Precipitation Hardening A.J. ARDELL The topic of precipitation hardening is critically reviewed, emphasizing the influence of precipitates on the CRSS or yield strength of aged alloys. Recent progress in understanding the statistics of dislocation-precipitate interactions is highlighted. It is shown that Pythagorean superposition for strengthening by random mixtures of localized obstacles of different strengths is rigorously obeyed in the limit of very weak obstacles; this had been known previously as a result of computer simulation experiments. Some experimental data are discussed in light of this prediction. All of the currently viable mechanisms of precipitation hardening are reviewed. It is demonstrated that all versions of the theory of coherency hardening are woefully inadequate, while the theory of order hardening is capable of accurately predicting the contribution of 3/' precipitates to the CRSS of aged Ni-A1 alloys. It is also convincingly shown that a new theory based on computer simulation experiments of the motion of dislocations through arrays of obstacles having a finite range of interaction cannot explain these same data, and is of doubtful validity in other instances for which its success has been proclaimed. A new theory of hardening by spinodal decomposition is proposed. It is based on the statistics of interaction between dislocations and diffuse attractive obstacles, and is shown to be in very good quantitative agreement with much of the limited data available. Some of the problems that remain to be addressed and solved are discussed.

I.

INTRODUCTION

A N alloy is said to be precipitation hardenable when its hardness or yield strength increases with time at a constant temperature (the aging temperature), after rapid cooling from a much higher temperature (the solution treatment temperature). The phenomenon of precipitation hardening was discovered in the first decade of this century by Wilm,l who observed that the hardness of aluminum alloys containing small amounts of Cu, Mg, Si, and Fe increased with time at room temperature, after having been quenched from a temperature slightly below its melting point. The first plausible explanation for this "aging" process was presented by Merica e t a l . 2 who postulated that age hardening occurred in alloys in which the solid solubility increased with increasing temperature, thereby enabling a new phase to form at the lower temperature by precipitation from an initially supersaturated solid solution. Wilm's discovery and the explanation of Merica e t a l . gave birth to an entirely new area of research in physical metallurgy. The main focus of the research throughout the decades of the 20's and 30's was with the mechanism of precipitation, or aging, rather than with the mechanism of strengthening p e r s e . Unraveling the mysterious nature of the decomposition process was quite an undertaking, because the precipitates were too small to be observed directly using instrumentation of that epoch. A history of the progression of thought on the mechanism of precipitation from solid solution through the late 30's is contained in a comprehensive review article by Mehl and Jetter. 3 It is perhaps a measure of the preoccupation with the mechanism of precipitation, rather than concern for the manner in which precipitates strengthen the matrix in which they reside, that the article by Mehl and Jetter makes no reference to the A.J. ARDELL is Professor and Chairman, Department of Materials Science and Engineering, University of California, Los Angeles, CA 90024. This paper is based on a presentation made at the symposium "50th Anniversary of the Introduction of Dislocations" held at the fall meeting of the TMS-AIME in Detroit, Michigan in October 1984 under the TMSAIME Mechanical Metallurgy and Physical Metallurgy Committees. METALLURGICAL TRANSACTIONS A

possible role of dislocations, which had been discovered in 1934 by Orowan, 4 Taylor, 5 and Polanyi. 6 In fact, no mention is made of dislocations in any of the papers in the symposium volume in which the paper of Mehl and Jetter is published. The earliest attempted explanation of age hardening using the concept of the dislocation was that of Mott and Nabarro,7 who suggested that strengthening arose through the interaction between dislocations and the internal stresses produced by misfitting coherent precipitates. Several years later Orowan 8 derived the famous equation relating the strength of an alloy containing hard particles to the ratio of the shear modulus of the dislocation and the average planar spacing of the particles. While the theory of Mott and Nabarro has been supplanted by more detailed theories of strengthening by coherent misfitting precipitates, the Orowan equation stands as a landmark achievement. It has been refined, to be sure, but it still serves as a basis for the theory of dispersion strengthening, or strengthening of alloys by nondeformable particles. Precipitation hardening, on the other hand, involves the strengthening of alloys by coherent precipitates which are capable of being sheared by dislocations. This was recognized by Mott and Nabarro, although progress in the understanding of the details of this process was relatively slow in coming throughout the 40's and 50's. For example, a review article on age hardening published by Smith9 in 1950 was devoted entirely to the (then) recent progress in the understanding of precipitation phenomena and made no reference whatsoever to the work of Mott and N a b a r r o 7 o r Orowan. 8 By the late 50's the emphasis shifted from research on precipitation mechanisms to exploratory studies of strengthening mechanisms. The early attempts at formulating theories of precipitation hardening were admirably reviewed in the classic treatise on precipitation hardening by Kelly and Nicholson. 10 They devoted nearly a third of their article to discussions of the mechanical behavior of precipitation hardened alloys, including theories of the yield strength. Thus, it took nearly 50 years to become comfortable at last with the complexities of the processes of precipitation from VOLUME 16A, DECEMBER 1985--2131

supersaturated solid solution. The understanding of this aspect of age hardening, while by no means complete to this day, was measurably accelerated by the application of transmission electron microscopy to the examination of the microstructures of aged alloys. The profound impact of this technique is beautifully illustrated in the review article by Kelly and Nicholson. Impressive contributions to the understanding of the mechanisms of precipitation hardening were made throughout the 60's. It was in this decade that the first quantitative theories of coherency strengthening, order strengthening, modulus hardening, and stacking-fault strengthening were proposed, and the magnitude of strengthening produced by spinodat decomposition was estimated. Most of these theories incorporated new ideas concerning the statistics of dislocation-precipitate interactions. Furthermore, computer simulation of the interaction between dislocations and large arrays of point obstacles was attempted for the first time. These advances in understanding of the mechanisms of precipitation hardening were thoroughly discussed in the seminal review article by Brown and Ham.~l Their paper was devoted entirely to the manner in which dislocations interact with precipitates, with minimal reference to the mechanisms of precipitation. In this paper we shall concentrate on the progress made over the past dozen years or so in understanding the statistics of dislocation-particle interactions and the mechanisms of age hardening. Overlap with the review article of Brown and Ham is regrettably unavoidable because the necessary groundwork must be laid. It turns out that while there has been considerable progress, our understanding is still in many ways imperfect. It is well known that the age hardening of metals and alloys can have profound effects on ductility, fracture toughness, susceptibility to stress corrosion cracking, etc. However, due to the voluminous literature on these topics, this paper will focus on the yield strength of age hardened polycrystalline material or the critical resolved shear stress (CRSS) of single crystals. The literature abounds with the results of experimental investigations of age hardening. Not many of these are suitable for testing the various theories of precipitation hardening that have become more-or-less accepted over the past decade. The data discussed herein have been selected with precisely that objective. II. T H E STATISTICS OF DISLOCATION-PARTICLE INTERACTIONS In this section we shall consider, from a phenomenological viewpoint, the manner in which a dislocation of Burgers vector b and line tension F interacts with obstacles in its glide plane. The problem is to determine how the dislocation, which moves forward under the influence of an applied shear stress, % overcomes the obstacles which impede its motion. Point obstacles can be overcome with the assistance of thermal activation, which is important in solidsolution strengthening, or by inertial motion of the dislocation, which can influence internal friction behavior. Ordinarily, neither of these processes is effectual in precipitation hardening. In the parlance of solid solution strengthening and internal friction, the results of this section describe the overdamped motion of dislocations through an array of penetrable point obstacles at 0 K. 2132--VOLUME 16A, DECEMBER 1985

If there are ns obstacles per unit area in the glide plane, a convenient measure of the spacing between them is the so-called square lattice spacing, Ls, defined by Ls = ns ~/~.

[1]

It is useful to keep in mind that Ls is not equal to other characteristic interparticle spacings, such as the average planar spacing between an obstacle and its nearest neighbor (which is Ls/2). Nor is it equal to the average spacing, L, of the obstacles along the dislocation line as it traverses the glide plane. This average spacing is, instead, a quantity that remains to be determined. The problem of dislocation-particle interactions can be subdivided into several categories, depending upon whether the obstacles can be considered as point obstacles, or whether they are finite in size. Furthermore, following Nabarro, 12the obstacles can be regarded as either localized or diffuse. Localized obstacles are those that interact with dislocations only when they are in direct physical contact. Diffuse obstacles, on the other hand, can interact with dislocations at a distance, characterized by a range, w, and an interaction energy, U0. Diffuse obstacles are repulsive when U0 > 0 and attractive when U0 < 0.

A. Localized Obstacles A dislocation interacting with point obstacles randomly distributed in its slip plane is depicted schematically in Figure 1. To characterize the mechanics of the interaction process it is necessary to describe the strength of the obstacle. For simplicity, it will be assumed that all the obstacles in the glide plane have identical strengths. In reality, the obstacle strengths will be distributed, but we will deal with this more realistic situation later on. When the dislocation encounters the obstacle it will bow out under the influence of the applied stress into an arc of constant radius R if F is independent of the character of the dislocation. Some of the consequences of a nonconstant F will also be examined later, but for now we shall use the constant line tension approximation because it is a very good approximation so long as the obstacles are relatively weak. The dislocation will bow out until the obstacle can no longer sustain the force acting upon it, whereupon the dislocation will break free and move until it encounters a new obstacle. The maximum force that the obstacle can sustain, F,,, is determined by the force balance depicted in Figure 1, which shows the dislocation configuration at the critical breaking angle, q'c. From Figure 1 it is apparent that the relationship

Fig. 1 - Schematic illustration of the penetration of a random array of point obstacles by a dislocation. METALLURGICAL TRANSACTIONS A

among F,,, F, and ~bc is

given by

F m / 2 r = cos(q~/2).

[2]

It is often convenient to describe the mechanical equilibrium in terms of the complementary angle Oc (see Figure 1) and the parameter/3c, which are related by /3c = F,,,/2F = sin(0~/2),

[3]

which, on substitution into Eq. [6] produces the result ~'c = 2F/33'2/bLs.

thereby producing the simple result

[4]

~'* = /33/2 .

Because the effective obstacle spacing, L, and Rc are related by sin(0~/2) = L / 2 R c ,

[5]

we have, on substituting Eqs. [3] and [4] into Eq. [5] 2F rc = ~-~ sin(0d2) = 2 F / 3 J b L .

[6]

In order to solve the problem completely it is necessary to determine how L depends upon the applied stress. The earliest approach to solving it was taken by Friedel, 13 who defined the geometry used in the statistical theory in a treatment of the penetration of forest dislocations by a glide dislocation during creep. The argument is that when the dislocation breaks free from the obstacle pinning it, it will bow out to a new length (its radius of curvature will remain unchanged so long as r does not change), and at the same time will encounter one new obstacle. Thus, as noted by Brown and Ham,l~ a steady state unzipping is implied, during which the average area swept out each time the dislocation breaks free from the obstacle pinning it contains one newly encountered obstacle. What this means in physical terms is that as the obstacle strength increases, the dislocation is forced to bow more, thereby increasing its length and the probability that it will encounter a new obstacle. The geometry of this process is idealized in Figure 2. A mathematical statement of the steady state unzipping condition was derived at about the same time by Friedel ~4 and Fleischer and Hibbard. ~5 It is equivalent to the expression Sens = 1,

[7]

where SF is the area swept out by the dislocation after breaking free of the obstacle. From the geometry of Figure 2 we find for small 0~ (= 2/3~) the equation 4R~/33~/L~ = 1,

[8]

where use has been made of Eq. [1]. Utilizing Eq. [5], and substituting the result into Eq. [8], we obtain a relationship between the effective obstacle spacing L = LF, and L~ /

/ / 2 fie

Fig. 2 - - I l l u s t r a t i n g the area, SF, swept out in the basic geometry of Friedel statistics. 13

METALLURGICAL TRANSACTIONS A

[10]

It is frequently convenient to express Eq. [10] in dimensionless form, using a dimensionless stress, ~'*, defined by

where 0~ = 7r - ~O~.The critical stress, ~'~, required for the dislocation to break free of the obstacle is related to the radius, R~, of the dislocation by 'rc = F / b 8 ~ .

[9]

L = LF = Ls//3~ '2 ,

,r* = 'rcbL,/2F ,

[11] [12]

It is emphasized that Eqs. [10] and [12] are valid only when/3c is small. In fact, it is impossible to satisfy Eq. [7] for values of/3r > 0.5 (values of 0c > 60 deg), because when/3c = 0.5 the dislocation, on its release from the central obstacle in Figure 2, becomes semicircular between the obstacles flanking it. In keeping with what has become more or less accepted terminology, we shall refer to the statistics of the unzipping process just described as Friedel statistics. Other analytical theories of the statistics of the unzipping process 16-19 all agree with Eqs. [9] and [10] or [12] in the limit of small tic. Significant advances in the understanding of the statistics of penetration of a dislocation through an array of point obstacles were made through the advent of simulation methods. The first of these was a graphical method used by Kocks. 2~ This was followed shortly by the first research to utilize computer simulation, in which Foreman and Makin 21 studied the movement of a dislocation of constant F through arrays of up to 10,000 obstacles of identical strength. In the computer simulation experiments of Foreman and Makin the CRSS is defined as the value of r* at which the dislocation penetrates the array in its entirety without the need for thermal activation. These simulation studies showed that Eq. [12] was quite accurate even at values of /3c approaching 0.6 (at which the geometry in Figure 2 cannot be satisfied). However, at larger values of/3c the deviations from Eq. [12] are substantial, and T* approaches the limiting value (at tic = 1) ~-* = 0.81,

[13]

which is a simple, but useful and widely accepted, form of the Orowan equation. 8 Foreman and Makin showed that at values of/3~ approaching unity the dislocation penetrates the array of strong obstacles through paths of easy movement. While the dislocation remains nearly straight when /3~ is small, its shape deviates considerably on average from that of a straight line when/3c is large. New computer simulation studies were carried out by Morris and Klahn, 22 who extended the work of Foreman and Makin using a different algorithm to study the distribution of obstacle strengths and the distribution of obstacle spacings along the dislocation line in the critical configuration (i.e., at values of the critical stress), as well as the dependence of r* on/3c. Their results, including additional results of computer simulation studies reported b y Hanson and Morris, 23 are shown in Figure 3, along with the results of Foreman and Makin. It is seen that the data on computer simulations are all in substantial agreement, and that all the data agree well with Eq. [12] at small values of/3~ (<0.5). VOLUME 16A, DECEMBER 1985--2133

1.00 1.0

0.97

0.9

0.8 0.7 0.6

0.4

0.2

0

0.9 0.96

t

9 FOREMAN AND MAKIN

0.8 0.7

-

\\ N

0.92

0.6 0.5

-

0.88

0.4 0.3 0.2

:

x,

0.84

0.80

9 FOREMAN AND MAKIN 0.1

I 0

0.0

I 0.2

1 0.4

I

I 0.6

[

I 0.8

I

I 1.0

/~c 2 0

20

40

60

80

100

120

140 160 180

% Fig. 3 - - T h e data of Foreman and Makin 2~ (10,000 obstacles) and Hanson and Morris 23 generated by computer simulation. The solid curve was calculated from Eq. [14] and the dashed curve from Eq. [12].

Morris and Klahn z2 and Hanson and M o r r i s , 23 however, found that the relationship between L and /3c predicted by Eq. [9] was not accurately obeyed, the theoretical values invariably exceeding those generated by computer simulation. If the data in Figure 3 are evaluated on a purely empirical basis it is found that even Eq. [12] is not obeyed precisely in the limit of small tic. In particular, if we plot these data in the form z c,/,~3/2. / p c vs_ tic,2 as is done in Figure 4, we see that a nearly linear relationship is observed, which is valid to values of /3c approximately equal to 0.9. The data in Figure 4 are adequately represented by the equation r* = 0.956fl3/2(1 - fl2/8)

[14]

over nearly the entire range of tic, the only serious deviation occurring for 0.9 < tic < 1 (even here the disagreement is less than 5 pct). It is emphasized that Eq. [14] is only an empirical representation of the data generated by computer simulation, and is intended primarily to highlight the observation that Eq. [12] slightly overestimates the experimental results, the discrepancy increasing as/3c increases. A more meaningful representation of the data in Figure 3 at large values of tic (small values of ~0~)is that suggested by Brown and Ham, n namely r~* = 0.81/3~,

[15]

which is accurate over the range of 0.5
2134--VOLUME 16A, DECEMBER 1985

Fig. 4 - - T h e data of Foreman and Makin 21 and Hanson and M o r r i s 23 replotted as "t*/fl~ 2 vs fl~, illustrating that Eq. [12] overestimates the CRSS at all values of tic. One data point from the work of Hanson and Morris has been omitted (the one at the smallest values of ~'* and tic).

The results of the computer simulation experiments clearly predict that the average spacing of the point obstacles along the dislocation line decreases as the strength of the obstacles increases, which is consistent with the physical picture represented by Eq. [9]. Though the average spacing of the obstacles is smaller than that predicted by Eq. [9], the resulting flow stress is only marginally smaller than that predicted by Eq. [12]. In the limit of strong obstacles the effective obstacle spacing becomes nearly independent of the obstacle strength and approaches the value L = 1.25 Ls. This is a simple manifestation of the fact that the probability that the dislocation interacts with an increasing number of obstacles as Rc gets smaller cannot continue to increase indefinitely. The computer simulation studies of Morris and Klahn 22 were followed by a new analytical approach, conceived by Hanson and Morris, 23 to the statistical problem of dislocation-particle interactions. We start with an array of point obstacles randomly distributed in the glide plane and imagine that the dislocation line, such as that seen in Figure 1, is part of a longer line that started at the left-hand boundary of the array. The line consists of circular segments of radius R and can be generated by "circle-rolling" from the left-hand boundary. Circle rolling involves pivoting a circle of radius R in an anti-clockwise direction around one of the points, or obstacles, on its circumference. Suppose that the dislocation has arrived at point k in Figure 5 by this process. The segments connecting k, k - 1, k - 2, etc. are stable because the stress, which determines R, is smaller than that required to satisfy Eq. [6]. The continuation of the dislocation line to the right of point k involves a search for new stable segments. The statistical theory of this search process was developed in approximate fashion by Hanson a n d M o r r i s . 23

METALLURGICAL TRANSACTIONS A

I

0 = 0r REGION

T :

0<4'<0 e

~< REGION

0<0 e

"IT

+

:

Oc - rr < c~ < O

0<0 REGION

<0 e

TIT :

- r , - < 4, < O c - ~-

0 < 0 < 4'-~r Fig. 5--111ustrating the search geometry used in the circle-rolling procedure of Hanson and Morris. 2a The shaded area, So, contains, on average, one obstacle at a stress corresponding to the CRSS.

The search geometry used by Hanson and Morris is also shown in Figure 5. If we ignore for the moment the fact the obstacle labeled k + 1 is present in the area labeled Sc in Figure 5, it is clear that the dislocation (circle) can be pivoted around point k without encountering another obstacle, and that the configuration will remain stable until the arc is rotated through angle G. At this point the critical breaking angle is reached and the dislocation will break free from obstacle k. In sweeping through angle 0c, the leading semicircle of the dislocation sweeps out the entire shaded area illustrated in Figure 6. Any point contained within this area, S, represents a possible point of interception of the dislocation which can give birth to a new stable segment of the dislocation line. One of the constraints of the problem, however, is that the dislocation must remain quasi-straight. This excludes points in the lower part of the search area in Figure 6, such as the point labeled k' + 1, because continuation of the dislocation from point k' + 1 to a point to the right of k' + 1 will cause the dislocation to deviate significantly from the quasi-straight line configuration. The requirement of quasistraightness therefore excludes part of the overall search area, S, generated by circle rolling to the much smaller area, So. Another way of looking at the quasi-straightness condition is as follows. Suppose point k ' + 1 were, in fact, the next point encountered. We continue the circle-rolling process, but at the same time demand that the new segment generated by pivoting the semicircle around k' + 1 produces an intersection that enables the dislocation to remain quasi-straight. It is practically impossible to do this, though, without exceeding the strength of obstacle k' + 1. Hanson and Morris determine Sc in the following way. The coordinates of points in S are prescribed by a coordinate system 0, 6. The coordinate 0 is a point on the leading semicircle of the dislocation line (the locus of points 0 = 0 defines the initial position of the semicircle), while 4 is a point on the trailing semicircle (the initial position of this line is the locus of points 4 = - I t ) . The advantage of utilizing this coordinate system is that 4 describes the angle between the tangential directions of the dislocation segments at successive obstacles. The condition of quasistraightness therefore is expressed by the equation (6) = 0. METALLURGICAL TRANSACTIONS A

[16]

Fig. 6--Illustrating the entire area, S, that can be generated by circlerolling a dislocation about an obstacle until the breaking strength, 0~, of that obstacle is reached. Regions I, II, and III are defined by the angles 0, ~b shown. The boundaries of these regions define the limits of the integrals in Eq. [21]. The heavily-outlined area has the same significance as the shaded area in Fig. 5.

In terms of the coordinates 0, 4 an element of area dS is given by dS = R 2 sin(0 - 6) dO d 4

[17]

which can be written in dimensionless form, utilizing Eq. [11], as dS* = (R*) 2 sin(0 - 6) dO d 4

[18]

where S* = S / L 2 = Sn~,

[19]

and R* = 1 / 2 ~ .

[20]

An alternative way of looking at Eq. [19] is that it is a measure of the number of points within S from which stable dislocation segments can be generated, enabling continuation of the dislocation line toward the right-hand side of the array. Although it is not stated in quite this way, the subarea from which allowable points are chosen, consistent with Eq. [ 16], is such that on average the dimensionless area defined by Eq. [ 19] contains one point. This places a lower limit on the value of 6, namely - 40, such that the equation Sc/L~ = S*~ = (R*)2 f f sin(0 - 6) dO d 4 = 1

[21]

is satisfied at a value of ~- corresponding to the CRSS (hence the subscript c). The limits on the integrals in Eq. [21] depend upon the region of integration, and are illustrated in Figure 6. In the theory of Hanson and Morris the area S generally contains n obstacles, such that S* = n, and each obstacle represents a neighboring point on a stable dislocation segment extending from obstacle k. If the stress is small, R is large and so is n; i.e., a large number of stable segments can originate at k. As T increases R decreases and the number of stable segments decreases. If ~" becomes too big R becomes so small that no stable segments can be found. The VOLUME 16A, DECEMBER 1985--2135

transition between stability and instability defines the CRSS, ~'*, and Hanson and Morris use the statistical theory of branching processes to demonstrate that the condition (n) = S* = 1, which is equivalent to Eq. [21], defines the most stable dislocation line. While the equations of the theory are developed for a circle-rolling procedure that progresses from left to right, the final results are independent of the initial conditions and the circle-rolling direction. For analytical purposes it is convenient to express Eq. [21] as (R*)2s0 = 1,

[22]

where so is a dimensionless radius-independent area. Using the appropriate limits on the integrals (Figure 6), Eqs. 116] and [21] yield the results, 24 So = 0~ + sin ~0 - sin(0~ + ~0),

[23]

and s0(~b) = 02/2 + 4~o[sin(0c + 4}o) - sin ~b0] - cos th0 + cos(0r + d%); ~b0 < 7r - 0,,. [24] These equations provide the necessary relationship between 0c and ~bo which is required to solve Eq. [22] for a-*. The numerically obtained solutions of these equations appear to underestimate slightly the results generated by computer simulation. They therefore provide semiquantitative rationalization for the discrepancy between these results and the prediction of Eq. [12] shown in Figure 4 and expressed in Eq. [14]. Equations [23] and [24] cannot be solved in closed form but, as Melander24 has shown, they reduce in the limit of small ~o and 0c (= 2/3c) to the equations So = 4/3~/3 + 2/3~b0 + /3Abg

[25]

0 = 2/3~ - 3flAb~- 2~b30.

[26]

and

The solution of Eq. [26] is ~b0 = 0.33880~ = 0.6776/3r which, on substitution into Eq. [22], taking into account Eq. [20], results in r* = 0.8871/33~2.

[27]

Hanson and Morris also derived explicit expressions for the average distance, (L), between pinning points along the dislocation line in the limiting configuration, and for the distribution of forces on the obstacles pinning the dislocation in the limiting configuration. For small values of 0c they show that Eq. [9] is replaced by (L) = 0.764L~//3~/2

[28]

Labusch25 has reconsidered the search probabilities in the theory of Hanson and Morris from a statistical mechanics lpproach, and has attempted to account for degeneracies ssociated with the generation of stable dislocation segents using the theory of branching processes. He obtained corrected value of the coefficient relating ~'* and 133/2, pducing the result r* = 0.949/3~ '2.

[291

numerical constant in Eq. [29] is very close to the value red empirically from Figure 4 in the limit/3c ~ 0 (see [/OLUME 16A, DECEMBER 1985

Eq. [14]). In addition, Labusch also derived equations for the distribution of the distances between pinning points along the dislocation line which are in better agreement with the results of computer simulation experiments than the equation derived by Hanson and Morris. Labusch also derived an equation for the distribution of forces acting on the pinning points which is in excellent agreement with the results of the computer simulation experiments. The advances in the theory of the statistics of dislocationpoint obstacle interactions, greatly assisted by the results of the computer simulation experiments, provide significant insight into the reason why Eq. [7] yields such a useful result. The simple geometry used in Figure 2 is, in a very real sense, justified by Eq. [16]. The arrangement of relatively weak obstacles in a straight line for the purpose of calculating the area swept out when the dislocation breaks free from the central obstacle is an alternative statement of the quasi-straightness condition. Equation [7] is, in fact, a variation of Eq. [21 ], although the geometry and the process of encountering the new obstacle are rather different. The analytical theory also predicts quite generally that the CRSS falls slightly below the value predicted by Eq. [12], although it is unable to explain the systematic departure displayed in Figure 4 and expressed by Eq. [14]. Either Eq. [29] or the empirical result, Eq. [14], provides the best estimate of strengthening by point obstacles at values of/3c below approximately 0.7. As the Orowan limit is approached, Eq. [15] provides the most reasonable estimate of r*. B. Diffuse Obstacles

The interaction between a dislocation and diffuse obstacles was first considered by Mott; 26Mott's treatment was, in fact, one of the earliest to recognize the role of dislocation flexibility in the determination of the spacing of obstacles along the dislocation line. Mott dealt with a single dislocation interacting with attractive obstacles (Uo < 0) at zero applied stress. The interaction causes the dislocation to assume a zig-zag configuration, with an average amplitude hi2. The geometry is depicted schematically in Figure 7. The effective obstacle spacing along the dislocation line (known now as the Mott spacing), L~, is determined partly by the condition that the area defined by the rectangle LMh contains on average one obstacle, i.e. n~LMh = 1.

[30]

A relation between h and the other parameters of the problem, mainly Uo and F, is found by minimizing the total energy of the dislocation line in the zig-zag array of obstacles. This results in the expression for h given by h 3 = Uo/2n~F,

[31]

which on utilizing Eqs. [1] and [30] produces the result LM = (2FL~/Uo) ~'3.

[321

We note that Eq. [301, which obtains for the case of socalled Mott statistics, is analogous to Eq. [7]. Under the influence of an applied stress the dislocation cannot break free of the obstacle until a maximum force F,, is applied, at which point we can write Uo = F,,to.

[33]

METALLURGICAL TRANSACTIONS A

To calculate rc we utilize Eqs. [31, [32], and [33], which in conjunction with Eq. [6] yields the result ~'c = 2Fw911304/3/t.r oc l o l - , ,1./3.

[34]

Defining a dimensionless range w* =-- ~o/L,, Eq. [34] can be expressed in dimensionless form, using Eq. [11], as r* = (oJ*)v3fl 4'3,

[351

which is the counterpart of Eq. [12] for the regime of Mort statistics. We note that the numerical coefficients in Eqs. [32] and [34] differ somewhat from previous treatmerits, H'26 but agree with the estimate given by Friedel, 14 who used the same geometry shown in Figure 7. The original treatment of Mott 26 has been reconsidered in more recent papers by Riddhagni and Asimow, z7 Labusch, 28'29and, most recently, Schindlmayr and Schlipf? ~ The theories of Labusch 2s'29 and Schindlmayr and Schlipf3~ differ in several respects. The principal equations of Labusch's theory are derived from consideration of the forces acting on a dislocation line by a distribution of neighboring obstacles in its slip plane. The theory of Schindlmayr and Schlipf, on the other hand, is, like the original theory of Mort, based on minimization of the energy of the dislocation in a zig-zag configuration. Their theory, however, is applicable to repulsive as well as attractive diffuse obstacles. In both theories the behavior of the solutions for the CRSS is governed by a parameter 770, defined by O)

r/0 = ~ ( 2 F / F ~ ) v~ = o)*/fl~ ~2.

[36]

In Labusch's papers this parameter appears in the form r/o2/2, while in the theory of Schindlmayr and Schlipf it appears as r/o z. The two most recent theories 29'3~both predict that as r/0 becomes large, i.e., in the regime of large ~o* or small fl~, the flow stress is governed by Mott statistics, Eq. [34] prescribing the value of r~. In the limit of small oJ*, when the obstacles are highly localized, the dislocationparticle interactions are governed by Friedel statistics and r~ is given by Eq. [10]. Whether or not Mort statistics can ever apply to a real situation is evidently debatable. The discussions by Brown and Ham 11 and NabarrC 2 are prime examples of the concerns expressed. At issue is whether the CRSS is governed by new obstacles as the dislocation bows out from its pinning points (Friedel statistics) or whether it is governed by

release of the dislocation from the obstacles attracting it under zero applied stress (Mott statistics). As noted by Nabarro ~z the precise limits on r/o for which Eqs. [10] and [34] can be expected to apply have not yet been determined. The definition of "diffuse" appears to be crucial in this context. If the obstacles can interact with a dislocation over a range comparable with Ls(~o* = 1), the encounter statistics descriptive of the Friedel process become less meaningful because the dislocation will interact with a new obstacle well in advance of clearing one that happens to be pinning it. The analytical theories of Labusch zs'29 and Schindlmayr and Schlipf3~ suggest that r/0 "> 1 is the regime of Mott statistics, while r/0 < 1 is the regime of Friedel statistics. For reasons that will become clearer later, it is suggested here that for Mott statistics to be valid we must also have ~o* >> 1 (according to Eq. [36] r/0 can become quite large for o9" ~ 1 in the limit tic --~ O, i.e., as the obstacles become very weak). While perhaps mathematically imprecise, these criteria are easy to visualize on a microstructural level, and are consistent with the idea 12that obstacles are localized when a dislocation encounters increasing numbers of them as it bows out under the influence of an applied stress, but diffuse when the dislocation interacts with the same obstacles at the CRSS that it interacted with at zero applied stress. These arguments are also consistent with the criteria suggested by Kocks et al. 31 The only example to date of computer simulation of the motion of a dislocation through an array of diffuse obstacles is that of Schwarz and Labusch. 32 They set up a dynamic equation of motion of the dislocation and, through a suitable normalization of the spatial coordinates, expressed their results in terms of a reduced stress ~'** defined by ,./.c~~

~ 3f2 9 = ~'c/tic

[37]

They find that for so-called "overdamped" dislocation motion (i.e., dislocation motion in which inertial effects are unimportant) ~'** is a unique function of rio. The nature of the solution depends upon the interaction profile between the obstacle and the dislocation. Schwarz and Labusch considered two types of force vs distance profiles, one of which is symmetrical and the other asymmetrical. The symmetrical profile was chosen to be representative of what they refer to as energy conserving interactions (insofar as energy in neither gained nor lost after the dislocation has completely bypassed the obstacle). The asymmetrical profile represents energy storing interactions (in which the energy gained, for example, by passage of the dislocation through the repulsive part of the interaction is stored because the attractive part of the interaction is nonexistent). Their results are reproduced in Figure 8. For the energy conserving interaction profile Schwarz and Labusch find that the data on computer simulation can be represented by the empirical equation r** = 0.94(1 + CsLr/o) ll3,

[38a]

while for the energy storing interaction profile the data are nearly linear over the entire range of r/0, and are consistent with the equation r** = 0.94(1 + C~Lr/o). Fig. 7 - - Illustrating the basic geometry of Mott statistics.56 The dislocation zig-zags among obstacles which interact with it through a range w.

METALLURGICAL TRANSACTIONS A

[38b]

In Eqs. [38] CsL ~-- 2.5 and C~L = 0.7 are empirical constants. At small values of r/0(<0.4) the behavior for both VOLUME 16A, DECEMBER 1985--2137

proaches zero only as co itself approaches zero while /3~ remains small; the theory cannot be expected to apply for strong obstacles of vanishingly small range of interaction.

2,0

1.8

[]

C. M i x t u r e s o f Obstacles o f D i s t i n c t Strengths

1~

1.6

[] *~

o

1.4

t

3

~

-~

o

1.2

1.0

0.0

,.~

0 ENERGY CONSERVING [] ENERGY STORING

0.2

0.4

0.6

0,8

1.0

1.2

"qo Fig. 8 - - T h e results of the computer experiments of Schwarz and Labusch 32on the movement of a dislocation through arrays of either energy conserving or energy storing obstacles when inertial effects on the dislocation motion are negligible.

types of obstacles is identical. Equation [38a] predicts that as r/0 becomes large (>>1) the regime of Mort statistics is approached, while Friedel statistics are obeyed when rl0 '~ I. Except for numerical factors, Eqs. [10] and [34] are recovered in the two limits. While the results of Schwarz and Labusch appear to provide, in a convincing manner, a continuous connection between the regimes of Mott and Friedel statistics, through the behavior of parameter r/0, it is easy to overlook the fact that their arguments are valid only for small values of/3c. The reason for this becomes clear when we examine the equation of motion they used. In their expression for the contribution to the force on the dislocation due to its line tension, they write the radius of curvature of the dislocation as R--I = d 2 y / d x z '

where y is the displacement of the dislocation line lying initially along the x direction. The exact expression for the radius of curvature for the dislocation is d2y/dx 2

[1 + (dy/dx)2] 3'2" So long as the dislocation line remains nearly straight we are justified in ignoring the denominator in this expression, 3ut when the obstacles are reasonably strong the con-ibution of this term to the radius of the dislocation starts to :come important. The maximum value of d y / d x is fl,, so at for values of/3~ > 0.3 (approximately), this factor will rt to influence the results of the calculations. For these ~nger particles the solution to the equations of the theory chwarz and Labusch no longer depends uniquely on r/0, must depend upon fl~ as well. We thus conclude that the ry of Schwarz and Labusch will be valid as "00 apVOLUME 16A, DECEMBER 1985

In any real precipitation hardened alloy the precipitates are never monodisperse because there is always a statistical distribution of particle sizes. Even if there were only a single particle size, there would still be a distribution of particle sizes in the slip plane of a dislocation, since the slip plane will intersect a particle of given shape at different positions, resulting in a distribution of cross-sectional areas in the glide plane itself. In recognition of this, it becomes important to estimate how a distribution of obstacle strengths affects the CRSS, anticipating that the cross-section of a particle in the slip plane is related to the resistance of that particle to the motion of a dislocation. An important related problem is how the strengths of the matrix and precipitate add to produce the strength of the alloy, This latter problem is not of concern in computer simulation experiments because there the CRSS of the matrix is zero. Several relationships have been proposed to account for the contributions to the CRSS of distinct obstacles of two or more strengths. Consider two types of obstacles of strengths /3cj and/3c2, and assume that when obstacles of strength fl~j are present by themselves they produce an increment in the strength of the material given by r,-1; a similar argument holds for obstacles of strength/3c2. We will ignore for the moment how to account for the flow stress of the matrix and assume that it is zero. The most common expressions for the flow stress for two types of obstacles are those discussed by Brown and Ham, ~ namely zc = rcj + r,,2,

[39a1

rc = r, jX~t '2 + rc2X~~2,

[39c]

where X~ and X2 are defined by the relationships Xj = n s t / n ,

X2 = n~2/ns = 1 - Xt

[40]

and n~ and n,2 are the number of obstacles per unit area of types one and two, respectively. Utilizing Eqs. II] and [11] the addition rules, Eqs. [39] become, respectively 9 vl,2

[41a]

(~.,)z = (r,)2Xj + (rcz)-X,

141b]

"/'c* = "gc~Xl -{- T , ~ X 2 .

[41c]

Of the three choices, the first (linear superposition) represents the simplest assumption one can make, the second is the so-called Pythagorean addition rule first proposed by Koppenaal and Kuhlmann-Wilsdorf," and the third is a law of mixtures. Other addition rules have been proposed. The theory of Labusch 28 suggests a superposition law of the type r /3;'~' =

3,.? ~',.i+ r i3;~; .

142]

This equation would appear to be restricted to mixtures of two different types of diffuse obstacles because it is in consideration of hardening by these types of obstacles that Eq. [42] was derived. Most recently Biittner and Nembach 34 METALLURGICAL TRANSACTIONS A

and Neite et al. 35 have proposed an addition rule of the type

1.0 ['-

r q = ~'ql + rq2,

o 9 L-

[43]

where q is an adjustable parameter. The principal justification for the use of one or another of these addition rules has been the extent to which they agree with the data generated by computer simulation. In particular, Foreman and Makin 36considered various combinations of ~'~ and r ~ over a range of values of X] and X 2 in their computer simulation experiments, and also investigated the effect of a square spectrum obstacle strength on the CRSS. They observed that for the weakest combinations Eq. [39b] (or [41b]) best represented their data for all possible combinations of XI and X2. The effect of a random mixture of distinct point obstacles was investigated theoretically by Hanson and M o r r i s , 37 who derived the results

-

0.2

-

A

0.809

0.260

O. 1

-

I'-I

0.530

0.260

0

1

(7*) 2 = ~- ~ Xisoi,

0

[44]

I 0.1

I 0.2

I 0.3

( rc~ )2X I + ( rc2 )2X 2 .....

I 0.4

l 0.5

:1 X1 + TC*2 X2

I 0.6

I 0.7

I 0.8

I 0.9

I 1.0

i"

X2

and 0 = ~ X, so,(49,),

[451

i

where Soi and (~bi) are given by Eqs. [23] and [24]. On substituting the results valid for small flci and ~boi given by Eqs. [25] and [26] into Eq. [44], we obtain the result (~.,)2 = ~:X~(~.,)2,

[46]

i

which is the generalization of Eq. [41b]. Thus the theory of Hanson and M o r r i s 37 provides, for the first time, justification for the Pythagorean addition rule first proposed by Koppenaal and Kuhlmann-Wilsdorf. 33 This fact was recognized by Louat, 38 but was not explicitly stated by Hanson and Morris themselves. It is instructive to compare the results of the various superposition laws (Eqs. [39] and [41]) with the results of the computer simulation experiments of Foreman and Makin. 36 This is done in Figures 9 and 10. It is evident in Figure 9 that Pythagorean superposition clearly provides the best agreement between theory and experiment for the three different combinations of obstacle strengths used. The agreement, as noted by Foreman and Makin, 36 is far better for the combination of the two weakest obstacles than for the other two combinations. It is also apparent in Figure 9 that the linear superposition law is a very bad approximation to the total CRSS for most combinations of obstacle strengths and concentrations. Since dr*/dXz approaches ~ as X2 ~ 0 and -oo as X2 ~ 1, it should come as no surprise that Eq. [41a] is generally a poor approximation. Yet when there are small numbers of strong obstacles mixed in with many weak ones (Xz > 0.95, z* = 0.809, ~'& = 0.260), linear superposition produces better agreement than Pythagorean superposition, as noted by Brown and Ham. 1~ For the continuous distribution of obstacle strengths used by Foreman and Makin, only comparison with Eqs. [41b] and [41c] is convenient. In constructing the curves in Figure 10, Eqs. [41b] and [41c] were expressed in terms of

METALLURGICALTRANSACTIONS A

Fig. 9 - - Results of the computer experiments of Foreman and Makin 36for random mixtures of obstacles of two distinct strengths, compared with the three superposition laws shown. X2 represents the areal concentration of the weaker obstacles.

0.9

(~bmax § Cmin)/2

0.8( ~ / 8 - ' ~ ~ , ~ ~ ( 9 ~/4 u 0.7 3~/8

9

~./~min = 0

~./ o

O.E ~'12

_n

0

~~

~

9

0.5 ( 0.4 0.3

~,,./'-

'/'max =

~"

0.2( 0.1

W4

W2

3 ~'/4

rr

q'w Fig. 10-- Results of the computer simulation experiments of Foreman and Makin36 for a continuous rectangular distribution of obstacle strengths, compared with the predictions of Pythagorean superposition (solid curves) and a generalized law of mixtures (dashed curves).

VOLUME 16A, DECEMBER 1985--2139

the continuous variable 6~ and written as either

0"*)~= f~m0xg(q'3 [~*(~031~dq,~

[47a1

qJrain or

r* =

f

qJmax

g(q,c)r*(q*~) &Oc,

[47b1

q* rain

where the distribution function is itself expressed in the form g(~b<) = 1/(6max- /]/min) = 1/6w.

[481

The definitions of qJmi,, qJ.... and ~bw are identical to those used by Foreman and Makin. 36 For the calculations Eq. [14] was used for 0 < / 3 c < 0 . 8 8 1 8 0 d e g > 6 c > 56.715 deg), while Eq. [15] was used for 6c < 56.715 deg (at this value of/3c Eqs. [14] and [15] yield identical values of r*). With the distribution chosen by Foreman and Makin (Eq. [48]), the integral in Eq. [47a] can be evaluated in closed form, but the integral in Eq. [47b] can be evaluated only numerically. The two points emerging most clearly from Figure 10 are that Eq. [41b] is in much better agreement with the results of the computer simulation experiments than Eq. [41 c]. The effect is most noticeable at large values of 4'c (i.e., for the weaker obstacles), as pointed out by Foreman and Makin. Contrary to what is stated by Brown and Ham, the generalized law of mixtures (Eq. [41c]) is generally not in good agreement with the results of the computer simulation, except for narrow distributions of strong particles. In nearly all circumstances, and for all practical purposes, the representation of the distribution of obstacle strengths by the obstacle of average strength is adequately justified. The discrepancy between Pythagorean superposition and the results of the computer experiments observed when the obstacles are strong is not terribly large. For weaker obstacles the agreement is essentially perfect. The data in Figures 9 and 10 provide experimental confirmation of the predictions of the theory of Hanson and M o r r i s 37 expressed in Eq. [46]. Pythagorean superposition is the correct law in the limit of weak obstacles of distinct or continuously distributed strengths. For mixtures of strong, or strong plus weak, distinct obstacles it is an excellent approximation under nearly all circumstances. It is worth pointing out in this connection that Eq. [43], which in terms of r~ and r~ can be written as (Tc~) q = ('l'cCr

Jr- (Tcetc2)qxq2/2 ,

[49]

provides excellent agreement with the data in Figure 9 if the value of q is chosen appropriately. For the combination %* = 0.809 and r~* = 0.530 q = 1.75, for re* = 0.809 and r~* = 0.260 q - 1.4, while for rc* = 0.530 and r~* = 0.260 q = 1.8. As the obstacle strength becomes smaller q clearly approaches 2, as expected theoretically. D. Effect of Finite Obstacle Size It has been customary in theoretical developments of precipitation hardening to regard precipitates of finite size as point obstacles, a practice which is justified so long as some reasonable measure of the average planar particle spacing is very large compared to the dimensions of the particles (or equivalently, when the volume fraction, f, is small). In 2140--VOLUME 16A, DECEMBER 1985

many circumstances this assumption is not justified, and it is necessary, therefore, to reconsider some of the previous equations in order to estimate the effect of finite particle size. It is useful to recall here the relationships, valid for finite particles, among the quantities Ls, the average particle radius, (r), the average planar radius, (rs), and the volume fraction, f. The appropriate relationships among these parameters are summarized below:* *There are some extremely long-standing misconceptions that are not easily disabused. Among these is the frequently used, but mistaken, relationship (rs)2 = ~3(r) 2. It is incorrect because (r~) :~ {rs)2.

rr(r2)/L 2 = f ;

[50a]

(rs) = 7r(r)/4;

[50b]

(r 2) = 32(rs)2/37r 2 .

[50c]

Substitution of Eq. [50c] into [50a] results in / 3 2 ~1/2 (27r) 1/2 L, = ~3-~f) ( r , ) = \ - ~ / (r).

[51]

We consider the simplest adaptation of the point particle approximation by allowing the configuration depicted in Figure 2 to be adjusted for the case of finite particles. Some of the possibilities are depicted in Figure 11. Figure 1l(a) represents the configuration for impenetrable particles which hug the boundary of the precipitates and cannot cut through them until the critical breaking angle is reached; this situation is akin to that which obtains during coherency hardening, although it is not quite equivalent. Figure 11(b) depicts the geometry expected when the dislocation is able to penetrate the particle and reaches a critical configuration when the arc of the dislocation line inside the particle subtends the planar diameter; this is the critical configuration in the problem of order strengthening. Figure 11(c) illustrates the geometry expected when the dislocation experiences its maximum resisting force after having penetrated the entire particle; this resembles somewhat a situation that can occur during stacking-fault strengthening. In each of the three cases depicted in Figure l 1 the calculation of SF is algebraically messy, but not too difficult. The result for Figure ll(a) leads, for small tic, to the equation (r*)-' = fl73a _ 0.814fl/2.

[521

On comparison with Eq. [ 10], we see that the flow stress for finite obstacles is larger than that for point obstacles of identical breaking strength, the extent of the increase in strength increasing with increasing f. The geometry in Figure l l(a) was chosen deliberately because the type of dislocation-precipitate interaction depicted is identical to that treated by Melander, 39 who investigated the effect of finite particle size using an adaptation of the search geometry of Hanson and Morris. 23 Melandete9 calculated the effect of finite obstacle size for only one value o f f over a range of values of/3c and found that the CRSS was always lower than that for point obstacles of equivalent strength. The reasons for the discrepancies between Melander's calculations and the simple result embodied in Eq. [52] is not understood, but may have to do with the rather large fraction of the search area excluded by METALLURGICAL TRANSACTIONS A

The consequence of finite obstacle size for impenetrable particles is to use Eq. [13], taking Ls as 11 Ls = nj -la - 2(r,) = [(32/37rf) 1/2 - 2](r,).

(a)

(b)

[53]

Through Eq. [53] the square lattice spacing becomes a measure of the free spacing between finite obstacles, rather than the center-to-center spacing. It will be assumed throughout the remainder of this paper that finite obstacles are spherical in shape. This is not generally true, of course, but greatly simplifies the discussion. It often happens that the shapes of real precipitates are equiaxed, so that we can speak of an effective particle "radius" defined appropriately on a case-by-case basis. Under such circumstances the modifications of equations derived for spherical particles are straightforward. If the particles are not equiaxed (e.g., disc or needle shaped), however, special problems can arise. These will not be dealt with in the following sections because generalizations are nearly impossible owing to the variety of situations that we can realistically imagine. III. FACTORS AFFECTING THE E V A L U A T I O N OF E X P E R I M E N T A L DATA

(c) Fig. 11 - - Illustrating various adaptations of the geometry of Friedel statistics for finite obstacles. In (a)the particles are impenetrable until the critical breaking angle is reached. In (b) the dislocation penetrates the particle until it is near the center, at which point it breaks free. In (c) the maximum resisting force occurs just as the dislocation is about to penetrate the particle completely.

him from consideration. Although the discrepancy remains unresolved, it is difficult to find fault with the arguments leading to Eq. [52]. They must be increasingly representative of the real physical situation as the particle radius becomes smaller. One other potentially important consequence of finite obstacle size is that if the particles are too weak, the dislocation may not clear the central obstacle after exceeding its breaking strength. Taking this as a necessary condition we find for the geometry in Figure ll(a) /~min ~ 2(rs)/L, which on using Eqs. [9] and [51] produces the result/3mr, --~ 3"trf/8 = 1.18f, valid for small f. Equation [52] cannot describe the CRSS for particles so weak that/3c < /3ram. Melande r39 also predicts a value of time, which differs from 1.18f because of different geometry he used. Applying this same argument to the geometry in Figure ll(b), we find /3min ~ (rs)/t, leading to the result/3min ~ 37rf/32 --~ 0.3f. There have been no other statistical studies of the effect of finite particle size on the CRSS. However, one interpretation of the results of Schwarz and Labusch 32 is that the CRSS of finite obstacles can be predicted provided that the value of (rs) is incorporated as part of the range of interaction, w; this has, in fact, been done by several investigators. Unfortunately, this idea is easily misused because there is a real distinction between diffuse obstacles of finite range and finite, but localized, obstacles. A composition fluctuation producing an elastic distortion is an example of the former, while a nonmisfitting ordered coherent precipitate exemplifies the latter. METALLURGICAL TRANSACTIONS A

A. Role of Dislocation Character No discussion of this topic can be complete without mention of the role of the character of the dislocations on the flow stress. All the situations considered to this point have made use of the assumption that F is constant. In reality F depends on the angle ~, between the dislocation line and its Burgers vector according to the formula H'4~ Gb2[-l/ + v - 3u sin:~.] r = -47L 1 "---v J In(A/r0)

[54]

where v is Poisson's ratio and A and r0 are outer and inner cut-off distances used in calculating the line energy of the dislocation. For a dislocation initially pure edge in orientation (~: = zr/2), r varies from (Tr - 0c)/2 at the obstacle to ~ / 2 midway between the obstacles in, e.g., Figure 2. If the dislocation is initially pure screw in character (~ = 0), varies from 0 J 2 to 0 over the same distance. Edge dislocations are more flexible than screws, the line tension being smaller by a factor of (1 - 2v)/(1 + v); this amounts to 1/4 for v = 1/3. An important consequence of Eq. [54] is that the dislocation shape is not truly circular as it bows out between two obstacles. Instead, its radius of curvature is given by Eq. [4] at every point on the line; Owing to their greater flexibility, edge dislocations will bow out more under a given stress, and consequently will interact with more obstacles than screws. For weak obstacles this can be formally accounted for by calculating the average line tension (F), over the length of the arc, then assuming that the arc is circular so that Eq. [4] is obeyed with F = (F). For dislocations initially pure edge in character interacting with weak obstacles ( G / 2 ~- /33, (Fe} = a[1 + v - 3v(1 - /32/3)]/(1 - v),

I55]

while for dislocations initially pure screw in character, (F,) = a(1 + v - /3~v)/(1 - v),

[561

VOLUME 16A, DECEMBER 1985--2141

where

the elastic response to a shear stress on {Ill} is

Gb 2

a = -~

In(A/r0).

[57]

Melander z4 has derived these results and presented them in a slightly different form. Equations [54] or [55] can be incorporated into estimates of zc by, for example, substitution into Eqs. [8] or [10] if the improvement in accuracy is warranted. An iteration procedure is required for calculation, though, since/3 is itself a function of F via Eq. [3]. Another important problem associated with the use of Eq. [54] is the evaluation of A and r0. The value of r0 is subject to minor uncertainty, and is usually chosen to lie within the range b < ro < 4b. It is always independent of the surroundings of the dislocation. On the other hand, A is for more problematical because as the dislocation bows out more when the obstacle strength increases, the adjacent arms at the obstacle interact more strongly. The interaction is attractive, thereby tending to reduce To. These difficulties have been thoroughly discussed by Brown and Ham. ~ They recommend taking A ~ L~ for weak obstacles and A - ~ 2(r,) for strong obstacles. The former suggestion creates some practical difficulties, since by virtue of Eqs. [3] and [9] A becomes dependent upon F, also necessitating calculation by iteration. Nevertheless, reasonable values of F are not too difficult to estimate. Most importantly, A depends on both (r) and f, thereby rendering F weakly dependent upon these parameters. A good approximation, especially for alloys aged to near the peak strength condition, is ln(A/ro) = 4, which corresponds to A/ro ~-55. For a typical fcc alloy, this implies peak strengthening at a value of (r) in the neighborhood of 10 nm. Despite the weak logarithmic dependence of F on A, the choice of A (and r0, for that matter) can have serious quantitative consequences. This has been recognized for quite some time, but is nevertheless nicely illustrated in a recent paper by Nembach. 41 Unfortunately, so much freedom is exercised in choosing A, that, when coupled with the typical uncertainties in the values of the variety of physical constants that enter into mechanistic theories of precipitation hardening, it becomes possible to "prove" almost anything one sets out to prove.

B. Elastic Anisotropy In many of the mechanistic theories of precipitation hardening it is necessary to calculate the elastic interaction between dislocations and precipitates, or between dislocations themselves. This is invariably done using linear isotropic elasticity theory because of its relative simplicity or because solutions to the anisotropic problems are unavailable, intractable, or both. Comparison between theory and experiment depends heavily on the validity of isotropic elasticity theory, even though it is well recognized that even the cubic alloys for which data exist are typically quite anisotropic. Since the single crystal elastic moduli, c o, are known for many pure metals and quite a few alloys, it seems sensible at least to address this issue to some extent by using appropriate combinations of elastic constants whenever possible. A case in point is that of {111}(1]-0) slip in fcc metals and alloys. Using the coordinate system transformations of Zener 42 or Turley and Sines, 43 the shear modulus describing

2142--VOLUME16A,DECEMBER1985

Gill :

3c44(ci1 - Cl9

cl, - Clz + 4c44

9

[58]

This quantity is independent of crystallographic direction in the (111) plane. 43 A frequently used "isotropic" shear modulus, G~o, derivable from the single crystal elastic constants is [59]

Cl2)C44/2] 1/2

ais o = [(Cll-

Values of Giso are comparable to the shear modulus determined experimentally using randomly oriented polycrystalline specimens: Using the data compiled in Table 5.1 of Kelly and Groves, 44values of Gill and G~sohave been calculated for six fcc metals and are shown in Table I. With the exception of A1, which is nearly elastically isotropic, GIll and G~sodiffer substantially. Since G tll is the appropriate shear modulus for use in calculations pertaining to fcc alloys, it has been used throughout the remainder of this paper, although the subscripts have been dropped; it is henceforth understood that G is identically Gill. Similar considerations apply for crystals of different symmetry. For example, for bcc crystals which slip on the system {ll0}(1T1), the appropriate shear modulus is Gli0,1rl, the subscripts indicating that Gll0 depends upon the direction in the {110} plane. 43 For the sake of completeness, values of v have been included in Table I. These were also calculated from the single crystal elastic constants using the formula44 [60]

v = Cl2/(Cll + Cl2).

It is evident that the common assumption v = '/3 is rarely valid for these metals. Only for A1 and Ni is it even a decent approximation.

C. Addition Rule for Experimentally Deformed Alloys We turn here to the problem of adding the contributions of the obstacles and the matrix to the CRSS of real crystals, i.e., when the flow stress of the matrix is nonzero. The superposition laws that have been used are variants of Eqs. [39], [42], and [43], and have been discussed recently by Lilholt. 45 In general, the matrix phase of a precipitation hardened alloy is itself a solid solution, in some instances dilute but more often than not relatively concentrated. The legitimate question then arises as to whether the atoms in solid solution should be regarded as point obstacle manifestations of the precipitates, or whether the mechanism of Table I. Values of GI,,, G,so and v Calculated for Six fcc Metals from Equations [58], [59], and [60], Respectively, Using Room Temperature Values of the Single Crystal Stiffnesses Tabulated by Kelly and Groves. 44 The Units for G Are GN/m 2.

Metal Ag A1 AU Cu Ni Pb

G,l, 19.7 24.9 18.6 30.5 62.1 4.8

Giso 26.6 25.9 24.7 42.1 78.7 7.2

p 0.430 0.362 0.458 0.419 0.374 0.461

Glll/Giso 0.741 0.961 0.753 0.724 0.789 0.667

METALLURGICALTRANSACTIONSA

strengthening by the solute atoms is of no concern whatsoever. This situation has been discussed by Lilholt45 and Kocks, 46 who argue that so long as one of the two contributions to the CRSS is due to relatively weak obstacles (solute atoms) and the other due to relatively strong obstacles (precipitates of any reasonable size), the former can be regarded as a frictional stress and the contributions of the two should be linearly additive. That is, Eq. [39a] is appropriate, consistent with the results of the computer simulation experiments on mixtures of a few strong and many weak obstacles. If the matrix is pure, linear superposition must obtain because the CRSS of the matrix is not even obstacle controlled. Such is the case in dispersion strengthened alloys prepared by internal oxidation, 47 for which Eq. [39a] has been verified. Ebeling and Ashby48have convincingly demonstrated that the linear addition rule works for the superposition of solid solution strengthening and dispersion strengthening in interally oxidized Cu-Si alloys containing Au in solid solution. In this alloy the CRSS is clearly given by the sum of the contributions due to the pure matrix, the increment due to solid solution hardening by Au, and the contribution due to the nondeformable SiO2 particles. An apparent exception to the linear superposition law was recently reported by Nembach and Martin, 49 who analyzed data on precipitation hardened Cu-Co alloys in which the Cu matrix was also solid-solution strengthened by the addition of Au. They claimed that Pythagorean superposition more accurately fitted their results, although a subsequent analysis by Bfittner and Nembach 34 of data on this same alloy system showed that the results were more consistent with Eq. [43], with q = 1.3. Both analyses49'34 depend on the viability of the theory of coherency strengthening, which is quantitatively inaccurate (see Section IV. D.2). Hence the data on Cu(Au)-Co still await final interpretation. Since there is experimental and theoretical justification for linear superposition of the contributions of the precipitates and the matrix to the total CRSS of a precipitation hardened alloy, and no convincing evidence to the contrary, all the experimental data in subsequent sections will be analyzed according to Eq. [39a]. This has the additional advantage of eliminating the addition rule as an adjustable "parameter" in the analysis of experimental data. For notational convenience the increment in the CRSS determined experimentally will be denoted by At, with a subscript added to specify the mechanism under consideration. Denoting the total experimentally measured CRSS by ~'r, and the contribution of the matrix by ~-s~, Eq. [39a] is readily rearranged as A'r = r r -

r,s.

[61]

(3) modulus hardening, which occurs when the shear moduli of the matrix and precipitate phases differ; (4) coherency strengthening, which arises by virtue of the elastic interaction between the strain fields of a coherent misfitting precipitate and the dislocation; (5)order strengthening, which operates when the crystal structure of a coherent precipitate is a superlattice and the matrix is a disordered solid solution. There is no reason why two or more of these mechanisms cannot be operative simultaneously, and in many real alloy systems this is probably the rule rather than the exception. Nevertheless, there are many examples of alloys in which one of the aforementioned mechanisms should dominate the interaction. For the most part, we will focus our discussion on those types of alloys in order to ascertain the extent to which theory agrees with experiment. In a large number of age hardenable alloys precipitation occurs by homogeneous nucleation, followed by particle growth and coarsening. The nucleation and growth stages normally transpire very rapidly, so that aging occurs, i.e., the strength increases with aging time, during the coarsening stage, in which the average particle size increases while the volume fraction of precipitates remains essentially constant. There are exceptions to this description, of course, particularly in Al-base alloys in which the aging sequences are often spectacularly complicated, ~~ but age hardening at nearly contant volume fraction of precipitate is probably the rule rather than the exception. Age hardening also occurs when the supersaturated matrix decomposes by spinodal decomposition. In the early stages of this process the microstructure consists of composition modulations, the spatial variations of which are roughly sinusoidal. Phase separation in the sense of matrix and precipitates separated by well-defined sharp interfaces does not occur until the later stages of aging. So long as the lattice constant changes with solute concentration there will be a periodic variation in elastic strains associated with the periodic variation in composition. It is these strains that can give rise to significant increases in the strength of alloys hardened by spinodal decomposition. In the following sections we shall consider the best theories available for strengthening by these various mechanisms from a critical point of view, and shall evaluate some of the available experimental data to determine the extent of the agreement between experiment and theory. Nearly all the precipitation hardening theories involve dislocationprecipitate interactions governed by Friedel statistics. It is those that are discussed in this section. Hardening by spinodal decomposition is quite different, and for that reason is dealt with separately in Section V. A. Chemical Strengthening

IV. PRECIPITATION HARDENING MECHANISMS Precipitate particles can impede the motion of dislocations through a variety of interaction mechanisms. Those for which theories have been developed include: (1) chemical strengthening, which results from the additional matrix-precipitate interface created by the dislocation when it shears through a coherent particle; (2) stackingfault strengthening, which occurs when the stacking-fault energies of the precipitate and matrix phases differ; METALLURGICAL TRANSACTIONS A

This mechanism is one of the earliest to have been considered theoretically. The passage of a dislocation through the precipitate creates two ledges of new precipitate-matrix interface of specific energy Ys. The original version of the theory of Kelly and Fine 5~has been updated by Brown and Ham n for consistency with Friedel statistics. The result, with Fm= 2y,b is, utilizing Eqs. [10] and [51] ~'cc = (6T3 b f / zrF)ll2(r) -l.

[62]

The theory therefore predicts that for fixed f the CRSS VOLUME 16A, DECEMBER 1985--2143

decreases as the particle size increases, contrary to what is normally observed age hardening behavior. An alternative theory of chemical strengthening was constructed by Harkness and Hren 5~ to explain their data on aged binary A1-Zn alloys, which clearly showed an increase of rc with increasing particle size. Their theory, which predicted behavior opposite to that described by Eq. [62], has been dismissed by Gerold 52 as incorrect. Gerold claims that when the theory of Harkness and Hren is properly formulated, the predicted values of rc decrease rather than increase with increasing particle size. It therefore appears that chemical strengthening is not an important mechanism, except perhaps for precipitates of very small size. Since the magnitude of the strengthening effect predicted under such circumstances is rather large compared to what is observed experimentally, it seems safe to conclude that this mechanism does not contribute significantly to the strength of aged alloys.

(a)

(b)

B. Stacking-Fault Strengthening When the stacking-fault energies of the precipitate and matrix phases differ, the motion of dislocations will be impeded because the separation of the partial dislocations varies depending upon the phase in which the dislocations reside. A theory for strengthening by this mechanism was first proposed by Hirsch and Kelly, 53 who considered a variety of different situations, determined by the relative values of (rs) and the ribbon widths, w~ and Wp, and stacking-fault energies, y # and 3'#, of the matrix and precipitate phases, respectively. Some of the possibilities for different combinations of wm and (r,) are illustrated in Figure 12 for the case yg,, > Y~. The CRSS was estimated by Hirsch and Kelly for the various configurations depicted in Figure 11 on taking the effective obstacle spacing along the dislocation line to be the Mott spacing. As Brown and Ham II have argued, Mott statistics are inappropriate to this problem. Gerold and Hartmann 54 later derived a formula based on the application of Friedel statistics, and demonstrated that it can be made to agree with their own data on aged A1-1.8 pct Ag single crystals. According to Hirsch and Kelly 53 and Gerold and Hartmann54 the maximum force experienced by the split dislocation is given by F,, = AT/,

[63]

where AT = IY~rz - Y4pl and l is the length of the chord inside the particle at the critical breaking condition depicted in Figure 12(c). As shown by Gerold and Hartmann, I is a function or (r,) wherever (r,) is greater than both Wp and Win. Alternatively, as discussed by Hirsch and Kelly, when (r,) is comparable to or less than w~, l is equal to 2(r,). From Eqs. [10], [51], and [63] the increment in the CRSS of underaged alloys can therefore be expected to vary as

"i'c~, = A y3/z(37r2f(r ) /32Fb2) v2

[64]

s6 long as the condition 2(r~)< Wm is satisified. Since Eq. [63] is independent of the character of the dislocation Eq. [64] is quite general. Edge dislocations will therefore produce higher values of ~-~vbecause of their lower values of F. Assuming that edge dislocations control the CRSS, Gerold and Hartmann demonstrated that the strengthening will fall below the value predicted by Eq. [64] as the particle 2144--VOLUME 16A, DECEMBER 1985

(c) Fig. 12--Illustrating some of the possible configurations involved in dislocation-precipitate interactions during stacking-fault strengthening when Y4m > y~(wm < wv). (a) w m > 2(r,), l = 2(rs); (b) w m = 2(r,), l = 2(rs); (c) w,, < 2(r,), l < 2(rs).

size increases. While it does not seem possible to derive simple expressions for r for all combinations of (rs), Wp, and Wr,, estimates can be made for the specific case 2(r,) >> Wp, which represents the regime of large particles. For this situation Hirsch and Kelly suggest the formula K [ ~__(r)(%:)] v2 l =(--~)L K- 1 j

[65]

where

K =

Gb2 (2 - 3v + 4v sin 2 ~), 87r(1 - v) (7~) = (y~p + y~m)/2,

[66] [67]

bv is the Burgers vector of the partial dislocations, and use has been made of Eq. [50b]. The predicted behavior then depends on whether the particles are "weak" or "strong", i.e., whether the values of/3c calculated on substitution of Eq. [65] in Eqs. [63] and then [3] are such that the flow stress should be calculated using Eq. [10] or Eq. [15]. If (r) >> K/qr(%f), Tc~ oc ( r ) -1/4 for weak particles, while zcr oc (r) -~/z for strong particles. Approximate expressions are

Ar z~ -~ 1.15---b---\ F /

r K ]3'4(r)_,: 4

L{-y"~}J

[68]

--Z'-_ (r) -'/2

[69]

for weak particles, and ":crY--

METALLURGICAL TRANSACTIONS A

for strong particles. In Eqs. [68] and [69] b and F refer to the perfect dislocation. While these two equations can be expected to provide only a rough estimate, at best, of rc~, they clearly demonstrate that stacking-fault strengthening produces overaging behavior for a wide range of particle sizes when (rs) >> Wm, i.e., when %r is large. The data of Gerold and Hartmann are shown in Figure 13, plotted as Arr vs ((r)f/b) 1/2. According to Eq. [64] the data should plot linearly, with a slope given by 0.96(Ay3/Fb) 1/2. As can be seen in Figure 13, the data on the samples aged at 140 ~ are reasonably linear and extrapolate to A,r~ = 0 (Gerold and Hartmann do not mention specifically that their reported values of the CRSS are values of Zr, but the solubility of Ag in A1 is ~1 pct, 54 and the solid-solution strengthening contribution to ~'ss is quite small). Assuming that screw dislocations control plastic flow, Eq. [54] with v = 0.37 and l n ( A / r 0 ) = 4 results in F = 0.692Gb 2. Analysis of the slopes of these curves, using the values G = 24.9 GN/m 2 (295 K, Table I), and G = 27.6 GN/m 2 (77 K), 54 yields the values A y = 0.158 J / m 2 and 0.185 J/m 2 for these two temperatures, respectively. The values of y~p calculated from these, assuming 0.2 J / m 2 as the most likely55 value of Y~r~ for pure A1, are 0.042 and 0.015 J/m 2 at 295 and 77 K. This temperature dependence of ygp is probably too strong, but the magnitudes compare favorably in absolute value with that estimated by Gerold and Hartmann themselves (ygp = 0.02 j/m2). It should be pointed out, however, that they assumed in their calculations that edge dislocations control the flow stress, with F = 0.5 Gb 2. If they had used the more realistic value F = 0.133 Gb 2, calculated from Eq. [54], the values of Ay derived from the data in Figure 13 would have been much smaller (0.091 and 0.107 J/m 2 at 295 and 77 K, to be precise) than the value they obtained, with a correspondingly larger value of %Ip. Evaluation of the dislocation microstructures of deformed underaged A1-Ag alloys would clearly be valuable to determine whether or not the good agreement between theory and experiment suggested by the analysis of the data in Figure 13 is fortuitous.

50

40

E

3O

Z

2o /

TEST T E M P (K)

/I i/ // I// / 9" ,, / I / / t ,, ,/ /

10

9

9

77

9

295

,,/'// I

0

0.1

L

I

0.2

i

[

0.3

t

I

0.4

J

l

0.5

L

1

0.6

I

I

0.7

(Q)flbl '/~

Fig. 1 3 - - T h e experimental data of Gerold and Hartmann ~4 on age hardened monocrystalline AI-1.8 pct Ag. The open and filled symbols represent data on samples aged at 140 and 225 ~ respectively. The dashed curves schematically illustrate expected behavior. METALLURGICAL TRANSACTIONS A

The data on the samples aged at 225 ~ are consistent with a smaller value of A y arising through a larger value of y~p, as claimed by Gerold and Hartmann. An increased value of y ~ , due presumably to the larger A1 content of the precipitate phase, 54is consistent with both a smaller value of Wp and departure from the behavior predicted by Eq. [64] at smaller particle sizes. In other words, the value of l becomes smaller than 2(rs) (Figure 12(c)), the strength drops below the value of rc~ (Eq. [64]), and the dependence of ~'c~ on (r) becomes weaker than (r) 1/2, as predicted by Gerold and Hartmann.

C. Modulus Hardening Modulus hardening has been one of the more difficult strengthening mechanisms to deal with theoretically. There are two interaction regimes, depending upon whether the dislocation is inside or outside the particle. The interaction energy between an infinitely long straight screw dislocation in a medium of shear modulus G and Poisson's ratio v external to a spherical precipitate of corresponding elastic constants Gp and vp has been calculated at increasingly accurate levels of approximation by Weeks et al.,56 Comninou and Dundurs, 57 and Gavazza and Barnett. 58 The force of interaction has also been calculated, but only in the plane containing the dislocation line and the center of the particle. This force is proportional to AG = IGp - G I and reaches a maximum at the particle-matrix interface. When the dislocation penetrates the particle the force of interaction increases, but a different calculation is required. The interaction energy between an infinitely long straight screw and dislocation passing through the interior of a precipitate having smaller elastic constants than the matrix has been calculated by Weeks et al., 56 Knowles and Kelly, 59 Melander and Persson, 6~and Nembach, 6~who has also made the only calculations to date for edge dislocations. The force of interaction is obtained from the equations for the interaction energy by differentiation. Since the interaction forces are greatest when the dislocation has entered the precipitate, only this case is relevant for estimating the maximum increment in the CRSS due to modulus hardening. Nevertheless, the interaction forces in the external interaction regime can, in principle, be very important under other circumstances. Knowles and Kelly, 59 Melander and Persson, 6~ and Nembach 61 have each constructed theories of precipitation hardening using their own calculations of the interaction forces. The age hardening theory of Knowles and Kelly is essentially a theory of overaging (for particles increasing in size at constant volume fraction), because they used a fixed obstacle spacing along the dislocation to calculate the CRSS. Melander and Persson used the results of their calculations of F~ in conjunction with the theory of Hanson and M o r r i s 37 to construct a theory which is capable of predicting normal age hardening response. The equations of their theory cannot normally be expressed in closed form, but they claim that for small particle sizes the increment in the CRSS can be approximated by [ A G \ 3/2

r~c = 0.9((r)f)l/21---f ~--G-) [2b ln(2(r)/fV2b)]-3/2. /9

[70] VOLUME 16A, DECEMBER 1985--2145

Nembach 6~ calculated the forces of interaction for different models of the dislocation core. His results are sensitive to the assumed structure of the core, but for most of the models he considered Fm varies with (r) according to an equation of the type

Fm= CoAGb2((r)/b) m,

[711

where Co and m are constants roughly equal to 0.05 and 0.85, respectively, for edge dislocations. Substitution of this result into Eq. [10], utilizing Eqs. [3] and [5l], results in the expression for the CRSS

rcC = O.O055(AG)3/2(f/F)'/2b((r)/b) -l+3m/z.

[72]

Nembach attempted to demonstrate using Eqs. [71] and [72] that the extent of strengthening by modulus hardening is small compared to other strengthening mechanisms, citing the comparison with coherency hardening in aged Cu-Co alloys49 as a particular example. Unfortunately, Nembach's claim that the theory of coherency hardening is in very good agreement with experiment is unfounded, as demonstrated in Section IV. D.2. That theory predicts values of the CRSS that are between two and three times larger than those measured experimentally. The fact that the theoretically predicted CRSS due to coherency hardening can be several-fold that due to modulus hardening cannot be used to dismiss the latter mechanism as insignificant. Melander and Persson 6~ used the results of their calculations to demonstrate that modulus hardening provides a satisfactory explanation for their experimental data on strengthening in AI-2 pct Zn-l.4 pct Mg alloys, as well as data on a similar alloy obtained by DiJnkeloh et al. 62 They find that their version of the theory of modulus hardening is in very good agreement with these sets of data, and conclude that modulus hardening is the dominant strengthening mechanism in these A1 alloys in the underaged and peakaged conditions. One of the major difficulties in comparing theory with experiment is that Ge is not known with certainty because the alloy is strengthened by metastable precipitates in both the underaged and peak-aged conditions. It is therefore impossible to obtain independent measurements of Gp from the bulk phases. Gp is, in essence, an adjustable parameter in this system, and while the value of Gv used by Melander and Persson is not unreasonable, the agreement between their theory and experiment may nevertheless be fortuitous. There is yet another theory of modulus hardening, proposed by Russell and Brown. 63Their theory exploits the fact that a dislocation is "refracted" when it crosses the boundary between two media of different elastic constants. Russell and Brown estimate the angles of refraction for a spherical precipitate shape, relate these angles to Oc, and estimate the CRSS from Eqs. [10] and [15]. Their equations predict a maximum strength at very small particle sizes. The theory thus describes overaging for precipitates growing at constant volume fraction. While the approach used by Russell and Brown appears to be promising, it is predicated on the hypothesis that the minimum value of ~b is also the critical value, which is attained (for Gp "(G) just as the dislocation exits the particle. There is no proof offered that this is the critical configuration, although the calculations of Nembach 61 show that Fm is reached when the dislocation is quite close to the particle2 1 4 6 - - V O L U M E 16A, DECEMBER 1985

matrix interface, providing some justification for the calculations of Russell and Brown. There are no experimental data ideally suited to testing any of the aforementioned theories of modulus hardening. Knowles and Kelly 59 showed that their equations could be fitted to data on overaged Fe-l.5 pct Cu, Fe-0.9 pct Cu, Cu-0.95 pct Fe, and Cu-1.45 pct Fe alloys, while Russell and Brown 63 showed that their equations could be fitted to data on peak-aged and overaged Fe-0.94 pct, 1.68 pct, and 3.95 pct Cu alloys. Melander and Persson, 6~ as already mentioned, used their theory to explain age hardening in AI-Zn-Mg alloys. Ibrahim and Ardel164 demonstrated that the theory of Knowles and Kelly best explained their data on overaged Cu3Au-l.5 pct Co alloys (these same data were shown to be in quite poor agreement with the theory of Russell and Brown). Despite the uncertainties, shortcomings, and limitations of the various theories, there is mounting evidence that modulus hardening may be an important mechanism of overaging in several alloys systems. Overaging is definitely not the result of the Orowan mechanism, since, as pointed out by Russell and Brown, low work-hardening rates are observed in the overaged alloys they studied. Additionally, Orowan loops and other microstructural evidence of hardening by impenetrable obstacles were not observed by Ibrahim and Ardel164 in overaged Cu3Au-l.5 pct Co alloys.

D. Coherency Strengthening 1. Theoretical considerations The strengthening of alloys by misfitting coherent precipitates occurs by virtue of the interaction between the stress fields of the precipitates and the dislocation. This is probably the oldest source of strengthening recognized, and it remains one of the most poorly characterized quantitatively. By far the most thoroughly modeled case is that of a pure edge dislocation interacting elastically with a spherical coherent precipitate of radius r with a misfit parameter, e, given by --181[1 + 2G(1 - 2ve)/Gp(1 + Vp)].

[73]

Here 8 is related to the difference between the lattice parameters ap and a of the precipitate and matrix phases, respectively, by the equation 6 = ap - a a

[74]

The problem is attacked first by calculating the interaction force per unit length acting on the slip plane of the edge dislocation and then integrating this result over the length of the dislocation to calculate the total force. The nature of the interaction force, i.e., whether it is attractive or repulsive, depends upon the sign of t$ and the nature of the edge dislocation, as depicted in Figure 14. It is obvious that, statistically, a given dislocation will experience equal numbers of attractive and repulsive particles. In theoretical treatments of the strengthening due to this effect, it is customary to consider only the effect that repulsive particles have on the dislocation insofar as calculations of the maximum force of interaction are concerned, In fact, the maximum interaction force is identical for attractive and repulsive particles,

METALLURGICAL TRANSACTIONS A

and Haberkorn and Brown and Ham. All these theories produce the result, valid for underaged alloys, given by "rc~ = x ( e G ) 3 / 2 ( < r > f b / F )

PL

Fig. 1 4 - - Schematic representation of the interaction between a spherical coherent precipitate, 6 > 0, and a positive edge dislocation on a slip plane at distance z from the center of the particle. In the hatched regions the elastic interactions are repulsive for this combination of 6 and b. Reversing the sign of either also reverses the quadrants of repulsive and attractive interactions.

the main difference between the two being that for repulsive particles the maximum force acts before the dislocation passes the center of the particle, while for attractive interactions the maximum force occurs after the dislocation has passed the center of the particle. Gerold and Haberkorn, 65 using isotropic elasticity theory, showed that the maximum interaction force, Fm~,between a spherical coherent precipitate and a straight edge dislocation gliding on a slip plane located at distance z from the center of the particle (see Figure 14) is ~2Fm~'(1 - ~2)1/2; Fm~ = [ 33~2F~/8~2 ;

~z < 3/4 ~2 > 3/4

[751

where ~" = z/r and Fm is

F~ = 4Gebr.

For values of if2 < 3/4 the maximum force the dislocation experiences is located at the particle-matrix interface, while for ~2 > 3/4 the maximum force is experienced when the dislocation is not in contact with the particle. Fmz =Fm at ~2 = 1/2 and Fmz = 0 when the slip plane contains the center of the precipitate (ff = 0). Due to the fact that the dislocation feels the strain field of the precipitates that physically intersect its slip plane as well as the strain fields of precipitates that do not, it is necessary to find some means of averaging the effects of all the particles. The original approaches taken by Gerold and Haberkorn 65 and Gleiter 66 differed in this regard, yielding somewhat different results, as discussed by Gerold67 and Brown and Ham. u Once the suitable method of averaging is selected an average value of Fm~ is calculated. This result is substituted into Eqs. [3] and [10], producing the final expression for the CRSS. The methods of averaging the effects of all of the precipitates in the microstructure on the CRSS have been discussed by Brown and Ham. u They themselves applied Pythagorean averaging based on Eq. [46] and produced a result differing only by a numerical factor from that of Gerold and Haberkom. 65 Jansson and Melander 68 also considered this problem from the viewpoint of the theory of Hanson and Morris, 37 i.e., Eq. [46], and derived yet another equation differing only by a numerical factor from those of Gerold METALLURGICAL TRANSACTIONS A

[771

where X is a constant which differs from one theory to the other, varying between 2 and 3. While the calculations of Brown and Ham u and Jansson and Melander purport to average the effect of all the precipitates in the crystal, it is demonstrated in the Appendix that this is not so. When coherent spherical precipitates are large, they are capable of interacting strongly enough with a dislocation to bend it through such large angles that the straight dislocation approximation is quite poor. Brown and Ham, u although using the interaction force between the spherical coherent precipitate and a straight edge dislocation, nevertheless estimated the radius at which this large bending can occur using the criterion/3c = Fm/2F = 1. With Fm given by Eq. [76], the expression for (r)m~x is (r)max =

F/2Gbe.

[78]

This value of (r) is, of course, a crude upper limit partly because Eq. [76] has been used to estimate F,,. It is crude because as the dislocation approaches the precipitate-matrix interface and bends under the influence of the applied stress, it does not have to bend very far until the segment of the dislocation between the two particles pinning it lies in a region in which the stress field of the precipitate actually assists the applied stress in moving the dislocation forward. Equation [76] therefore overestimates the maximum force actually experienced by the dislocation. This difficulty notwithstanding, it is possible to estimate the maximum value of 7c~ due to coherency strengthening simply by substituting Eq. [78] into Eq. [77], producing the result (with X = 2.6) 7". . . . .

[76]

1'2,

= 1.84Gef

1/2.

[79]

An alternative method of estimating 7"c. . . . is to apply the integral form of Eq. [46], with 7"* given by Eq. [15] and tic by Eqs. [3] and [75]. Integration over twice the range 0 < ~" < ~ produces Eq. [79], while limiting the integration to the range 0 < ~"< 1 produces a result about 10 pct smaller. Equation [79] is strikingly similar to the result derived originally by Mott and Nabarro, 7 except that they predicted 7". . . . . OCf, while Eq. [79] predicts 7.. . . . . OCfl/2. Brown and Ham u recognized that a large coherent precipitate can produce bending forces sufficiently large that tic = 1 even when the particle does not physically intersect the glide plane of the dislocation. This can happen over the range

F/4Geb < z < (33/2Geb(r)3/4F) ~/2. Brown and Ham compute the CRSS produced by such particles using the formula

dz/(r),

(7",)2= (0.81)2 Zmin

yielding the result 7"c~ = 1.2fl/z( F3 Ge/b3(r)3) TM,

[80]

with Zmin = 0 and Zm~x= (33/2Geb(r)3/4F) 1/2. Equation [80] follows on integration after making use of Eqs. [11] and [51]. With F = Gb2/2 we recover Eq. [2.49] of Brown and Ham. VOLUME 16A, DECEMBER 1985--2147

Equation [80] is unquestionably of limited quantitative significance because when Zmax> (r) the mathematical fraction of particles between z and z + dz is zero (as demonstrated in the Appendix, this fraction, d z / ( r ) , is defined only over the range 0 < z <- (r), and must satisfy the equation f~or)dz/(r) = 1). Despite this shortcoming, Eq. [80] suggests that overaging can occur prior to intervention of the Orowan process. Equations [75] to [80] were all derived on the basis of the rigid dislocation approximation. This is not a bad approximation when the particles interact weakly with edge dislocations, i.e., when the precipitates are small, but it is clearly a poor approximation when they are strong and capable of bending the dislocation through large angles. Gleiter66 recognized this in his early theory and later suggested 69 that the incorporation of dislocation flexibility in it is the source of disagreement between his theory and that of Gerold and Haberkorn, 65 an argument subsequently invalidated by Brown and Ham. 11 The role of dislocation flexibility has been reconsidered most recently by Gerold and Pham. 7~ They used numerical solutions to the nonlinear differential equation developed by Wiedersich 72 to explore the influence of the interaction force on a dislocation initially pure edge in character when the dislocation is allowed to bend. In these calculations the dislocation is treated as an elastic string of constant line tension, but the variation of F with ~ (see Eq. [54]) was not taken into account. Gerold and Pham 7~ discuss their resuits in terms of a parameter V = Gebr/F,

which is exactly half the value of/3c obtained on substituting Eq. [76] into Eq. [3]. They show that for V < 0.2 the rigid dislocation approximation is quite accurate, and consequently recover Eq. [77] in this limit. For larger values of V the rigid dislocation approximation becomes increasingly bad, and the CRSS fails below the value predicted by Eq. [77]. It is not possible to extract an analytical equation for the CRSS from the analysis.

2. Evaluation of experimental data There are several alloys in which coherency strengthening is purported to be the predominant strengthening mechanism. The alloy systems C u - C o 73-76 and C u 3 A u - C o 64'70'71'77 have been the most thoroughly and probably the most carefully studied. Data are also available on the C u - F e 78'79 and Cu-Mn 8~ alloy systems. The early data of Livingston 73 and Phillips TMon polycrystalline Cu-Co alloys have been supplanted by several investigations on single crystals, and we shall concentrate our analysis on the data from these. The data on Cu3Au-Co alloys were obtained from single crystal samples, as were the available data on Cu-Fe alloys. Only the data on Cu-Mn were obtained from polycrystalline samples, and for comparison with the predictions of theory it is necessary to divide by the Taylor factor, which will be assumed to equal 3 for conversion to values of the CRSS. The data from all four alloy systems are presented in Figure 15, where experimentally measured values of A ~ ' J e 3/2 are plotted against ((r)f/b) 1/2. The values of the parameters used in the analysis are given in Table II and Figure 15, and are the most reasonable estimates available. The data are plotted in this way because the values of G for the Cu-rich and Cu3Au-rich matrixes are all within a few percent of each other (Table II), and if Eq. [77] were valid the data on the underaged alloys would superimpose on a single curve, irrespective of the value of X. It can be plainly seen that this does not happen. The representation of the data in Figure 15 clearly demonstrates that the theory of coherency hardening, no matter which version is chosen, is quite inadequate. It has long been recognized that Eq. [77] seriously overestimates the strengthening observed. This prompted Ibrahim and Ardel177 to analyze their data on CuaAu-Co using the theory of Gerold and Haberkorn, 65 modified for the effect of screw dislocations, which were observed in the deformation microstructure of thin foils cut parallel to the slip plane of the alloys (see Figure 16). While theory and experiment can be made to agree under these conditions, as demonstrated by Ibrahim and Ardell, the theory itself is a gross approximation. This was clearly recognized and stated by Gerold

Table H. Values of G, v, and e Used in Analyzing the Data on Cu Base Alloys According to the Theory of Coherency Hardening. Estimates of Gp Are Also Included.

Alloy Cu-Co Cu-Fe Cu-Mn Cu3Au-Co

G (GN/m 2)

u

e • 102

Gp (GN/m 2)

AG/G

30.5 a 30.5 a 33.6 b 31.5 c

0.419 a 0.419 a 0.420 d 0.423 ~

- 1.49 f -0.97 ~ + 1.05 h -4.28 i

74.T 57.3 k ~ 74.T

+ 1.449 +0.879 -+ 1.371

a. Table 1 b. Estimated from the data of Nakajima and Numakura82from their quoted value of G~o multiplied by 0.73 by analogy with Cu (Table I). c. Calculated from the data of Flinn et a l . 83 using Eq. [58]. d. Assumed equal to the value for pure Cu. e. Calculated from the data of Flinn et al. 83 using Eq. [60]. f. Value reported by Gerold and Pham.7~ g. Estimated from extrapolation of the data of Basinski et al. 84 h. Estimated from extrapolation of the data of Basinski and Christian85 and the lattice parameters of the equilibrium Cu-33 pct Mn alloy compiled by Pearson.86 i. Value reported by Ibrahim and Ardell. 77 j. Calculated from Eq. [58] using the single crystal stiffness of fcc Co at room temperatureestimated by Dragsdorf. 87 k. Obtained from multiplying the value reported by Wendt and Wagner79by 0.79, analogous to the value of G,I/G~o for Ni (Table I). 1. Yeomans and McCormicks~cite a value of Gi~ofor fcc Mn attributed to Nakajima and Numakura,s2 although there is no such value reported in the latter paper. 2148--VOLUME 16A,DECEMBER1985

METALLURGICALTRANSACTIONSA

Fig. 1 5 - - A compilation of precipitation hardening data on a variety of Cu base alloys tested at room temperature. The sources of the data are: Cu1.40 pct Co, 75 2.00 pct Co, 76 1.36 pct Fe, TM 1.15 pct Fe, 79 38 pct Mn; 8~ CuaAu-l.5 pct C o , 64'77 1.6 pct Co. 7~ The bold solid line represents the behavior theoretically predicted by Eq. [77] with X = 2.6. The values of f are those reported by the authors cited, except for the Cu-38 pct Mn alloy, s~ In this case f was estimated from the equilibrium diagram in Hansen, sl assuming the precipitates are pure Mn.

Fig. 16--Transmission electron micrograph of the slip plane section of a monocrystalline specimen of underaged ordered CusAu-l.5 pct Co deformed to about 2 pct shear strain in compression. 77 The long straight dislocation segments running diagonally are in screw orientation. Similar results were found for the aged disordered CusAu-Co alloys.

and Haberkorn themselves. The problem with the calculation based on the interaction between a straight screw dislocation and a spherical coherent precipitate is that the total force of interaction is zero. The only reason a nonzero value of the CRSS is predicted is because the value of F,,~ is calculated by integration of the expression for the force per unit length over only one-half the length of the dislocation. In addition, the difficulty observed microstructurally, i.e., that screw dislocations appear to control the flow stress, is not unique to the Cu3Au-Co alloy system. This behavior has also been observed in AIZn 88 and Cu-Cr 89alloys. Quite good experimental data exist on the A1-Zn system, 9~ but these also badly disagree with the predictions of Eq. [77]. Gerold and Pham 7~ have recently suggested that the anisotropy of Cu might be responsible for the relatively poor agreement between theory and experiment, which remains even after the effect of dislocation flexibility is included in estimates of F,,~. However, the values of G (= G,11, Table I) have already been incorporated into the theoretical estimate represented by the bold line in Figure 15. A candidate strengthening mechanism for the alloys under discussion is modulus hardening, because the values of AG (Table II) are quite large for the Cu-Co and Cu2Au-Co alloys, which show the largest strengthening effect, and smaller for Cu-Fe, in which the strengthening is not so great. Unfortunately, there is no convincing quantitative correlation between the extent of the strengthening observed and the values of AG. Whatever the source of the discrepancies noted in Figure 15 may be, it does not appear to be rooted in the method of averaging or in the statistics of dislocation-particle interactions. There is a fundamental problem in our current level of comprehension of the coherency strengthening problem which precludes quantitative prediction of the magnitude of the expected strengthening effect for underaged alloys. The refinements discussed by Gerold and Pham7~ are not helpful in this regard.

For the alloys aged to peak strength and beyond, the situation is no better. Comparison of the available data with. the predictions of Eqs. [78] and [79] is shown in Table III. It is evident that the experimentally observed peak values of the CRSS are substantially smaller than those predicted theoretically, while the values of (r) at which peak strength is observed are far greater than those predicted. The theory thus fails to evaluate the magnitude of the maximum strengthening observed experimentally and the particle size at which this is expected. Given the nature of the approximations used in the derivations of Eqs. [78] and [79], small discrepancies would probably not be surprising. The sizes of the discrepancies are so large, however, that it is tempting to conclude that it is the theory itself, and not the approximations, which is sorely inadequate. Overaging of these Cu base alloys does not appear to be due to the Orowan mechanism. Neither Ibrahim and Ardell,64 Matsuura et al., 78 nor Yeomans and McCormick 8~ found evidence for the formation of Orowan loops or loss of coherency on examination of the microstructures of their alloys aged to peak strength and slightly beyond. Furthermore, the data of Ibrahim and Ardel164 and Yeomans and McCormick 8~ were found to disagree quantitatively with the predictions of the strengthening expected using refined modifications of the Orowan equation. 47'92 Ibrahim and Ardell, 64 Matsuura et al. ,78 and Yeomans and McCormick 8~also compared their results on the overaged alloys with the calculation of Brown and Ham 1~(Eq. [80]), with no success. This also is not surprising, given the nature of the assumptions used to derive that equation. We are thus forced to conclude that the theories of coherency strengthening are incapable of providing us even with good ball-park estimates of the strengthening expected in the underaged, peak-aged, and overaged hardening regimes. This is all the more distressing because most of the alloy systems that have been investigated with the purpose of testing the theories are quite well characterized.

METALLURGICAL TRANSACTIONS A

VOLUME 16A, DECEMBER 1985--2149

Table III. Comparison between the Values of (r),~x (nm) and ~'c. . . . . (MN/m 2) Calculated from Equations [78] and [79], Respectively, with the Experimentally Measured Values for Several Cu Base Alloys. In the Calculations of (r) .... th,o, F Was Taken as 0.089 Gb 2, Which Follows from Equation [54] with ~ = ~ / 2 and in(A/ro) = 4.

Alloy Cu-l.4 pct Co Cu-1.36 pct Fe Cu-38 pct Mn Cu3Au-l.5 pct Co

(r) . . . . . p 6 to 7 5 to 7 10 1.6

(F) ....

E. Order Strengthening 1. Theoretical considerations Strengthening by ordered coherent precipitates occurs when a matrix dislocation shears an ordered precipitate and creates an antiphase boundary (APB) on the slip plane of the precipitate phase. The APB energy per unit area on the slip plane, 7oeb, represents the force per unit length opposing the motion of the dislocation as it penetrates the particle. It is characteristic of alloys strengthened by ordered precipitates that dislocations travel in groups, the number of dislocations in the group being determined by the number required to restore perfect order in the precipitate. Among the most important alloys strengthened by ordered precipitates are nickel base superalloys and stainless steels strengthened by 7' precipitates. These have a crystal structure of the type L12 (the Cu3Au structure). In these alloys the dislocations typically travel in pairs, because the passage of a pair of matrix dislocations (b = a(110)/2) through a 7' particle restores perfect order on the {111} slip plane. The first quantitative theory of order strengthening was that of Gleiter and Hornbogen. 93 Their theory included the effect of paired dislocations, but, as Brown and Ham u have pointed out, they used an effective particle spacing along the leading dislocation that differed from the Friedel spacing. The first theories to employ Friedel statistics were those of Castagn694 and Ham. 95 Their result (for single dislocations) is readily derived by noting that the maximum force of interaction is given by the expression F m = 27avb(r,)

:kr ..... 55 to 60 35 to 40 50 113

theo

0.76 1.16 1.12 0.03

":co = - ~

Ln/ '

"/'cO = ~

[84a]

(37r27ovbf(r)/32F) v2.

[84b]

When the particles are large enough (or strong enough) so that Eq. [15] governs the flow stress, substitution of Eq. [81] into [3] and [15], incorporating Eq. [51] and taking the force balance into account, yields the result Yco, m a x = 0 . 8 1 - ~7apb ' - ([3~.f/8]1/2 _ f ) ,

[85a]

which also assumes that the second dislocation remains straight. On the other hand, if dxi = 0, we have "Yapb

[85b]

~'~0,m~ = 0.81--~- (37rf/8) 1/2.

0

dl =

2(rs>

] I

N

O_}_QLI= LF

0 l

dll

O

@

[82]

0

0

!~

(a)

0

0

0

O

LI1 0

0

o

0

0

0

0

n

I

du = 0 0

0

0

0

(b) [83]

where the ratios dl/Lr and dll/LII are defined in Figure 17. A 2150--VOLUME 16A, DECEMBER 1985

[ ( 37r27,pbf (r ) / 32F) 1/2 - f]

This is equivalent to the equation derived by Raynor and Silcock. 97The first term in Eq. [84a] is identical to Eq. [82] divided by two. If dH = 0, as suggested in Figure 17(b), only this first term remains, and the CRSS is

[811

A result similar to Eq. [82] was obtained by Guyot, 96 using the statistical theory of Dorn et al. 19 The influence of the second dislocation on the CRSS is profound. Its effect is difficult to calculate because it depends upon the statistical interaction between the trailing dislocation and the particles that have already been sheared by the leading dislocation. Many of the intricacies of t h i s problem have been discussed by Brown and Ham." The most important configuration for underaged alloys is that of relatively weakly coupled pairs, as illustrated in Figure 17. A balance of the forces acting on the dislocations in the pair

("i "q 2rcob = "Yapb

Reference 75 78 80 77

simple and common assumption is that the second dislocation remains straight while the first dislocation bows out between the obstacles and acquires the spacing L~ = Lr. In this case the ratio di/Li = 2(rs)/Le, d l i / L n = f and the CRSS is given by

which, on substitution into Eqs. [3] and [10], utilizing Eq. [51], yields

"rco = "Yapb b (37r27apbf(r)/32F)V2

zc. . . . 94 60 178 280

Fig. 17 - - Schematic illustration, adapted from Gleiter and Hombogen, 93 of the shearing of ordered coherent precipitates by a pair of dislocations. In (a) du is finite, while in (b) dn = 0. METALLURGICAL TRANSACTIONS A

Gleiter and Hornbogen93 and Brown and Ham u realized that when the particles become large, so that the condition fl~ --- 1 is satisfied, the coupling of the dislocations in the pair would be particularly strong and both dislocations might reside in the same particle. Hiither and Reppich 98have recently analyzed this situation for spherical ordered precipitates and have derived a formula in which the CRSS decreases relatively slowly with increasing particle size according to - r f 1/2/ 7./-2~r~ ---- 0 4 4 / ) J ( ~ HYapb

--

b-77;-

4-b-f

~ 1/2 1

where u (= 2(rs)/Lv) is determined from the solution to the equation (4B + B 2 / 3 ) 1/2 u =

[86]

where the second expression obtains w h e n 7r2(r)~apb/ 4pF >> 1. In Eq. [86] the parameter p accounts for the repulsion of the dislocations within the precipitate, and is essentially an adjustable parameter which is required to produce agreement between theory and experiment. The physical problem treated by Hfither and Reppich 98 is similar to that of stacking-fault strengthening, except that the critical configuration is encountered when the dislocations enter rather than exit the particle. This accounts for the (r) -~/2 dependence of r~0 (HiJther and Reppich neglected the factor 0.81 in their calculation). If both dislocations do not penetrate the same particles but remain separated, irrespective of whether the alloy is underaged or aged to near the peak strength condition, there is no compelling reason why the second dislocation should be straight. Under these circumstances neither diI nor L. are easy to specify. There is presently no criterion for determining the value of dii/Lii, and there is no theory for the case of weak pair coupling which enables this ratio to be evaluated for various combinations of (r) and f. If the crystal structure of the precipitates is more complex than that of Cu3Au, more complicated coupling can be expected and the procession of dislocations may exceed two. Examples which have been reported are those of coherent y" precipitates (which have the D022 crystal structure, isomorphous with Ni3Nb) in the Ni-base superalloy INCONEL* 718 and ordered coherent/3' (CuaTi) precipi*INCONEL is a trademark of the INCO family of companies.

tates in an aged Cu-l.2 pct Ti alloy. In the former system, Oblak et al. 99 observed processions of four dislocations, which are required to restore order fully in y" particles. Greggi and Soffa ~~176 observed processions of five dislocations in their Cu-Ti alloy aged to contain/3' precipitates which have the Dla crystal structure. In such cases force balances analogous to Eq. [83] can be derived, As discussed by Oblak et al. ,99 the balance of forces is influenced by the extent to which order in the deformed precipitates is restored by the passage of successive dislocations. This factor can result in a different coefficient in front of the ratio di/Li in the force balance equations. The most obvious consequence of a procession of j dislocations is that the coefficient yapb/2b in Eq. [84] is replaced by Y,eo/jb, irrespective of the values of di/Li for the other (j-I) dislocations. Ni-base alloys, stainless steels, and Co-base superalloys are often strengthened by large volume fractions of 7' particles. Indeed, it is possible to prepare binary or ternary alloys containing y ' volume fractions in excess of 0.3 to

[87]

"r~o = ~/apbU/2b,

)

~-- 0.69(pFfyopb/b2(r)) ~/2 ,

METALLURGICAL TRANSACTIONS A

0.4. Under such conditions the point obstacle approximation can hardly be expected to apply. Modifications of the theoretical equations for the CRSS are then required. The most straightforward modification is an adaptation of Ham's 95 approach, used by Ardell et al.~~ and based upon the geometry in Figure ll(b). The result is

2(1

-

-

-

B

B/6)

'

[88]

where B = 37r2yapof(r)/32F.

[891

Equation [87] deliberately excludes the effect of the trailing dislocation in the pair, but incorporates the factor of two reduction in the predicted CRSS. It is expected to be valid for relatively weak particles, and therefore to be able t o describe the CRSS of underaged alloys. It is easy to show that u approaches B ~/2 when B "~ 1 so that Eqs. [87] and [84b] are equivalent w h e n f or (r) are small. For the conditions under which Eq. [87] determines the CRSS, the equations describing the balance of the forces u can be rearranged to solve for the spacing of the dislocations (D in Figure 17) in the pair that shears the y ' particles. The result is, with dH - 0, D = Gb2/'n'u(1 -

V)Yapb.

[90]

If the trailing dislocation is straight, u is replaced by u + f in the denominator of Eq. [90]. In a later paper Ardell ~~ attempted to refine the approach of Ham even further by including the variation of F with/3 in the calculations; these calculations employed Eq. [55]. The results of those calculations are difficult to present in a transparent form, but they predict greater strengthening than Eq. [87] because an initially pure edge dislocation becomes increasingly difficult to bow out as /3 increases and the dislocation acquires more screw character. 2. Evaluation o f e x p e r i m e n t a l data

The literature on age hardening of alloys containing ordered coherent precipitates is large and growing. A survey of representative Ni, Fe, and Co rich alloys strengthened by Ni3(AI, Ti) type 7' precipitates is presented in Table IV, and a compilation of alloys strengthened by other ordered precipitates (including non-nickel-containing L12 precipitates) is found in Table V. These two tables identify alloys for which data exist on yield strength or CRSS as a function of aging time and particle size. It is seen in Tables IV and V that only a limited amount of work has been done on monocrystalline samples. For purposes of illustration, the subsequent evaluation of data will be restricted to the work on single crystals of Ni-A1. The data of Munjal and Ardell 1~ and Ardell et al. ~o~ on binary Ni-A1 alloys containing a large spectrum of y' volume fractions are shown in Figure 18. These data were taken from samples tested at several different temperatures. The values of A~'o were obtained using Eq. [61]. The binary Ni-A1 alloy system has the particular advantage that the kinetics of y ' particle coarsening have been thoroughly VOLUME 16A, DECEMBER 1985--2151

Table IV. Nickel, Iron, and Cobalt Base Alloys, Strengthened by 1/' Precipitates, for Which Data Are Available on Yield Strength or CRSS as a Function of Aging Time and/or Particle Size (m = monocrystalline, p = polycrystalline).

Alloy Sample Type Ref. Ni-A1 m 101,103,104,106 Ni-A1 p 105 Ni-Ti p 107 Co, Ni, Cr-Ti p 108 Fe, Ni, Cr-Ti p 109 Ni, Cr-Ti, A1 p 110 (NIMONIC 80A)* Ni, Cr, Mo-Ti, A1 p 111 Fe, Ni, Cr-Ti, A1 p 97 Ni, Cr, Co, Mo-Ti, A1 p 94,112,114 Ni, Fe, Cr, Mo-Ti, A1 m 113 (PE-16) Ni, Fe, Cr, Mo-Ti, A1 p 97, 114 (PE-16) Fe, Ni, Cr, Mn, Mo-Ti, A1 p 97,115 (A-286) *NIMONIC is a trademark of the INCO family of companies.

The data in Figure 18 are presented as plots of A~'o v s (r) 1/2 ((r) is taken as half the edge length of the cuboidal particles) for the sole purpose of demonstrating that the point obstacle representation of order hardening theory, embodied in Eq. [83], appears to be reasonable for the 12.2 pct

100 -

6O 4O a)

20

0'

~1

I

I

I

I

I

I

I

180 160

O

zx

140

documented 125'~26so that the 7' particle size is known quite accurately as a function of time and temperature for all the alloys. In addition, magnetic measurements can be used to monitor the solute content of the matrix with considerable accuracy, 127 thereby enabling the volume fraction of 3" to be determined quite precisely for individual mechanically tested samples without the need to perform tedious quantitative metallographic measurements. All these data were obtained from compression testing of single crystal samples oriented for single slip. A minor disadvantage of these alloys for testing the theory of order strengthening is that the 3" particles are cuboidal in shape rather than spherical (although the tendency toward this shape increases only as the particle size increases), and there is a small lattice mismatch (e = +0.35 pct), leading to the possibility that coherency hardening might contribute to the overall strengthening observed. However, the shear moduli of the precipitate and matrix phases are nearly identical, and most of the parameters which enter into the theory are reasonably well known. The major exception to this, as is the case for all other 3'/3" alloys, is that the values of 3',.ob are not known with precision.

120 100 8O 6O 4O

(b)

20

0 ~0

I

I

I

I

I

I

l

I

I

200 180

<>

8

_ ~5

160

120 100

Table V. Alloys Strengthened by Ordered Precipitates That Differ from ~/', and Non-Ni-Base Binary Alloys Strengthened by L l z Type Precipitates for Which Data Exist on Yield Strength or CRSS as a Function of Aging Time and/or Particle Size (m = monocrystalline, p = polycrystalline).

Alloy AI-Li Pb-Na Ni-Mo Cu-Ti MgO-Fe203 Co, Ni, Fe, Cr-Nb Fe-Si, Ti Fe-Ni, A1, Ti Fe, Cr-Ni, A1

Type p m p m m p p p p

Precip. A13Li(8') Pb3Na Ni,Mo03) Cu4Ti(/3') MgFe204 Ni3Nb(3/')

Fe2TiSi Ni(A1Ti) NiA1

2152--VOLUME 16A, DECEMBER 1985

Structure L12 L12 Dla Dla spinel 0022 L21 B2 B2

Ref. 116 117 118 100 119 120 121,122 123 124

T _TEM__P_P __ (K)

/

80 60

/

zx O O []

40 -

77 183 297 373

(c)

200

I 0

1

i

I 2

i

1 3

i

I

[

4

5

(r~> 89 (nm) 89

Fig. 1 8 - - Data on the age hardening of monocrystalline Ni-A1 alloys tested in compression. (a) Ni-12.2 pct A1~~ aged at 625 ~ (b) Ni-15.6 pct AI l~ aged at 625 ~ (c) Ni-14.8 to 16.1 pct A11~ aged at 600 ~

METALLURGICAL TRANSACTIONS A

A1 alloy insofar as a negative intercept is observed by extrapolation to (r) = 0. This is obviously not the case for the more concentrated alloys. Furthermore, it is apparent on inspection of the data that the values of A ~'o for the underaged alloys, for all values o f f , are essentially temperature independent, whereas this is not true for the alloys aged to peak strength. Equations [84b] to [87] state that ~'~o is a universal function of (r}f/F, approaching a square-root dependence as the product (r)f becomes small. Accordingly, the data in Figure 18 (297 K only) on the underaged samples have been replotted in Figure 19 as Ar vs ((r)f/F) 1/2. In the calculation of F, Eq. [54] was used with ~e = 7r/2, ro = 2b and A given by Eq. [9]. The edge dislocation was chosen because Ham 95 has shown that it is more resistant to motion than screws. The value of G used was 59.3 G N / m 2 which seems to be a reasonable compromise between the results calculated from Eq. [58] using the values of the single crystal elastic constants of concentrated Ni-A1 solid solutions measured by Vintaikin ~28 and Pottebohm et al. 129 The data in Figure 19 cover a very broad range of values of (r) (0.85 to 5.85 nm) a n d f (0.041 to 0.357), over which F varies from about 0.23 to 0.48 nN.* Nevertheless, the *These values are about 20 pct smaller than those calculated previously by Ardell et al. Jo~ because of the smaller values of G used here.

data points vary quite smoothly with ((rff/F) it2, suggesting that A ro is indeed a universal function of this parameter and implying that d l i / t u is much smaller than f over the entire ranges of (r) and f. At small values of ((r)f/F) ~/2 the data are sensibly linear (Figure 19) and are consistent with Eq. [84b] provided 7~pb = 0.14 m J / m 2. Using this value of 7apb, Eq. [87] predicts the solid curve in Figure 19, which is seen to be in excellent agreement with the experimental data over the entire range of ((r)f/F) 1/2. Thus, using the newly calculated values of F, which are certainly the best estimates available, all the data agree quantitatively with Eq. [87]. There is no need to invoke different values of d , / L n for different ranges of f, as had been done previously by Ardell et al. lOl and Ardell. 102 The values of A rco . . . . for the alloys aged to peak strength are compared with the theoretically calculated values predicted by Eq. [85b] in Table VI. The experimental values increase with increasing test temperature, as noted previously, which has been interpreted as convincing evidence

Table VI.

160

00

0 140

120

~

100

Z

8O

P 6O

40

//

AGING TEMP (~

//~

20

A MUN,JAL AND ARDELL 625 9 ARDELL et al. 625

/

0

[] ARDELL et aL J

I

J

1

0

I

*

=

=

600 J

I

1

j

% AI 12.2 15.6 14.8-16.1 I

I

I

2 (<~r~>f/F)

y2

(m/N)

89

Fig. 19--The data taken at room temperature on the underaged monocrystalline Ni-AI alloys tested in compression by Ardell et al. lOl and Munjal and Ardell, ~~ plotted according to the prediction that A'ro is a universal function of ( r ) f / F . The solid curves show the linear dependence predicted by Eq. [84b] and the theoretical prediction of Eq. [87] with Y,~b = 0.14 J/m 2,

that coherency strengthening plays an important role in alloys aged to peak strength, ~3~ as concluded by other investigations. ~3~'132 The theoretically calculated values of A~'cO .... are generally slightly smaller than the measured values, but are clearly in the right neighborhood. Finally, we turn to measurements of the dislocation spacings for comparison with the predictions of Eq. [90]. Representative microstructures taken from sections cut parallel to the slip plane of deformed samples 1~ containing large volume fractions of 3/' are shown in Figure 20. Measured values of D from these samples and from a sample of the deformed 12.2 pct A1 alloy ~~ are reported in Table VII, where they are compared with the predictions of Eq. [90]. While the calculated values slightly overestimate the measured ones in two out of the three cases, the agreement is nevertheless satisfactory.

Comparison between the Experimentally Measured Values of Ar . . . . (MN/m 2) at the Four Test Temperatures with Those Predicted Theoretically by Equation [85b]. The Data on the Ni.12.2 Pct A! Alloy Are from Munjal and Ardell. 1~ The Rest Are from Ardell e t al. 1~ m To, max

Pct A1 12.2 15.6 15.6 16.1 15.8 16.1 15.8 16.1

Aging Temp. (~ 625 625 625 600 600 600 600 600

METALLURGICAL TRANSACTIONS A

f 0.07 0.33 0.34 0.38 0.36 0.39 0.36 0.39

77 86 155 --142 158 149 158

I

3

183 91 164 --150 166 156 167

297 92 172 164 158 159 175 168 176

373 (~ 92 176 --166 179 169 182

"Coo. . . . 65 140 141 150 146 152 146 152

VOLUME 16A, DECEMBER 1985--2153

Fig. 20--Transmission electron micrographs illustrating the dislocation coupling observed in samples of Ni-15.8 pct A1 aged at 600 ~ for (a) 24 h ((r) = 3.92 nm) and (b) 48 h ((r) = 4.94 rim). Both thin foils were cut parallel to the slip plane of monocrystallinesamples deformedin compression.~01 3. Order strengthening and the theory of Schwarz and Labusch There has been a tendency in recent years to apply the theory of Schwarz and Labusch 3z to the analysis of data on strengthening in 7 / ' / ' alloys. This trend was started by Haasen and Labusch 133 who reanalyzed the data of Ardell et al. 101 according to the equations of the Schwarz and Labusch theory. They were able to demonstrate good agreement between theory and the data of Ardell et al., although they required an unrealistically large value of the line tension to do so. For some reason Haasen and Labusch chose not to analyze the data of Munjal and Ardell.l~ The rationale for the approach of Haasen and Labusch was provided by the large volume fraction of 3/' in the aged alloys of ArdeU et al., coupled with the apparent improbability that the sequence of events characterizing Friedel statistics could transpire as required by their theory when f is so large. It would appear that the aforementioned sequence of events is hardly more probable for point obstacles than it is for finite obstacles, and that a more stringent criterion is that discussed in Section II.D, i.e., that the condition j[~min ~> 0.3f apply (we note that Figure ll(b) is identical to the geometry used by Ardell et al. 101). With ~min = Table VII.

YapbTr(r)/4F, we find, using `/apb 0.14 J / m 2 and F = 0.25 nM, that (r) > 0.68f (nm). With f = 0.35, (r) > 0.24 nm, which is satisfied for all the combinations of alloy content and aging times used by Ardell et al. On this basis there is little to object to conceming the viability of the theory of Ardell et al. Nevertheless, in view of its growing popularity and potential importance, the application of the theory of Schwarz and Labusch to data on precipitation hardened y / y ' alloys is discussed below. The analysis proceeds in the following fashion. Equation [38b] is the appropriate one to use because order strengthening is an energy storing interaction mechanism. 133 Experimental values of ~-** must be calculated using measured values of A~-o and dividing these by either of Eqs. [84], whichever is deemed appropriate. It is also necessary to calculate r/0 from Eq. [36] using estimates for w and Fro. The former is proportional to (rs) and the latter to the product (rs)y~pb. The parameters are then adjusted so that the subsequent plot of ~-** vs ~7o yields a straight line of slope C'sL and intercept 0.94. The approaches taken by various investigators who have analyzed their data accordingly are summarized in Table VIII. It is obvious that there is no unanimity in the =

Comparison between the Experimentally Measured Values of D (nm) in Deformed Ni-Ai Monocrystals with Those Predicted Theoretically by Equation [90].

Alloy (Pct A1) 12.2 15.6 15.6 2154--VOLUME16A,DECEMBER1985

(r) (nm) 8.40 3.92 4.94

f 0.07 0.35 0.35

D~p 38.1 23.8 22.2

D~heo 38.5 16.0 19.5

Reference 103 101 101

METALLURGICALTRANSACTIONSA

Table VIII. Illustrating the Variety of Values of the Parameters and Equations Used by Various Investigators to Analyze Data on the Strengthening of 7 / 7 ' Alloys According to the Theory of Schwarz and Labusch. 32 1)o Is Presented in Units of (I'f/y~b(r)) ~/2 and I" Is in Units of G~ob 2.

Alloy Ni-16 pctA1 PE-16

~/o 1.27 1.27

C'sL 1.57 0.202

PE-16 A-286 Ni-13.3, 14,16 pct A1

0.89 0.89 2.10

3.0 3.5 0.78

F ]

88 (G = Gm) variable 3 8~ 88

calculation of ri0 and F, and the addition rule used to estimate Azo varies considerably. In addition, the values of C~L emerging from the various analyses are also quite variable; only one of them is even close to the value 0.7 derived from the results of the computer simulation studies (Figure 8) which gave birth to the analysis in the first place. It is difficult to pinpoint the reasons for the values in Table VIII chosen for some of the parameters, except to note that with these choices the authors have all demonstrated the efficacy of the theory. In view of the acknowledged validity of Eq. [54] there is certainly no excuse for using F = Gb2/2 or F = Gb2/4. Of the selections in Table VIII, only the variable values used by Reppich et al. 114 and the average value chosen by Thompson and Brooks H5 are reasonable. Reppich et al. used a superposition law that not only is clearly inappropriate to precipitation hardening, but was also used incorrectly, since the obstacle-controlled contributions to the CRSS were not separated from the frictional (matrix) contribution. Finally, for the commercial alloy PE-16 the values of 3/~pb required to produce satisfactory agreement between theory and experiment are rather low. This assessment is based on the observation of Raynor and Silcock97that 3/~b increases with increasing Ti content of the 3/' phase. Since the lowest acceptable value of 3/apb for "pure" Ni3A1 appears to be 0.135 J/m 2, used by Ardell, ~~ Haasen and Labusch, 133 and Gr6hlich et al. ,106a larger value of 3/oebfor Ti-containing 3/' in PE- 16 would be expected. The value arrived at by Raynor and Silcock themselves was 0.240 J/m 2, which seems a bit high. An evaluation of the data on Ni-A1 in light of the theory of Schwarz and Labusch, using acceptable values of the parameters characteristic of this alloy system, is presented in Figure 21. Figure 21(a) is based upon the use of Eq. [84a], while Figure 21(b) utilizes Eq. [84b]. In both figures ri0 was estimated with to = (rs) (following Reppich~3S), 3/apb = 0.14 J/m 2, Fm given by Eq. [81], and the same values of F used in the analysis producing Figure 19. With these parameters the current values of ri0 are approximately a factor or two smaller than those of Haasen and Labusch.133 This hardly matters, however, since it is obvious that the representation of the data in Figure 21 is meaningless in the context of the theory of Schwarz and Labusch. The data points in Figure 21(a) are scattered about a crudely constant value, which is not surprising since A~'o has already been shown to be a function of (r)f/F (Figure 20). The plot in Figure 21(b) suggests that Azo divided by ~'co defined by Eq. [84a] is a function of rl0. That

METALLURGICAL TRANSACTIONS A

7apb (J/m 2) 0.135 0.1

Addition Rule Linear Pythagorean

Reference 133 134

0.12 0.16 0.133

~ Linear Pythagorean

114 115 106

function, though, is not even approximately linear, as demanded by the theory of Schwarz and Labusch (Eq. [38b]). On the basis of the analysis above, it can be concluded that the theory of Schwarz and Labusch can be made to agree with experimental data only by judicious manipulation of those data, combined with carefully selected (but not necessarily realistic) values of many of the other parameters,* particularly F (see Table VIII). It is conceivable that *For example, attempts have been made in this study to duplicate the analysis of Haasen and Labusch, 133using their values of F, 3'opb,etc. and each of Eqs. [84] applied to the data of Ardell et al. Jo, These attempts have resulted in total failure to reproduce Figure 3 of Haasen and Labusch. Moderate success was achieved using Eq. [82] instead of Eqs. [84], but this is absurd because the sheafing of 3/' particles by single dislocations cannot possibly govern the CRSS of the underaged alloys. Since it is reasonable to assume that Haasen and Labusch cannot have been mistaken about this, the inability to reproduce their results is mystifying.

1.0

ZS A

,~f-.

/X

[]

/X

0

[]

~o

0.8

rn

[]

C~3FI [3 0.6 (a)
J

0.4

I

=

I

J

I

I

AGING TEMP. (~ i

%

t

% A~

MUNJAL

AND ARDELL

625

12.2

ARDELL

et al.

625

15.6

[~ A R D E L L

et al.

600

. -

9

4

I

,/~ .

2

(b) 0

I 0

I 0.1

t

I 0.2

I

I 0.3

t

I 0.4

I

I 0.5

~o

Fig. 2 1 - - T h e data of Ardell et al. 1Ol and Muajal and Ardell j~ in Fi~., 19 replotted according to the theory of Schwarz and Labusch. 32 In (a) 70 is calculated from the data using Eq. [84b], while in (b) it is calculated using Eq. [84a]. The ordinate scale in (a) is five times that in (b).

VOLUME 16A, DECEMBER 1985--2155

the theory may have some utility and validity in describing order strengthening in real alloys, but the analyses published to date are far from convincing.

K The Effect of Distributed Particle Sizes In any isothermally aged alloy the particle size is never monodisperse. There is always a distribution of particle sizes, the breadth of which varies from one alloy system to another, but which is generally within a factor of two or so of the distribution predicted by the theory of particle coarsening due to Lifshitz and Slyozov 136and Wagner. ~37As mentioned previously, there will always be a distribution of values of rs even if the distribution of r were monodisperse because the slip plane will intersect a particle of diameter 2r anywhere from z = - r to z = r with equal probability (that probability is dz/2r). Thus, for a given distribution of r, that of rs is even broader, leading to a broad distribution of obstacle strengths. The distribution of obstacle strengths can be affected for other reasons. In alloys strengthened by metastable precipitates, such as the vast majority of commercial A1 base alloys, isothermal aging can produce microstructures containing more than one type of precipitate, each of which can strengthen the matrix by a separate mechanism. The distribution of obstacle strengths under these circumstances will generally be far broader than that due to the presence of a single type of precipitate. Also, some commercial alloys are subjected to dual aging treatments or to non-isothermal annealing cycles which can produce particle size distributions, hence strength distributions, that are bimodal or even trimodal. Very few analyses or experiments have been performed to explore these aspects of precipitation hardening. Melander and his co-workers have published several papers (see Melander and Jansson 13sfor a summary) in which the distribution of obstacle strengths arising naturally due to the particle size distribution has been accounted for theoretically, using as a foundation the theory of Hanson and M o r r i s . 37 This approach is certainly on a firm theoretical basis, and the accuracy of its predictions is limited primarily by the various formulae used to calculate/3c (or F,,), which depends on the assumed strengthening mechanism. A minor disadvantage of this method is that estimates of the CRSS are made numerically, so that simple equations are not always available for elucidating the role of the variety of parameters that enter into the theory. A major conclusion of the computer experiments of Foreman and Makin 36 (Figure 10) is that for all but quite weak obstacles the CRSS of a broad distribution is within a few percent of that calculated as if all the particles had the same average planar radius. It is difficult to test this prediction experimentally, and attempts have been made only by Munjal and Ardel1139 and Nembach and Chow. 140 Munjal and Ardell aged specimens of a Ni-12.2 pet A1 alloy at two temperature 25 ~ apart to produce unimodal y' particle size distributions with standard deviations different from that corresponding to isothermally aged samples. The first aging treatment was chosen to produce a value of (r) corresponding to the peak strength condition. Since the CRSS is insensitive to {r) in this regime, small changes in {r) produced by the two-step aging treatment would be expected to have little influence on it. However, f also 2156--VOLUME 16A, DECEMBER 1985

changed slightly during the two-step aging treatments, and since rco ~f~s2 for alloys aged to near the peak strength condition (Eq. [85]), the experimentally measured values of Azo were corrected for these effects by dividing them by flJ2. The aging treatments produced 7' particle size distributions about 30 pct broader than those resulting from isothermal aging, this change resulting from the addition of small (weak) particles to the distribution. The corresponding values of A'ro/f 1~2, which were nearly constant for the twelve measurements made, were approximately 8 pct smaller than the value obtained for isothermally aged samples. Munjal and Ardel1139concluded that while this change was small, it was nevertheless much larger than that predicted by the results of the computer simulation experiments of Foreman and Makin, 36 basing this on what would be obtained from a rectangular distribution of obstacle strengths having a standard deviation equivalent to that of the experimental distribution. They also suggested that the modeling of finite particles by point obstacles may have contributed to the discrepancy. It is possible to test these suggestions now in a manner that was impossible when the experiments were done because the appropriate theory had not yet been formulated. Nembach and Chow ~4~performed experiments similar to those of Munjal and Ardell on polycrystalline samples of NIMON1C* PE-16. They chose aging temperatures and *NIMONIC is a trademark of the INCO family of companies.

times which produced changes in (r) in samples in the underaged regime. They found that their aging treatments produced small changes in the standard deviation of the particle size distributions, compared to those of isothermally aged samples, but that the yield strength increased significantly, even after correcting for variations of (r) by the theoretically expected (r) vz dependence of the CRSS (Eqs. [84]). A similar increase in strength was also observed for specimens double-aged to produce average particle sizes characteristic of the peak strength and overaged regimes. Nembach and Chow suggested that the observed increases in strength were due somehow to an effect of double aging on the distribution of particle spacings, although these were not measured for comparison with those of isothermally aged samples. Additionally, Nembach and Chow could not measure changes in f resulting from the double aging treatments (this was done in the binary Ni-A1 alloys using magnetic measurements of the Curie temperatureS39), and were forced to assume that the values of f in these specimens were equilibrium values. At this juncture a critical assessment of these two sets of experiments is difficult. The conjecture here is that if all the necessary corrections for {r) a n d f to the measured values of the CRSS were made, Pythagorean superposition applied to the continuous distribution of particle strengths in conjunction with the theory of order strengthening embodied in Eqs. [85] and [87], would accurately predict the CRSS. The other experiments done on the effect of particle size distributions addressed the issue of bimodal distributions of y' particle sizes in Ni-A1 alloys TM and PE-1614~ Bimodal particle size distributions are also produced by double aging, but with a difference in temperatures large enough so that a new population of particles nucleates, grows, and coarsens independently at the lower aging temperature. METALLURGICAL TRANSACTIONS A

Chellman and Ardell TM studied this problem in a concentrated binary Ni-A1 alloy aged at 800 ~ for a fixed time, followed by aging at 600 ~ for variable times. The particle size resulting at 800 ~ (Class 1) was such that the alloy was overaged, deformation proceeding by the motion of single dislocations. In the doubly-aged alloys the dislocations traveled, as usual, in pairs because the small y' particles (Class 2) were sheared. Chellman and Ardell measured (r) a n d f characterizing the Class 2 particles and calculated the expected values of Aro2 using Eq. [87] (albeit with Giso and a higher value of 7~pO than those used to construct Figure 19). They then argued that since the measured CRSS of the Class 1 particles, Arm, would have been halved had the dislocations attacked them in pairs, the measured CRSS of the doubly-aged samples should be compared to an appropriate superposition of Aro2 and A r o l / 2 . By far the best agreement was obtained for linear superposition, i.e., A r o --- Arol/2 + A r o 2 ,

in clear contradiction to the predictions of Eq. [46]. It is now evident that the argument of Chellman and Ardell is incorrect because the flow stress due to the large particles should be doubled (not halved) when they are attacked by paired dislocations. None of the superposition laws works when 2rm and to2 are substituted into the appropriate equations. Nembach and C h o w 14~performed a similar experiment on a single sample of polycrystalline PE-16. They used a correction procedure to determine A ro2, and demonstrated that Aro was given accurately by A r o = (Ar2~ + Ar~2) '/2,

consistent with Pythagorean superposition. They observed dislocation pairs (as well as single dislocations) in electron micrographs of samples containing only Class 1 particles, and concluded that double aging did not alter the dislocation mechanism responsible for deformation. This is a curious result because the value of ro~ was that of an overaged sample, and the dislocations are typically uncoupled in the Orowan regime. In a somewhat different type of experiment, Long et al. 89 diffused Cr into internally oxidized Cu-Si single crystals, thereby producing specimens that were dispersion strengthened by SiO2 particles and precipitation hardenable by Crrich precipitates. Unfortunately the volume fraction of SiO2 particles was too small to produce substantial strengthening (the authors estimated a contribution of ~ 10 MN/m 2, compared to the nearly 200 MN/m 2 contribution of the Cr precipitates), and the addition rule could not be reliably determined. In conclusion it is fair to state that the appropriate superposition rule in alloys containing obstacles of more-or-less two distinct strengths has yet to be verified by reliable and conclusive experiments. V. H A R D E N I N G BY SPINODAL DECOMPOSITION A. Theoretical Considerations

There are currently two theories of hardening by spinodal decomposition which seem to have survived the test of time, according to a recent review article by Wagner. 142The first METALLURGICALTRANSACTIONS A

of these is due to C a h n , 143 who calculated the stresses required to liberate either edge or screw dislocations from entrapment by the periodic strain field produced by the sinusoidal composition modulations resulting from spinodal decomposition. His theory predicts an increment in the CRSS given by %s = Jc(A rly)2bh / F

[91 ]

where h is the wavelength of the composition modulation, A is the amplitude of that modulation (in atom fraction), r/ is defined by ~7 = a -l d a / d C

= d(ln a ) / d C ,

[92]

where d a / d C is the variation of the lattice constant, a, with composition, C, (in atom fraction), Y is related to the elastic constants cn and Cl2 by the equation Y = (Cn + 2Clz)(Cll - c12)/Cll

[93]

for cubic crystals of normal anisotropy, and Jc is a constant equal to 1/zr63~2 for screw dislocations and 1/zr23a for edges. In the derivation of Eq. [91] it is assumed that the decomposition process is perfectly periodic in nature, in that the wavelength and amplitude are not statistically distributed throughout the material. The critical stress is that for which stable solutions for the dislocation shape can no longer be found. An alternative theory of spinodal decomposition was formulated by Kato et a1.,~44 who argued that a dislocation of mixed character is far harder to move than either a pure edge or screw dislocation in a periodic stress field and therefore represents the most realistic dislocation orientation controlling the flow stress. Kato et al. claim that since dislocations of either pure edge or pure screw character move relatively easily they will rapidly reorient themselves into the mixed configuration, which resists motion at larger stresses. In other respects, the analysis of Kato et al. is similar to that of Cahn. Their equation for the flow stress is %s = ArlY/61/2.

[94]

Other theories of hardening by spinodal decomposition have been formulated to explore the effects of other sources of strengthening. Ghista and Nix 145suggested that the variation of the elastic constants accompanying the composition modulations would interact with dislocations and produce considerable resistance to their movement. Hanai et al. 146 developed a theory based on interface creation by slip, i.e., a kind of chemical hardening theory of spinodal decomposition, while Ditchek and Schwartz 147 derived equations for strengthening based on the lattice mismatch accompanying spinodal decomposition. Kato et al. ~44 claim that the extent of strengthening predicted by all three theories is too small to account for experimental observations. Furthermore, they specifically criticize the theories of Ghista and Nix (as do Brown and Hamn), Hanai et al., and Ditchek and Schwartz individually and leave little doubt that all three are inadequately formulated. In comparing Eqs. [91] and [94] it is obvious that their predictions are very different, the reasons for which have been discussed in another paper by Kato et al. 148 In particular, in the theory of Kato et al. ~44 the increment in the CRSS is independent of A, while in Cahn's theory m the dependence on h is linear. These differences between the two VOLUME 16A, DECEMBER 1985--2157

theories notwithstanding, it seems here quite peculiar that any theory of hardening of underaged alloys should predict a CRSS that is entirely independent of the characteristics of the dislocation line. Since the CRSS in the theory of Kato et al. 144 (Eq. [94]) is independent of b and F, it suggests implicitly that the dislocation samples the stress field produced by the composition modulations entirely independently of its own flexibility. The theories of Cahn 143and Kato et al. 144 are both primitive in the sense that they both predict the CRSS of an ideal microstructure. The counterpart of this in the case of point obstacles, is to place each and every obstacle on the lattice point of a square lattice. We can then easily calculate the flow stress required to move the dislocation parallel to the edge of the unit cell. The CRSS for this configuration is simply Eq. [6] with L --- L, representing the lattice constant. Application of this equation to a real alloy in which the obstacles are randomly distributed in the glide plane obviously yields the wrong result, the reason being that in a real crystal a flexible dislocation interacts with an increasing number of obstacles the more it bows. Given that the microstructures of alloys hardened by spinodal decomposition are modulated, but are not perfectly periodic, it seems reasonable to construct a theory which explicitly acknowledges this aspect of the microstructure. Such a theory exists. It is the theory for hardening by strong diffuse obstacles, reviewed in Section II.B and expressed by Eq. [34]. This theory is conceptually similar to that of C a h n and Kato et al. in that the CRSS is the stress required to liberate the dislocation from the obstacles trapping it (the stress field produced by the composition modulations). The main difference, however, is that Eq. [34] already has the statistical aspects of the microstructure built into it. To apply Eq. [34] to this problem, it is necessary to obtain reasonable estimates for tic, to, and L,. The last of these is the simplest, i.e., we take L,-~ A (following Nabarro~2), and note that in a nonideal microstructure the spacing of the "obstacles" along the dislocation line is given by the Mott spacing, Eq. [32]. To estimate fl~ and to we shall assume that the dislocation takes the sinusoidal shape it would acquire in the ideal, perfectly periodic microstructure. To this end we can use the solutions for the shape of the dislocation presented by either Cahn or Kato et al. The values of fl~ and to used in the following were obtained from the solutions of Cahn ~43for the periodic shape of the edge dislocation. The reason for choosing the edge dislocation is that the resulting value of r~s provided the best agreement with the limited data available. The principles of the calculation can be applied to dislocations of any character. The value of tic is taken as the maximum slope of the dislocation line, while to is taken as its amplitude. From Cahn's solutions* the values of/3c and to, respectively, are *As noted by Kato et al. ~48 there are errors in Cahn's Eqs. [29]. They provided the correction for the calculation of the shape of the screw dislocation, but not for that of the edge. On applying Galerkin's method 143 to the edge dislocation, it turns out the Cahn's expression for B 22is too small by a factor of 2(27r/A) 2, In the notation of this article to = 82 and /3~ (which is the maximum value of d x / d y ) = 2~'B2/6~%. The values of B~ were chosen as the minimum values for a given applied stress.

tic = A r l Y b A /21/2~'F

2158--VOLUME 16A, DECEMBER 1985

[95]

and to = 3tlZArlYbAZ/27r z.

[96]

Substitution of Eqs. [95] and [96] into Eq. [34], with Ls = A, results in Zcs = O. 122(A rty)5/3(Ab/F) 2/3.

[97]

The numerical coefficient in Eq. [97] depends upon the character of the dislocation, being 0.041, 0.026, and 0.122 for screw, mixed (~ = 60 deg) and edge dislocations, respectively. Since F is smallest for edges and largest for screws, rcs is significantly higher for edges than for mixed and screw dislocations. Equation [97] predicts a dependence of the CRSS on A and ~7 which is somewhat less strong than that predicted by Cahn's theory, and a much weaker dependence on A (but not independent of it as predicted by the theory of Kato et al.). Furthermore, Eq. [97] at least contains the important parameters characterizing flexibility of the dislocation line. It is worth pointing out that the solutions to the shape of the dislocation determined by both Cahn and Kato et al. are virtually unaffected by the applied stress. Thus, the dislocation interacts with the same obstacles under zero applied stress that it does at the CRSS, which is consistent with the criteria 12 for the application of Mott statistics. B. Analysis o f E x p e r i m e n t a l Data

There are a large number of alloy systems that age harden by spinodal decomposition for which data on the yield stress or CRSS are available as a function of aging time. Representative examples are found from the following alloy systems; A1-Zn; 149 Au-Pt; 15~ Cu-Ni-Sn; 15I'm Cu-Ni-Fe; 153'154'155 CuTi. 156.157.158As noted by Wagner, 142the typical age hardening response is a rapid increase in the yield strength followed by a plateau region, where the strength remains nearly constant as a function of aging time. The Cu-Ti system is quite interesting in this regard, since dilute alloys containing between 0.8 and 1.2 pct Ti exhibit a relative maximum in the yield strength at short aging times, followed by a minimum and then an increase to even higher strength levels as aging proceeds 159'16~at 300 ~ Thompson and Williams 161 have also observed similar behavior in a 3.3 pct Ti alloy aged at 500 ~ Explanations of this aging "anomaly" have been offered by Greggi and Soffa ~62 and Kratochv~l and Haasen.163 Both explanations involve replacement of the spinodal microstructure by the ordered Cu4Ti (fl') precipitates, although the details differ. Thompson and Williams, 161 on pointing out that the precipitation sequence is not yet fully understood, have concluded that there is no convincing explanation for the double aging peak. The magnitude of the strengthening in the alloys mentioned above is considerable. Unfortunately, quantitative tests of the extant theories are virtually impossible for most of them due to lack of sufficient data on both amplitude and wavelength during aging. The most notable exceptions to this are the data of Butler and Thomas ~53 and Livak and Thomas TM on spinodally decomposed Cu-Ni-Fe alloys; these are analyzed below. The data on Cu-Ti and Cu-Ni-Sn alloys, which have been compared by Kato et al. 144 with their theory are not being deliberately ignored here; it is simply that the raw data used by them, which have also

METALLURGICAL TRANSACTIONS A

been reported in various forms previously by Ditchek and Schwartz ~64in another context, are not accessible from the published figures and tables. Butler and Thomas and Livak and Thomas used magnetic measurements of the ferromagnetic Curie temperature to determine the Cu contents of the Ni + Fe rich regions of the spinodally decomposed alloys, thereby determining A. Measurements of h were made from transmission electron micrographs and from sideband spacings in electron diffraction patterns. The data of Butler and Thomas 153 and Livak and Thomas ]54 are compared with the predictions of Eq. [97] in Figure 22. In analyzing the data the results on the yield stress of the polycrystalline samples tested were converted to values of the CRSS using a Taylor factor of 3. The CRSS was taken as the difference between those of the aged and quenched alloys. Values of ~7 were determined from the lattice parameters measured by Bradley et al., 165and values of G, v, and Y were obtained from Eqs. [58], [60], and [93] by taking weighted averages of the single crystal elastic constants of the pure metals,44 assuming that G is identical for Ni and 7-Fe. Values of h and A were obtained directly from the published curves, although it was necessary to convert Curie points to concentrations for data on the alloy Butler and Thomas aged at 775 ~ The values of the parameters used are summarized in Tables IX and X. With the exception of the 32 pct Cu alloy of Livak and Thomas, the results in Figure 22 are in good agreement with the behavior theoretically predicted by Eq. [97]. The reasons for the discrepancy in that one instance are unknown. A comparison between the data and the other theories is presented in Table X, along with the values of all the variables and other parameters used in the calculations. It is seen that the values of Zcs predicted by Cahn's theory (Eq. [91]) flank the experimentally measured values of A z s (except for the 32 pct Cu alloy of Livak and Thomas), the values calculated for screw dislocations generally being considerably lower and those for edge dislocations somewhat higher than the measured values. The values of ~'cs calculated from the theory of Kato et al. (Eq. [94]) far exceed the measured values of A~-s. In some instances they do not vary appreciably with aging time, owing undoubtedly to the predicted independence of ~'cs on h. The theories and analyses of the data previously presented are restricted to the underaged situation in which wave squaring effects, i.e., the tendency of the matrix and the precipitate phases to be separated by sharp interfaces, are not yet important. Theories of strengthening during the latter stages of spinodal decomposition have not yet been developed, although Dahlgren ~55derived an equation which predicts that the yield strength is dependent only on the product A~?Y. In this sense, his theory is not too different from that Table IX.

120

AGING

100

Cu

N1

Fe

CI 51.5 0 51.5

33.5 33.5 45.5 27,0

15.0 15.0 22.5 9.0

032.0 64.0

~ 8O

TEMP (~ 625

775 625

5

i

6o

D

40

~

20

~

o

~ 0

0

0

20

1 40

I

I

L

60

I

I

80

I

J

100

I 126

A'r s (MN/m 2) Fig. 22--Comparison between experimental results on hardening by spinodal decomposition and the theoretical predictions of Eq. [97]. The circles and squares represent the data of Butler and Thomas. 153The other data are from Livak and Thomas. ~54

of Kato et al.; TM in fact, the CRSS of Dahlgren's theory is exactly a factor of six greater than theirs. Dahlgren's equation therefore predicts very large plateau values of the CRSS and vastly overestimates the strength of the alloys in this regime. VI.

DISCUSSION AND SUMMARY

It should be apparent that considerable advances have occurred over the past dozen years in our level of understanding of the details of precipitation hardening. This is particularly true of the theoretical aspects of the statistics of the interaction between point obstacles and dislocations. It also seems reasonably safe to conclude that when the microstructure of an age hardened alloy closely approximates the model serving as the basis for a theory, the agreement between theory and experiment is good. The cases in point are order hardening and stacking-fault strengthening (exemplifying the strengthening due to localized obstacles) and hardening by spinodal decomposition (exemplifying strengthening by diffuse obstacles). The former mechanisms involve interactions between dislocations and only those precipitates that physically intersect their glide planes, while the spinodal microstructure is ideally diffuse and capable of

Values of the Parameters Used in Estimating the Theoretical Strengthening by Spinodal Decomposition in Cu-Ni-Fe Alloys. U Was Calculated from Equation [54] with ~ = n/2 for Edge Dislocations, = 0 for Screws, and ~ = ~r/3 for Mixed Dislocations, Using In(A/to) = 4.

Alloy

F in Units of Gb 2

Cu

Ni

Fe

-q

Y ( G N / m 2)

G ( G N / m 2)

u

Edge

Screw

Mixed

51.5 32.0 64.0

33.5 45.5 27.0

15.0 22.5 9.0

0.0130 0.0127 0.0184

166.2 186.0 153.3

45.9 52.0 41.9

0.394 0.386 0.399

0.111 0.118 0.107

0.732 0.718 0.741

0.267 0.268 0.266

METALLURGICAL TRANSACTIONS A

VOLUME 16A, DECEMBER 1985--2159

Table X. Values of the Parameters Used to Calculate ~'cs in This Study (Equation [97]), the Theory of C a h n 14a (Equation [91]), and the Theory of Kato et al. T M (KMS Equation [94]). The Data in the First Two Rows Were Taken from Butler and Thomas. is3 The Rest Are from Livak and Thomas. T M A Is (C - Co)/3, Where Co Is the Initial Cu Concentration of the Alloy.

Cu 51.5

Alloy Ni 33.5

Fe 15.0

51.5

33.5

15.0

775

32.0

45.5

22.5

625

64.0

27.0

9.0

625

Aging Temp. (~ 625

A (nm) 6.1 8.2 11.0 16.3 15.2 21.5 25.9 44.4 5.0 8.2 7.1 12.0 7.0 7.0 5.1 13.5

trapping dislocations so that the CRSS is governed by the statistics of the trapping process rather than by encounters with new obstacles. Strengthening by discrete precipitates which interact with dislocations over a range which substantially exceeds the particle size is poorly understood. Coherency hardening and, to some extent, modulus hardening belong in this category. This statement may be somewhat unfair, because the ideal alloy system for testing the theories of strengthening by either of these mechanisms has not been produced. What is needed are model alloys in which only the mechanism under consideration can contribute significantly to the strengthening observed. It can be argued that the misfit is so large in Cu3Au hardened by Co precipitates (Table II) that the role of other strengthening mechanisms should be minimal. Nevertheless, even in this case the theory of coherency strengthening fails miserably. Despite the failure of the theory, there are numerous alloy systems in which coherency strains are large enough to produce significant strengthening. The theories as presently formulated incorporate the interaction between straight pure edge dislocations and spherical coherent precipitates because this combination results in the largest interaction force resisting dislocation motion. In many instances, as previously noted, the microstructures reveal that the precipitates interact primarily with screw dislocations. This poses an immediate dilemma, because the interaction force between a straight infinitely long screw dislocation and a spherical coherent precipitate in an isotropic medium is zero. 65Zero strengthening, therefore, must be predicted on the basis of such a simplified model. Now, the force of interaction between a spherical precipitate of shear modulus Gp and an infinitely long straight screw dislocation is by no means negligible. While the interaction force has been calculated only for a slip plane passing through the center of the precipitate, rendering the solution to the problem incomplete at the present time, it is reasonable to assume that an appropriately calculated average force will cause an initially straight screw dislocation to bend. As 2160--VOLUME 16A,DECEMBER1985

A 0.107 0.115 0.121 0.131 0.090 0.090 0.089 0.095 0.037 0.042 0.052 0.069 0.059 0.091 0.114 0.124

A'rs 23.2 38.5 56.2 61.7 24.5 36.3 42.1 43.4 8.2 19.3 46.4 74.5 14.7 35.0 67.6 79.7

This Work 30.1 41.0 54.9 80.6 41.0 51.8 57.5 91.9 4.5 7.8 10.3 23.0 20.8 42.9 50.5 110.1

~'cs (MN/m 2) Cahn Screw Edge 0.83 28.6 1.28 44.0 1.92 66.0 3.31 113.4 1.46 49.9 2.06 70.4 2.42 83.0 4.73 t62.2 0.09 2.9 0.18 6.1 0.24 8.2 0.71 24.3 0.54 18.8 1.30 44.4 1.49 50.8 4.59 157.1

KMS 94.7 101.0 107.0 115.3 79.0 79.0 78.3 83.7 35.3 40.3 50.0 66.3 68.3 105.3 131.7 142.3

the screw dislocation bends it will acquire edge character and, in so doing, will immediately enable the dislocation to interact with the coherency strains of that precipitate. In the copper base alloys investigated to date it is conceivable that this occurs during the early stages of aging. This scenario is consistent with the microstructural observations, and intuitively accounts for the fact that the magnitude of the strengthening observed is far smaller than that predicted by theories invoking only the interaction between edge dislocations and spherical coherent precipitates. In the absence of detailed calculations, this must remain speculative, but it is clear that such calculations are warranted if the mystery of coherency strengthening is ever to be solved. A potentially fruitful approach to solving this problem may involve the theory of Schwarz and Labusch. 32 This theory is clearly more applicable to a system of precipitates that can interact with dislocations over a range that exceeds the particle dimensions than it is for localized obstacles. In this latter context, the theory seems misapplied to data on hardening of alloys by ordered coherent precipitates. As seen in Figure 21, the theory of Schwarz and Labusch is meaningless when applied to the data on monocrystalline Ni-A1 alloys. Furthermore, despite the determined efforts of Haasen and Labusch, 133 Reppich et a1.,114 and Grfhlich et al. to6 to demonstrate the validity of this theory by comparison with results on y ' strengthening in various Nibase alloys, the agreement claimed is too contrived to be acceptable. Viewed dispassionately, the straightforward modifications of Friedel statistics required to incorporate the effect of finite obstacle size (Section II.D) are quite adequate for explaining the strengthening observed in the binary Ni-AI alloys studied to date. The major source of uncertainty in this development is the manner in which the influence of the trailing dislocation is accounted for theoretically. This requires an analysis that has not yet been made with any degree of satisfaction. It is anticipated, however, that a properly constructed theory will predict values of dH/LH that are generally much smaller than f. METALLURGICAL TRANSACTIONS A

While the data on underaged alloys in a large variety of y'containing alloys is convincingly due to order strengthening alone, there is a serious question as to the strengthening mechanism associated with alloys aged to the maximum strength condition. As demonstrated by Ardell et al. ,130 the data on peak-aged Ni-A1 alloys show a definite trend in that the observed values of A~-o.... increase with increasing temperature, and that this increase correlates quite well with the fact that the misfit parameter, e, also increases in these alloys with increasing temperature. A similar correlation in aged Cu-Co alloys had been noted earlier by Phillips.74 It has been suggested by several investigators ~31'~32that coherency strengthening is an important mechanism contributing to the strength of the y / y ' alloys. There is little doubt that this is true for alloys aged to the peak strength condition. The problem of superposition of strengthening mechanisms has not been dealt with so far in any detail for the simple reason that little progress has been made on this important aspect of age hardening. The likely combination of coherency hardening and modulus hardening in the Cu-base alloys studied to date is an obvious prospect for study. The combination of coherency hardening and order strengthening is another. Lee and Ardel1166studied this problem by solving the differential equation of Wiedersich 72 numerically, but also including the effect of the variable ~ in the expression for F (Eq. [54]). They used parameters relevant to y ' in Ni-A1 alloys and calculated the forces of interaction for the value of z at which a random slip plane intersects a sphere of radius r on average (z/r = 0.619). For weak particles the straight dislocation approximation is accurate, as expected. For strong particles the total resisting force was observed to be smaller than the maximum force due to order hardening alone, because strong particles cause the dislocation to bend so much that the coherency stress actually assists the applied stress in helping the dislocation shear the particle. This result is not consistent with the data in Table VI, which show that the experimental values of Azco.max already exceed the theoretically predicted ones. This final example illustrates the nagging inconsistencies that appear when theory is compared with experiment. Perfect agreement is probably too much to hope for at this stage, but it seems that what is needed now are advances in theory. Reliable data on well-characterized systems are available, but analytical theories covering the spectrum of dislocation precipitate interaction mechanisms, and computer simulations of dislocation motion through arrays of localized, but finite, obstacles are not. Until this effort has been made the uncertainties will remain with us.

APPENDIX

r2(z) dz -

~

nv

rc2 =

F3z(Z/r) dz,

F3z(~") d• -

1 b2L22F

fo

F3z(~") d~'.

(r*)2 = f;/33(ff)-~ = lo[T*(~')]2-~ .

[A3]

The calculation of Brown and Ham purports to include the effect of all the particles in the crystal in calculating the CRSS. Equation [A3], however, is the counterpart of Eq. [46] for a continuous distribution of obstacle strengths; i.e., it is equivalent to Eq. [47a] with g(t#c)&bc = dz/r. Now, dz/r in Eq. [A3] is simply the probability that a plane intersects a spherical particle of radius r at distance between z and z + dz from its center. Thus, the limits on the integral in Eq. [A1] notwithstanding, the averaging method of Brown and Ham is valid only over a range equivalent to the particle radius, because it is only over this range that the probability dz/r is finite. (The factor of 2 in Eq. [All actually doubles the range but halves the probability, producing the same result.) The integration over the range 1 < z/r < oo is mathematically possible, due to the inverse square dependence of Fmz o n ~" at large ~" (Eq. [75]), but in the context of this problem it has no physical significance (although it has a numerical consequence). Similar arguments apply to the calculations of Jannson and Melander. The difference between their result and that of Brown and Ham is that they chose their probability as dz/~s where ~ is the edge length of a cube which on average contains one particle, i.e., ~ 3 n v ---- 1. If we take Eq. [2] of Jannson and Melander, write ~3 = 2rL~, and take into account the factor of 0.8871 that appears in Eq. [27], we find that the values of X in Eq. [77] calculated by Brown and Ham and Jannson and Melander agree exactly. While their calculations appear to circumvent the difficulty of choosing a range of interaction, they fail to do so. LIST OF SYMBOLS A

a(ap) B

C(Co) C~ CsL, C'SL

cU

METALLURGICAL TRANSACTIONS A

fo

By virtue of Eqs. [3], [11], and [12], Eq. [A2] can be expressed as

[A1]

where nv is the number of particles per unit volume. Incorporated implicitly in Eq. [A1] is the value of the line tension F = Gb2/2 used by Brown and Ham. Assuming that the particles are monodisperse, so that r -- (r) and

3f Fb24~r 2

[A21

b(bp)

We examine here the approach taken by Brown and Ham" and Jannson and Melander 68 toward the averaging process used in calculating Eq. [77]. Brown and Ham start with the equation (expressed in the notation of this paper)

"cce= 2

47rr3nv/3 = f, Eq. [A1] can be rewritten as

D dl, dll

Amplitude of a composition modulation during spinodal decomposition. a(ap) Lattice parameter of the matrix (precipitate). Dimensionless variable in the order strengthening theory of Ardell, Munjal, and Chellman. Burgers vectors of total (partial) dislocations. Solute concentration (initial value) in atom fraction. A constant in an expression for the maximum interaction force in modulus hardening. Empirical constants in the theory of Schwarz and Labusch for energy conserving and energy storing interactions, respectively. Single crystal elastic moduli. Spacing of the dislocation pair in order strengthening. Effective particle diameters for the leading (I) and trailing (II) dislocations of a pair at the VOLUME 16A, DECEMBER 1985--2161

G.

critical configuration during order strengthening. Maximum force of interaction that an obstacle can withstand. Maximum force of interaction between a precipitate and a dislocation on a slip plane at distance z from the particle center. Volume fraction of precipitate. Shear modulus on the slip plane in the slip direction of the matrix of an fcc crystal. Shear modulus of the matrix of an isotropic crystal. Shear modulus of the precipitate.

AG

t G . - ~1.

F~ F~ f G, Gnl Giso

g( ) h

J~ J K k L Le L~t Ls LI, LII

l

m

n

n$ #iv

P q

R(R*) R<(R*) r ro r$

s(s*)

Distribution function of the variable contained within the parentheses. Twice the amplitude of a zig-zag dislocation in the presence of attractive obstacles. A constant in Cahn's theory of spinodal decomposition. The number of dislocations in a procession during the shearing of ordered precipitates. Force of interaction between two partial dislocations of Burgers vector bp. Index representing successive obstacles encountered by a dislocation in circle rolling. Effective spacing of obstacles along a dislocation at the critical configuration. The Friedel spacing. The Mott spacing. The square lattice spacing. Effective spacings of obstacles along the leading and trailing dislocations of a pair at the critical configuration in order strengthening. Edge length of a cube containing one obstacle on average. Effective particle diameter for the trailing partial at the critical configuration in stackingfault strengthening. An exponent appearing in equations for the maximum force and CRSS due to modulus hardening. Number of obstacles contained within a dimensionless search area generated by circle rolling. Number of obstacles per unit area in the glide plane. Number of obstacles or precipitates per unit volume. An adjustable parameter in the Hfither and Reppich theory of order strengthening by strong pair coupling. An exponent in an empirical addition rule. Radius (dimensionless radius) of a curved dislocation. As above, at the critical breaking stress. Radius of a spherical precipitate. Inner cut-off distance in the expression for the dislocation line energy. Planar radius of a spherical precipitate. Area (dimensionless area) swept out by a dislocation in circle rolling.

2162--VOLUME 16A, DECEMBER 1985

s~* SF So Soi

u0 u

V

w.(w.) x~ x

Y Y z t~

t3< F Fe, Fs

Ys "Yapb

Ygm Yee

( ~',s) A,), 6 E

Dimensionless critical area swept out by a dislocation in circle rolling, containing one obstacle. Area swept out by a dislocation in Friedel statistics. Dimensionless, radius-independent search area at the critical configuration in circle rolling. As above, but containing on average one obstacle of type i in a random mixture of distinct obstacles. Energy of interaction between an obstacle and a dislocation. A dimensionless variable in the order hardening theory of Ardell, Munjal, and Chellman, equal to 2(rs)/LF. A parameter related to the maximum force of interaction between a flexible edge dislocation and a spherical coherent precipitate. Ribbon width of a stacking-fault in the matrix (precipitate). Fraction of obstacles of type i in the slip plane; an areal concentration. Spatial coordinate parallel to an initially straight dislocation line. An elastic modulus resisting lattice deformation during spinodal decomposition. Spatial coordinate measuring the displacements of a dislocation from a straight line. Distance between the center of a spherical precipitate and the slip plane of a dislocation. A coefficient in the expression for the dislocation line tension. Dimensionless critical force exerted by a dislocation on an obstacle. Line tension of a dislocation. Line tension of a pure edge, screw dislocation. Energy of a matrix-precipitate interface created by slip. Antiphase boundary energy on the slip plane of an ordered precipitate. Stacking-fault energy of the matrix. Stacking-fault energy of the precipitate. Average stacking-fault energy of the matrix and precipitate phases. I ")',,'Sin -- ~':spl 9

Fractional misfit between the lattice parameters of the matrix and precipitate phases. Constrained strain; the fractional misfit between an in situ coherent precipitate and the matrix. z/r.

~7 r#o O< A tt

A measure of the lattice strains produced during spinodal decomposition. Parameter measuring the ratio of the obstacle range and the square root of its breaking strength. Critical angle through which the dislocation turns at an obstacle in, e.g., circle rolling. Outer cut-off distance in the expression for the dislocation line energy. Wavelength of a composition modulation during spinodal decomposition.

METALLURGICAL TRANSACTIONS A

Poisson's ratio of the matrix (precipitate). Angle between the dislocation line and its Burgers vector. CRSS (dimensionless CRSS) predicted theoretically. z** Reduced theoretical CRSS; T_./03/2 c/Pc 9 I"** Experimentally determined value of r** relevant to data on order hardening. CRSS (dimensionless CRSS) predicted theoretically for obstacles of type i in a random mixture of distinct obstacles. Theoretically predicted CRSS due to chemical TcC hardening. Theoretically predicted CRSS due to modulus TcG hardening. Theoretically predicted CRSS due to order Tco hardening. Theoretically predicted by CRSS due to harden%s ing by spinodal decomposition. Theoretically predicted CRSS due to coher"l"ce ency hardening. Theoretically predicted CRSS due to stackingTc7 fault strengthening. Contribution of the solid solution matrix to the "Fss CRSS. Contributions of Class 1 and Class 2 precipi7"O1, '/'02 tates to the CRSS in alloys aged to contain bimodal T' particle size distributions. Experimentally measured CRSS of a crystal. TT hz Contribution of precipitates to the CRSS determined experimentally. As above, in order strengthening. AT o As above, in hardening by spinodal ATs decomposition. A'/'~ As above, in coherency hardening. As above, in stacking-fault strengthening. AT~ Angle between tangential directions of the 4, dislocation line at successive obstacles in circle rolling. Similar to ~b, but for obstacles of type i in a random mixture of distinct obstacles. Lower limit on ~b which defines a boundary of 4,0 the search area in circle rolling that contains one obstacle on average. A constant in the theory of coherency X strengthening. g,c Critical breaking angle (cusp angle) included between adjacent arms of the dislocation at an obstacle. The difference between the maximum and minimum values of g'c. o~(o~*) Range (dimensionless range) of interaction between an obstacle and a dislocation. (> Symbols denoting the average value of the quantity contained within. max, min, Subscripts denoting maximum, minimum, experimental, and/or theoretical values of a exp, parameter or variable. theor ACKNOWLEDGMENTS The author is grateful to the Department of Energy, Office of Basic Energy Sciences and the National Science FoundaMETALLURGICALTRANSACTIONSA

tion for supporting his experimental research on this topic throughout much of the 70's (DOE was then, of course, AEC and ERDA, in that order). He was helped immeasurably in preparing this paper by Mr. J.C. Huang, and also thanks K. Ono and U.F. Kocks for several helpful discussions. He is particularly grateful to L. H. Schwartz and J.W. Cahn for extremely valuable correspondence and discussions on spinodal decomposition.

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81. M. Hansen: Constitution of Binary Alloys, 2nd ed., McGraw-Hill Book Co., New York, NY, 1958, p. 596. 82. K. Nakajima and K. Numakura: Phil. Mag., 1965, vol. 12, p, 361. 83. P.A. Flinn, G.M. McManus, and J.A. Rayne: J. Phys. Chem. Solids, 1960, vol. 15, p. 189. 84. Z.S. Basinski, W. Hume-Rothery, and A. L. Sutton: Proc. Roy. Soc. (London) A, 1955, vol. 229, p. 459. 85. Z.S. Basinski and J. W. Christian: J. Inst. Metals, 1952, vol. 80, p. 659. 86. W.B. Pearson: A Handbook of Lattice Spacings and Structures of Metals andAlloys, Pergamon Press, Oxford, England, 1958, p. 733. 87. R.D. Dragsdoff: J. Appl. Phys., 1960, vol. 31, p. 434. 88. H. Haberkom: Mater. Sci. and Engr., 1968/69, vol. 3, p. 302. 89. N.J. Long, M.H. Loretto, and C.H. Lloyd: Acta MetaU., 1980, vol. 28, p. 709. 90. H. Haberkorn: Phys. Stat. Sol., 1966, vol. 15, p. 153. 91. H. Haberkom and V. Gerold: Phys. Stat. Sol., 1966, vol. 15, p. 167. 92. P.B. Hirsch and F. J. Humphries: Physics of Strength and Plasticity, A.S. Argon, ed., M.I.T. Press, Cambridge, MA, 1969, p. 189. 93. H. Gleiter and E. Hornbogen: Phys. Stat. Sol., 1965, vol. 12, p. 235. 94. J.L. Castagn6: J. de Physique, 1966, vol. 27, pp. C3-233. 95. R.K. Ham: Trans. Japan. Inst. Metals, 1968, vol. 9 supplement, p. 52. 96. P. Guyot: Phil. Mag., 1971, vol. 24, p. 987. 97. D. Raynor and J. M. Silcock: Metal Sci. J., 1970, vol. 4, p. 121. 98. W. Hiither and B. Reppich: Z. MetaUkde, 1978, vol. 69, p. 628. 99. J.M. Oblak, D. F. Paulonis, and D. S. Duvall: Metall. Trans., 1974, vol. 5, p. 143. 100. J. Greggi and W. A. Soffa: Strength of Metals andAlloys, Proc. 5th Int. Conf. on the Strength of Metals and Alloys, P. Haasen, V. Gerold, and G. Kostorz, eds., Pergamon Press, Oxford, England, 1979, vol. 1, p. 651. 101. A.J. Ardell, V. Munjal, and D. J. Chellman: Metall. Trans. A, 1976, vol. 7A, p. 1263. 102. A.J. Ardell: Metal. Sci. J., 1980, vol. 14, p. 221. 103. V. Munjal and A.J. Ardell: Acta Metall., 1975, vol. 23, p. 513. 104. N.T. Travina and G.I. Nosova: Phys. Met. Metallogr., 1970, vol. 29, p. 119. 105. V.A. Phillips: Acta Metall., 1966, vol. 14, p. 1533. 106. M. Gr6hlich, P. Haasen, and G. Frommeyer: Scripta Metall., 1982, vol. 16, p. 367. 107. M.M. Dawance, D.H. Ben Israel, and M.E. Fine: Acta MetaU., 1964, vol. 12, p. 705. 108. M.C. Chaturvedi, D.J. Lloyd, and D.W. Chung: Metal Sci. J., 1976, vol. 10, p. 373. 109. L.K. Singhal and J. W. Martin: ActaMetaU., 1968, vol. 16, p. 947. 110. A. Melander and P./~. Persson: Metal Sci. J., 1978, vol. 13, p. 391. 111. J.L. Castagn6, E Lecroisey, and A. Pineau: C. R. Acad. Sci. Paris, 1968, vol. 266, p. 510. 112. A. Pineau, E Lecroisey, J.L. Castagn6, and M. Sindzingre: Acta Metall., 1969, vol. 17, p. 905. 113. V. Martens and E. Nembach: Acta Metall., 1975, vol. 23, p. 149. 114. B. Reppich, P. Schepp, and G. Wehner: Acta Metall., 1982, vol. 30, p. 95. 115. A. Thompson and J.A. Brooks: Acta Metall., 1982, vol. 30, p. 2197. 116. B. Noble, S. J. Harris, and K. Dinsdale: Metal Sci. J., 1982, vol. 16, p. 425. 117. M. Rembges, P. Haasen, and L. Schultz: Z. Metallke, 1976, vol. 67, p. 330. 118. J.W. Goodrum and B. G. LeFevre: Metall. Trans. A, 1977, vol. 8A, p. 939. 119. W. Hiither and B. Reppich: Mater. Sci. and Engr., 1979, vol. 39, p. 247. 120. M.C. Chaturvedi and D.W. Chung: Metall. Trans. A, 1981, vol. 12A, p. 77. 121. G.G. Brown and J.A. Whiteman: J. Aust. Inst. Metals, 1969, vol. 14, p. 217. 122. D.H. Jack and R. W. K. Honeycombe: Acta Metall., 1972, vol. 20, p. 787. 123. I.O. Smith and M.G. White: Metall. Trans. A, 1976, vol. 7A, p. 293. 124. R. Taillard and A. Pineau: Mater. Sci. and Engr., 1982, vol. 56, p. 219. 125. A.J. Ardell and R.B. Nicholson: J. Phys. Chem. Solids, 1966, vol. 27, p. 1793. 126. D.J. Chellman and A. J. Ardell: Acta MetaU., 1974, vol. 22, p. 577. METALLURGICAL TRANSACTIONS A

127. A.J. Ardell: Acta MetaU., 1968, vol. 16, p. 511. 128. E.Z. Vintaikin: Sov. Phys. Dokl., 1966, vol. 11, p. 91. 129. H. Pottebohm, G. Neite, and E. Nembach: Mater. Sci. and Engr., 1983, vol. 60, p. 189. 130. A.J. Ardell, D. J. Chellman, and V. Munjal: Proc. 4th Int. Conf. on the Strength of Metals and Alloys, Lab. de Phys. du Solide, eds., Nancy, France, 1976, vol. 1, p. 209. 131. R.F. Decker and J. R. Mihalisin: Trans. ASM, 1969, vol. 62, p. 481. 132. B.A. Parker: J. Aust. Inst. Metals, 1969, vol. 14, p. 321. 133. P. Haasen and R. Labusch: Strength of Metals and Alloys, Proc. 5th Int. Conf. on the Strength of Metals and Alloys, P. Haasen, V. Gerold, and G. Kostorz, eds., Pergamon Press, Oxford, England, 1979, vol. 1, p. 639. 134. E. Nembach: Z. Metallke, 1981, vol. 72, p. 401. 135. B. Reppich: Acta Metall., 1982, vol. 30, p. 87. 136. I.M. Lifshitz and V.V. Slyozov: J. Phys. Chem. Solids, 1961, vol. 19, p. 35. 137. C. Wagner: E. Elektrochem., 1961, vol. 65, p. 581. 138. A. Melander and B. Jansson: Strength of Metals and Alloys, Proc. 5th Int. Conf. on the Strength of Metals and Alloys, P. Haasen, V. Gerold, and G. Kostorz, eds., Pergamon Press, Oxford, England, 1979, vol. 1, p. 627. 139. V. Munjal and A.J. Ardell: Acta Metall., 1976, vol. 24, p. 827. 140. E. Nembach and C.-K. Chow: Mater. Sci. and Engr., 1978, vol. 36, p. 271. 141. D.J. Cheltman and A. J. Ardell: Proc. 4th Int. Conf. on the Strength of Metals andAlloys, Lab. de Phys. du Solide, eds., Nancy, France, 1976, vol. 1, p. 219. 142. R. Wagner: Czech. J. Phys. B, 1981, vol. 31, p. 198. 143. J.W. Cahn: Acta Metall., 1963, vol. 11, p. 1275. 144. M. Kato, T. Moil, and L. H. Schwartz: Acta Metall., 1980, vol. 28, p. 285. 145. D.N. Ghista and W.D. Nix: Mater. Sci. and Engr., 1969, vol. 3, p. 293. 146. Y. Hanai, T. Miyazaki, and H. Moil: J. Mater. Sci., 1979, vol. 14, p. 599. 147. B. Ditchek and L. H. Schwartz: Proc. 4th Int. Conf. on the Strength of Metals andAlloys, Lab. de Phys. du Solide, eds., Nancy, France, 1976, vol. 2, p. 593.

METALLURGICALTRANSACTIONS A

148. M. Kato, T. Moil, and L. H. Schwartz: Mater. Sci. andEngr., 1981, vol. 51, p. 25. 149. D.L. Douglass and T.W. Barbee: J. Mater. Sci, 1969, vol. 4, p. 121. 150. R.W. Carpenter: Acta Metall., 1967, vol. 15, p. 1297. 15 I. L.H. Schwartz and J. T. Plewes: Acta Metall., 1974, vol. 22, p. 91 I. 152. B.G. LeFevre, A.T. D'Annessa, and D. Kalish: Metall. Trans. A, 1978, vol. 9A, p. 577. 153. E.P. Butler and G. Thomas: Acta Metall., 1970, vol. 18, p. 347. 154. R.J. Livak and G. Thomas: Acta Metall., 1971, vol. 19, p. 497. 155. S.D. Dahlgren: MetaU. Trans. A, 1977, vol. 8A, p. 347. 156. T. Miyazaki, E. Yajima, and H. Suga: Trans. Japan. Inst.Metals, 1971, vol. 12, p. 119. 157. R. Wagner: Strength of Metals and Alloys, Proc. 5th Int. Conf. on the Strength of Metals and Alloys, P. Haasen, V. Gerold, and G. Kostorz, eds., Pergamon Press, Oxford, England, 1979, vol. I, p. 645. 158. A. Datta and W. A. Sofia: Acta Metall., 1976, vol. 24, p. 987. 159. P. Kratochv~l, M. Saxlov;i, and J. Pe~i~ka: Strength of Metals and Alloys, Proc. 5th Int. Conf. on the Strength of Metals and Alloys, P. Haasen, V. Gerold, and G. Kostorz, eds., Pergamon Press, Oxford, England, 1979, vol: 1, p. 687. 160. V.I. Dotsenko, P. Kratochvil, J. Pe~i~ka, and B. Wielke: Czech. J. Phys. B, 1981, vol. 31, p. 209. 161. A.W. Thompson and J.C. Williams: Metall. Trans. A, 1984, vol. 15A, p. 931. 162. J. Greggi and W.A. Sofia: Scripta MetaU., 1980, vol. 14, p. 649. 163. P. Kratochv~l and P. Haasen: Scripta MetaU., 1982, vol. 16, p. 197. 164. B. Ditchek and L.H. Schwartz: Ann. Rev. Mater. Sci., R.A. Huggins, R. H. Bube, and D. A. Vermilyea, eds., Annual Reviews Inc., Palo Alto, CA, 1979, vol. 9, p. 219. 165. A.J. Bradley, W. E Cox, and H.J. Goldschmidt: J. Inst. Metals, 1941, vol. 67, p. 189. 166. E.J. Lee and A.J. Ardell: Strength of Metals and Alloys, Proc. 5th Int. Conf. on the Strength of Metals and Alloys, P. Haasen, V. Gerold, and G. Kostorz, eds., Pergamon Press, Oxford, England, 1979, vol. 1, p. 633,

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