*} C = [a/4#0. + 2#)][(3. + 3#)(2#e* + 2e*) - (~ + #)(2~ex + ae*)].

(6)

The displacements and stresses can now be found; for the region x > 0, we obtain u = e-*~{A + B x } cos ay, and v = e -*={C +

B x } sin ,~y.

e = e - ~ { - ~ A + ~Bx + B} cos oty

Acknowledgement--This research was supported by the

Natural Sciences and Engineering Research Council of Canada.

(2)

where f ( x ) is odd, and g ( x ) is even. The problem becomes two-dimensional, and the following equilibrium conditions are found using the basic equations of elasticity

CONCLUSIONS Several quantitative aspects of the effects of applied tensile stress on the growth and morphology of grain-boundary initiated discontinuous precipitates in the C u - C d system have been investigated. It is found the system develops a spiked morphology for applied stresses above a critical level, and that the separation, or "wavelength" of the protuberances becomes smaller for higher stresses. An analysis is presented in which the instability is considered driven by the virtual mechanical forces associated with the interaction of the transformation strain with the applied stress, and opposed by surface free energy. It is thought (but not proven) that the transformation strain will tend to partition in the direction of the applied stress, so as to more effectively accommodate the stress.

(1)

(7)

then ax = 2e + 2#du/dx

REFERENCES

1. M. S. Sulonen, Acta polytech, scand., Ser. no. 28, p. 5 (1964). 2. M. S. Sulonen, Acta metall. 12, 749 (1964). 3. M. Hillert, Metall. Trans. 3, 2729 (1972). 4. J. R. Dryden and G. R. Purdy, Acta metall. 38, 1255 (1990).

% = ae + 2#dv/dy a~ = 2e Xxy = # {B - ,,[,4 + C -- 2Bx]}e -*~ sin e,y.

(8)

In particular, the expression for the stress in the y direction is cry= {).(C -- A + S / e Q + 2 # C } ~ t c o s c t y . (9)