A dislocation network theory of Harper-Dorn creep—I. Steady state creep of monocrystalline Al

,411~ mr/u//. Vol. 34, No. II. pp. 2411 2423. 1986 Printed in Great Britcrm. All rights reserved

C‘opynght

000 I-h I60 X0 $3.00 + 0.00 I.td

t I9)x6PergmonJournals

A DISLOCATION NETWORK THEORY OF HARPER-DORN CREEP-I. STEADY STATE CREEP OF MONOCRYSTALLINE Al A. J. ARDELL Department

and S. S. LEE

of Materiais Science and Engineering, School of Engineering and Applied University of California, Los Angeles. CA 90024. U.S.A.

Science.

(Receiaed 6 Junuur_v 1986)

Abstract-A recently developed dislocation network theory of high temperature creep is moditied for consistency with the equally recent experimental observation that the dislocation link length distribution Al deformed in compression at 920 K that develops during Harper-Dorn (H-D) creep of monocrystalline contains no segments that are long enough to glide or climb freely. H-D creep is therefore a phenomenon in which all the plastic strain in the crystal is generated by the process of stress-assisted dislocation network coarsening, during which the glide and climb of dislocations is constrained by the requirement that the forces acting on individual links are balanced by the line tension of the links. It is not possible to test all the predictions of the theory because not all the required input information is available. In particular. the kinetic law govermng the growth rate of individual links is not known. Nonetheless, using an established and well-known equation as a first guess, it is possible to simulate the experimentally observed distribution of dislocation link lengths with reasonable semiquantitative accuracy. In other respects, especially regarding the steady state creep rate, the theory is in excellent agreement with the experimental resuIts. R&urn&-Now modifions une theorie de disiocations d&veloppt-e r&emment pour le &age Li haute tempirature afin de la rendre compatible avec I’observation exptrimentale tout aussi r&ente qui suit: ia distribution des longueurs de segments de dislocation qui se dtveloppe durant le fluagc de type Harper -Darn (H-D) de monocristaux d’aluminium d&form&s en compression ri 920 K ne contient pas de segments assez longs pour glisser ou monter librement. Le fluage H- D est done un ph&nomkne dans lequel toute la dtformation plastique du monocristal est engendrte par le phbnomene de grossissement aid& par la contrainte du rkseau de dislocations; durant ce grossissement ie glissement et la montte des disiocattons ne sont pas libres du fait que les forces agissant sur les segments individuels doivent Ptre Cquilibrkes par la tension de ligne des segments. 11 n’est pas possible de tester toutes les prCdictions de La theorie parce que 1’0” ne dispose pas de toute I’information de depart ntcessaire. En particulier, on ne connait pas ia loi cinbtique qui regit la vitesse de croissance des segments individuels. Cependant, en utilisant en premiPre hypothgse une Equation bien connue, il est possible dc simuler la rkpartition de longueurs de segments de dislocation observCe expCrimentalement avec une prBcision semi-quantative raisonnable, A d’autres ri-gards, surtout en ce qui concerne la vitesse de fluage stationnaire, la thiorie est en excellent accord avec les &sultats expkrimentaux. Zusammenfassung-Eine vor kurzem auf der Basis von Versetzungsnetzwerken entwickelte Theorie des Hochtemperaturkriechens wird so modifiziert, da13 sie mit neuen Beobachtungen vertraglich ist. Diese Bcobachtungen beteffen die Lgngenverteilung der Versetzungssegmente, die such wihrend des Harper Dorn-Kriechens von Al-Einkristallen unter Druckbelastung bei 920 K einstelit: danach gibt es keine Segmente, die fiir freies Gleiten oder Klettern lang genug w6ren. Das Harper-Dorn-Kriechen ist daher ein Vorgang, bei dem die gesamte pkdstische Dehnung des Kristalles iiber eine spannungsu~~terst~tzte Vergr~berung des Ve~tzungsnetzwerkes entsteht. Wghrend dieser Vergr~bcrung ist das Gleiten und Klettern der Versetzungen dadurch eingeschrankt, da13 die auf die enzetnen Segmente einwirkenden Spannungen durch die Linienspannung der Segmente kompcnsiert wird. Es kiinnen nicht samtliche Aussagen der Theorie gepriift werden, da nicht alle Eingangsdaten bekannt sind. Insbesondere ist das kinetische Gesetz, welches die Wachstumsrate der einzelnen Segmente bestimmt, nicht bekannt. Immerhin lal3t sich mit einer bekannten Gleichung in erster Naherung die experimentell beobachtetc Verteilung der Langen der Versetzungssegmente mit hinreichender halbquantitativer Genauigkeit abschatzen. In anderer Hinsicht, insbesondere hinsichtlich der stationaren Kriechrate. stimmt die Theorie mit den Expcrimenten ausgezeichnet iiberein

1. INTRODUCTION

melting temperature) and very low reduced stresses (a/G < 5 x IO 6, where G is the shear modulus) by Harper and Dorn [I]. has now been observed by nnmcrons other investigators to occur in a wide variety of materials over a fairly large spectrum of is the

The unexpectedly rapid rate of Newtonian steady state creep of Al, observed originally at very high homologous temperatures (T/T,, > 0.95,where T, 2411

2412

ARDELL and LEE: THEORY OF HARPER-DORN CREEP-I.

experimental conditions. Specifically, Harper-Dorn (H-D) creep, as it is now known, has been observed under high-temperature, low-stress conditions in the pure metals Al [14], Pb [3], Sn [3] and Ag [5], several Al-Mg alloys [6-91, and the alloy Pb-9%Sn [lo], at somewhat lower homologous temperatures (T/T, = 0.87) in the inorganic compound KZnF, [1 11, and at considerably lower homologous temperatures (T/T,,, around 0.5) in the pure metals cc-Ti [12], /?-Co [13], E-Fe [14] and cc-Zr [15] and the oxide CaO [16]. The microstructural characteristics of H-D creep have been investigated and summarized by Yavari et al. [9] and Mohamed and Ginter [4]. The main observations are that the steady state dislocations density, ps, is low (the order of lo* mm-*) and independent of Q, and that the steady state creep rate, i,, is independent of the grain size. Subgrain formation is rare in the steady state microstructures, although dislocation boundaries are observed during primary creep. These conclusions, based upon observations made on pure Al [4] and Al-5%Mg [9] and evaluation of earlier work, also appear to be characteristic of the deformation microstructure of a-Zr [15]; the value of p, in that metal was observed to be about 4 orders of magnitude higher than in Al. In all the materials in which the dislocation microstructure has been examined, either by etch-pitting techniques [2,4,5,9] or transmission electron microscopy [4,9, 161, it can be convincingly described as a three-dimensional network. There is to date no unanimity regarding the mechanism of H-D creep, although it is generally accepted that it involves the diffusion-controlled glide or climb of dislocations. At high homologous temperatures both of these processes are controlled by lattice diffusion, while at lower temperatures dislocation pipe diffusion has been invoked as the rate controlling process, at least in a-Fe [14, 17, 181and cr-Zr [15]. A detailed evaluation of the mechanisms of H-D creep postulated prior to 1982 was performed by Langdon and Yavari [19], who pointed out the shortcomings of these early suggestions and proposed themselves that H-D creep was controlled by the climb of edge dislocations saturated by vacancies. Subsequent to the work of Langdon and Yavari there have been other suggestions regarding the mechanism of H-D creep. Weertman and Blacic [20] have argued that H-D creep may be an artifact of the temperature cycling produced by any temperature controller during elevated temperature testing. Wu and Sherby [21] have demonstrated that the data on Al, Pb and Sn can be fitted to a universal curve spanning several orders of magnitude stress by using Garofalo’s [22] steady

of reduced state creep

Tit is vital to elaborate on the significance of the terminology “free glide”. In fact, it is essential to consider the meaning of “free climb” as well, but we postpone this to the next section.

STEADY STATE

equation, modified for the effect of an internal stress. In their view there is nothing special about H-D creep; it is simply a special case of normal (i.e. power law) creep that arises when the applied stress becomes smaller than the internal stress. Finally, Raman and Raj [23] have noted a correlation between specimen thickness (relative to the grain size) and i, during H-D creep, i, increasing as the ratio of thickness to grain size increases. This observation has prompted Raj [24] to suggest that surface dislocation sources may play an important role in the H-D creep process. Whatever the merits or shortcomings of these various proposed mechanisms [19-21,241, they are all concerned with the steady state mechanical behavior and microstructure representative of H-D creep. Virtually no effort has been made to construct a model consistent with the observed primary creep behavior, which is characterized by a creep rate that decreases to the final steady state value, resembling in this sense the behavior obeyed by samples deformed at higher stresses in the power law creep regime. Such behavior is typical of samples deformed at high [l-3,8,9], intermediate [1l] and low [14-16,251 homologous temperatures. Furthermore, there has been no attempt to describe the H-D creep process in light of the clearly recognized and acknowledged network nature of the dislocation microstructure. Lee and Ardell [26] recently reported the results of measurements of the distributions of dislocations link lengths in specimens of monocrystalline Al deformed in compression in the H-D creep regime. We showed that in both the primary and steady state stages of creep the distribution of dislocation link lengths is such that the longest link in the distribution, L,, is shorter than the length, Lc, of the arc that can be released from a Frank-Read (F-R) source. It was also shown that the transition from H-D to power law creep occurred at a stress large enough so that some links of length L exist in the distribution satisfying the condition L > L,; only then does the stress exponent rise to about 4.5, typical of power law creep. This observation is consistent with the earlier suggestion [4, lo] that the transition stress between the H-D and power law regimes is the value of Q required to operate a F-R source. The implication of this observation is profound, because it means that plastic flow in the H-D creep regime cannot be controlled by the free glide of dis1ocations.t Instead, H-D creep must be a consequence of the strain generated entirely by coarsening of the dislocation network, steady state obtaining via chance collisions between moving links that refine the network and further stimulate coarsening. The purpose of this paper (Part I) is to develop the theoretical apparatus enabling us to describe quantitatively the H-D creep process in terms of a dislocation network model, but we limit the scope to discussion of the steady state creep behavior. Primary creep is dealt with in Part II.

ARDELL

and

LEE:

THEORY

OF

HARPER-DORN

2. THEORY

The theory of H-D creep developed in this section is an adaptation of the recently published dislocation network theory of Ardell and Przystupa (A-P) [27], modified to accommodate the observation that no link in the network is long enough to glide freely. It is necessary here to be rather more careful than were Ardell and Przystupa in defining the meaning of “free” dislocation motion, and we begin with this in mind. 2.1. Significance

of the critical link length, L,

As before [27], we claim that under the action of an applied stress, u, each and every link bows out to the same radius of curvature, r, which is inversely related to 0, i.e. YCCC’. The distribution of link lengths, 4(L, t), is defined in such a manner that q5(L, t)dL represents the number of bowed-out links per unit volume with arc length between L and L + dL at time t. In the A-P model coarsening of the network enables the node spacing, 1, of the larger links to increase until A:= 2r, at which point L = L, = 7Lr. In the A-P theory L, was treated as a critical length for “free” dislocation glide. By “free” we mean here (as did A-P) the length beyond which the dislocation can glide without the need to remain in equilibrium with its own line tension, r. L, is precisely the length a dislocation segment acquires at the stress required to operate a F-R source. At low temperatures L, is related only to the shear stress on the glide plane because climb, which can also cause the dislocation to bow out from two pinning points, is unimportant. At high temperatures, however, climb is a recognizably important process which can also contribute to the arc length of a dislocation. It is well known, of course, that the critical stress to operate a pure climb (Bardeen-Herring) source [28] is also related to L, by an equation of the type L,KU -‘. Thus, even if the resolved shear stress on the glide plane of a pinned dislocation is zero, the dislocation will bow out under the influence of the climb force acting on it. If the temperature is high enough, the supply of vacancies to the dislocation will enable it to remain in equilibrium with its line tension until the arc acquires length L,. It is now possible to propose a rigorous definition of L, valid for elevated temperature deformation. It is the arc length above which a dislocation can move by either climb or glide, or a combination of both, without the need to remain in equilibrium with its line tension. Its motion is “free” in this sense, but it would perhaps be more precise to refer to such motion as unconstrained. It is equally important to recognize that when L < L, the dislocation must move somehow if it is to adjust its length when its pinning points (the nodes in the network) move as the network coarsens. This movement must necessarily also occur by small increments of glide or climb, or a combina-

CREEP-I.

STEADY

STATE

2413

tion of both, but the major difference is that when L < L, such motion is constrained by the requirement that the forces acting on the dislocation and impelling its motion are always in equilibrium with its line tension. A major premise of the A-P theory is that network coarsening continually produces a supply of links that grow and exceed length L,. These links can then move in an unconstrained manner until they collide with shorter links in the network, thereby stimulating the coarsening process anew. Steady state deformation is possible when network coarsening and refinement balance each other in such a way that q5(L, t) becomes independent of time. The observation of Lee and Ardell [26] is obviously inconsistent with the A-P theory as formulated because if L, > L always, there can never be unconstrained motion of the dislocations in the network. Therefore the scenario treated by A-P could never occur; it would appear that there is no physical process capable of producing network refinement, and no vehicle for ultimately arriving at steady state flow on the basis of this model. It turns out that such is not the case, as will be demonstrated in the next section. Before proceeding to that, however, it is useful to discuss the factors involved in estimating L, because it is a crucial parameter in the theory upon which the entire explanation of H-D creep depends.

2.2. Calculation

of L,

There is no formula that can be used to predict precisely the critical stress required to operate a mixed dislocation source, by which we mean a source activated potentially through a combination of glide and climb. It should be possible to do this for certain specified geometries, i.e. certain combinations of the Burgers vector, b, and the glide plane orientation with respect to the direction of C. However, for a randomly oriented dislocation segment in a three dimensional network, a detailed calculation is not justified, because the number of available slip systems increases at elevated temperatures. In particular, in Al it is known that in addition to the normal { Ill} (I f0) slip systems, the { 1 IO} (I TO) [29] and {OOI} (110) [30] slip systems are activated at relatively low homologous temperatures (T/T,,, = 0.50.55). It is partially for this reason that attempts to incorporate the distribution of slip systems in network models of creep [3 I] are unlikely to lead to real improvement in the predictive capabilities of the theory. Another reason is, of course, failure to recognize the role of dislocation climb in the bow out process. These factors explain why the data on H-D creep of Al are independent of whether the samples are monocrystalline or polycrystalhne, and independent of crystallographic orientation of the former. Given that the precise plane containing the bowedout dislocation arc cannot be specified, the value of L, can be estimated only by appealing to reasonable

ARDELL and LEE:

2414

physical arguments formula

through

THEORY OF HARPER-DORN

CREEP-I.

STEADY STATE

the use of the established

r = r/mob,

(1)

where m is an average orientation factor. In our calculations of L, = xr we have used the value m = l/2, reasoning that the average shear component of the force on the dislocation must be less than ab/2, while the average climb component of the force on the same dislocation can be greater than ob/2. Allowing for the variation of r with orientation of the dislocation [32, 331 by choosing r = (redgerwreW)“2, we obtain for r the expression r=

Gb2 4X(1 -v)

[(l + v)(l - 2v)]‘121n (A/r&

(2)

where v is Poisson’s ratio and A and r,, are outer and inner cut-off distances in the expression for the dislocation line energy. The values of ,4 and r, are also sources of uncertainty, but it seems reasonable to take n = (L) (the average value of L) and r, = 2b by analogy with similar problems [33]. With these choices the expression for L, becomes Gba-‘[(l Lc = 2(1 - v) which is the formula

+ v)(l - 2v)]‘!2 In ((L)/2b),

the flux, J(L, t), due to network coarsening and the contributions of the source terms Q, (gain) and Q, (loss).

We start with equation

a465 t) -= at

(16) of the A-P

a.qL, t) -7

k

+

c Q”l(L ,=I

theory

t)

(5)

where J(L, t) = 4(L, t)g(L, t) = 4 dL/dt

theory

2.3.1. Evolution of the distribution function. The A-P theory was formulated assuming that the only important collision process was that occurring between freely moving (unconstrained) and network (constrained) links. We believe that this is the most important process of network refinement under ordinary creep testing conditions. In reality, the rate per unit volume, M, with which links collide consists of three terms: M8, the rate per unit volume of collisions between unconstrained links; M,,, the rate per unit volume of collisions between unconstrained and constrained links; M,, the rate per unit volume of collisions between constrained links. M is thus given by M=M,+M,,+M,.

L

Fig. 1. Schematic curve of the distribution of dislocation link lengths illustrating the change in $(L, t) resulting from

(3)

used in this work.

2.3. Modification of the A-P

dL

(4)

In the A-P model MW was assumed to dominate, i.e. M = MBn>>Mg + M,. In the absence of unconstrained links, however, the only possibility for network refinement is that collisions occur among the constrained network links, i.e. IV, = Mgn = 0, M=M,. In what follows we treat M,, as a phenomenological constant. We suppose that a first-principles calculation of M, would require an estimate of the probabilities of collisions between network links of various L growing at different rates, g = dL/dt, in random directions. We have attempted no such calculations, but we show later that the values of M, required to fit our data can be rationalized sensibly using simple dimensional arguments.

(6)

is the flux of links leaving the interval L, L + dL at time t due to network coarsening, and Qai(L, t) is the ith source term contributing to refinement of the network in the interval L, L + dL at t by collisions between links. When plastic strain is generated exclusively by the constrained motion of dislocations that takes place during network coarsening there are only two possible source terms. These are illustrated schematically in Fig. 1, which shows that the interval of width dL can acquire links of length between L and L + dL only by the intersection of links of length L’ > L, whereas links can be lost from the same interval if a link already in it suffers a collision. The acquisition process gives rise to the contribution Q,, and the loss process to Q,,. We consider Qn2 first. Since the network links are intersected by other network links at rate M, per unit volume, the number of network links per unit volume intersected during a time interval 6t is 2M,&. The factor of two enters because each collision between two links results in their destruction and the creation of four new links, producing a net increase of two new links [27]. Let the number of newly intersected links per unit volume with lengths between L and L + dL at time t be $(L, t) dL. The probability that the length of a newly intersected link lies in the interval L, L + dL at time t is $(L, t) dL/2M&. We now make the same assumption used by Ardell and Przystupa [27] that this probability is identical to the original probability of finding a link in the same interval, i.e.

ARDELL

4(L, t) dL/N,,

and

LEE:

THEORY

OF

HARPER-DORN

where N,, =

CREEP--I.

STEADY

while the condition Lmc$(L, t) dL j0

(7)

is the total number of links in the network. Equating these two probabilities produces the result $(L_, t) = 24(L, t)M&lN,.

Q,z(-Lt) = - 24 CL,tPfn/nh.

(8)

LmQ,z(L’, t) dL’ (9) L’ jL where the factor of two arises from the fact that there are two points at which the link of length L’ can be intersected to produce a new link in the interval L, L + dL. Substitution of equation (8) into (9) yields the result Lm4 (L’, t)dL’ (10) Q,,(L> t) = L’ n I_

p = 0 produces

jO’mB.y,dL

= jo’mJsdL

2415

the result =O.

[27] (15)

2.3.2. The strain rate. According to Ardell and Przystupa [27] the contribution to plastic strain due to network coarsening alone is governed in general by the equation

The rate per unit volume at which network links are removed from the interval is thus Qn2 = - $/&, or

So long as a network link of length L’ (> L) can be intersected anywhere along its length with equal probability, the probability that its involvement in a collision produces a new link in the interval L, L + dL is dL/L’. Since all links in the interval L < L’ < L, can suffer intersection, they can all potentially contribute new links to the interval L, L + dL. The rate per unit volume at which this happens is thus

STATE

L‘ i = zbr

j0

J(L, t) sin’(L/2r)

dL.

Since L, > L, during H-D creep, this expression accounts for the entire strain rate under H--D creep conditions. Accordingly, we can write &n J,(L) sin2(L/2r) j0

i, = rbr

dL

(16)

the form of which is valid during primary as well as steady state creep, though it is expressed here specifically for steady state conditi0ns.t It is shown in Appendix I that equation (5) can be integrated (with @~/at = 0) to yield the result

Q,,(L, t) dL = - 2 dL

“+

s

It is easy to show that the constraint

i

j’”0

conditions

Q,,,(L, t) dL = 2M,

[27] (11)

and LQ,,(L, i

t) dL = 0

(12)

I:

are satisfied on direct substitution of equations (8) and (IO) into (11) and (12). During steady state creep &$(L, t) at = 0, dN,,/dt = 0 and dpjdt = 0 = 0. Substituting equations (8) and (10) into (5) and differentiating once with respect to L produces the second order differential equation d2J, ztp-

2Mns dA + 4Mm 6s o (13) --= Nns dL Nns L where the subscript s indicates the steady state values of the parameters. Equation (13) can be used to calculate 4,(L), keeping in mind equation (6) provided g,(L) is known. The condition dN,/dt = 0 leads, via equations (7). (5) and (II), to lim J, = - 2M,, (14) L-0 ~~ ~~ ?This equation was presented in Lee and Ardell[26] with the r in front of the integral inadvertently omitted. We also note that an apparently straightforward generalization of equation (16), i.e. equation (41) of Ardell and Przystupa [27], cannot be used here because (g,) = 0.

J,W)=4x,=+

2M

-

jo’“‘4,(L’)dL’~.

(17)

Therefore, numerical integration of the experimentally measured link length distributions provides a method of estimating J,(L). Division of these derived values of J,(L) by the experimental vahles of 4,(L) then enables us to obtain useful information on the form of g(L). We shall explore the consequences of this procedure later. For the moment, however, we pursue the consequences of substituting equation (17) into (l6), for it provides a means of predicting the steady-state creep rate under H-D creep conditions. An immediate consequence, as shown in Appendix I, is that equation (16) can be written in the alternate form

~&NM = 2r’ 2abM,,,

s

Lm4, sin*(L/2r)

dL

L

0

r2 --j L

Lrn 4,sin(Ljr)dL 0

(18)

which is more convenient for numerical analysis than the combination of equations (16) and (17). Also, since L, = nr >>L for most values of CJ in the H-D creep regime, we can let sin(x) N x - .u’/3! in the integrals in equation (18) discard all terms of higher order than L4, and readily perform the integrations to yield the result

i N b

rMn,b2
where use has been made of equation m = l/2

and

(19)

24r

(Lz)N,,,=

(I) with

&n L3c$,(L) dL. s0

2416

ARDELL

and LEE:

THEORY OF HARPER-DORN

CREEP-I.

STEADY STATE

Equation (I 9) predicts that steady-state H-D creep is Newtonian in the limit of small applied stress, as observed ex~rimentally. The tem~rature dependence of i, is contained primarily in the parameter M,,, which is logically governed by diffusion. Some contribution from the factor (Lz) can be expected if the shape of the link length distribution is temperature dependent during steady state H-D creep.

3. RESULTS AND ANALYSIS

The link length distributions at several values of LT are shown in Fig. 2.t These curves were obtained using the method of Lin et al. [34] applied to etch pit photographs. This method is an adaptation of that used by Oden et al. [35] to obtain b,,(L) from transmission electron micrographs of thin foil samples. Additional details concerning the experimental procedures used in these investigations of H-D creep in monocrystalline Al are described by Lee [36]. The major conclusions to be drawn from the curves in Fig. 2 are: (1) a significant change in the behavior of 4,(L), accompanied by a sharp reduction in N, [the area under the curves of 4,(L) vs L], occurs at a stress of -0.1 MPa at 920 K [Fig. 2(a)]; L, N L, at this value of o; (2) in the H-D creep regime L, > L, and 4,(L) is independent of cr [Fig. 2(b)]. The data in Fig. 2(b) serve as the cornerstone of the conclusion that plastic strain during H-D creep is generated exclusively by the constrained motion of dislocations accompanying network growth. The steady state creep rates obtained by compression testing of Al single crystals and the comparison with the results of previous studies are shown in Fig. 3. It is seen that the results of this study and the earlier investigations are in excellent agreement. The transition from H-D to power law creep occurs at a value of o/G N 10m5. This corresponds to d u 0.1 MPa and is consistent with the transition in the behavior of (p,(L) seen in Fig, 2. For a complete quantitative comparison between theory and experiment we need to know the constant M,, and the function g,(L). Knowledge of these enables us to calculate 4,(L) via equation (13), J,(L) from the relationship f,(L) = ~~(~)g~(~) and subsequently i, from equation (18). Unfortunately, neither of these is known, so that IV,,, must be regarded as an adjustable parameter and the form of g,(L) ascertained experimentally. Comparison between theory and experiment is then evaluated on

tTbe data in Fig. 2(b) differ slightly from those published in Fig. 5 of Lee and Ardell [26] due to the subsequent acquisition of additional data points. Also, the values of L, are a little different because the logarithmic factor in equation (3) was defined in a less satisfactory manner at that stage of the research. The difference between the calculated values of L, then and now is small, and the main conclusion that H-D creep occurs when L, > 4, is entirely unaffected.

(4

rr(MPa)

8 0

0.05

A 0.06 l

0

0.05

0.10

0.15

0.20

0.08

0.25

0.30

(b) L (mm) Fig. 2. The dislocation link length distributions in monocrystalline samples of Al deformed to steady state creep in compression at 920K. &b,(L)at stresses in the transition region from H-D to power-law creep are shown in (a), while #,(L) at stresses in the H-D creep regime (c < 0.1 MPa) are shown in (b). The arrows in (a) and (b) indicate the values of Lc calculated using equation (3) for the indicated values of u. The curves inset in (a) are plotted on a logarithmic scale to illustrate more clearly that H-D creep is associated with the condition L, > L,.

the basis of self-consistency and by appealing to straightforward physical and dimensional arguments. We begin by noting that values of M,,, can be obtained from equation (18), computing the integrals numerically using the data in Fig. 2(b), equations (1) and (2) to evaluate r, and equation (7) to obtain values of N,,, also from the data in Fig. 2(b). For these calculations we used the single crystal elastic constants at 920 K measured by Sutton [37] to estimate v and G using the formulas

and

ARDELL and LEE:

THEORY OF HARPER-DORN

CREEP-I.

0 Harper and Darn 5

3417

STEADY STATE

Ill

/

/

a Barrett et al. I21 .

t

00 0

Thas Study

/

0.04

0.02

0.06

0.10

0.08

0.12

Fig. 3. Data on the steady state creep of Al, compensated for temperature and stress through normalization by the self-diffusion coefficient, D, and the shear modulus, G.

Fig. 4. A linear plot of the steady state creep rate, i, vs the applied stress, (T,at 920 K in the H-D regime (CT< 0.1 MPa). The values of i, at the transition stress, u = 0.1 MPa, are also included.

respectively. The value of b at 920 K was estimated from the data compiled by Pearson [38]. The relevant numbers are v = 0.38, G = 15.5 GPa and b = 0.2906 nm. The value of t[ was taken as l/2. The values of M,,, calculated in this way are summarized in Table 1, along with all the other relevant parameters, including those that can be obtained from analysis of the data in Fig. 2(b). It is evident that for stresses below 0.1 MPa the values of (L,), N,,,psand M,, are constant. Particularly noteworthy is that the values of pS in this study compare favorably with those obtained by other investigators [3,4], which provides us with considerable confidence that the method of measuring 4,(L) described by Lin ef al. [34] and used by Lee and Ardell [26] is quite reliable. We shall return later to the physical significance of the average value of M,, obtained in this study, i.e. M,, = 343 f 14 mm-j s’. We note here that M,, can be obtained in a simpler way by taking advantage of equation (19) which predicts that the slope of a plot of (, vs 0 is aM,,b2(L:)/24r. The data in the H-D regime on e, vs o obtained in this work, as well as the data of Harper and Dorn [l] and Barrett et al. [2] at 920 K, are plotted in Fig. 4. It is seen that the data are generally in excellent agreement, those of Barrett et al. lying slightly lower. Using the value (Li) = 10m3mm’ obtained from analysis of the dis-

tribution functions in Fig. 2(b), the value of M,,, extracted from the slope in Fig. 4 is 335 mm-% ‘. This value is in excellent agreement with those in Table 1 and suggests that M,, is a parameter that can be evaluated without recourse to knowledge of 4,(L), provided that 4,(L) at different values of r~ and Tcan be obtained through suitable normalization, e.g. of the kind used by Lin et al. [34] to compare link length distributions generated during power law creep of NaCl [39] and MgO [40]. Using the values of M,, in Table 1 it is possible to derive the expected behavior of J,(L) by numerical integration of equation (17). The expected behavior of g,(L) can then be examined via equation (6) after dividing the calculated values of J,(L) by the experimentally measured values of 4,(L). The results of these procedures are shown in Figs 5 and 6. Figure 5 illustrates the behavior of J,(L) and displays several important features: (1) the flux in the H-D creep regime (a < 0.1 MPa) is independent of stress; (2) there is a critical link length, L,*, which has the significance that J,(L) < 0 for L < Lb while J,(L) > 0 for L r L,*; (3) the curves of J,(L) extrapolate to a finite negative value at L = 0. The last observation is a consequence of equation (14), and the curves for o < 0.1 MPa in Fig. 5 extrapolate to a value of J, = - 690 mm ’ sm‘, consistent

Table

I. Values

of the various value

g (MPa)

(where

i, x IO” (s

‘)

quantities relevant)

relevant

to H-D

representative

.L (mm)

P. (mm

0.05

2.52

0.50

86

0.06

3.04

0.41

0.07

3.62

0.36

0.08

4.23

0.10

5.66

creep

of H-D ‘1

N,,

of Al at 920 K. The last entry creep

(mm

(data ‘)

for

o > 0.08 Ml%

U,,, (mm

‘s

‘)

in each column

1s the average

are excluded) CL,)

(mm)

L:

(mm)

1040

361

0.0830

0.0365

83

9x0

344

0.0845

0.0410

90

1040

328

0 OX60

0.0320

0.31

76

xx0

339

0.0872

0.0430

0.24

165

-

X3.8 + 5.9

9x5 +70

343 ?I4

0.0630 0.0852 +0.0018

0.0381 t 0.0049

ARDELL

2418

and

LEE:

THEORY

OF HARPER-DORN

-4

0.05

A 0

0.06 0.07

.

0.08

0

0.10

s L:

-6

0

STATE

(20)

2

-5

-7 c

STEADY

always small values of M,, and dominate the collision process. The values of L: in the H-D region are also tabulated in Table 1. Its average value for the four test conditions is L: = 0.0351 + 0.0049 mm. That there must be a value of L: follows from equation (16) which predicts J,(L$) = 0, yielding the expression &n ML) dL L,* = s LrLm

cr(MPa) 0

CREEP-I.

,

,

I

1

I

0.05

0.10

0.15

0.20

0.25

L (mm) Fig. 5. Illustrating the behavior of the steady state dislocation link flux, J,(L), vs L in Al monocrystals deformed at 920 K at various stresses in the H-D creep regime. J,(L) at the transition stress is also shown.

the average value of M, = 343 mmd3 SK’. This is neither an accident nor evidence of self-consistency; it is simply a manifestation of the procedure used to estimate M,,. The curves do illustrate, however, that the value of M,, in the transition region increases rapidly. This is to be expected because dislocations can move in an unconstrained manner when L, < L,, leading to large values of Mngs which swamp the

ML ) dL

Since 4,(L) is independent of u in the H-D regime, L,* must also be independent of 0. The behavior of g,(L) is illustrated in Fig. 6. It is seen that g,(L) < 0 for L CL,* while g,(L) > 0 for L > L,*. The physical significance of L$ is therefore clarified; links in the region L 0), while links in the region L > L: are growing (dL/dt > 0). The data in Fig. 6 are plotted in Fig. 7 as g,(L) vs l/L to test the suggestion of Ostrom and Lagneborg [41] that g,(L) obeys the equation

gs(L)=K

with

(&-; >

(21)

where K is a rate constant. Equation (21) is appealing because it leads, under stress-free conditions, to a static recovery law of the type (L)20c t. Some of its consequences under conditions of power-law creep 0.1

1

0 0.05 -0.1 0

-0.2 u(MPa)

-0.05 '; Ln E

0

0.05

A 0.06 Cl 0.07 .

-0.10

‘;

0.08

E 3 x

-0.4 i

3

-0.15

-0.5

,_ -0.6

-0.20

-0.7

-0.25

A 0 .

-0.8

0.06 0.07 0.08

-0.9

-0.30 0

0.05

0.15

0.10

0.20

0.25

L (mm1

Fig. 6. Illustrating the behavior

of the steady state rate of growth of individual dislocation links, g,(L), as a function of the link length, L, in Al monocrystals deformed at 920 K at various stresses in the H-D creep regime.

-1.0 0

0.2

0.4

0.6

0.8

1.0

1.2

l/Lx

10-2

hllm-1)

1.4

Fig. 7. The data in Fig. 6 plotted as g,(L) significance of the dashed curve is explained

1.6

1.8

2.0

vs l/L. The in the text.

ARDELL

and

LEE:

THEORY

OF HARPER-DORN

have been discussed by Ardell and Przystupa [27]. Despite its attraction, it is obvious that the linear behavior predicted by equation (21) in a plot of g,(L) vs l/L is not observed in Fig. 7. It can, in fact, be shown analytically that equation (21) cannot be correct provided that the theory has been formulated properly, i.e. that equation (13) (or its first integration) is correct. This follows from the linear dependence on L at small values of L. and is demonstrated explicitly in Appendix II. Despite the failure of equation (21) to describe correctly the growth rate of individual links of length L, it is instructive to use it so that equation (13) can be solved numerically. In this way J,(L) and d,(L) can be simulated, even though we know at the outset that the simulation must be imperfect. Using equation (6) and letting L/L,* = 1 +.I-. equation (13) is readily transformed to the equation

CREEP-I.

STEADY

STATE

2419

consequences of the failure of equation (21) to describe the rate of growth of individual links. Nevertheless, the fact that substitution of equation (21) into (13) produces a function that displays the correct behavior semiquantitatively suggests that a relatively straightforward modification of equation (21) is all that is needed to yield a quantitatively accurate description of the process of network growth under an applied stress. 4. DISCUSSION

The modified A-P theory of steady state H--D creep developed in this research effort provides a clear, straightforward and self-consistent quantttative

“r

7

.\-‘$+[.X(.X

+ I)%

+;(2.u

- l)J,=O

(22)

where ( =

2M,,L~‘IN,,,K.

(23)

In principle, equation (22) can be solved numerically subject to the boundary conditions J,(O) = 0 and J&J = 0. where .Y”,+ I = L,/L,*. The solution must also be consistent with the conditions imposed by equations (14) and (I 5). In practice it proved impossible to simulate the curves in Fig. 2(b) accurately because a solution to equation (22) satisfying all these conditions could not be found. Nevertheless, it was determined by trial and error [36] that a reasonable result could be obtained with [ = 0.992 and x,, = 6.1. The value of K derived from [ via equation (23) and the parameters in Table I is 1.02 x IO ‘mm- * s- ‘. If equation (21) were physically significant the data would plot linearly, with a slope indicated by the dashed line in Fig. 7. The poor agreement observed underscores the inadequacy of equation (21). The value of n,, however, implies L, = 7. I L: which, with L: = 0.0381 mm, results in L, = 0.271 mm, in reasonable agreement with the data in Fig. 2(b). Even with these values of { and .u,,, it is still necessary to assign one arbitrary constant, the value of which was chosen to match the average of the maximum values of $,(L) in Fig. 2(b). The simulated values of 4,(L) and J,(L) are compared in Fig. 8 with the data on the sample tested at 0.08 MPa. In overall appearance the simulated functions match the experimentally determined ones reasonably well. The major discrepancies are, however, clearly noticeable. In particular, the initial slope of the simulated curve of 4,(L) is roughly twice as large as the experimental slope [Fig. 8(a)], and the fitted curve does not match the experimental one very well at values of L close to L,. Also, the behavior of the simulated function J,(L) in the limit L+O produces a value of M,, much greater than that observed experimentally [Fig. 8(b)]. These discrepancies are

0

0.05

0.10

0.15

0.20

0.25

L imm) (a)

3

5

-4 t/. -6 L 0

I

I

I

I

I

I

0.05

0.10

0.15

0.20

0.25

L (mm) (b)

Fig. 8. Results of the computer simulation study. (a) Illustrating the comparison between the simulated curve of 4,(L) vs L (solid line) and the data on the sample deformed at 0.08 MPa. (b) Illustrating the comparison between the simulated curve of J,(L) vs L (solid line) and the data on the sample deformed at 0.08 MPa.

2420

ARDELL and LEE:

THEORY OF HARPER-DORN

picture of a creep process that Mohamed and Ginter [4] not so long ago referred to as “baffling”. The principal shortcoming of this description is that we have introduced the parameter M,, which is central to the theory, yet which is new and therefore unfamiliar. Hence, it is essential to assess its physical significance, which we do now using simple dimensional arguments. Since M,, is a rate per unit volume we can define an average time, z,, spent between collisions of network links at steady state by Z, = NJ&f,,. Using the values in Table 1, we find Z, = 2.87 s, meaning that on average two links collide every 2.87 s. The collision frequency, v, = r,‘, is thus 0.348 SC’. We now ask how far a link moves on average before suffering a collision, and with what velocity, a,, does it move? It is reasonable to expect that a link will travel a distance q(L,) where q is some number smaller than unity. Then z+= q(L,),k,. It is with some trepidation that we call tt, a velocity because it is easy to confuse it with an unconstrained glide or climb velocity, which it most definitely is not. Instead, we recognize that during network coarsening the links must move as the nodes connecting them move, and even though they move in this constrained manner must do so with a certain velocity; this velocity is v,. Now, we suppose that a reasonable approximation of the strain rate during steady state H-D creep can be obtained from the simple equation i, = or&?: - p;)a,,

(24)

which is an adaptation of the Taylor-Orowan equation. The partitioning of the (network) dislocation density into two parts, p: and p;, has the following significance. During network growth under an applied stress the population of dislocation segments is biased so that the motion of the links produces a net plastic strain in the appropriate direction, i.e. a tensile stress causes the sample to elongate, a compressive stress causes it to contract. It is evident that not all the links can contribute to flow in the same direction because some are shrinking and some are growing. Only the growing links contribute to positive strain while the shrinking links contribute negatively to the strain. To envision this more clearly imagine a link that is growing (L > L:) just prior to a collision which produces two segments, one satisfying L > L,* and the other L < L:. The link of length L > L: will continue to grow and produce positive strain while the other must shrink and produce negative strain until it disappears. Analytical expressions for p: and p; are therefore L: (25a) J%(L) dL P; = s0 and

s b

P;'

=

LcP,(L)dL

Wb)

G

with p; + pt = ps. Average values of p;

and pt

CREEP-I.

STEADY STATE

are easy to estimate from the data in Fig. 2(b) and the average value of L,* = 0.038 mm; we find p; N lOmm-* and p$ N 74mm-*. Using the average value of i,= 3.35 x lo-* s-’ over the range 0.05-0.08 MPa, equation (24) indicates that the value of q required to produce this strain rate is q N 0.12. On average, then, a growing link moves a distance roughly equal to 1 pm in slightly less than 3 s before it collides with another segment. These numbers appear quite reasonable to us, and provide considerable confidence in the physical picture of our de~ription of H-D creep. The failure of equation (21) to describe the data in Fig. 7 deserves further comment. As mentioned earlier this equation predicts the correct kinetic behavior for network coarsening (static recovery) in the absence of an applied stress. We perceive two major differences between network coarsening under static and dynamic conditions that may be responsible for the shortcomings of equation (21) under creep deformation. First, under an applied stress the dislocations bow out into arcs, and it is likely that the nodes connected by the curved links migrate differently than they do under zero applied stress. Secondly, the collisions responsible for network refinement must produce nodes at which four dislocation segments intersect and struggle to attain a local equilibrium configuration. We do not know the details of this process but anticipate that it will influence the local rate of dislocation link growth in a manner that equation (21) is unable to predict; equation (21) is expected to apply to a network in which typically three dislocations meet at a node. It is likely that an expression for g(L, t) valid for dislocation link growth under an applied stress will not be forthcoming until the dynamics of node movement under stressed conditions are better understood. It remains to discuss the applicability of the recently advanced explanations of H-D creep. We start with the explanation of Wu and Sherby [21], who invoke the existence of an internal stress to explain the phenomenon of H-D creep. The source of internal stress is hypothesized to be the dislocations in the microstructure (i.e. the three-dimensiona dislocation network). As noted many years ago by Friedel [42], the average value of the long range internal stress due to a three-dimensional dislocation network is zero. The maximum value the internal stress can attain locally is given approximately by the formula used by Wu and Sherby to calculate the average internal stress. We believe that even this maximum value will be a considerable overestimate at elevated temperatures because diffusion can readily supply the dislocations with their equilibrium concentration of vacancies. These vacancies must, in turn, partially relieve the stress fields of the dislocations. On average, then, all the material between the dislocations must be stress free; the only significant stress a moving dislocation can experience at high temperatures is the external applied stress. For these

ARDELL and LEE:

THEORY OF HARPER-DORN

reasons we fail to see how internal stresses in a single phase material can develop to the point that they are large enough to influence any steady state deformation process at temperatures high enough for diffusion to control the rate at which dislocations move. We turn next to the recent suggestion of Raj [24] that surface sources of dislocations play an important role in H-D creep. This suggestion was offered as an explanation of the observation of Raman and Raj [23] that specimen size influences the transition stress from H-D to power law creep. On examination of Fig. 1 of Raman and Raj it is obvious that one data point, that generated by Dixon-Stubbs and Wilshire [I61 on CaO, has an inordinately large influence on the analysis of data on a large number of samples. If we were to examine the data on pure Al as one group, or the data on the solid solutions Al-Mg and Pb-Sn (and throw in the data on pure PB and Sn) as another, we would conclude with no reservations or equivocation at all that there was no effect whatsoever of specimen size on the transition stress. Viewed in this light there is really no effect that requires explanation, and we remain totally unconvinced by the analysis of the data presented. Also, if surface sources of dislocations were important enough to influence H-D creep significantly, we would expect grain size to play an important role in H-D creep because grain boundaries are important sources of dislocations. It is well known that H-D creep is grain size independent above a critical grain size, providing additional evidence that surface sources cannot play an important role during H-D creep. Finally, we address the suggestion of Weertman and Blacic [20] that H-D creep may be an artifact of low-amplitude temperature cycling. Their calculations suggest that a perfectly normal temperature control variation of k 1 K over a period ranging from 5 to 50 min can produce a constant dislocation density in the sample and a chemical stress that can become as large as 5 MPa (two orders of magnitude higher than the smallest value of Q used in our experiments). We have already pointed out [26] and demonstrated [36] that the temperature cycle in our experiments was the order of 25 h, meaning that according to the criterion of Weertman and Blacic there should be no influence from this source on our results. Also, while we have not made calculations to verify the point, it would appear that the observations of H-D creep at the low homologous temperatures used in the studies of Malakondaiah and Rama Rao [13, 141, Fiala et al. [15] and Novotny et al. [16] would rule out the possibility of H-D creep being an artifact. Nevertheless, the question posed by Weertman and Blacic regarding the independence of p, on g in the H-D regime has never been satisfactorily explained. The explanation of this observation may be related to the fact that since L CL, for all qb(L, t) in the H-D

CREEP-I.

STEADY STATE

2421

regime, there is no obvious reason why the density of links should be influenced by o. We suppose there could be some relationship but are not surprised that there is none. On the other hand, in the power law creep regime the applied stress produces a relatively initial large dislocation density through dislocation mutliplication. It is dissipated during primary creep, but since L, < L, there are always unconstrained links in the distribution. Because of them Mn8 is expected to be strongly stress dependent, and consequently the shape of &J(L,t) and the density of dislocations will also depend on u. The foregoing discussion is not intended to provide a definitive solution to the problem of the stress dependence of ps through the H-D and power law creep regimes. It is intended, instead, to emphasize the fundamental distinction between the relationship of L, to &(L, t) in the two regimes, and to note that on this basis the independence of ps and (r in the H-D and the dependence of pS on (Tin the power law regime, are not particularly surprising. In closing, we touch briefly on the issues addressed by Langdon and Yavari [19]. Much of their discussion is concerned with the factors that control the rate of dislocation motion. In the context of the current model their arguments are relevant to the conditions that govern M,, through its relationship to v,. Many of the mechanisms discussed by them could be rate controlling under appropriate circumstances, and we find that they neither conflict with nor support the basic features of the phenomenological theory developed in this paper.

5. SUMMARY AND CONCLUSIONS 1. The A-P theory has been modified successfully to account for the experimental observation that none of the links in the distribution of dislocation link lengths in monocrystalline Al deformed in compression at 920 K is long enough to glide or climb in an unconstrained manner. 2. The modified A-P theory predicts that the steady state creep rate in the H-D creep regime is Newtonian at low stresses and is also proportional to M,,, which is the only unknown parameter in the theory. 3. The values of M,, required to bring theory and experiment into quantitative agreement are physically reasonable. 4. The kinetic law governing the rate of growth of individual links, equation (21), is not obeyed by the data on steady state H-D creep of Al. Nevertheless, it can be used to predict curves of 4,(L) vs L which are in reasonable agreement with the experimentally measured curves. Acknowledgements-The authors express their gratitude to the National Science Foundation for financial support of this research under Grant No. DMR-81-I 1133.

2422

ARDELL

and

LEE:

THEORY

OF HARPER-DORN

REFERENCES 1. J. G. Harper and J. E. Dorn, Acfa metall. 5,654 (1957). 2. C. R. Barrett, E. C. Muehleisen and W. D. Nix, Mater. Sci. Engng 10, 33 (1972). K. L. Murty and J. W. Morris Jr, 3. F. A. Mohamed, Metall. Trans. 4, 135 (1973). 4. F. A. Mohamed and T. J. Ginter, Acta metall. 30, 1869 (1982). 5. K. K. Zilling, Fizika Metal/ 22, 931 (1966). 6. K. L. Murty, F. A. Mohamed and J. E. Dorn, Acta metall. 20, 1009 (1972). I. F. A. Mohamed, Metall. Trans. A 9A, 1342 (1978). 8. P. Yavari, F. A. Mohamed and T. G. Langdon, Acta meraIl. 29, 1495 (1981). 9. P. Yavari, D. A. Miller and T. G. Langdon, Acra melall. 30, 871 (1982). 10. K. L. Murty, Mater. Sci. Engng 14, 169 (1974) 11. J. P. Poirier, J. Peyronneau, J. Y. Gesland and G. Brebec, Phys. Earth Planet Int. 32, 213 (1983). 12. G. Malakondaiah and P. Rama Rao, Acta metall. 29, 1263 (1981). 13. G. Malakondaiah and P. Rama Rao, Mater Sci. Engng 52, 207 (1982). 14. J. Fiala, J. Novotny and J. Cadek, Mafer. Sci. Engng 60, 195 (1983). 15. J. Novotny, J. Fiala and J. Cadek, Acta metall. 33, 905 (1985). 16. P. J. Dixon-Stubbs and B. Wilshire, Phil. Mag. 45, 519 (1982). 17. A. H. Chokshi, Scripta metall. 19, 529 (1985). J. Fiala and J. Cadek, Scripta merall. 19, 18. J. Novotny, 867 (1985). 19. T. G. Langdon and P. Yavari, Acta melall. 30, 881 (1982). 20. J. Weertman and J. Blacic, Geophys. Res. Lett. 11, 117 (1984). 21. M. Y. Wu and 0. D. Sherby, Acta metall. 32, 1561 (1984). 22. F. Garofalo, Trans. Am. Inst. Min. Engrs 221, 351 (1963). 23. V. Raman and S. V. Raj, Scripta metall. 19, 629 (1985). 24. S. V. Raj, Scripta metall. 19, 1069 (1985). 25. G. Malakondaiah and P. Rama Rao, Metal Sci. J. 15, 442 (1981). 26. S. Lee and A. J. Ardell, Strength of Metals and Alloys (edited by H. J. McQueen et al.), Vol. 1, p. 671. Pergamon Press, Oxford (1985). 21. A. J. Ardell and M. A. Przystupa, Mech. Mater. 3, 319 (1984). 28. J. Bardeen and C. Herring, Imperfections in Nearly Perfect Crystals (edited by W. Shockley et al.), p. 261. Wiley, New York (1952). 29. R. Le Hazif and J. P. Poirier, Acta metall. 23, 865 (1975). 30. M. Carrard and J. L. Martin, Strength of Metals and Alloys (edited by H. J. McQueen et al.), Vol. 1, p. 665. Pergamon Press, Oxford (1985). 31. P. Ostrom and R. Lagneborg, Res. Mech. 1, 59 (1980). 32. G. DeWit and J. S. Koehler, Phys. Rev. 116, 1113 (1959). 33. L. M. Brown and R. K. Ham, Strengthening Methods in Crystals (edited by A. Kelly and R. B. Nicholson), p. 9. Halsted Press Div., Wiley, New York (1971). 34. P. Lin, M. A. Przystupa and A. J. Ardell, Strength of Metals and Alloys (edited by H. J. McQueen et al.), Vol. 1, p. 595. Pergamon Press, Oxford (1985). 35. A. Oden, E. Lind and R. Lagneborg, Creep Strength in Steel and High-Temperature Alloys, p. 60. Metals Sot., London (1974). 36. S. S. Lee, Ph.D. dissertation, Univ. of California, Los Angeles (1985). 37. P. M. Sutton, Phys. Rev. 91, 816 (1953).

CREEP-I.

STEADY

STATE

38. W. B. Pearson, Handbook of Lattice Spacings and Structures of Metals, p. 311, Pergamon Press, Oxford (1958). 39. P. Lin, PhD. dissertation, Univ. of California, Los Angeles (1984). 40 J. B. Bilde-Siirensen, Acta metall. 21, 1495 (1973). 41. P. &trom and R. Lagneborg, J. Engng Mater. Tech., Trans. A.S.M.E., Ser. H 98, 114 (1976). 42. J. Friedel, Internal Stresses and Fatigue in Metals (edited by 0. M. Rassweiler and W. L. Grube), p. 220. Elsevier, Amsterdam (1959).

APPENDIX The steady stale link ,flux and H-D

I creep rate

We start the derivation of equation (17) with (S), after inserting the expressions for Q.,, equations (8) and (lo), and setting @/at = 0. On first integration this produces the result

J,(L)

-J,(O) =

2 -jjoL [lL;ds(L;!,dL’]dL. - joL &(L’)

dL’}

(A.I. 1)

where the variables under the integrals are dummy variables. We next define a function F(L) by the equation dF(L)/dL = 4&)/L, whereby the first integral in equation (A.I.1) can be written

s

Lmc$,(L”) dL” L”

I_

L, dF(L”) = F(L,)

=Ls

Substitution of this result produces the expression 2[LF(L,) the second follows

into the first integral

is readily

L

-

0

integrated

by parts

as

s

L’ dF(L’)

II

= LF(L) With this, the entire becomes 2L[F(L,)

Now,

in (A.I.1)

L

F(L’) dL’ = LF(L)

Recalling as

(A.I.2)

- joLF(L’) dL$

term of which

s

- F(L’).

equations

term in brackets

-F(L)]

+

-

’ &(L’) dL’. s0

in equation

(A.I.1)

L +,(L’) dL’. s0

(A.I.2) and (14), we can rewrite (A.I.1)

since

2M,, = +

Lm4,(L) dL ns s 0 2M = -[s’ &(L’) dL’ + jLLrn+,(L’) dL’] N, o

(A.I.4)

by virtue of equation (7), equation (17) follows immediately on substitution of equation (A.I.4) into (A.I.3).

ARDELL To derive producing

equation

and

LEE:

TIIEORY

(18) we first insert

OF HARPER.-,DORN

(17) into

(16).

CREEP--I.

STEADY

of the curves in Fig. 2(b) are linear almost value of t#,(L). We can thus write d,(L) = liL,

_

$,(L’) dL’1 sin’(Lj2r)dL.

Jo LJL

(A.1.5)

1

STATE

2423 to the maximum

lit.1l.I)

(0 < L < t:,

where p is a proportionality constant. We now rewrite equation (17) in a manner that enables us to evaluate it in the limit of small L

We shall consider the integrals in equation (A.1.5) in turn. Using equation (A.I.2) the first integral can be expressed by straightforward integration as [Lh - 2rL,

sin(L,,/r)

- 2r’cos(L,/r)

+ 2r*]F(L,)/2 1,

-2

LF(L) sin’(L/2r) dL. 10 The second term in this expression can be readily integrated by parts, which when added to the first term produces the result 4 + r*[F(L,)

- F(O)] - r*

s s

L, #,(L) cos(L/r)

0

pLg, = 2+2L
- 3/tL$!

- N,,,).

tA.11.3)

On substituting kquation (21) into (15) we tind that for that particular growth law (L,‘) must satisfy the equation

dL Substitution yields

#J,(L) sin (L/r) dL.

0

But since

(A.II.2)

ri.nL’dL’ (A.II.2) !0 1 in the alternative form

(L,

‘) = l/L:.

of equations

(21) and

tA.11.4)

L

1.m

-r

Equation

-r&Ljdr.+ 0 can be expressed

/&L/L;

-pK

(A.II.4)

into

tA.II.3)

-4M,,LtL: - 2M,, - ~~~~“‘L~~!~~~~. (A.11.5)

F(L,)

- F(0) =

‘,$,(L) s0 to write the first integral

it is possible

s

Lm4,(L) sin*(L/2r)

!+.2

At small L we can ignore the term in L’ and equate the coefficients of like powers of L, since equation (A.II.5) must be valid for all L. This procedure produces the entirely incompatible results

as

dL

L

0

dL,L,

FtK = 2M,,

s L”,

and

4,(L) sin (L/r) dL.

-r

0

To dG(L)

M = 4M,,

evaluate the second integral we = 4,(L) dL, resulting in the expression

first

let

Lm

[L, -r

sin (L,/r)]G(L,)/2

-

G(L) sin*(L/2r) dL. s0 by parts we obtain for the

integrating the second term second integral the result 1.m P, r 4,(L) sin (L/r) dL. 2 2, ---s On subtracting the second integral in equation the first we obtain equation (18). APPENDIX Analysis

of the growth fitnction.

We start

the analysis

(A.I.5) from

by noting

g,(L) = K’h(L/L,*),IL will satisfy

equation

(14) provided iim h(L,/L:) ,‘.-0

II equation

demonstrating that equation (21) is incompatible with the linear behavior of the function 4,(L) at small L expressed in equation (A.11. I). While equation (21) does not correctly- describe the growth rate of individual links, the limiting behavior predicted by it is of the correct form, i.e. the product 4,gs = c$,KL -‘(L/L* - 1) is finite in the limit L-0 so long as equation (A.II.l) is applicable in this limit. It seems that a function

(21)

that the initial

portions

that

= const.

and equation (A.ll.l) is satisfied. The function however. remains to be determined.

h(L/L,*),