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An investigation of Harper-Dorn creep—II. The flow process
COOI-616&82 040881-07Y)3.00,0 Pcrpmon Press Lrd
_(cru mernll. Vol. j0. pp. 881 IO 887. 1982 Pnnrcd m Great Bntain
AN INVESTIGATION OF HARPER-DORN THE FLOW PROCESS TERENCE
G. LANGDON
and PARVIZ
CREEP-II.
YAVARI
Departments of Materials Science and Mechanical Engineering, University of Southern California, Los Angeles, CA 9COO7,U.S.A. (Received 13 August 1981) Abstract-Several
deformation mechanisms have been proposed to explain the flow process of HarperDorn creep. The implications of these various mechanisms are compared with the available experimental data, and it is concluded that flow occurs by the climb of edge dislocations under saturated conditions. Using the theoretical model for this process, it is possible to specify the experimental criteria for observations of Harper-Dorn-creep in terms of the upper limiting stress and the lower limiting grain size. R&m&-Plusieurs micanismes de deformation ont bte proposes pour expliquer I’tcoulement plastique au tours du fluage de Harper et Dorn. Nous comparons les implications de ces divers mecanismes avec les r&hats experimentaux disponibles; nous en concluons que l’ecoulement se produit par montee de dislocations coins dans des conditions de saturation. A l’aide du modtle theorique de ce phenomtde. on peut specifier les crittres exptimentaux pour observer le fluage de Harper et Dorn. sous forme dune contrainte limite superieure et dune taille de grains limite inferieure. Zusammenfassung-Verschiedene
Mechanismen wurden schon zur Erkllrung des Harper-Dorn-Kriechens vorgeschlagen. Die Folgerungen aus den einzelnen Mechanismen werden mit den verftigbaren experimentellen Ergebnissen verglichen. Das Ergebnis ist, daB das FlieDen herriihrt von dem Klettern der Stufenversetzungen unter Wtigungsbedingungen. Mit dem theoretischen Model1 fur diesen Mechanismus ist es moglich, die experimentellen Bedingungen fur das Auftreten des Harper-Dorn-Kriechens als obere Grenzspannung und untece KorngrijBe anzugeben.
1. INTRODUCI’ION
In an earlier report, hereafter designated I [l], it was demonstrated that a number of metals exhibit Harper-Dorn creep at low stress levels. The major distinguishing feature of Harper-Dorn creep is a steady-state strain rate which is significantly larger than the value predicted for Nabarrcl-Herring diffusional creep [2,3]. For several metals tested at high homologous temperatures (typically in the vicinity of 0.98 T,, where r, is the absolute melting point) and with large gram sixes (of the order of 1-4mm), the steady-state shear strain rate in the Harper-Dorn region, Jo,is given by
where D, is the diffusion coefhcient for lattice selfdilfusion, G is the shear modulus, b is the Burgers vector, k is Boltzmann’s constant, T is the absolute temperature, r is the shear stress, and A& is a dimensionless constant. As indicated by the summary of experimental data given in Table 1 of I, the value of A’,, is typically about 5 x 10-r’. In a series of detailed creep experiments on the Al-S% Mg solid solution alloy, described in I, it was shown that the behavior at low stress levels, up to r x 0.25 MPa, obeyed equation (1) with
Ai, = 4.5 x lo-“. In addition, microstructural observations in this region revealed the presence of a predominance of edge dislocations and a dislocation density, determined from etch pit measurements, which was independent of stress and equal to (4.7 C 0.9) x IO3 cm-‘. By combining these results with earlier reports of Harper-Dom creep, it was possible to establish the primary mechanical and microstructural characteristics for this tylze of flow. Several mechanisms have been proposed to explain the flow process in Harper-Dorn creep, and these were considered briefly in several early reports [4-6l and examined in detail in the review of Harper-Dom creep by Mohamed er al. [7-J. The purposes of the present paper are two-fold. First, to examine the implications of each of the suggested deformation mechanisms with respect to the established experimental characteristics listed in I and to thereby determine the most probable flow process. Second, to use this information to establish the criteria for observations of Harper-Dorn creep in terms of the upper limiting stress and the lower limiting grain size.
2. THE DEFORMATION HARPER-DORN
MECHANISM CREEP
FOR
In this section, each mechanism is considered sep881
882
LANGDON
arately with reference presented in I.
AND
YAVARI:
HARPER-DORN
to the experimental
data
2.1 The motion of jogged screw dislocations In the early work of Harper et al. (8,9], the Harper-Dom process was attributed to the motion of jogged screw dislocations and the associated production of vacancies, using a concept developed earlier by Mott [lo]. Following the more recent analysis of this process by Hirth and Lothe [ll], this mechanism leads to a steady-state shear strain rate which is given by DtGb j = 12zpljab F
T ‘.’ c
( )O
(21
where p is the dislocation density and 1, is the jog spacing on the screw dislocations. This mechanism is not appropriate for HarperDorn creep for two reasons. First, it requires a predominance of screw dislocations, which is not consistent with the experimental evidence as demonstrated by Fig. 7 in I. Second, by putting Ah, = 127cplj,b = 4.5 x IO-I1 and p = 5 x 103cm-* from Fig. 8 in I, it may be shown that this mechanism requires a jog spacing which is unrealistically small (I, z 0.3b). In addition, this discrepancy wouid be further enhanced if, as seems likely (see section 2.6), the experimentat value of p from etch pit measurements is an under-estimation of the true value. 2.2 NabarrcF-Herring creep associated with subgrain boundaries
FriedeI[12] proposed that the experimental creep rate may be faster than the value predicted by the Nabarro-Herring process if the vacancies diffuse between subgrain boundaries. In this case, the steadystate shear strain rate is given by the relationship for Nabarro-Herring creep [2,33 with the grain size, d, replaced by the subgrain size, %,so that
CREEP-II.
THE FLOW PROCESS
where by 5 10. Substitution of equation (4) into equation (3) gives $ x r3, which is inconsistent with experimental results in the Harper-Dorn region. Furthermore, even If it is assumed, as concluded by Muehleisen [16], that the subgrain size reaches an essentially constant value at low stresses, the average experimental value of 2 2: 2mm predicts creep rates through equation (3) which are about two orders of magnitude slower than the observed rates?. 23 The climb-controlled generation of dislocations from a fixed density of sources This mechanism was proposed by Barrett et al. [ 133 to explain their results on pure Al, based on the inference from data on Al with FeAI, preci’pitates that Harper-Dom creep is cobtrolled by a dislocation process. The principle of the mechanism may be briefly summarized in the context of the present work as follows. If dislocation generation occurs by diffusion-controlled climb at a fixed number of sources per unit volume which is independent of the applied stress, the dislocation generation rate, i+, is expressed as Pe = PO&/l-
(5)
where p. is the fixed dislocation source length per unit volume, u, is the climb velocity, and r is the Taking v, = 2Dlb2t/kT and climb distance. r = Gb/2r, i+ is given by b+ = 4poD,bT2/kTG.
(6)
An important assumption in the model of Barrett et ai. [I33 is that d~~ocation a~ihi~ation occurs only at s&rain boundaries. In this case, the annihilation rate, b_, is given by &i_ = P&/A:
(7)
where pm is the mobile dislocation density and ue is the glide velocity. Taking ve = 2u0r, where u. is a constant, and R = Gb[/z from equation (4), leads to i-
= 2p, vor2/Gb&
(‘3)
The rate of change of density with time, 4, is there-
where A:VH- 84. This mechanism also leads to two inconsistencies with the experimental data. First, subgrains are not always observed, as in the experimental work described in I or in the earlier investigations by Harper et al. [8,9] : it was conclude-d in I that the condition for subgrain formation is d > 1.
Second, if subgrains form in the Harper-Dom region, the data of Barrett et al. El33 indicate that, as in dislocation climb, there is a variation in subgrain size with stress such that [14,15] i/b = j(q’G)- 1 7 This discrepancy may be appreciated position of the line for NabarreHerring d = 3.3 mm in Fig. 1 of 1.
(4) by noting the creep when
fore 4poD, bs’ 2PIn@*9 . . p* - p_ = kTG Gbj
(31
so that, at steady-state when i, = 0, pm = 2poD,b2[/kTvo.
Substituting equation (10) into the standard pression for $ ( = p,bu& leads to
(10)
ex-
(11) This mechanism has the advantage of predicting reasonable values for the creep rates: for example, taking p. z lo3 cms2 as a typical annealed dislocation density and c ^u IO, equation (11) gives AHD = 4pob2C z 3.3 x lo-“, which is close to the
LANGDON
AND
YAVARI:
HARPER-DORN
experimental value of A& = 5.0 x lo- l1 listed for Al in Table I of I. However, the derivation of equation (11) is based spectically on the assumption that the dislocations are annihilated only at subgrain boundaries. If subgrains do not form, as in the experiments described in I or the work of Harper et al. [S, 93, the dislocations will be annihilated at the grain boundaries, so that equation (8) is replaced by fi_ = Zj,c,r/d.
W)
Performing the same analysis, the steady-state shear strain rate is then given by 3 = 4p,b’(!$($(;)2.
(13)
Equation (13) predicts a stress exponent of 2 and a linear dependence on grain size, and this is clearly inconsistent with the available experimental data for Harper-Dorn creep (see. for example, Fig. 1 of part I). It is therefore concluded that this mechanism is unable to account for the flow process in the HarperDorn region. 2.4 The ciscous glide of dislocations pheres
with solute armos-
Murty [6] observed Harper-Dorn creep in a Pb-9% Sn solid solution alloy at values of r/G below r 2 x lo- ‘. Based on an earlier observation by Stang ez al. [17] of a constant dislocation density from etch pit measurements on Fe-3% Si at r/G s lo-‘, Murty [63 suggested that Harper-Dorn creep in alloys may arise from a modified viscous glide process in which the dislocation density is constant and dislocation movement is restricted due to the presence of solute atmospheres. This concept was later developed in a quantitative manner by Mohamed [lS] to explain the results obtained on an Al-20/, Mg solid solution alloy. At high stresses, many solid solution alloys deform by viscous glide where the movement of dislocations is impeded by the dragging of solute atom atmospheres. Following Takeuchi and Argon [19]. the steady-state shear strain rate for a homogeneous distribution of dislocations is given by
where e is the solute-solvent size difference, c is the concentration of solute atoms, and 0’ is the diffusion coefficient for the solute atom in the solvent lattice. In general, the dislocation density in high temperature creep is given by ~cp*b= (r/G)‘.’
(15)
where a is a constant close to unity. Thus, equation (14) usually leads to 9 x TV. However, as noted by Murty [6] and Mohamed [lS], the dislocation density is independent of stress in the Harper-Dorn region, so that the dragging of solute atmospheres leads to a stress exponent of n = 1 at low stress levels.
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883
Taking p = 5 x lo3 cm-’ from Fig. 8 in I, e = 0.121 for the Al-Mg system [ZO], c = 0.058 for the Al-j% Mg alloy used in I, and 0’ = DI for pure . I Al [21], equation (14) gtves AHD2 6.0 x 10-l’ which is similar to, but about one order of magnitude lower than, the experimental result. An alternative calculation may be undertaken by combining the approach of Cottrell and Jaswon [22] for the interaction of dislocations with solute atom atmospheres and the microcreep mechanism developed by Weertman [23]. In this case, the steady-state shear strain rate for viscous glide is given by [21] 4= ‘(l --$Tpb’
(&>‘($!)(d””
(16)
where v is Poisson’s ratio. Taking v = 0.34 and ti = 1, equations (16) and (14) differ only by a factor of 0.7 although they are based on different theoretical concepts. From equation (16), the equivalent value for Ahe is -4.2 x 10-i’. There are two significant points of agreement between this modified viscous glide mechanism and the experimental observations. First, subgrains do not form during viscous glide with n c 3 and there is an essentially uniform distribution of dislocations [24-271: this is consistent with the lack of subgrains in Al-j’% Mg in the Harper-Dorn region, as documented in I. Second, the theory of Takeuchi and Argon [19] is based on control by the slower-moving edge dislocations, and this is in agreement with the preponderance of edge dislocations observed in I by transmission electron microscopy. Furthermore, there have been several experimental reports of a majority of edge dislocations in Al-Mg alloys after creep testing under viscous glide conditions with n 5: 3 [24,26,28]. However, closer inspection shows three important inconsistencies in this mechanism. First, in Al-Mg solid solution alloys exhibiting viscous glide behavior, the transition with decreasing stress is not from viscous glide with n c 3 to Newtonian viscous flow with n = 1.0. as required both in the initial concept and in the subsequent development of this mechanism. On the contrary, the transition is from viscous glide with n = 3 to dislocation climb with n z 4.5 and then to Newtonian Row with n = 1.0, as shown in Fig. 2 of I. Furthermore, the transition from n z 3 (class A behavior) to n ‘z 4.5 (class M behavior) with decreasing stress has been firmly established in Al-Mg alloys in both polycrystal[4,27,29-311 and single crystal [32] form, including in earlier work on the Al-2% Mg alloy by Mohamed [33], and there is also good evidence for a similar transition in the Pb-9% Sn alloy used by Murty [34]. Thus, it is unrealistic to assume that Harper-Dom creep arises from a simple modification of the viscous glide model. Second, if Harper-Dom creep in solid solution alloys is due to viscous glide through equations (14) or (16), the experimental value of the dimensionless
884
LANGDON
AND
YAVARI:
HARPER-DORN
constant, .&,, should decrease with increasing solute concentration because Ah, x l/c. This trend is not supported by the three sets of values for A& shown in Table 1 in I for three different Al-MO alloys, nor is it supported by the data in I on Al-W0 Mg where the experimental value of Ah, (4.5 x lo- Ii) is almost identical to the tabulated value for pure Al (5.0 x lo-“). Third, a mechanism involving the dragging of solute atmospheres would necessarily apply only to solid solution alloys, and the development of a restrictive mechanism of this type seems unreasonable when it is noted that there is a very close similarity in the Harper-Dorn region between the results obtained on alloys and on the appropriate pure metal (see Fig. 9 of I).
In an earlier review of Harper-Dom creep, Mohamed et al. [7] considered the possibility of control by the climb of jogged edge dislocations. For this process, Hirth and Lothe [11] have shown that the steady-state shear strain rate is given by 12npb3 je
D,Gb
t
( - kT >o G
I.’
(17)
where fje is the jog spacing on the edge dislocations. Putting A>, = 12xpb3/lje = 4.5 x lo-” for Al-S% Mg, and taking p = 5 x lo3 cm-’ from Fig. 8 in I, equation (17) gives lje 2: 3bt. This value of lie is very low, although it is interesting to note that it is comparable to the equilibrium spacing (~2.56) of solute atoms in the dislocation core in the Al-5% Mg alloy [35]. However, it is necessary to compare the predicted value for fj= with the thermal jog spacing in Al at a temperature of 823 K. The thermal jog spacing is of the order of [36] lje cv (b/lO)exp(Uj/kT)
(18)
where I_Jjis the formation energy for a jog in an edge dislocation. Taking Uj CC0.4eV [37] for aluminum, equation (18) gives ljL z 306, which again indicates a discrepancy of about one order of magnitude with the experimental results. There are two points of interest with respect to this mechanism. On the one hand, the process requires a large number of reasonably straight edge dislocations, and this is consistent with the analysis by electron microscopy as shown in Fig. 7 of I. On the other hand, as noted earlier by Mohamed et al. [7], it is anticipated that an array of dislocations containing a very high jog density would exhibit saturation. It is therefore necessary to consider climb under saturated conditions, and this is examined in the following section. t Earlier estimates of I, were of the order of these values were obtained by taking p z 104cm-2 and Ako 5 1.5 x IO-“. -2Ob [7, 181:
THE
FLOW
PROCESS
2.6 The climb of edge dislocations under saturated conditions
Dislocation jogs may become saturated with vacancies at high temperatures, and the dislocation velocity is then controlled by the rate of diffusion of vacancies to and from the dislocation line. This mechanism was first developed by Friedel [Cl, and it was subsequently presented as a possible process for Harper-Dorn creep by Mohamed et al. [7]. Following Hirth and Lothe Cl!], the steady-state shear strain rate for this process is given by .
6Kpb2
‘/ = ln(l/p1i2b) Taking
the
(!q$“. kT
experimental
(19)
value
of
p = 5 x 10’ cmv2 from I, equation (19) gives A;Io ‘v 6 x lo-l2 which is again almost one order of
2.5 The climb of jogged edge dislocations
j=I
CREEP-II.
magnitude smaller than the experimental value. However, despite the apparent discrepancy in Ah,, close inspection shows that there are numerous points of agreement between the requirements of this mechanism and the extensive experimental observations. For convenience, the points of agreement are listed below using the same format as in the list of the primary characteristics of Harper-Dom creep given in section 4.2 of I: (i) The mechanism gives a stress exponent of n = 1.0. (ii) The process is intragranular and the creep rate is therefore independent of grain size. (iii) The activation energy is equal to the value for lattice self-diffusion. (iv) It is likely that there would be a short primary stage of creep as the dislocations adjust to an equilibrium configuration. (v) The mechanism specifically requires a low dislosaturation cation density for (typically, p 2 lo5 cmd2 [12]), and this density is independent of the stress level. (vi) The climb of edge dislocations under saturated conditions would not preclude the formation of subgrains, procided 1. < d. (vii) Due to interference with dislocation motion in the presence of a dispersion of precipitates, the mechanism leads to a slower creep rate when precipitates are present. (viii) The mechanism requires a random and reasonably uniform distribution of dislocations. (ix) This process is consistent with the experimental observation of a predominance of dislocations in edge orientation. (x) The process takes place in a similar manner in pure metals and solid solution alloys, and there is no dependence on the solute concentration. To investigate this mechanism in more detail, it is necessary to calculate the limiting condition for vacancy saturation. Dislocation jogs become saturated with vacancies when the average distance between the jogs is less
LASGDON
A40
YAVARI:
HARPER-DORN
than the characteristic length, A, for vacancy diffusion. At high temperatures, saturation occurs when [12] ln(n/b) > 2.5.
(20)
For the climb of randomly dispersed dislocations, A = b In(r/b) where r is the separation between (= l/p+). Thus, the saturation condition is In(l/pfb) > 12.2
(21) dislocations
(22)
which, for Al or Al-Mg alloys, requires an upper limiting dislocation density of p c 3 x 104cmm2. This is equivalent to a limiting condition for saturation of p*b z 5 x lo-’ in Fig. 8 of I. Alternatively, taking the experimental value of A’,, = 4.5 x lo-‘* in I and setting this equal to [6npb’/ln (l/p*b)] in equation (19X the value required for the dislocation density is p 5 3 x lo4 cm-‘. Thus, the value calculated for p by substituting the experimental data into equation (19) is identical to the limiting value estimated for p for jog saturation through equations (20-22). Despite this apparent agreement, there is an obvious discrepancy, by a factor of six times, between the value of p z 3 x lo4 cm-’ estimated from equation (19) and the average experimental value of p z 5 x lo3 cm-l given in Fig. 8 of I. It is possible, and indeed seems likely, that this is due to the difficulties associated with measuring p in aluminum and aluminum alloys by etch pitting techniques. At reasonably high dislocation densities @)b 2: 10e4), there is a consistent discrepancy in several metals between the low values of p obtained by etch pitting and the significantly higher values obtained for p by transmission electron microscopy (see, for example, the plot of ptb vs o/G in Fig. 20 of Bird er al. [14]t). In addition, it is well established in pure aluminum that the values of the subgrain sizes estimated from etch pitting are consistently higher than the values estimated at the same stress levels using other procedures [38,39-J, thereby also inferring that etching may not fully reveal the dislocation substructure in aluminum. Based on the foregoing analysis, it seems probable that Harper-Dorn creep arises from the climb of edge dislocations under saturated conditions. At reasonably high stresses in pure metals, the creep behavior is due to dislocation climb with n = 4.5. As the stress level is reduced, the dislocation density also decreases, as indicated in equation (15). Jog saturation occurs at a limiting dislocation density defined by equation (22), and at lower stress levels the dislocation density remains constant and the creep rate is given by equat Unfortunately. Fig. 20 of Bird et al. [14] contains no etch pit measurements of p in pure aluminum or aluminum alloys.
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885
tion (19) with n = 1.0. Similar behavior occurs also in solid solution alloys obeying class M behavior with n 2: 4.5 [27], or in alloys exhibiting a class A-class M transition so that there is a change in the stress exponent from -3 to -4.5 with decreasing stress [27]. Examples of the latter behavior are provided by a wide range of Al-Mg alloys [4,27,29-331 and Pb-Sn alloys with solute concentrations above --7 at.% [21,34]. To date, no experiments have been conducted to determine whether Harper-Dom creep is also observed in those solid solution alloys where the grain size is large and the material exhibits class A behavior (n z 3) to very low stress levels. An interesting implication of the present approach arises by reference to Figs 8 and 9 in I. The creep behavior of pure Al and Al-5% Mg is essentially identical in the Harper-Dorn region (Fig. 9 of I), but the transition to dislocation climb occurs at a higher value of r/G for the solid solution alloy. The normalized transition stress for pure Al is r/G = 3 x 10e6, and this corresponds, from the plot of p*b vs r/G (Fig. 8 of I), to a constant in equation (15) of u 2: 0.6 at the value of p*b z 5 x 10e6 estimated for climb under saturated conditions. By contrast, the normalized transition stress of r/G z lo-’ in Al-5% Mg corresponds, at the same level of p*b, to a constant of x 1 2.0. The only experimental value obtained for pure Al is SL= 0.7 by Daily and Ahlquist [40] using single crystals; in polycrystalline Al-j% Mg, Oikawa et al. [41] reported dislocation density data which were subsequently shown by Takeuchi and Argon [15] to give z = 1.5. Thus, the limited experimental values available for r in Al and Al-5% Mg are in excellent agreement with the predicted values, and with the experimental trends depicted in Fig. 9 of I, prouided pfb is set equal to -5 x 10m6. As noted earlier, this corresponds to a dislocation density of p = 3 x 104cm-V. From this analysis, and the demonstrated agreement in terms of the values of CL,it is therefore concluded that Harper-Dorn creep is satisfactorily explained by the climb of edge dislocations under saturated conditions. 3. THE CRITERIA FOR OBSERVATIONS OF HARPER-DORN CREEP In an earlier analysis [42], based on a review of the available experimental data, it was shown that a condition for the observation of Harper-Dorn creep was a grain size typically greater than -5OOpm. It is now possible to specify the criteria for observations of Harper-Dorn creep more precisely. 3.1 The upper limiting stress In the Harper-Dorn creep region, it is assumed that creep arises from the climb of edge dislocations under saturated conditions with a constant dislocation density. The steady-state shear strain rate is given by equation (19).
886
LANGDON
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HARPER-DORN
CREEP-II.
THE FLOW PROCESS
It is possible to check equation (25) by noting that an analysis by Bird et al. [14] of data for pure Al in the dislocation climb region gives n 1 4.5 and A; z 8.5 x 10’:. Substituting these values into equation (25) the upper limiting normalized shear stress for Harper-Dorn creep in pure Al is estimated as r/G 1 5 x 10e6, which is in good agreement with the data shown in Fig. 9 of I. 3.2 The lower limiting grain size Soturoted chmb:
There is a lower limiting grain size for HarperDorn creep because a polycrystalline material having a small grain size will deform by Nabarro-Herring diffusional creep [2,3] at high temperatures and low stress levels. In this case, the steady-state shear strain rate is given by
3 = 84E)G)‘(;)‘”
(26)
so that, from equations (19) and (26), the grain size requirement for Harper-Dorn creep may be specified as Fig. 1. Schematic illustration of temperature compensated shear strain rate versus normalized shear stress, showing the transition from dislocation climb with a stress exponent of n at high stresses to saturated climb with a stress exponent of 1 at low stresses.
For saturation, equation (27) reduces to d > 1.5 x 106b.
At high stresses, in the dislocation climb regime, the steady-state shear strain rate is given by j =
A;pE)(‘6>’
(23)
where Ai is a dimensionless constant and n is typically -4.5. The constant Ai incorporates the variation in dislocation density with stress through equation (15). These two processes operate independently [43], and the behavior is illustrated schematically in Fig. 1 in a logarithmic plot of jkT/D,Gb vs r/G at constant grain size. From equations (19) and (23), the stress requirement for Harper-Dorn creep is given by
which, for the limiting saturation equation (22), reduces to ; < (3.9 x;-lt)ll(l-“_
condition given in
(25)
t If p*b is put equal to the experimental value of 2 x 10e6 in Fig. 8 of I, the corresponding values of a, based on the transition values of r/G shown in Fig. 9 of I, are -1.5 and -5.0 for pure Al and AI-5% Mg, respectively. The latter value exceeds the highest value of z (- 2.6 for Cu) obtained in any of the many investigations tabulated for pure metals and alloys by Takeuchi and Argon [ 151. $ Bird er al. [14] quote a dimensionless constant of
2.5 x lo6 under tensile conditions: the value of 8.5 x 10’ applies to shear conditions.
(28)
This result is very close to the earlier estimate, based only on an analysis of published data for Harper-Dorn creep, of d > 1.7 x lo6 b [42]. Using the anticipated value of p z 3 x 104cm-‘, equation (28) gives a lower limiting grain size for Harper-Dom creep of -430pm for pure Al and Al-Mg alloys. This value is in excellent agreement with the limiting grain sizes of 450 and 400 pm estimated for pure aluminum by Murty [6] and Mohamed [44], respectively, from analyses of the experimental data of Mohamed et al. [S], Harper and Dorn [8], Barrett er al. Cl33 and Burton [45]. Alternatively, at low stress levels and temperatures in the vicinity of 0.5 T,, a polycrystal with a small gram size will deform by Coble diffusional creep [46]. The steady-state shear strain rate is then given by J =
‘+
(;)(!!$!)(!$(;)“”
(29)
where 6 is the width of the grain boundary and D, is the coefficient for grain boundary diffusion. Under conditions where Coble creep is faster than Nabarr-Herring creep, the grain size requirement, obtained from equations (19) and (29), may be specified as d,
*!3
(30)
which, for saturation, is equivalent to (31)
LANGDON
AND
Y.AVARI:
HARPER-DORN
From equations (26) and (29), the condition that Coble creep is faster than NabarreHerring creep requires an upper limiting grain size of
Inspection of equations (27), (30), and (32) shows that the lower limiting grain size for Harper-Dorn creep increases to higher values in the Coble creep regime. Thus. a slightly larger grain size is needed to observe Harper-Dorn creep at temperatures of the order of 0.5 T, when Coble creep is the dominant mechanism of diffusional flow. 4. SUMMARY
AND CONCLUSIONS
1. The various possible deformation mechanisms for Harper-Dorn creep are compared with the available experimental data. 2. It is concluded that Harper-Dom creep is due to the climb of edge dislocations under saturated conditions. This process is in excellent agreement with all of the experimental evidence prorided the dislocation density in pure Al and Al-Mg alloys is a factor of six times higher than the value estimated experimentally from etch pit measurements. 3. Using the theory of climb under saturated conditions, it is shown that (i) the upper limiting normalized shear stress for Harper-Dorn creep is given by (3.9 X IO- rr/A;)“(“- 1’ where A; and n are, respectively, the dimensionless constant and stress exponent in the equation for dislocation climb, and (ii) the lower limiting grain size for Harper-Dorn creep is given by 1.5 x lo6 b if Nabarro-Herring creep is dominant in the diffusional creep range or by 1.4 x IO4 b2,3 (6D2/D1)“3 if Coble creep is dominant, where b is the Burgers vector, 6 is the width of the grain boundary, and D, and Dt are the coefficients for grain boundary and lattice diffusion, respectively. Acknowledgement-This work was supported by the United States Department of Energy under Contract DEAM03-76SFOO113 PA-DE-AT03-76ER10408.
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YS7
10. N. F. Mott. Creep and Fracture of.Wetals at High TrmDeratures. D. 21. H.M.S.O.. London (1956). 11. J. P. HiAh and J. Lothe, Theory of Dislocations.
McGraw-Hill, New York (1968). 12. J. Friedel, Dislocations. Pergamon Press. Oxford (1964). 13. C. R. Barrett, E. C. Muehleisen and W. D. Nix. ,Water. Sci. Engng 10, 331 (1972). 14. J. E. Bird. A. K. Mukherjee and J. E. Darn, Quantitative Relation Between
Properties
and .Microstructure
(edited by D. G. Brandon and A. Rosen). p. 255. Israel Universities Press, Jerusalem (1969). 15. S. Takeuchi and A. S. Argon, J. Mater. Sci. 11. 1542 (1976).
16. E. C. Muehleisen, Ph.D. Dissertation, Department of Materials Science, Stanford University (1969). 17. R. G. Stang, W. D. Nix and C. R. Barrett, .Metall. Trans. 4, 1695 (1973). 18. F. A. Mohamed, .Cletall. Trans. A 9, 1342 (1978). 19. S. Takeuchi and A. S. Argon, Acta metail. 24, 853 (1976).
20. H. W. King, J. hlater. Sci. 1. 79 (1966). 21. F. A. Mohamed and T. G. Langdon, Acta metall. 22, 779 (1974). 22. A. H. Cottrell and IM. A. Jaswon, Proc. R. Sot. A 199, 104 (1949).
23. J. Weertman, J. appl. Phys. 28, 1185 (1957). 24. R. Horiuchi and M. Otsuka, Trans. Japan Inst. .Vetals 13, 284 (1972).
25. N. Matsumo, H. Oikawa and S. Karashima.
.Vippon
Kinroku Gakkai Shi (J. Japan Inst. Metals) 38. 1071 (1974).
26. A. Orlova and J. Cadek. Z. i\fetalfk. 65, 200 (1974). 27. P. Yavari. F. A. Mohamed and T. G. Lanedon. Acta metall. 29, 1495 (1981). 28. A. Orlova and J. Cadek, ,Vater. Sci. Engng 38, 139 (1979). 29. K. Kuchaiova and J. Cadek, Korore’ Materidly 10, 240 (1972).
30. H. Oikawa, K. Sugawara and S. Karashima. Trans. Japan Inst. Metals i!$ 611 (1978). 31. M. Pahutova and J. Cadek. Phvsicn status solidi (a) 56. 305 (1979). 32. S. Kikuchi, Y. Motoyama and M. Adachi, Keikinzoku (J. Japan Inst. Light Metals) 30, 449 (1980). 33. F. A. Mohamed. MetalI. Trans. A 9. 1013 (1978). 34. K. L. Murty, Phil. Mug. 29, 422 (1974). 35. K. Kuchaiova, I. Saxl and J. Cadek, Acta metall. 22, 465 (1974). 36. A. J. Ardell, H. Reiss and W. D. Nix. J. appl. Phys. 36, 1727 (1965). 37. A. Seeger, Report of a Conference on Defects in Crystalline Solids, p. 391. The Physical Society. London (1955). 38. P. W. Chen, C. T. Young and J. L. Lytton, Rate Processes in Plusric Deformation of Materials (edited by J. C. M. Li and A. K. Mukherjee), p. 605. American Society for Metals, Metals Park, Ohio (1975). 39. R. B. Vastava and T. G. Langdon, Advances in Materials Technology in the Americas-1980 (edited by I. Le Mav). . Vol. 2. D. 61. American Societv of Mechanical Engineers, New York (1980). . 40. S. Daily and C. N. Ahlquist, Scripta metoll. 6, 95 (1972). 41. H. Oikawa. N. Matsuno and S. Karashima, .\fetal Sci. 9, 209 (1975). 42. P. Yavari and T. G. Lanedon. Scrinta metail. 11, 863 I.
(1977).
43. T. G. Langdon and F. A. Mohamed, J. .Australasian Inst. Metals 22, lg.9 (1977). 44. F. A. Mohamed, Mater. Sci. Engng 32, 37 (1978). 45. B. Burton, Phil. Mug. 25, 645 (1972). 46. R. L. Coble, J. appl. Phys. 34, 1679 (1963).
_(cru mernll. Vol. j0. pp. 881 IO 887. 1982 Pnnrcd m Great Bntain
AN INVESTIGATION OF HARPER-DORN THE FLOW PROCESS TERENCE
G. LANGDON
and PARVIZ
CREEP-II.
YAVARI
Departments of Materials Science and Mechanical Engineering, University of Southern California, Los Angeles, CA 9COO7,U.S.A. (Received 13 August 1981) Abstract-Several
deformation mechanisms have been proposed to explain the flow process of HarperDorn creep. The implications of these various mechanisms are compared with the available experimental data, and it is concluded that flow occurs by the climb of edge dislocations under saturated conditions. Using the theoretical model for this process, it is possible to specify the experimental criteria for observations of Harper-Dorn-creep in terms of the upper limiting stress and the lower limiting grain size. R&m&-Plusieurs micanismes de deformation ont bte proposes pour expliquer I’tcoulement plastique au tours du fluage de Harper et Dorn. Nous comparons les implications de ces divers mecanismes avec les r&hats experimentaux disponibles; nous en concluons que l’ecoulement se produit par montee de dislocations coins dans des conditions de saturation. A l’aide du modtle theorique de ce phenomtde. on peut specifier les crittres exptimentaux pour observer le fluage de Harper et Dorn. sous forme dune contrainte limite superieure et dune taille de grains limite inferieure. Zusammenfassung-Verschiedene
Mechanismen wurden schon zur Erkllrung des Harper-Dorn-Kriechens vorgeschlagen. Die Folgerungen aus den einzelnen Mechanismen werden mit den verftigbaren experimentellen Ergebnissen verglichen. Das Ergebnis ist, daB das FlieDen herriihrt von dem Klettern der Stufenversetzungen unter Wtigungsbedingungen. Mit dem theoretischen Model1 fur diesen Mechanismus ist es moglich, die experimentellen Bedingungen fur das Auftreten des Harper-Dorn-Kriechens als obere Grenzspannung und untece KorngrijBe anzugeben.
1. INTRODUCI’ION
In an earlier report, hereafter designated I [l], it was demonstrated that a number of metals exhibit Harper-Dorn creep at low stress levels. The major distinguishing feature of Harper-Dorn creep is a steady-state strain rate which is significantly larger than the value predicted for Nabarrcl-Herring diffusional creep [2,3]. For several metals tested at high homologous temperatures (typically in the vicinity of 0.98 T,, where r, is the absolute melting point) and with large gram sixes (of the order of 1-4mm), the steady-state shear strain rate in the Harper-Dorn region, Jo,is given by
where D, is the diffusion coefhcient for lattice selfdilfusion, G is the shear modulus, b is the Burgers vector, k is Boltzmann’s constant, T is the absolute temperature, r is the shear stress, and A& is a dimensionless constant. As indicated by the summary of experimental data given in Table 1 of I, the value of A’,, is typically about 5 x 10-r’. In a series of detailed creep experiments on the Al-S% Mg solid solution alloy, described in I, it was shown that the behavior at low stress levels, up to r x 0.25 MPa, obeyed equation (1) with
Ai, = 4.5 x lo-“. In addition, microstructural observations in this region revealed the presence of a predominance of edge dislocations and a dislocation density, determined from etch pit measurements, which was independent of stress and equal to (4.7 C 0.9) x IO3 cm-‘. By combining these results with earlier reports of Harper-Dom creep, it was possible to establish the primary mechanical and microstructural characteristics for this tylze of flow. Several mechanisms have been proposed to explain the flow process in Harper-Dorn creep, and these were considered briefly in several early reports [4-6l and examined in detail in the review of Harper-Dom creep by Mohamed er al. [7-J. The purposes of the present paper are two-fold. First, to examine the implications of each of the suggested deformation mechanisms with respect to the established experimental characteristics listed in I and to thereby determine the most probable flow process. Second, to use this information to establish the criteria for observations of Harper-Dorn creep in terms of the upper limiting stress and the lower limiting grain size.
2. THE DEFORMATION HARPER-DORN
MECHANISM CREEP
FOR
In this section, each mechanism is considered sep881
882
LANGDON
arately with reference presented in I.
AND
YAVARI:
HARPER-DORN
to the experimental
data
2.1 The motion of jogged screw dislocations In the early work of Harper et al. (8,9], the Harper-Dom process was attributed to the motion of jogged screw dislocations and the associated production of vacancies, using a concept developed earlier by Mott [lo]. Following the more recent analysis of this process by Hirth and Lothe [ll], this mechanism leads to a steady-state shear strain rate which is given by DtGb j = 12zpljab F
T ‘.’ c
( )O
(21
where p is the dislocation density and 1, is the jog spacing on the screw dislocations. This mechanism is not appropriate for HarperDorn creep for two reasons. First, it requires a predominance of screw dislocations, which is not consistent with the experimental evidence as demonstrated by Fig. 7 in I. Second, by putting Ah, = 127cplj,b = 4.5 x IO-I1 and p = 5 x 103cm-* from Fig. 8 in I, it may be shown that this mechanism requires a jog spacing which is unrealistically small (I, z 0.3b). In addition, this discrepancy wouid be further enhanced if, as seems likely (see section 2.6), the experimentat value of p from etch pit measurements is an under-estimation of the true value. 2.2 NabarrcF-Herring creep associated with subgrain boundaries
FriedeI[12] proposed that the experimental creep rate may be faster than the value predicted by the Nabarro-Herring process if the vacancies diffuse between subgrain boundaries. In this case, the steadystate shear strain rate is given by the relationship for Nabarro-Herring creep [2,33 with the grain size, d, replaced by the subgrain size, %,so that
CREEP-II.
THE FLOW PROCESS
where by 5 10. Substitution of equation (4) into equation (3) gives $ x r3, which is inconsistent with experimental results in the Harper-Dorn region. Furthermore, even If it is assumed, as concluded by Muehleisen [16], that the subgrain size reaches an essentially constant value at low stresses, the average experimental value of 2 2: 2mm predicts creep rates through equation (3) which are about two orders of magnitude slower than the observed rates?. 23 The climb-controlled generation of dislocations from a fixed density of sources This mechanism was proposed by Barrett et al. [ 133 to explain their results on pure Al, based on the inference from data on Al with FeAI, preci’pitates that Harper-Dom creep is cobtrolled by a dislocation process. The principle of the mechanism may be briefly summarized in the context of the present work as follows. If dislocation generation occurs by diffusion-controlled climb at a fixed number of sources per unit volume which is independent of the applied stress, the dislocation generation rate, i+, is expressed as Pe = PO&/l-
(5)
where p. is the fixed dislocation source length per unit volume, u, is the climb velocity, and r is the Taking v, = 2Dlb2t/kT and climb distance. r = Gb/2r, i+ is given by b+ = 4poD,bT2/kTG.
(6)
An important assumption in the model of Barrett et ai. [I33 is that d~~ocation a~ihi~ation occurs only at s&rain boundaries. In this case, the annihilation rate, b_, is given by &i_ = P&/A:
(7)
where pm is the mobile dislocation density and ue is the glide velocity. Taking ve = 2u0r, where u. is a constant, and R = Gb[/z from equation (4), leads to i-
= 2p, vor2/Gb&
(‘3)
The rate of change of density with time, 4, is there-
where A:VH- 84. This mechanism also leads to two inconsistencies with the experimental data. First, subgrains are not always observed, as in the experimental work described in I or in the earlier investigations by Harper et al. [8,9] : it was conclude-d in I that the condition for subgrain formation is d > 1.
Second, if subgrains form in the Harper-Dom region, the data of Barrett et al. El33 indicate that, as in dislocation climb, there is a variation in subgrain size with stress such that [14,15] i/b = j(q’G)- 1 7 This discrepancy may be appreciated position of the line for NabarreHerring d = 3.3 mm in Fig. 1 of 1.
(4) by noting the creep when
fore 4poD, bs’ 2PIn@*9 . . p* - p_ = kTG Gbj
(31
so that, at steady-state when i, = 0, pm = 2poD,b2[/kTvo.
Substituting equation (10) into the standard pression for $ ( = p,bu& leads to
(10)
ex-
(11) This mechanism has the advantage of predicting reasonable values for the creep rates: for example, taking p. z lo3 cms2 as a typical annealed dislocation density and c ^u IO, equation (11) gives AHD = 4pob2C z 3.3 x lo-“, which is close to the
LANGDON
AND
YAVARI:
HARPER-DORN
experimental value of A& = 5.0 x lo- l1 listed for Al in Table I of I. However, the derivation of equation (11) is based spectically on the assumption that the dislocations are annihilated only at subgrain boundaries. If subgrains do not form, as in the experiments described in I or the work of Harper et al. [S, 93, the dislocations will be annihilated at the grain boundaries, so that equation (8) is replaced by fi_ = Zj,c,r/d.
W)
Performing the same analysis, the steady-state shear strain rate is then given by 3 = 4p,b’(!$($(;)2.
(13)
Equation (13) predicts a stress exponent of 2 and a linear dependence on grain size, and this is clearly inconsistent with the available experimental data for Harper-Dorn creep (see. for example, Fig. 1 of part I). It is therefore concluded that this mechanism is unable to account for the flow process in the HarperDorn region. 2.4 The ciscous glide of dislocations pheres
with solute armos-
Murty [6] observed Harper-Dorn creep in a Pb-9% Sn solid solution alloy at values of r/G below r 2 x lo- ‘. Based on an earlier observation by Stang ez al. [17] of a constant dislocation density from etch pit measurements on Fe-3% Si at r/G s lo-‘, Murty [63 suggested that Harper-Dorn creep in alloys may arise from a modified viscous glide process in which the dislocation density is constant and dislocation movement is restricted due to the presence of solute atmospheres. This concept was later developed in a quantitative manner by Mohamed [lS] to explain the results obtained on an Al-20/, Mg solid solution alloy. At high stresses, many solid solution alloys deform by viscous glide where the movement of dislocations is impeded by the dragging of solute atom atmospheres. Following Takeuchi and Argon [19]. the steady-state shear strain rate for a homogeneous distribution of dislocations is given by
where e is the solute-solvent size difference, c is the concentration of solute atoms, and 0’ is the diffusion coefficient for the solute atom in the solvent lattice. In general, the dislocation density in high temperature creep is given by ~cp*b= (r/G)‘.’
(15)
where a is a constant close to unity. Thus, equation (14) usually leads to 9 x TV. However, as noted by Murty [6] and Mohamed [lS], the dislocation density is independent of stress in the Harper-Dorn region, so that the dragging of solute atmospheres leads to a stress exponent of n = 1 at low stress levels.
CREEP-II.
THE
FLOW
PROCESS
883
Taking p = 5 x lo3 cm-’ from Fig. 8 in I, e = 0.121 for the Al-Mg system [ZO], c = 0.058 for the Al-j% Mg alloy used in I, and 0’ = DI for pure . I Al [21], equation (14) gtves AHD2 6.0 x 10-l’ which is similar to, but about one order of magnitude lower than, the experimental result. An alternative calculation may be undertaken by combining the approach of Cottrell and Jaswon [22] for the interaction of dislocations with solute atom atmospheres and the microcreep mechanism developed by Weertman [23]. In this case, the steady-state shear strain rate for viscous glide is given by [21] 4= ‘(l --$Tpb’
(&>‘($!)(d””
(16)
where v is Poisson’s ratio. Taking v = 0.34 and ti = 1, equations (16) and (14) differ only by a factor of 0.7 although they are based on different theoretical concepts. From equation (16), the equivalent value for Ahe is -4.2 x 10-i’. There are two significant points of agreement between this modified viscous glide mechanism and the experimental observations. First, subgrains do not form during viscous glide with n c 3 and there is an essentially uniform distribution of dislocations [24-271: this is consistent with the lack of subgrains in Al-j’% Mg in the Harper-Dorn region, as documented in I. Second, the theory of Takeuchi and Argon [19] is based on control by the slower-moving edge dislocations, and this is in agreement with the preponderance of edge dislocations observed in I by transmission electron microscopy. Furthermore, there have been several experimental reports of a majority of edge dislocations in Al-Mg alloys after creep testing under viscous glide conditions with n 5: 3 [24,26,28]. However, closer inspection shows three important inconsistencies in this mechanism. First, in Al-Mg solid solution alloys exhibiting viscous glide behavior, the transition with decreasing stress is not from viscous glide with n c 3 to Newtonian viscous flow with n = 1.0. as required both in the initial concept and in the subsequent development of this mechanism. On the contrary, the transition is from viscous glide with n = 3 to dislocation climb with n z 4.5 and then to Newtonian Row with n = 1.0, as shown in Fig. 2 of I. Furthermore, the transition from n z 3 (class A behavior) to n ‘z 4.5 (class M behavior) with decreasing stress has been firmly established in Al-Mg alloys in both polycrystal[4,27,29-311 and single crystal [32] form, including in earlier work on the Al-2% Mg alloy by Mohamed [33], and there is also good evidence for a similar transition in the Pb-9% Sn alloy used by Murty [34]. Thus, it is unrealistic to assume that Harper-Dom creep arises from a simple modification of the viscous glide model. Second, if Harper-Dom creep in solid solution alloys is due to viscous glide through equations (14) or (16), the experimental value of the dimensionless
884
LANGDON
AND
YAVARI:
HARPER-DORN
constant, .&,, should decrease with increasing solute concentration because Ah, x l/c. This trend is not supported by the three sets of values for A& shown in Table 1 in I for three different Al-MO alloys, nor is it supported by the data in I on Al-W0 Mg where the experimental value of Ah, (4.5 x lo- Ii) is almost identical to the tabulated value for pure Al (5.0 x lo-“). Third, a mechanism involving the dragging of solute atmospheres would necessarily apply only to solid solution alloys, and the development of a restrictive mechanism of this type seems unreasonable when it is noted that there is a very close similarity in the Harper-Dorn region between the results obtained on alloys and on the appropriate pure metal (see Fig. 9 of I).
In an earlier review of Harper-Dom creep, Mohamed et al. [7] considered the possibility of control by the climb of jogged edge dislocations. For this process, Hirth and Lothe [11] have shown that the steady-state shear strain rate is given by 12npb3 je
D,Gb
t
( - kT >o G
I.’
(17)
where fje is the jog spacing on the edge dislocations. Putting A>, = 12xpb3/lje = 4.5 x lo-” for Al-S% Mg, and taking p = 5 x lo3 cm-’ from Fig. 8 in I, equation (17) gives lje 2: 3bt. This value of lie is very low, although it is interesting to note that it is comparable to the equilibrium spacing (~2.56) of solute atoms in the dislocation core in the Al-5% Mg alloy [35]. However, it is necessary to compare the predicted value for fj= with the thermal jog spacing in Al at a temperature of 823 K. The thermal jog spacing is of the order of [36] lje cv (b/lO)exp(Uj/kT)
(18)
where I_Jjis the formation energy for a jog in an edge dislocation. Taking Uj CC0.4eV [37] for aluminum, equation (18) gives ljL z 306, which again indicates a discrepancy of about one order of magnitude with the experimental results. There are two points of interest with respect to this mechanism. On the one hand, the process requires a large number of reasonably straight edge dislocations, and this is consistent with the analysis by electron microscopy as shown in Fig. 7 of I. On the other hand, as noted earlier by Mohamed et al. [7], it is anticipated that an array of dislocations containing a very high jog density would exhibit saturation. It is therefore necessary to consider climb under saturated conditions, and this is examined in the following section. t Earlier estimates of I, were of the order of these values were obtained by taking p z 104cm-2 and Ako 5 1.5 x IO-“. -2Ob [7, 181:
THE
FLOW
PROCESS
2.6 The climb of edge dislocations under saturated conditions
Dislocation jogs may become saturated with vacancies at high temperatures, and the dislocation velocity is then controlled by the rate of diffusion of vacancies to and from the dislocation line. This mechanism was first developed by Friedel [Cl, and it was subsequently presented as a possible process for Harper-Dorn creep by Mohamed et al. [7]. Following Hirth and Lothe Cl!], the steady-state shear strain rate for this process is given by .
6Kpb2
‘/ = ln(l/p1i2b) Taking
the
(!q$“. kT
experimental
(19)
value
of
p = 5 x 10’ cmv2 from I, equation (19) gives A;Io ‘v 6 x lo-l2 which is again almost one order of
2.5 The climb of jogged edge dislocations
j=I
CREEP-II.
magnitude smaller than the experimental value. However, despite the apparent discrepancy in Ah,, close inspection shows that there are numerous points of agreement between the requirements of this mechanism and the extensive experimental observations. For convenience, the points of agreement are listed below using the same format as in the list of the primary characteristics of Harper-Dom creep given in section 4.2 of I: (i) The mechanism gives a stress exponent of n = 1.0. (ii) The process is intragranular and the creep rate is therefore independent of grain size. (iii) The activation energy is equal to the value for lattice self-diffusion. (iv) It is likely that there would be a short primary stage of creep as the dislocations adjust to an equilibrium configuration. (v) The mechanism specifically requires a low dislosaturation cation density for (typically, p 2 lo5 cmd2 [12]), and this density is independent of the stress level. (vi) The climb of edge dislocations under saturated conditions would not preclude the formation of subgrains, procided 1. < d. (vii) Due to interference with dislocation motion in the presence of a dispersion of precipitates, the mechanism leads to a slower creep rate when precipitates are present. (viii) The mechanism requires a random and reasonably uniform distribution of dislocations. (ix) This process is consistent with the experimental observation of a predominance of dislocations in edge orientation. (x) The process takes place in a similar manner in pure metals and solid solution alloys, and there is no dependence on the solute concentration. To investigate this mechanism in more detail, it is necessary to calculate the limiting condition for vacancy saturation. Dislocation jogs become saturated with vacancies when the average distance between the jogs is less
LASGDON
A40
YAVARI:
HARPER-DORN
than the characteristic length, A, for vacancy diffusion. At high temperatures, saturation occurs when [12] ln(n/b) > 2.5.
(20)
For the climb of randomly dispersed dislocations, A = b In(r/b) where r is the separation between (= l/p+). Thus, the saturation condition is In(l/pfb) > 12.2
(21) dislocations
(22)
which, for Al or Al-Mg alloys, requires an upper limiting dislocation density of p c 3 x 104cmm2. This is equivalent to a limiting condition for saturation of p*b z 5 x lo-’ in Fig. 8 of I. Alternatively, taking the experimental value of A’,, = 4.5 x lo-‘* in I and setting this equal to [6npb’/ln (l/p*b)] in equation (19X the value required for the dislocation density is p 5 3 x lo4 cm-‘. Thus, the value calculated for p by substituting the experimental data into equation (19) is identical to the limiting value estimated for p for jog saturation through equations (20-22). Despite this apparent agreement, there is an obvious discrepancy, by a factor of six times, between the value of p z 3 x lo4 cm-’ estimated from equation (19) and the average experimental value of p z 5 x lo3 cm-l given in Fig. 8 of I. It is possible, and indeed seems likely, that this is due to the difficulties associated with measuring p in aluminum and aluminum alloys by etch pitting techniques. At reasonably high dislocation densities @)b 2: 10e4), there is a consistent discrepancy in several metals between the low values of p obtained by etch pitting and the significantly higher values obtained for p by transmission electron microscopy (see, for example, the plot of ptb vs o/G in Fig. 20 of Bird er al. [14]t). In addition, it is well established in pure aluminum that the values of the subgrain sizes estimated from etch pitting are consistently higher than the values estimated at the same stress levels using other procedures [38,39-J, thereby also inferring that etching may not fully reveal the dislocation substructure in aluminum. Based on the foregoing analysis, it seems probable that Harper-Dorn creep arises from the climb of edge dislocations under saturated conditions. At reasonably high stresses in pure metals, the creep behavior is due to dislocation climb with n = 4.5. As the stress level is reduced, the dislocation density also decreases, as indicated in equation (15). Jog saturation occurs at a limiting dislocation density defined by equation (22), and at lower stress levels the dislocation density remains constant and the creep rate is given by equat Unfortunately. Fig. 20 of Bird et al. [14] contains no etch pit measurements of p in pure aluminum or aluminum alloys.
CREEP--II.
THE FLOW PROCESS
885
tion (19) with n = 1.0. Similar behavior occurs also in solid solution alloys obeying class M behavior with n 2: 4.5 [27], or in alloys exhibiting a class A-class M transition so that there is a change in the stress exponent from -3 to -4.5 with decreasing stress [27]. Examples of the latter behavior are provided by a wide range of Al-Mg alloys [4,27,29-331 and Pb-Sn alloys with solute concentrations above --7 at.% [21,34]. To date, no experiments have been conducted to determine whether Harper-Dom creep is also observed in those solid solution alloys where the grain size is large and the material exhibits class A behavior (n z 3) to very low stress levels. An interesting implication of the present approach arises by reference to Figs 8 and 9 in I. The creep behavior of pure Al and Al-5% Mg is essentially identical in the Harper-Dorn region (Fig. 9 of I), but the transition to dislocation climb occurs at a higher value of r/G for the solid solution alloy. The normalized transition stress for pure Al is r/G = 3 x 10e6, and this corresponds, from the plot of p*b vs r/G (Fig. 8 of I), to a constant in equation (15) of u 2: 0.6 at the value of p*b z 5 x 10e6 estimated for climb under saturated conditions. By contrast, the normalized transition stress of r/G z lo-’ in Al-5% Mg corresponds, at the same level of p*b, to a constant of x 1 2.0. The only experimental value obtained for pure Al is SL= 0.7 by Daily and Ahlquist [40] using single crystals; in polycrystalline Al-j% Mg, Oikawa et al. [41] reported dislocation density data which were subsequently shown by Takeuchi and Argon [15] to give z = 1.5. Thus, the limited experimental values available for r in Al and Al-5% Mg are in excellent agreement with the predicted values, and with the experimental trends depicted in Fig. 9 of I, prouided pfb is set equal to -5 x 10m6. As noted earlier, this corresponds to a dislocation density of p = 3 x 104cm-V. From this analysis, and the demonstrated agreement in terms of the values of CL,it is therefore concluded that Harper-Dorn creep is satisfactorily explained by the climb of edge dislocations under saturated conditions. 3. THE CRITERIA FOR OBSERVATIONS OF HARPER-DORN CREEP In an earlier analysis [42], based on a review of the available experimental data, it was shown that a condition for the observation of Harper-Dorn creep was a grain size typically greater than -5OOpm. It is now possible to specify the criteria for observations of Harper-Dorn creep more precisely. 3.1 The upper limiting stress In the Harper-Dorn creep region, it is assumed that creep arises from the climb of edge dislocations under saturated conditions with a constant dislocation density. The steady-state shear strain rate is given by equation (19).
886
LANGDON
AND
YAVARI:
HARPER-DORN
CREEP-II.
THE FLOW PROCESS
It is possible to check equation (25) by noting that an analysis by Bird et al. [14] of data for pure Al in the dislocation climb region gives n 1 4.5 and A; z 8.5 x 10’:. Substituting these values into equation (25) the upper limiting normalized shear stress for Harper-Dorn creep in pure Al is estimated as r/G 1 5 x 10e6, which is in good agreement with the data shown in Fig. 9 of I. 3.2 The lower limiting grain size Soturoted chmb:
There is a lower limiting grain size for HarperDorn creep because a polycrystalline material having a small grain size will deform by Nabarro-Herring diffusional creep [2,3] at high temperatures and low stress levels. In this case, the steady-state shear strain rate is given by
3 = 84E)G)‘(;)‘”
(26)
so that, from equations (19) and (26), the grain size requirement for Harper-Dorn creep may be specified as Fig. 1. Schematic illustration of temperature compensated shear strain rate versus normalized shear stress, showing the transition from dislocation climb with a stress exponent of n at high stresses to saturated climb with a stress exponent of 1 at low stresses.
For saturation, equation (27) reduces to d > 1.5 x 106b.
At high stresses, in the dislocation climb regime, the steady-state shear strain rate is given by j =
A;pE)(‘6>’
(23)
where Ai is a dimensionless constant and n is typically -4.5. The constant Ai incorporates the variation in dislocation density with stress through equation (15). These two processes operate independently [43], and the behavior is illustrated schematically in Fig. 1 in a logarithmic plot of jkT/D,Gb vs r/G at constant grain size. From equations (19) and (23), the stress requirement for Harper-Dorn creep is given by
which, for the limiting saturation equation (22), reduces to ; < (3.9 x;-lt)ll(l-“_
condition given in
(25)
t If p*b is put equal to the experimental value of 2 x 10e6 in Fig. 8 of I, the corresponding values of a, based on the transition values of r/G shown in Fig. 9 of I, are -1.5 and -5.0 for pure Al and AI-5% Mg, respectively. The latter value exceeds the highest value of z (- 2.6 for Cu) obtained in any of the many investigations tabulated for pure metals and alloys by Takeuchi and Argon [ 151. $ Bird er al. [14] quote a dimensionless constant of
2.5 x lo6 under tensile conditions: the value of 8.5 x 10’ applies to shear conditions.
(28)
This result is very close to the earlier estimate, based only on an analysis of published data for Harper-Dorn creep, of d > 1.7 x lo6 b [42]. Using the anticipated value of p z 3 x 104cm-‘, equation (28) gives a lower limiting grain size for Harper-Dom creep of -430pm for pure Al and Al-Mg alloys. This value is in excellent agreement with the limiting grain sizes of 450 and 400 pm estimated for pure aluminum by Murty [6] and Mohamed [44], respectively, from analyses of the experimental data of Mohamed et al. [S], Harper and Dorn [8], Barrett er al. Cl33 and Burton [45]. Alternatively, at low stress levels and temperatures in the vicinity of 0.5 T,, a polycrystal with a small gram size will deform by Coble diffusional creep [46]. The steady-state shear strain rate is then given by J =
‘+
(;)(!!$!)(!$(;)“”
(29)
where 6 is the width of the grain boundary and D, is the coefficient for grain boundary diffusion. Under conditions where Coble creep is faster than Nabarr-Herring creep, the grain size requirement, obtained from equations (19) and (29), may be specified as d,
*!3
(30)
which, for saturation, is equivalent to (31)
LANGDON
AND
Y.AVARI:
HARPER-DORN
From equations (26) and (29), the condition that Coble creep is faster than NabarreHerring creep requires an upper limiting grain size of
Inspection of equations (27), (30), and (32) shows that the lower limiting grain size for Harper-Dorn creep increases to higher values in the Coble creep regime. Thus. a slightly larger grain size is needed to observe Harper-Dorn creep at temperatures of the order of 0.5 T, when Coble creep is the dominant mechanism of diffusional flow. 4. SUMMARY
AND CONCLUSIONS
1. The various possible deformation mechanisms for Harper-Dorn creep are compared with the available experimental data. 2. It is concluded that Harper-Dom creep is due to the climb of edge dislocations under saturated conditions. This process is in excellent agreement with all of the experimental evidence prorided the dislocation density in pure Al and Al-Mg alloys is a factor of six times higher than the value estimated experimentally from etch pit measurements. 3. Using the theory of climb under saturated conditions, it is shown that (i) the upper limiting normalized shear stress for Harper-Dorn creep is given by (3.9 X IO- rr/A;)“(“- 1’ where A; and n are, respectively, the dimensionless constant and stress exponent in the equation for dislocation climb, and (ii) the lower limiting grain size for Harper-Dorn creep is given by 1.5 x lo6 b if Nabarro-Herring creep is dominant in the diffusional creep range or by 1.4 x IO4 b2,3 (6D2/D1)“3 if Coble creep is dominant, where b is the Burgers vector, 6 is the width of the grain boundary, and D, and Dt are the coefficients for grain boundary and lattice diffusion, respectively. Acknowledgement-This work was supported by the United States Department of Energy under Contract DEAM03-76SFOO113 PA-DE-AT03-76ER10408.
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