Development of a theoretical equation for steady-state dislocation creep and comparison with data

DEVELOPMENT OF DISLOCATION

A THEORETICAL CREEP AND J.

H.

EQUATION COMPARISON

FOR STEADY-STATE WITH DATA*

GITTUSt

For nineteen materials, including silver, tantalum and gamma iron, Ashby found that an empirical equation C, = d(D,$/kT)(a/;l)” gives a reasonable fit to Steady-state creep data where -4 and n are It is nom shown that if the materials obey the Bailey-Orowan equation, Taylor’s materials constants. work-hardening model and Friedel’s network-climb recovery-equation then 8n identical creep equation is predicted ITith -4 = 89c, and n = 3. Here c., = jog-concentration. In the cases of nickel, cadmium, beta thallium and stoichiometric ITO? the predlcted and actual creep strengths differ by less than a fector TWO. ET_1BLISSE>IEST D’USE EQC_1TIOS THEORIQL7E POCR LE FLC-1GE ST.1TIO>YAIRE DE DISLOCATIONS ET COJIPARAISOX _VEC LES RESULTATS ESPERIXEST_IUX Pour dis-neuf met&iaus. y compris l’argent, le tsntale et le fer gamma, Ashby a trouv6 que 1’6quation empirique 8, = d(D,,ub/kT)(a/p)” est en assez bon accord avec les r6sultats exp&imentaus du fluage stationnaire (-4 et n sent cles constantes des mat&iaux). L’auteur montre que si les met&iaux suivent l‘bquation de Bailey-Orowan, le mod&le d’Bcrouiss8ge de Taylor et 1’6qU8tiOn de revenu de Friedel relative $ la montee des dislocations, on peut 810~5 prbvoir une bquation identique pour le flusge, oh -4 = 8a3c, et n = 3. Ici, cI = concentration des trans. Dens les cas du nickel, du cadmium, du thallium beta et du bioxyde d’uranium stoechiom&rique U02, les valeurs p&we3 pour le fluage diff&rent des \-aleurs obse&es de moins d’un facteur deux. _mLEITUSG EISER THEORETISCHEN GLEICHUNG VERSETZL?;GSKRIECHE?u’ L7XD VERGLEICH

FCR XIT

STATIOS:%RES DATEX

Ashv hat gezeigt, da13 die empiriache Gleichung .5, = B(D,~cb/kT)(o/~c)” fti neunzehn Xaterialien, einschheBlich Silber, Tantel und Gammaeisen, mit den Jlaterialkonstanten 1 und 71 eine verniinftige -1npassung an Datan des stationhren Kriechens erlaubt. Wenn die AIaterialien der Bailey-OrowanGleichung dem Verfestigungsmodell ron Taylor und der Friedel-Gleichung (Xetzwerk-Erholung durch Klettern) gehorchen, kann eine identische Gleichung fiir Kriechen mit 4 = SW%, und n = 3 vorhergesagt. werclen. Dabei ist cj die Konzentration der Versetzungsspriinge. In den Fiillen Nickel, Kadmium, BetaThallium uncl stiichiometrisches UO, differieren vorhergesagte und gemessene Kriechfestigkeit m-n n-eniger als einen Faktor zn-ei. 1. INTRODUCTION

Ashby, in a recent paper on deformation maps, remarked (folloGng earlier workers,‘2*3)) that the steady-state rate of creep (k,) due to the diffusioncontrolled movement of crystal dislocations, obeys the constitutive relation: &=A---

D,,ub (T.a kT 0,u

where -4 and n are materials-constants, D, is the bulk self-diffusion coefficient, ,u the shear modulus, b Burgers vector: k Boltzmann’s constant, and T the absolute temperature. He remarked”) that no convincing theory can be said to esist which is capable of explaining equation (1), although it is w-e11documented and data exist for the constants 3 and n for a wide variety of materials. A contemporary review confirms this vie+) and shows that there are half a dozen theories (some of them specific to certain materials, structures, conditions of stress, temperature, etc.) none of which can really be said to have gained wide acceptance.

VOL.

21, JCXE

19i1

2. DEVELOPMENT THEORETICAL CREEP

OF THE EQUATION

Bailev1 and Orowan’gJo) formulated t,he follodng equation for the balance between work-hardening and recovery-processes for a material undergoing steady-state creep :

do=($)ds+(z)dt.

Equation (2) expresses the change produced by a small plastic strain a brief time interval dt.

* Received Sovember 7, 1953; revisecl December 25, 19i3. t Cnited Kingclom Atomic Energy _&thority Springfields, Saln-ick, Preston, Lancashire, England. ACTA METALLURGIC_%,

Amongst the better known are those of Keert,man(“-7) and Nabarro.‘8) In the course of this review it became evident that there is one comparatively simple explanation of the steady-state creep process, Khich although it relies only on long established equations for the recovery and hardening of the structure, has not received attention. Accordingly this paper seeks to remedy that deficiency, presenting a simple theory for steady-state creep in crystalline materials and comparing its predictions with the actual behavior of the materials whose relevant properties were summarised so conveniently in Ashby’s paper.(l)

589

(du) in flow-stress de applied during

.%CTA

i90

METALLURGICA,

Taylor(ll) eras the first to develop a theory of workhardening, based on the long-range stress-fields of dislocations. Evans and Williams have recently argued”*) that Taylor’s theory is applicable to a material undergoing creep and that the mean-free glide-path should for such a case, be set at the spacing between networkdislocations (l/z/p = r, where p = the dislocation density). Evans and Williams’ equation for the work-hardening coefficient is (3) Xow the magnitude of the work-hardening coefficient defined by equation (3) is larger by a factor of about 50 than that observed during rapid tensile tests on metal crystals below half their absolute temperature of melting (T,). However, as Evans and Williams point out,‘12’ work-hardening coefficients measured by changing the stress during a creeptest (characteristically above T,/2) are in fact of the order of magnitude predicted by equation (3). In the present context this agreement gives us some confidence in the applicability of equation (3) to our theory of dislocation creep. Turning to recovery: Friedel (Ref. 13, equation (8.32), p. 239) shows that the rate at which a dislocation-network will coarsen due to jog-controlled climb is, theoretically:

ars D,pb3cj

-=-.at

1

kT

(4)

rs

where rs is the network-spacing, and cj is the jogconcentration given, he shows (Ref. (13), p. 23) by cj=exp where

-z$%T)

02 > % 5

(5)

aa

-,ub -. 27rr,’

(6)

Combining equations (4) and (6) and substituting for r,: au -4D bc .TT%~ -= at

;kI:

’

99,

19i4

constant external stress, a constant rate of creep will occur. From equation (2) for that condition (i.e. for do/d1 = 0) : (S) Substituting

equations

(7) and (3) into equation

(8) :

Equation (9) is the required equation for the theoretical steady-state creep-rate (Psi), produced by a stress (TV. 3. COiMPARISON

WITH

(7

When the rate of network-refinement due to straining equals the rate of coarsening due to climb, the network spacing n-ill be constant and, at a

CREEP

DATA

The theoretical equation (9) is algebraically identical with the equation which Ashby found to be a good empirical representation of experimental data for the materials listed in Table 1. Ashby tabulated values of n and of 8. Generally he found that 16> 3 (typically n = 4-6) and his values of B are considerably at variance with the value (9 = %r3cj) indicated by equation (9). However, to some extent this latter disparity is a consequence of the former, for the larger the value of n which we choose to represent the trend of scattered data points plotted on axes of log P, and log (a/p) the larger will be the associated value of d. A better basis of comparison, therefore, is the creepstrength or creep-rate calculated from equation (1) with Ashby’s values of B and n or from equation (9). Thus from equations (1) and (9) for G = Go: R, =

!Et_ sx3cj *-l

5

0 5

c3-n)

(10)

P

and from equation (1) : R, = R;l”

-$ .

Sow coarsening, where it occurs: will reduce the flow stress of the net,work$ for (14) G = ,ub/(25rr,) and so -= ars

VOL.

(11)

Table 1 shoKs the results obtained by evaluating equations (10) and (11). The first two columns compare the actual and theoretical creep-strengths (equation 11). For example the entry R, = 1.89 for silver in column (1) means that at a stress of 1.89 x 10e3 ,U the real material (equation (1) with Ashby’s values of A and n) creeps at the same rate as the “theoretical material” (equation 9) does at a stress of 1.00 x 1O-3 ~1. The second column in the table deals with stresses an order of magnitude loner. The third and fourth columns present the alternative comparison on the basis of creep-rates. For example the entry R, = 0.77 in Column 3 for gold means that at a stress of 10-3 ,u t,he theoretical creep-rate is 0.77 of the actual creep-rate.

GITTUS: TABLE

THEORETIClL 1. Comparison

of

EQC’_1TIOS actual

and

FOR

STEdDY-ST_iTE

theoretical creep-strengths (equation 10)

R,, ratio of creep strengths for: Element

DISLOC_~TIOS (equation

11)

and

CREEP creep-races

Rc, ratio of creep rates for: a, = lo-‘,u

(it = lo-s/&

fYt = lo-‘,lc

(Ti = 10-3;‘

Silver %$er

1.59 0.95 1._.w

5.12 2.88 2.71

29.0 0.77 2.56

Nickel Sluminium Lead Gamma Iron Cadmium Zinc _uphe Thallium Molybdenum Tantalum Tungsten Beta Thallium ;Upha Iron Delta Iron Germanium MgO Stoichiometric CO*

0.57 0.41 1.29 0.38 0.97 0.67 1.09 0.37 0.47 0.202 0.60 0.39 0.37 0.092 0.004 0.55

1.71 0.85 2.48 1.15 1.94 2.17 2.95 0.74 0.91 0.62 1.83 1.43 1.36 0.043 0.005 1.18

8;; 2.88 0.004 0.87 0.09 1.56 0.014 0.043 9.4 x 10-j 0.053 1.51 x 10-S 1.05 x 10-Z 1.13 x lo-’ 9.17 x 10-g 0.068

11.7 0.5 -15.6 2.25 17.35 113.0 311.0 0.98 0.68 0.06 33.44 I”.0 8.34 1.79 X lOma 1.82 x lo-’ 2.15

1

2

3

4

Column So.

cj=e.up-_TT;

pb=

d _ A D,Gb -_ .-LT

D n theoretically 0au ’

In columns 1 and 2, the comparison between theory and experiment is, in a number of cases, poor. Thus MgO at 4 x 10d6 ~1 creeps at the rate which is predicted to occur under a stress of 10e3 ,u: for Germanium the fit is better but still unacceptable. However, there are a number of other cases where the fit is more respectable (for example p-thallium, stoichiometric UO,, nickel and cadmium) and it would, in any case, be over-optimist.ic to hope that a single theoretical creep relationship would represent such a disparate group of materials in their entirety. A temperature of T,/2 was chosen for the calculation of cj, the jog density with x = &r. If x > l/(&r) then the comparison is appropriate to some other temperature above T$ (equation 5). 4. CONCLUSIONS

A theoretical equation has been derived (equation 9) for a material containing a dislocation-network in which a balance exists between the work-hardening due to dislocation-generation and the recovery due to network climb. Algebraically the theoretical equation is identical with one which, Ashby shows, is an acceptable empirical representation for the steady state creep of silver, tantalum, gamma iron and sixteen other crystalline metals, semi-conductors and ceramics. In

791

5.79 143.0 161.5

A = 8a3c, and R = 3.

several cases the predicted and actual creep strengths differ by less than a factor 2 (columns 1 and 2, Table 1). The fit is poor in the case of germanium and MgO, and reasonably good for nickel, cadmium, beta thallium and stoichiometric UO?. ACKNOWLEDGMENT

I am indebted to Nr. R. V. Moore, U.K.A.E.A. member for reactors, for permission to publish this paper. REFERENCES 1. 31. F. Asaau,dcta Met. 20, 887 (1972). 2. A. K. MCEEERJEE, J. E. BIRD and J. E. DORS, Trans. Am. Sot. Xetals 62, I65 (1969). 3. J. WEERTW. Tram. Xet. Sot. _-lI~IE 227. 1475 (1963). ’ 4. J. H. GITTUS, Creep, Vi8cwZasticity and Creep-rupture in solids. Elsevier (1974). 5. J. WEERT~I~V. J. aml. Phus. 21. 1213 (1955). 6. J. WEERTW; J. a&L P&8. 23; 362 (i95i).’ 61, 7. J. WEERT~~, Am. Sot. NetaZ.8. Trans. Quarterly 681-694 (1968). 8. F. R. X. XABSUZO, in The Application of _%dern Physica to the Earth and Planetary Interiors, edited by S. K. RUSCORX, p. 251. Wiley-Interscience (1959). 9. E. O~owa-v, J. W. Scotl. Iron and Steel Inst. 54, 45 (1946-47).

10. R. W. B-Y, 11. G. I. TAYLOR, 12. H. E. Evsvs

J. Imt. Xet. 35, 27 (19-‘6). Proc. Roy. Sot. A145, 362 (1934). and K. R. WILLLXM, Phil. Nag.

28, 227

1% ~~"FI%DEL, Di8~ocations. Pergemon Press (1964). 14. R. W. Cams (ed.) Physical Metallurgy. (See esp. pp. 756 and 757). Sorth-Holland (1965).