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Reexamination of a 1955 theory of climb-controlled steady-state creep
Acta metall, mater. Vol. 39, No. 1, pp. 105-110, 1991 Printed in Great Britain. All rights reserved
0956-7151/91 $3.00 + 0.00 Copyright © 1991 Pergamon Press plc
REEXAMINATION OF A 1955 THEORY OF CLIMB-CONTROLLED STEADY-STATE CREEP P. A. TIBBITS Center for Naval Analyses, 4401 Ford Avenue, Alexandria, VA 22302, U.S.A. (Received 23 October 1989; in revised form 25 May 1990) Abstract--Observed steady-state creep rates of many metal class materials show stress dependence i o~ z n, where n ~ 5. Many models of power-law creep predict n = 3 or 4, and require ad hoc assumption of the stress dependence of some geometric parameter to explain the observed n value. A 1955 climb-controlled model of steady-state creep assumes constant climb stress at the lead dislocation of an obstructed pileup to allow analysis. This analysis predicts n = 4. Invocation of a stress-dependent geometric parameter allows prediction of n ~ 5. In the present work, numerical simulation of the model shows assuming constant climb stress to be incorrect and assuming stress-dependent geometry to be unnecessary to predict n ~ 5. High and low stress limiting behaviors of the simulation agree with the results of earlier analytical models. Slip of pileup dislocations during climb of the lead dislocation is suggested as the physical explanation for the non-constant climb stress on the lead dislocation. Both this work and previous analytical models based on pileup mechanisms are subject to criticism based on the lack of observations of dislocation pileups in steady state creep. Rrsumr---Les vitesses de fluage observres en rrgime permanent pour beaucoup de matrriaux m&alliques r6vrlent une drpendance en fonction de la contrainte ~ ocz n, off n ~ 5. Plusieurs modrles de fluage avec loi de puissance prrvoient n = 3 ou 4 et exigent ainsi l'hypothrse d'une drpendance en fonction de la contrainte d'un certain paramrtre gromrtrique pour expliquer la valeur observre de n. propos6 en 1955, un modrle, contr616 par la montre, du fluage en rrgime permanent suppose une contrainte de montre constante sur la dislocation de t~te d'un empilement bloqur. Cette analyse prrvoit n = 4. En invoquant un paramrtre grom&rique drpendant de la contrainte, on peut prrvoir n ~ 5. Dans le prrsent travail, une simulation numrrique du modrle montre que l'hypothrse d'une contrainte de montre constante est incorrecte et que l'hypothrse d'une grom&rie drpendant de la contrainte n'est pas nrcessaire, pour prrdire que n ~ 5. Les resultats de la simulation, pour les limites des fortes et des faibles contraintes, sont en accord avec ceux de modrles analytiques antrrieurs. On suggrre le glissement de dislocations empilres pendant la montre de la dislocation de t&e pour expliquer physiquement une contrainte de montre non constante sur la dislocation de trte. Ce travail, comme les modrles analytiques antrrieurs basrs sur les mrchanismes d'empilement, est sujet ~ la critique par suite de l'absence d'observations d'empilements de dislocations dans le fluage en rrgime permanent.
Znsammenfassung--Die beobachteten station/iren Kriechraten vieler metallischer Materialien weisen eine Spannungsabh/ingigkeit der Form E~ , mit n ~ 5, auf. Viele Modelle des Potenzgesetzkriechens sagen n = 3 oder 4 voraus und erfordern die ad-hoc-Annahme der Spannungsabh/ingigkeit einiger geometrischer Parameter, um den beobachteten n-Wert zu erkl/iren. Ein Modell des Kletter kontrollierten station/iren Kriechens von 1955 nimmt eine konstante Kletterspannung an der fiihrenden Versetzung einer behinderten Aufstauung an, um die Analyse zu erm6glichen. Diese Analyse ergibt n = 4. Die Beriicksichtigung eines spannungsabh/ingigen geometrischen Parameters erm6glicht die Voraussage von n ~ 5. In der vorliegenden Arbeit wird das Modell numerisch simuliert; es zeigt sich, dab die Annahme einer konstanten Kletterspannung nicht richtig ist und die Annahme einer spannungsabhhngigen Geometrie fiir die Voraussage yon n ~ 5 unn6tig ist. Das Grenzverhalten des Modelles bei hoher und niedriger Spannung stimmt mit frtiheren analytischen Modellen iiberein. Als physikalische Erkltirung der auf die fiihrende Versetzung wirkenden nicht-konstanten Kletterspannung dient die Gleitung der aufgestauten Versetzungen wtihrend des Kletterns. Sowohl diese Arbeit wie auch fr/ihere analytische Modelle, die auf Aufstauungen beruhen, sind kritisch zu sehen im Hinblick auf die Tatsache, dab Beobachtungen von Aufstauungen beim station/iren Kriechen fehlen.
1. INTRODUCTION Observed steady-state creep rates o f m a n y metal class materials, in which glide mobilities far exceed climb mobilities, show stress dependence ~ ocz n, where n ~ 5 is the stress e x p o n e n t and z is the applied shear stress. M a n y models o f power-law creep predict n ~ 3 or 4. E x p l a n a t i o n o f observed stress exponents re-
quires ad hoc assumption o f the stress dependence o f some geometric parameter, giving rise to m u c h discussion [2, 3]. A 1955 climb-controlled model o f steady state creep [1] consists o f a dislocation source which generates a pileup o f positive infinitely long edge dislocations obstructed by a L o m e r dislocation. A s s u m i n g c o n s t a n t climb stress during climb o f the 105
106
TIBBITS: CLIMB-CONTROLLED STEADY-STATE CREEP
lead dislocation obviates modelling the detailed evolution of the pileup and allows analytical solution of the problem. This analysis predicts n = 4. Invocation of stress dependence of the source to obstacle distance allows prediction of n ~ 5. In the present work, a numerical technique calculates an approximate solution without assuming either constant climb stress on the lead dislocation or stress dependence of the source to obstacle distance. A model very similar to that proposed in 1955 but without ad hoc assumptions then predicts n ~ 5. 2. THE 1955 MODEL
Figure 1 diagrams the 1955 model, consisting of two F r a n k - R e a d sources, F R I and FR2, their slip planes S1 and $2, and the grain boundary GB which obstructs dislocation movement on S1. F R I operates when it experiences net positive shear stress. The screw segments of the loop produced by FR1 move at much higher velocity than the edge segments, leaving a pair of edge dipoles. The left, negative, limb of the dipole moving leftward on S1 under z is unobstructed, moves far from FR1, and is neglected. FR2 also produces infinite straight positive edge dislocations when the stress field from the pileup of dislocations from F R I on SI at GB becomes sufficiently strong. Dislocations from FR2 then move along $2 until a dislocation reaction occurs at the slip plane intersection O to form a Lomer-Cottrell lock. This sessile dislocation obstructs dislocation motion on S1, causing a pileup of N dislocations to the left of O, where N oc ~. The lead dislocation of this pileup must climb above S1 before the dislocations can slip further. The number of Lomer dislocations forming between FR1 and GB by reaction with dislocations from intersecting slip planes is stress dependent, leading to a stress dependence of the source to obstacle distance, Xob,(z). In metal-class materials, climb of the lead dislocation of the pileup is the rate-limiting step, i.e. the strain rate is under climb control. The strain rate is
inversely proportional to the climb time, which is found from the height to which the lead dislocation must climb and the rate at which it climbs, ~xy ~ 1/tclimb= /)climb/ hcyanb• 2.1. Assumptions o f the 1955 model
Experimental observations justify assuming dislocation glide and climb velocities linear in the glide and climb forces acting on the dislocations. The constants of proportionality are the glide and climb mobilities, M s and Me. Assuming constant climb stress of magnitude Nz on the lead dislocation during climb leads to constant climb velocity/)climb = m c m z . Because N oc z [1], this assumption leads to VcmbOCZ2. 2.2. Predictions o f the 1955 model
Knowledge of the height to which the lead dislocation must climb allows derivation of the stress dependence of the shear strain rate. At bypass, glide stress exerted by the obstacle on the lead dislocation just equals glide stress exerted by the applied stress and the pileup dislocations, Zobs= T g l i d e • Because stress exerted between dislocations is inversely proportional to their separation, %b, ocl/h~,,mb. NOW z#d,= Nz ~ ~2, so hclimb oc 1/'c2, and exyoc/)climb/hclimb0£ ,~4. This expression arises solely from dislocation dynamics, ascribing no stress dependence to Xobs or other structural parameters. Using the Xob~ stress dependence, n ~ 5. 3. SIMULATION OF THE 1955 MODEL
The detailed evolution with time of the positions of all the dislocations of a pileup during climb of the lead dislocation over an obstacle can be simulated by a system of coupled ordinary differential equations relating the velocity of each dislocation to the position of all the dislocations. Assuming unit Burgers vector and velocities proportional to force p = vy = Vclimb= Mcaxxb = Mcaxx Yc = vx = V~ide = Mg Zxr b = Mg rxr.
GO S2
.J_
sl
X
i
i
~j Ms Y'.N (Xi _ Xi)[(Xi _ Xi)2 _ i71 [(xj -- xi) 2 + (yj
(Yi -- Yi) 2] "F M g z -
-
yi)212
i#j
_1_ /
/ o
FR1
Using the stress exerted by one dislocation on another [4], these equations of motion are for dislocation j
Yj Mo i=/-'t
--(yj -- yi) [3(xj -- xi) 2 + (yj -- yi) 2] [(xj -- x,) 2 + (YY-- y~)212
i#j
I~
x.
-I
Fig. 1. Geometry of the 1955 model. S1 and $2 are slip planes, FR1 and FR2 are Frank-Read sources, GB is a barrier to dislocation movement on S1, and O is a Lomer dislocation formed by reaction of dislocations from FR1 and FR2 at the intersection of S1 and $2.
Units of stress are #/[2n(1 - v)] where/~ is the shear modulus and v "the Poisson's ratio. All distances are normalized by b. The summations over i include the sessile obstacle dislocation. The coordinates [xj(t), yj(t)] of the N ( t ) dislocations as a function of time constitute the solution of these coupled ordinary
TIBBITS: CLIMB-CONTROLLED STEADY-STATE CREEP differential equations. A dislocation sink X~nkto the right of the source limited the number of dislocations to be followed and the number of equations to be solved.
A pair of infinite straight edge dislocations of the same sign on parallel slip planes separated by perpendicular distance y bypass one another only if [7]
,fk y(ky2-y2)
3.1. Numerical solution of the simulation Many methods exist for the solution of such initial value problems. A dual order Runge-Kutta formulation with adaptive stepsize based on estimated error [5] solved the set of equations posed above. The equations required a moderate number of algebraic operations for derivative evaluation, and showed no excessive stiffness.
3.2. Input-output structure of the simulation The simulation required as input the locations of the obstacle (Xob~,Yo~) and sink Xsittk, plus the climb mobility M c and applied stress z. Using Yob~= - 0 . 2 5 obviated simulating thermally activated displacement of the lead dislocation from the climbmetastable position left of the obstacle, were the obstacle exactly in the source slip plane. All coordinates are relative to the source position. Glide mobility was assumed to be one. The simulation output the number N(t), locations [xi(t),yi(t)], and velocities [~i(t), ~fi(t)] of the dislocations as a function of time. These provided the instantaneous strain and strain rate [6]
exy(t) = ~ Xg(t) f"~l X2ink '= '
~ ~i(t) and
ExY ( t ) =
2 " i= 1 Xsink
The net glide and climb stresses at any of the dislocations were available as a function of time, but were output only for the lead dislocation of the pileup, Zxy l"adtt~ lead(t). The Boltzmann ,~,, and trx~ average
~
f
z > zcdt = -v .- (ky2 -b y2)2 ,
where
k = 3 +x/@
Increasing z over several runs in which N - 1 and which were allowed to go to long times resulted in a value for the minimum stress at which bypassing occurred which was very close to %fitComparison of results generated by the simulation accorded well with those generated by an earlier simulation [8] of the dynamic formation of a pileup. In this problem, climb is not allowed to occur, and the pileup equilibrates at an obstacle. The equilibrium positions of the dislocations, the instantaneous position of the lead dislocation and the instantaneous stress exerted by the pileup dislocations on the obstacle dislocation were very similar in the two simulations. Two runs of the simulation identical except for decreasing error tolerance showed by the very close agreement in their results that the error tolerance specified for the Runge-Kutta routine was sufficiently small. 4. RESULTS
4.1. Dislocation positions A series of plots of the dislocation positions provides a "movie" of events occurring in a run of the simulation. Figure 2 shows in such a movie a sequence of events in qualitative agreement with the events assumed in Weertmann's 1955 model. Figure 3 shows the lead dislocation of the pileup climbing alternately above and below the obstacle, an unanticipated sequence of events.
t=tIN(t) dt
dl=ti
t y - ti gave the expected number of dislocations for a given z. The slope of a line drawn through similar features of the quasi-steady-state portion of the cyclic E~y(t) vs t curve provided the average strain rate ~y at given z. A series of runs over increasing z provided a log(~y) vs log(z) curve the slope of which gave the n vs log(z) curve.
3.3. Calibration of the simulation Comparison to analytical solutions for two simple cases and to a previous simulation of pileup formation proved the simulation accurate. For a single infinite straight dislocation in unobstructed glide, ~y = z/x~i,k. 2 By inspection, n = [O log(~y)]/ [~ log(z)] = 1. Omitting the obstacle, the simulation calculated strain rates for several z. A backward finite difference formula, [log(-~y)2 - log(~-xy)l]/[log(z2)-log(z~)], approximated n, which was very close to one.
107
4.2. Results of a typical run Figures 4(a)-(e) show, as functions of time, the number of dislocations, the shear strain, shear strain rate, and glide and climb stresses exerted on the lead dislocation of the pileup. The plots shown are typical of the results from all runs. All of the plots show an initial transient behavior followed by a cyclic quasisteady-state region.
4.3. Average number of dislocations Table 1 shows N, the average number of pileup dislocations present in the quasi-steady state as a function of z for the runs made at Xobs=90, Mc = 0.01, 0.001, and 0.0001. Surprisingly, N is independent of Me. Where Kr > 1. N" is proportional to z.
4.4. Average strain rate, ~xy As Fig. 4(b) illustrates, -gxeis the slope of the Exy(t) vs t curve. Sets of log(~-~y) vs log(z) curves were obtained for Xobs = 10, 30, 90, and 270. Figures 5(a) and (b) show the sets of curves obtained at Xobs = 30
108
TIBBITS: CLIMB-CONTROLLED STEADY-STATE CREEP .k >±
O )
).1_
.L-IO
.L O 8
.L
>.L
O >.L ± O
1
)
9
J.
2
.L
>.L
>a.
O
_L
10
O
>.L >.L
O .1..
3
11
.L
>.L )_L
.L O
and :Cob,= 90, respectively. Each curve in a set was obtained at the noted value of Me. Runs made under special conditions investigated limiting behaviors of ~-xy. These limiting behaviors determined the range o f z investigated, and showed the z range of applicability of some earlier models. 4.4.1. Behavior o f - ~ in absence o f obstacle. Figs 5(a) and (b) each show a labelled l o g ( ~ ) vs log(x) curve for the no-obstacle case. A t low z, where only one dislocation exists, the slope of this curve is n = 1. At high z, the slope becomes 2, taking on intermediate values in a smooth fashion at intermediate stress levels. The strain rate in this case is independent of Me, is proportional to the glide mobility, and is under glide control [7]. 4.4.2. High-stress limiting behaviour of-~xy. In Figs 5(a) and (b), the log(~xy) vs log(x) curves for all values of Mc converge in the high stress region to the curve for the no-obstacle case, which is in this region a
O
.L ...L 12
±
±
O
.1_ .L O
Fig. 2. For T = 0.2, Xob~= 30, M c = 0.001, the sequence of events agrees with the 1955 model. Inverted "T" symbols denote the dislocations. A diamond denotes the obstacle dislocation. The source, denoted by half a diamond, has horizontal slip plane at its center. Frame 1. Three dislocations have formed a pileup to the left of the obstacle. The lead dislocation has begun climb. Frame 2. The lead dislocation has climbed appreciably. The other two pileup dislocations laave glided a significant distance during the lead dislocation climb, causing significant change in the climb stress on the lead dislocation during climb. Frame 3. The lead dislocation has bypassed the obstacle. A dislocation has been emitted to replace it in the pileup. The sequence of events repeats in steady state deformation.
>
O
O .L
-L >
O
2.
Fig. 3. "Movie" of simulation for z = 0.1, Mc = 0.1, and xo~ = 30. The climb stress on the lead dislocation changes in sign as well as in magnitude. Frame 1. Emission of first dislocation, D1. Frame 2. D2, the second dislocation is emitted after D1 has begun climb. Frame 3. The dislocations repel. D2 is now below the slip plane of the source. Frame 4. D1 has bypassed the obstacle. Frame 5. D1 continues glide, D2 has been pushed by D1 below the plane of the obstacle. Climb stress from the obstacle is negative. Frame 6. Continued glide of D I and D2, slow negative climb of D2. Frame 7. Continued glide of D 1 and D2, slow negative climb of D2. Frame 8. D1 annihilated at crystal boundary. D3 emitted. Negative climb stress exerted by D3 on D2. Positive climb stress exerted by D2 on D3. Frame 9. D3 remains above plane of obstacle, D2 undergoes visible negative climb. Frame 10. D3 is already visibly above slip plane of the source. D2 continues climb. Frame 11. D2 bypassing obstacle underneath. Frame 12. D2 has bypassed the obstacle, and will glide to annihilation at right, leaving system in some condition as in Frame 1. straight line of slope 2. A 1957 model [9] predicts this behavior in the glide-controlled limit. The point of convergence of the curves gives z above which ~ is glide controlled. N o maximum z was encountered above which investigation could not be carried out. Extension of the results past the point of convergence of the curves showed no new behavior. 4.4.3. Low-stress limiting behavior Of ~xy. The lowstress behavior of the log(~~y) vs log(x) curves also shows a tendency toward straight line behavior and slope 2. This is a consequence of having only one dislocation in the pileup below a certain Zl. Restricting N = 1, the log(~xy) vs log(x) curve was a straight line of slope 2. This curve coincided with the z < Zl region of the log(~-~y) vs log(z) curve obtained in
TIBBITS: CLIMB-CONTROLLED STEADY-STATE CREEP d
-zoo
109
(a)
-2.70
OHL
-3.40
-4.10 -4.80
% No o b s t a c l e S ,
b ~
.-'7
f
-8.~:~
~7 0.05 °11[
-6.90
0.01 ]
-7.~
0.001 J
-8.30 --
-9.00
'~1
I
I
I
I
I
I
I
I
I
I
I
I
-3.00 -2.60 -2.20 -1.~ -1.40 -1.1~1 -0.60 -0.20 0.20 0.60 1.~
C
Log (Xl
-5,40 -8,10
Fig. 4. The basic output of the simulation, plotted against time. All plots show initial transient, followed by repeated cycles (quasi-steady-state). (a) N, number of dislocations. (b) E~y, shear strain, with line drawn for calculation of average strain rate i'~y. (c) Instantaneous shear strain rate, ~y, (d) glide stress exerted on lead dislocation, ~xy , remains nearly constant. (e) climb stress exerted on the lead dislocation, cr~d, increases sharply during climb, in disagreement with an assumption of the 1955 model.
-8.80
?.p
I
No obst
0 j//j
-8.e9
.0.,0
0.
00;1 ~
r J
ordinary simulation at the same Me. Figure 5(b) shows the N = 1 curve for Mc = 0.01 and Xob,= 90. The m i n i m u m z of interest is z _ ~ = 1/Xob~, below which the source will not operate because of the negative glide stress exerted at the source by the obstacle dislocation. A practical limit on the minim u m z investigated resulted from the model's simplification of the F r a n k - R e a d source. When, at low r, climb of an emitted dislocation brings it above a line of slope one through the source, it exerts positive stress on the source. U n d e r net positive stress, the source emits a second dislocation. The second dislocation climbs downward, away from the first, and causes emission of a third dislocation when it passes below a line of slope - 1 through the source. This process continues until a nearly vertical wall of dislocations, a low-angle tilt boundary, exists near the Table 1..N for Xob, = 90 vs z and M e
Mo 0.01 0.0125 0.01875 0.022 0.025 0.05 0.1 0.2 0.4
0.001 1 1-2 1-2 1-2 2-3 5~ 9-10 17-18
0.0125 1 . . . . . . 0.025 1-2 0.05 2-3 0.1 5~i 0.2 9-10 0.4 17-18
0.0001 . . . 0.025 0.05 0.1 0.2 0.4
0.00001
.
. . . 1 2-3 5-6 9-10 17-18
. . . -0.05 0.1 0.2 0.4
-2-3 5 10 17.5
-1.10+1 /
./
0.0001 "f 0.00001
-I.03+IF I
I
I
I
I
I
-3.00 -2.70 -2.40 -2.10 -1.80 -1.50 -1.20 -0.99 -0.60 -0.30 0.00
Log (~)
Fig. 5. (a) L o g ( ~ ) vs log(r) for Xob,= 30, x,~k = 100, and Mc marked. Data points are marked by symbols. Curves fitted to data points are cubits at low z blended into straight lines with slope 2 at high z. z. . . . Zl, %it, and zbsp~ denote threshold stresses for source operation, N > 1, tiltwa11 emission, and bypass of obstacle in the Me = 0 case. (b) Log(~y) vs log(r) for xot, = 90, x~,k = 270, and climb mobilities marked. The case for which N = 1 is labelled. source. The stress below which this can occur, Ztilt, is near z , ~ for small Mc [7]. zl, z . . . . . . and Ztilt are marked on Fig. 5(a). If a dislocation sink exists just to the left of a source tilt wall emission may occur in real materials. 4.4.4. Low Mc limiting behaviour of ~xy. Me depends on temperature through its dependence on diffusivity [1], so this section also discusses low temperature limiting behavior. The log(~-~y) vs log(z) curve was observed to fall below this one. The low z end of this curve is at the stress below which the lead dislocation of the pileup cannot bypass the obstacle. This threshold is labelled as Zbyp, in Fig. 5(a). 4.4.5. High Mc limiting behavior of ~xy. Figs 5(a) and (b) show approach of the log(~xy) vs log (z) curves to the no-obstacle case. U n d e r very high Mc as in the no-obstacle case, ~xy is glide-controlled.
I10
TIBBITS: CLIMB-CONTROLLED STEADY-STATE CREEP Table
2.
nmax vs Me ~< 0.01
Xobs
for
results. This agreement in the case N = 1 suggests that continued slip of the dislocations in the pileup Xobs Me tlmax during climb of the lead dislocation is the physical 10 0.01 4.7 explanation for the increase in Cr~xadresponsible for 0.001 5.8 the disparity in n values at higher z. 30 0.001 4.5 0.0001 6.2 The relation trxxlead~r-~Nz is valid for static pileups. 90 0.01 3.7 Because climb of the lead dislocation reduces the 0.001 4.4 0.0001 5.3 glide stress opposing movement of the remaining 0.00001 11.0 pileup dislocations, they glide toward the obstacle. 270 0.01 3.0 Because Ms>>M¢, adjustment of the pileup dislo0.001 3.8 0.0001 5.6 cations to maintain glide equilibrium greatly increases tr~ d. The lead dislocation climbs under nonequilibrium conditions and al~d>>Nz, leading to 4.5. Stress exponent the higher n values observed in the simulation and in Taking the slope of a l o g ( ~ ) vs log(z) curve yields experiments. an n vs log(z) curve. The sigrnoidal log(~xy) vs log(z) Third power laws arising from many analytical curves have maximum slope near (%ou_~+ %yp~)/2. models are "natural" only in the sense that they arise The n vs log(z) curves resemble parabolas of negative logically from the assumptions made in constructing curvature with maxima near (Zsource+Zbypass)/2. the model. This work shows the fallacy of certain of Table 2 shows the maximum value of the stress those assumptions. exponent, n . . . . for each value of Xob~and values of The lack of experimental observation of disloM¢ ~< 0.01. Table 2 shows dependences of nmax on Me cation pileups in steady-state creep must be noted. and Xob~. Increasing M¢ decreases nm~x, as does in- This criticism applies equally to earlier analytical creasing Xob,. Larger stress exponents are therefore to models and the present simulation. be expected at low temperatures and in materials with finely distributed obstacles. 6. CONCLUSIONS 5. DISCUSSION Because z >>(z. . . . + Zbyp~)/2 produces glidecontrolled deformation, and z <<(%ou~ + Zbypass)/2 produces little deformation, n observed in climbcontrolled steady state creep should be close to nm~x. nm~xexceeds in many cases the value of 4 predicted by the earlier analysis of this model. No stress dependence of any structural parameter is needed to raise n to values observed in metal-class materials. The assumption in the earlier analysis of dislocation velocity proportional to stress on the dislocation cannot explain the disparity between n obtained from the earlier analysis and nm~ from the simulation, since the simulation also assumes this relationship. Using N vc z in the earlier analysis cannot be responsible for the disparity in n values, since Kr results show this to be true in the simulation. Figure 3 shows that both magnitude and sign of lead can vary. Figure 4(e) shows trxx trxx l~ad(t) to increase at an increasing rate to a maximum just before bypass even when the sign does not change. These results lead is incorrect. show that assuming constant trxx Suggesting this as explanation for the disparity between n obtained from the 1955 model and n ~ obtained from the simulation supports the applicability of the 1955 model, while exposing an incorrect assumption made to allow analysis of the model. When N = 1, and is not dependent on z, V¢mbOCNz o¢ z, and hdimbOC1/Nz oC 1/Z. Then ~xy OC Vclimb/heVlmb OCT 2. The 1955 model therefore predicts n = 2 at r < Zl, agreeing with the simulation
Assuming constant climb stress on the lead dislocation of the pileup during its climb to surmount the obstacle is incorrect. Using an improved solution method, and dropping lead the assumption of constant axx , no invocation of stress dependence of structural parameters is needed to obtain n = 5 and above, values typically observed in climb-controlled creep in metal-class materials. The 1955 model predicts the limiting behavior seen in the simulation results for low z. An early glide control model [9] predicts the limiting behavior seen in the simulation results for high Me and high z. Simulation results at low stress suggest a mechanism for the genesis of low angle tilt boundaries. REFERENCES
1. J. Weertmann, J. appl. Phys. 26, 1213 (1955). 2. J. Weertmann, Rate Processes in Plastic Deformation of Materials (edited by J. C. M. Li and A. K. Mukherjee), p. 315. Am. Soc. Metals, Metals Park, Ohio (1975). 3. J. Wccrtmann, Natural Fifth Power Creep Law for Pure Metals, Conf. on Engng Mater. Struct., Swansea, Wales. Pineridge Press, Swansea (1984). 4. Hirth and Lothe, Theory of Dislocations, 2nd edn. Wiley, New York. 5. J. C. Butcher, The Numerical Analysis of Ordinary Differential Equations, pp. 304-308. Wiley, New York (1987). 6. N. P. Suh and P. L. Turner, Elements of the Mechanical Behavior of Solids. McGraw-Hill, New York. 7. P. Tibbits, Dissertation, University Microfilms (1988). 8. M. F. Kanninen and A. R. Rosenfield, Phil. Mag. 20, 569 (1969). 9. J. Weertmann J. appL Phys. 28, 1165 (1957).
0956-7151/91 $3.00 + 0.00 Copyright © 1991 Pergamon Press plc
REEXAMINATION OF A 1955 THEORY OF CLIMB-CONTROLLED STEADY-STATE CREEP P. A. TIBBITS Center for Naval Analyses, 4401 Ford Avenue, Alexandria, VA 22302, U.S.A. (Received 23 October 1989; in revised form 25 May 1990) Abstract--Observed steady-state creep rates of many metal class materials show stress dependence i o~ z n, where n ~ 5. Many models of power-law creep predict n = 3 or 4, and require ad hoc assumption of the stress dependence of some geometric parameter to explain the observed n value. A 1955 climb-controlled model of steady-state creep assumes constant climb stress at the lead dislocation of an obstructed pileup to allow analysis. This analysis predicts n = 4. Invocation of a stress-dependent geometric parameter allows prediction of n ~ 5. In the present work, numerical simulation of the model shows assuming constant climb stress to be incorrect and assuming stress-dependent geometry to be unnecessary to predict n ~ 5. High and low stress limiting behaviors of the simulation agree with the results of earlier analytical models. Slip of pileup dislocations during climb of the lead dislocation is suggested as the physical explanation for the non-constant climb stress on the lead dislocation. Both this work and previous analytical models based on pileup mechanisms are subject to criticism based on the lack of observations of dislocation pileups in steady state creep. Rrsumr---Les vitesses de fluage observres en rrgime permanent pour beaucoup de matrriaux m&alliques r6vrlent une drpendance en fonction de la contrainte ~ ocz n, off n ~ 5. Plusieurs modrles de fluage avec loi de puissance prrvoient n = 3 ou 4 et exigent ainsi l'hypothrse d'une drpendance en fonction de la contrainte d'un certain paramrtre gromrtrique pour expliquer la valeur observre de n. propos6 en 1955, un modrle, contr616 par la montre, du fluage en rrgime permanent suppose une contrainte de montre constante sur la dislocation de t~te d'un empilement bloqur. Cette analyse prrvoit n = 4. En invoquant un paramrtre grom&rique drpendant de la contrainte, on peut prrvoir n ~ 5. Dans le prrsent travail, une simulation numrrique du modrle montre que l'hypothrse d'une contrainte de montre constante est incorrecte et que l'hypothrse d'une grom&rie drpendant de la contrainte n'est pas nrcessaire, pour prrdire que n ~ 5. Les resultats de la simulation, pour les limites des fortes et des faibles contraintes, sont en accord avec ceux de modrles analytiques antrrieurs. On suggrre le glissement de dislocations empilres pendant la montre de la dislocation de t&e pour expliquer physiquement une contrainte de montre non constante sur la dislocation de trte. Ce travail, comme les modrles analytiques antrrieurs basrs sur les mrchanismes d'empilement, est sujet ~ la critique par suite de l'absence d'observations d'empilements de dislocations dans le fluage en rrgime permanent.
Znsammenfassung--Die beobachteten station/iren Kriechraten vieler metallischer Materialien weisen eine Spannungsabh/ingigkeit der Form E~ , mit n ~ 5, auf. Viele Modelle des Potenzgesetzkriechens sagen n = 3 oder 4 voraus und erfordern die ad-hoc-Annahme der Spannungsabh/ingigkeit einiger geometrischer Parameter, um den beobachteten n-Wert zu erkl/iren. Ein Modell des Kletter kontrollierten station/iren Kriechens von 1955 nimmt eine konstante Kletterspannung an der fiihrenden Versetzung einer behinderten Aufstauung an, um die Analyse zu erm6glichen. Diese Analyse ergibt n = 4. Die Beriicksichtigung eines spannungsabh/ingigen geometrischen Parameters erm6glicht die Voraussage von n ~ 5. In der vorliegenden Arbeit wird das Modell numerisch simuliert; es zeigt sich, dab die Annahme einer konstanten Kletterspannung nicht richtig ist und die Annahme einer spannungsabhhngigen Geometrie fiir die Voraussage yon n ~ 5 unn6tig ist. Das Grenzverhalten des Modelles bei hoher und niedriger Spannung stimmt mit frtiheren analytischen Modellen iiberein. Als physikalische Erkltirung der auf die fiihrende Versetzung wirkenden nicht-konstanten Kletterspannung dient die Gleitung der aufgestauten Versetzungen wtihrend des Kletterns. Sowohl diese Arbeit wie auch fr/ihere analytische Modelle, die auf Aufstauungen beruhen, sind kritisch zu sehen im Hinblick auf die Tatsache, dab Beobachtungen von Aufstauungen beim station/iren Kriechen fehlen.
1. INTRODUCTION Observed steady-state creep rates o f m a n y metal class materials, in which glide mobilities far exceed climb mobilities, show stress dependence ~ ocz n, where n ~ 5 is the stress e x p o n e n t and z is the applied shear stress. M a n y models o f power-law creep predict n ~ 3 or 4. E x p l a n a t i o n o f observed stress exponents re-
quires ad hoc assumption o f the stress dependence o f some geometric parameter, giving rise to m u c h discussion [2, 3]. A 1955 climb-controlled model o f steady state creep [1] consists o f a dislocation source which generates a pileup o f positive infinitely long edge dislocations obstructed by a L o m e r dislocation. A s s u m i n g c o n s t a n t climb stress during climb o f the 105
106
TIBBITS: CLIMB-CONTROLLED STEADY-STATE CREEP
lead dislocation obviates modelling the detailed evolution of the pileup and allows analytical solution of the problem. This analysis predicts n = 4. Invocation of stress dependence of the source to obstacle distance allows prediction of n ~ 5. In the present work, a numerical technique calculates an approximate solution without assuming either constant climb stress on the lead dislocation or stress dependence of the source to obstacle distance. A model very similar to that proposed in 1955 but without ad hoc assumptions then predicts n ~ 5. 2. THE 1955 MODEL
Figure 1 diagrams the 1955 model, consisting of two F r a n k - R e a d sources, F R I and FR2, their slip planes S1 and $2, and the grain boundary GB which obstructs dislocation movement on S1. F R I operates when it experiences net positive shear stress. The screw segments of the loop produced by FR1 move at much higher velocity than the edge segments, leaving a pair of edge dipoles. The left, negative, limb of the dipole moving leftward on S1 under z is unobstructed, moves far from FR1, and is neglected. FR2 also produces infinite straight positive edge dislocations when the stress field from the pileup of dislocations from F R I on SI at GB becomes sufficiently strong. Dislocations from FR2 then move along $2 until a dislocation reaction occurs at the slip plane intersection O to form a Lomer-Cottrell lock. This sessile dislocation obstructs dislocation motion on S1, causing a pileup of N dislocations to the left of O, where N oc ~. The lead dislocation of this pileup must climb above S1 before the dislocations can slip further. The number of Lomer dislocations forming between FR1 and GB by reaction with dislocations from intersecting slip planes is stress dependent, leading to a stress dependence of the source to obstacle distance, Xob,(z). In metal-class materials, climb of the lead dislocation of the pileup is the rate-limiting step, i.e. the strain rate is under climb control. The strain rate is
inversely proportional to the climb time, which is found from the height to which the lead dislocation must climb and the rate at which it climbs, ~xy ~ 1/tclimb= /)climb/ hcyanb• 2.1. Assumptions o f the 1955 model
Experimental observations justify assuming dislocation glide and climb velocities linear in the glide and climb forces acting on the dislocations. The constants of proportionality are the glide and climb mobilities, M s and Me. Assuming constant climb stress of magnitude Nz on the lead dislocation during climb leads to constant climb velocity/)climb = m c m z . Because N oc z [1], this assumption leads to VcmbOCZ2. 2.2. Predictions o f the 1955 model
Knowledge of the height to which the lead dislocation must climb allows derivation of the stress dependence of the shear strain rate. At bypass, glide stress exerted by the obstacle on the lead dislocation just equals glide stress exerted by the applied stress and the pileup dislocations, Zobs= T g l i d e • Because stress exerted between dislocations is inversely proportional to their separation, %b, ocl/h~,,mb. NOW z#d,= Nz ~ ~2, so hclimb oc 1/'c2, and exyoc/)climb/hclimb0£ ,~4. This expression arises solely from dislocation dynamics, ascribing no stress dependence to Xobs or other structural parameters. Using the Xob~ stress dependence, n ~ 5. 3. SIMULATION OF THE 1955 MODEL
The detailed evolution with time of the positions of all the dislocations of a pileup during climb of the lead dislocation over an obstacle can be simulated by a system of coupled ordinary differential equations relating the velocity of each dislocation to the position of all the dislocations. Assuming unit Burgers vector and velocities proportional to force p = vy = Vclimb= Mcaxxb = Mcaxx Yc = vx = V~ide = Mg Zxr b = Mg rxr.
GO S2
.J_
sl
X
i
i
~j Ms Y'.N (Xi _ Xi)[(Xi _ Xi)2 _ i71 [(xj -- xi) 2 + (yj
(Yi -- Yi) 2] "F M g z -
-
yi)212
i#j
_1_ /
/ o
FR1
Using the stress exerted by one dislocation on another [4], these equations of motion are for dislocation j
Yj Mo i=/-'t
--(yj -- yi) [3(xj -- xi) 2 + (yj -- yi) 2] [(xj -- x,) 2 + (YY-- y~)212
i#j
I~
x.
-I
Fig. 1. Geometry of the 1955 model. S1 and $2 are slip planes, FR1 and FR2 are Frank-Read sources, GB is a barrier to dislocation movement on S1, and O is a Lomer dislocation formed by reaction of dislocations from FR1 and FR2 at the intersection of S1 and $2.
Units of stress are #/[2n(1 - v)] where/~ is the shear modulus and v "the Poisson's ratio. All distances are normalized by b. The summations over i include the sessile obstacle dislocation. The coordinates [xj(t), yj(t)] of the N ( t ) dislocations as a function of time constitute the solution of these coupled ordinary
TIBBITS: CLIMB-CONTROLLED STEADY-STATE CREEP differential equations. A dislocation sink X~nkto the right of the source limited the number of dislocations to be followed and the number of equations to be solved.
A pair of infinite straight edge dislocations of the same sign on parallel slip planes separated by perpendicular distance y bypass one another only if [7]
,fk y(ky2-y2)
3.1. Numerical solution of the simulation Many methods exist for the solution of such initial value problems. A dual order Runge-Kutta formulation with adaptive stepsize based on estimated error [5] solved the set of equations posed above. The equations required a moderate number of algebraic operations for derivative evaluation, and showed no excessive stiffness.
3.2. Input-output structure of the simulation The simulation required as input the locations of the obstacle (Xob~,Yo~) and sink Xsittk, plus the climb mobility M c and applied stress z. Using Yob~= - 0 . 2 5 obviated simulating thermally activated displacement of the lead dislocation from the climbmetastable position left of the obstacle, were the obstacle exactly in the source slip plane. All coordinates are relative to the source position. Glide mobility was assumed to be one. The simulation output the number N(t), locations [xi(t),yi(t)], and velocities [~i(t), ~fi(t)] of the dislocations as a function of time. These provided the instantaneous strain and strain rate [6]
exy(t) = ~ Xg(t) f"~l X2ink '= '
~ ~i(t) and
ExY ( t ) =
2 " i= 1 Xsink
The net glide and climb stresses at any of the dislocations were available as a function of time, but were output only for the lead dislocation of the pileup, Zxy l"adtt~ lead(t). The Boltzmann ,~,, and trx~ average
~
f
z > zcdt = -v .- (ky2 -b y2)2 ,
where
k = 3 +x/@
Increasing z over several runs in which N - 1 and which were allowed to go to long times resulted in a value for the minimum stress at which bypassing occurred which was very close to %fitComparison of results generated by the simulation accorded well with those generated by an earlier simulation [8] of the dynamic formation of a pileup. In this problem, climb is not allowed to occur, and the pileup equilibrates at an obstacle. The equilibrium positions of the dislocations, the instantaneous position of the lead dislocation and the instantaneous stress exerted by the pileup dislocations on the obstacle dislocation were very similar in the two simulations. Two runs of the simulation identical except for decreasing error tolerance showed by the very close agreement in their results that the error tolerance specified for the Runge-Kutta routine was sufficiently small. 4. RESULTS
4.1. Dislocation positions A series of plots of the dislocation positions provides a "movie" of events occurring in a run of the simulation. Figure 2 shows in such a movie a sequence of events in qualitative agreement with the events assumed in Weertmann's 1955 model. Figure 3 shows the lead dislocation of the pileup climbing alternately above and below the obstacle, an unanticipated sequence of events.
t=tIN(t) dt
dl=ti
t y - ti gave the expected number of dislocations for a given z. The slope of a line drawn through similar features of the quasi-steady-state portion of the cyclic E~y(t) vs t curve provided the average strain rate ~y at given z. A series of runs over increasing z provided a log(~y) vs log(z) curve the slope of which gave the n vs log(z) curve.
3.3. Calibration of the simulation Comparison to analytical solutions for two simple cases and to a previous simulation of pileup formation proved the simulation accurate. For a single infinite straight dislocation in unobstructed glide, ~y = z/x~i,k. 2 By inspection, n = [O log(~y)]/ [~ log(z)] = 1. Omitting the obstacle, the simulation calculated strain rates for several z. A backward finite difference formula, [log(-~y)2 - log(~-xy)l]/[log(z2)-log(z~)], approximated n, which was very close to one.
107
4.2. Results of a typical run Figures 4(a)-(e) show, as functions of time, the number of dislocations, the shear strain, shear strain rate, and glide and climb stresses exerted on the lead dislocation of the pileup. The plots shown are typical of the results from all runs. All of the plots show an initial transient behavior followed by a cyclic quasisteady-state region.
4.3. Average number of dislocations Table 1 shows N, the average number of pileup dislocations present in the quasi-steady state as a function of z for the runs made at Xobs=90, Mc = 0.01, 0.001, and 0.0001. Surprisingly, N is independent of Me. Where Kr > 1. N" is proportional to z.
4.4. Average strain rate, ~xy As Fig. 4(b) illustrates, -gxeis the slope of the Exy(t) vs t curve. Sets of log(~-~y) vs log(z) curves were obtained for Xobs = 10, 30, 90, and 270. Figures 5(a) and (b) show the sets of curves obtained at Xobs = 30
108
TIBBITS: CLIMB-CONTROLLED STEADY-STATE CREEP .k >±
O )
).1_
.L-IO
.L O 8
.L
>.L
O >.L ± O
1
)
9
J.
2
.L
>.L
>a.
O
_L
10
O
>.L >.L
O .1..
3
11
.L
>.L )_L
.L O
and :Cob,= 90, respectively. Each curve in a set was obtained at the noted value of Me. Runs made under special conditions investigated limiting behaviors of ~-xy. These limiting behaviors determined the range o f z investigated, and showed the z range of applicability of some earlier models. 4.4.1. Behavior o f - ~ in absence o f obstacle. Figs 5(a) and (b) each show a labelled l o g ( ~ ) vs log(x) curve for the no-obstacle case. A t low z, where only one dislocation exists, the slope of this curve is n = 1. At high z, the slope becomes 2, taking on intermediate values in a smooth fashion at intermediate stress levels. The strain rate in this case is independent of Me, is proportional to the glide mobility, and is under glide control [7]. 4.4.2. High-stress limiting behaviour of-~xy. In Figs 5(a) and (b), the log(~xy) vs log(x) curves for all values of Mc converge in the high stress region to the curve for the no-obstacle case, which is in this region a
O
.L ...L 12
±
±
O
.1_ .L O
Fig. 2. For T = 0.2, Xob~= 30, M c = 0.001, the sequence of events agrees with the 1955 model. Inverted "T" symbols denote the dislocations. A diamond denotes the obstacle dislocation. The source, denoted by half a diamond, has horizontal slip plane at its center. Frame 1. Three dislocations have formed a pileup to the left of the obstacle. The lead dislocation has begun climb. Frame 2. The lead dislocation has climbed appreciably. The other two pileup dislocations laave glided a significant distance during the lead dislocation climb, causing significant change in the climb stress on the lead dislocation during climb. Frame 3. The lead dislocation has bypassed the obstacle. A dislocation has been emitted to replace it in the pileup. The sequence of events repeats in steady state deformation.
>
O
O .L
-L >
O
2.
Fig. 3. "Movie" of simulation for z = 0.1, Mc = 0.1, and xo~ = 30. The climb stress on the lead dislocation changes in sign as well as in magnitude. Frame 1. Emission of first dislocation, D1. Frame 2. D2, the second dislocation is emitted after D1 has begun climb. Frame 3. The dislocations repel. D2 is now below the slip plane of the source. Frame 4. D1 has bypassed the obstacle. Frame 5. D1 continues glide, D2 has been pushed by D1 below the plane of the obstacle. Climb stress from the obstacle is negative. Frame 6. Continued glide of D I and D2, slow negative climb of D2. Frame 7. Continued glide of D 1 and D2, slow negative climb of D2. Frame 8. D1 annihilated at crystal boundary. D3 emitted. Negative climb stress exerted by D3 on D2. Positive climb stress exerted by D2 on D3. Frame 9. D3 remains above plane of obstacle, D2 undergoes visible negative climb. Frame 10. D3 is already visibly above slip plane of the source. D2 continues climb. Frame 11. D2 bypassing obstacle underneath. Frame 12. D2 has bypassed the obstacle, and will glide to annihilation at right, leaving system in some condition as in Frame 1. straight line of slope 2. A 1957 model [9] predicts this behavior in the glide-controlled limit. The point of convergence of the curves gives z above which ~ is glide controlled. N o maximum z was encountered above which investigation could not be carried out. Extension of the results past the point of convergence of the curves showed no new behavior. 4.4.3. Low-stress limiting behavior Of ~xy. The lowstress behavior of the log(~~y) vs log(x) curves also shows a tendency toward straight line behavior and slope 2. This is a consequence of having only one dislocation in the pileup below a certain Zl. Restricting N = 1, the log(~xy) vs log(x) curve was a straight line of slope 2. This curve coincided with the z < Zl region of the log(~-~y) vs log(z) curve obtained in
TIBBITS: CLIMB-CONTROLLED STEADY-STATE CREEP d
-zoo
109
(a)
-2.70
OHL
-3.40
-4.10 -4.80
% No o b s t a c l e S ,
b ~
.-'7
f
-8.~:~
~7 0.05 °11[
-6.90
0.01 ]
-7.~
0.001 J
-8.30 --
-9.00
'~1
I
I
I
I
I
I
I
I
I
I
I
I
-3.00 -2.60 -2.20 -1.~ -1.40 -1.1~1 -0.60 -0.20 0.20 0.60 1.~
C
Log (Xl
-5,40 -8,10
Fig. 4. The basic output of the simulation, plotted against time. All plots show initial transient, followed by repeated cycles (quasi-steady-state). (a) N, number of dislocations. (b) E~y, shear strain, with line drawn for calculation of average strain rate i'~y. (c) Instantaneous shear strain rate, ~y, (d) glide stress exerted on lead dislocation, ~xy , remains nearly constant. (e) climb stress exerted on the lead dislocation, cr~d, increases sharply during climb, in disagreement with an assumption of the 1955 model.
-8.80
?.p
I
No obst
0 j//j
-8.e9
.0.,0
0.
00;1 ~
r J
ordinary simulation at the same Me. Figure 5(b) shows the N = 1 curve for Mc = 0.01 and Xob,= 90. The m i n i m u m z of interest is z _ ~ = 1/Xob~, below which the source will not operate because of the negative glide stress exerted at the source by the obstacle dislocation. A practical limit on the minim u m z investigated resulted from the model's simplification of the F r a n k - R e a d source. When, at low r, climb of an emitted dislocation brings it above a line of slope one through the source, it exerts positive stress on the source. U n d e r net positive stress, the source emits a second dislocation. The second dislocation climbs downward, away from the first, and causes emission of a third dislocation when it passes below a line of slope - 1 through the source. This process continues until a nearly vertical wall of dislocations, a low-angle tilt boundary, exists near the Table 1..N for Xob, = 90 vs z and M e
Mo 0.01 0.0125 0.01875 0.022 0.025 0.05 0.1 0.2 0.4
0.001 1 1-2 1-2 1-2 2-3 5~ 9-10 17-18
0.0125 1 . . . . . . 0.025 1-2 0.05 2-3 0.1 5~i 0.2 9-10 0.4 17-18
0.0001 . . . 0.025 0.05 0.1 0.2 0.4
0.00001
.
. . . 1 2-3 5-6 9-10 17-18
. . . -0.05 0.1 0.2 0.4
-2-3 5 10 17.5
-1.10+1 /
./
0.0001 "f 0.00001
-I.03+IF I
I
I
I
I
I
-3.00 -2.70 -2.40 -2.10 -1.80 -1.50 -1.20 -0.99 -0.60 -0.30 0.00
Log (~)
Fig. 5. (a) L o g ( ~ ) vs log(r) for Xob,= 30, x,~k = 100, and Mc marked. Data points are marked by symbols. Curves fitted to data points are cubits at low z blended into straight lines with slope 2 at high z. z. . . . Zl, %it, and zbsp~ denote threshold stresses for source operation, N > 1, tiltwa11 emission, and bypass of obstacle in the Me = 0 case. (b) Log(~y) vs log(r) for xot, = 90, x~,k = 270, and climb mobilities marked. The case for which N = 1 is labelled. source. The stress below which this can occur, Ztilt, is near z , ~ for small Mc [7]. zl, z . . . . . . and Ztilt are marked on Fig. 5(a). If a dislocation sink exists just to the left of a source tilt wall emission may occur in real materials. 4.4.4. Low Mc limiting behaviour of ~xy. Me depends on temperature through its dependence on diffusivity [1], so this section also discusses low temperature limiting behavior. The log(~-~y) vs log(z) curve was observed to fall below this one. The low z end of this curve is at the stress below which the lead dislocation of the pileup cannot bypass the obstacle. This threshold is labelled as Zbyp, in Fig. 5(a). 4.4.5. High Mc limiting behavior of ~xy. Figs 5(a) and (b) show approach of the log(~xy) vs log (z) curves to the no-obstacle case. U n d e r very high Mc as in the no-obstacle case, ~xy is glide-controlled.
I10
TIBBITS: CLIMB-CONTROLLED STEADY-STATE CREEP Table
2.
nmax vs Me ~< 0.01
Xobs
for
results. This agreement in the case N = 1 suggests that continued slip of the dislocations in the pileup Xobs Me tlmax during climb of the lead dislocation is the physical 10 0.01 4.7 explanation for the increase in Cr~xadresponsible for 0.001 5.8 the disparity in n values at higher z. 30 0.001 4.5 0.0001 6.2 The relation trxxlead~r-~Nz is valid for static pileups. 90 0.01 3.7 Because climb of the lead dislocation reduces the 0.001 4.4 0.0001 5.3 glide stress opposing movement of the remaining 0.00001 11.0 pileup dislocations, they glide toward the obstacle. 270 0.01 3.0 Because Ms>>M¢, adjustment of the pileup dislo0.001 3.8 0.0001 5.6 cations to maintain glide equilibrium greatly increases tr~ d. The lead dislocation climbs under nonequilibrium conditions and al~d>>Nz, leading to 4.5. Stress exponent the higher n values observed in the simulation and in Taking the slope of a l o g ( ~ ) vs log(z) curve yields experiments. an n vs log(z) curve. The sigrnoidal log(~xy) vs log(z) Third power laws arising from many analytical curves have maximum slope near (%ou_~+ %yp~)/2. models are "natural" only in the sense that they arise The n vs log(z) curves resemble parabolas of negative logically from the assumptions made in constructing curvature with maxima near (Zsource+Zbypass)/2. the model. This work shows the fallacy of certain of Table 2 shows the maximum value of the stress those assumptions. exponent, n . . . . for each value of Xob~and values of The lack of experimental observation of disloM¢ ~< 0.01. Table 2 shows dependences of nmax on Me cation pileups in steady-state creep must be noted. and Xob~. Increasing M¢ decreases nm~x, as does in- This criticism applies equally to earlier analytical creasing Xob,. Larger stress exponents are therefore to models and the present simulation. be expected at low temperatures and in materials with finely distributed obstacles. 6. CONCLUSIONS 5. DISCUSSION Because z >>(z. . . . + Zbyp~)/2 produces glidecontrolled deformation, and z <<(%ou~ + Zbypass)/2 produces little deformation, n observed in climbcontrolled steady state creep should be close to nm~x. nm~xexceeds in many cases the value of 4 predicted by the earlier analysis of this model. No stress dependence of any structural parameter is needed to raise n to values observed in metal-class materials. The assumption in the earlier analysis of dislocation velocity proportional to stress on the dislocation cannot explain the disparity between n obtained from the earlier analysis and nm~ from the simulation, since the simulation also assumes this relationship. Using N vc z in the earlier analysis cannot be responsible for the disparity in n values, since Kr results show this to be true in the simulation. Figure 3 shows that both magnitude and sign of lead can vary. Figure 4(e) shows trxx trxx l~ad(t) to increase at an increasing rate to a maximum just before bypass even when the sign does not change. These results lead is incorrect. show that assuming constant trxx Suggesting this as explanation for the disparity between n obtained from the 1955 model and n ~ obtained from the simulation supports the applicability of the 1955 model, while exposing an incorrect assumption made to allow analysis of the model. When N = 1, and is not dependent on z, V¢mbOCNz o¢ z, and hdimbOC1/Nz oC 1/Z. Then ~xy OC Vclimb/heVlmb OCT 2. The 1955 model therefore predicts n = 2 at r < Zl, agreeing with the simulation
Assuming constant climb stress on the lead dislocation of the pileup during its climb to surmount the obstacle is incorrect. Using an improved solution method, and dropping lead the assumption of constant axx , no invocation of stress dependence of structural parameters is needed to obtain n = 5 and above, values typically observed in climb-controlled creep in metal-class materials. The 1955 model predicts the limiting behavior seen in the simulation results for low z. An early glide control model [9] predicts the limiting behavior seen in the simulation results for high Me and high z. Simulation results at low stress suggest a mechanism for the genesis of low angle tilt boundaries. REFERENCES
1. J. Weertmann, J. appl. Phys. 26, 1213 (1955). 2. J. Weertmann, Rate Processes in Plastic Deformation of Materials (edited by J. C. M. Li and A. K. Mukherjee), p. 315. Am. Soc. Metals, Metals Park, Ohio (1975). 3. J. Wccrtmann, Natural Fifth Power Creep Law for Pure Metals, Conf. on Engng Mater. Struct., Swansea, Wales. Pineridge Press, Swansea (1984). 4. Hirth and Lothe, Theory of Dislocations, 2nd edn. Wiley, New York. 5. J. C. Butcher, The Numerical Analysis of Ordinary Differential Equations, pp. 304-308. Wiley, New York (1987). 6. N. P. Suh and P. L. Turner, Elements of the Mechanical Behavior of Solids. McGraw-Hill, New York. 7. P. Tibbits, Dissertation, University Microfilms (1988). 8. M. F. Kanninen and A. R. Rosenfield, Phil. Mag. 20, 569 (1969). 9. J. Weertmann J. appL Phys. 28, 1165 (1957).