Scaling laws in dislocation creep

. Acta metall, mater. Vol. 39, No. 4, pp. 599-608, 1991 0956-7151/91 $3.00 + 0.00 Printed in Great Britain. All rights reserved Copyright © 1991 Pergamon Press pie

SCALING LAWS IN DISLOCATION CREEP

D. S. S T O N E

Department of Materials Science and Engineering, University of Wisconsin-Madison, Madison, WI 53706, U.S.A.

(Received 2 February 1990; in revised form 31 July 1990)

Abstract--A micromechanical model for creep is devised in which the macroplastic flow properties depend solely upon the diameters of subgrains generated during deformation. The diameters fall within distributions, and various possible distributions obey a law of self-similarity. Self-similarity introduces a well-defined statistics. It gives rise to familiar scaling laws and an internal state variable, the "hardness," or tr*, the high strain rate lime of flow stress, tr* is inversely proportional to the average subgrain diameter DA. Dislocation glide and subgrain boundary migration operate as parallel deformation mechanisms. Glide is governed by a law in which the flow stress decreases with increasing subgrain diameter and is independent of strain rate. Flow governed by migration of subgrain walls obeys a law similar to that for Nabarro-Herring creep. The continuous distribution of subgrain diameters implies a mixture of the two mechanisms operates, and that the partitioning of mechanisms among subgrains is determined by strain rate and temperature. The mixture of mechanisms gives rise to a statistical basis for the isostructural (constant ~r*) strain rate sensitivity. Based on the model a rate parameter d* can be introduced, that obeys the scaling relationship d* oz tr *t/'. For A1 at room temperature, 1/# = 5. Workhardening and dynamic recovery cause D A (or a*) to evolve, eventually leading to a steady state. At steady state, D A oc tr-m2 where a is the stress and m2 ~ 1. Power law creep is obeyed, % oc a m, in which m increases slowly with stress from a value of 3 at low stresses. The steady state strain rate depends on stacking fault energy. Power law breakdown occurs. The average velocity of migrating subgrain boundaries obeys <v > oc tr m, with m 4 ~ m - 1.

R t s u m ~ n propose un modtle micromtcanique pour le fluage, dans lequel les proprittts d'tcoulement microplastique dtpendent uniquement du diamttre des sous-grains cr66s pendant la dtformation. Ces diam~tres se rtpartissent selon des distributions, les difftrentes distributions possibles obtissant/t une loi d'auto-similitude. L'auto-similitude introduit une statistique bien dtfinie. Elle dtbouche sur des lois d'tchelle courantes et sur une variable d ' t ta t interne, la "durett", ou a*, ou limite de la contrainte d'ecoulement pour des vitesses de dtformation 61~vtes. a* est inversement proportionnelle au diamttre moyen des sous-grains D A. Le glissement des dislocations et la migration des sous-joints sonte deux mtcanismes de dtformation parall6les. Le glissement obtit ~ une loi dans laquelle la contrainte d'tcoulement diminue lorsque le diam&re des sous-grains augmente, et est indtpendante de la vitesse de dtformation. L'tcoulement, qui est r~gi par la migration des parois de sous-grains, obeit / t une loi semblable 5. la loi du fLuage de Nabarro et Herring. La distribution continue des diamttres des sous-grains implique que les deux mtcanismes optrent fi la fois, et que la rtpartition de ees mtcanismes parmi les sous-grains est d&ermin~e par la vitesse de dtformation et la temptrature. Le mtlange des mtcanismes donne lieu ~i une base statistique pour la sensibilit6 isostructurale ~i la vitesse de dtfonnation (~i tr* constante). En se basant sur ce modtle, on peut introduire tan paramttre de vitesse ~* qui obtit ~t la relation ~*octr**/#. Pour l'aluminium fi la temptrature ambiante, 1/# = 5. L'tcrouissage et la restauration dynamique son ~ l'origine de l'tvolution de DA (ou de a*), ce qui conduit en fin de compte ~ un 6tat stationnaire. A l ' t tat stationnaire, DA ~ tr -m2 oh a est la contrainte et m 2 ~ 1. I1 s'agit d'un fluage en loi de puissance, dss oc tr 'n, darts lequel m augmente lentement avec la contrainte ~i partir de la valeur 3 pour les faibles contraintes. La vitesse de dtformation stationnaire dtpend de l'tnergie de faute d'empilement. La loi de puissance cesse d'ttre suivie. La vitesse moyenne des sous-joints qui migrent obtit ~i la loi <v> oc cr m', avec m 4 ~ m - 1.

Zusammenfassung--Ein mikromechanisches Modell des Kriechens wird abgeleitet, bei dem das makro- plastische FlieBverhalten nur von den Durchmessern der Subktrner, die w/ihrend der Verformung entsteben, abhiingt. Die Durchmesser haben eine Verteilung und verschiedene mtgliche Verteilungen gehorchen einem Gesetz der Selbst/ihnlichkeit. Diese Selbst/ihnlichkeit fiihrt zu einer wohldefinierten Statistik, die wiederum zu bekannten Skalierungsgesetzen und zu einer Variablen des inneren Zustandes, der "H/irte" oder tr*, f/ihrt, die Grenze fiir die FlieBspannung bei hoher Dehnungstrate. Dieses a* h/tngt umgekehrt proportional vom mittleren Subkorndurchmesser DA ab. Versetzungsgleitung und Wanderung von Subkorngrenzen sind gleichzeitig ablaufende Verformungsmechanismen. Die Gleitung wird bestimmt von einem Gesetz, bei dem die FlieBspannung mit zunehmendem Subkorndurchmesser sinkt und nicht yon der Dehnungsrate abh/ingt. Das durch Wanderung von Subkorngrenzen auftretende FlieBen gehorcht einem Gesetz, welches dem Nabarro-Herring-Gesetz/thnelt. Die kontinuierliche Verteilung der Subkorn- durchmesser bedeutet, daft eine Mischung der beiden Mechanismen vorliegt uind dab die Aufteilung der beiden Mechanismen auf die Subktrner vonder Dehnungsrate und der Temperatur bestimmt wird. Diese Mischung der Mechanismen ergibt eine statistische Basis ffir die isostrukturelle (konstantes a*) Dehnungsratenempfindlichkeit. Aufbauend auf dem Modell kann ein Ratenparameter ~* eingefiihrt werden, der der Skalierungsbeziehung ~* ~ t r *~/' gehorcht. Bei Aluminium ist 1/# = 5 bei Raum- temperatur. Verfestigung und dynamische Erholung bedingen die Entwicklung von D A (oder or*), wodurch

599

600 STONE: SCALING LAWS IN DISLOCATION CREEP

schlieglich ein station/irer Zustand erreicht wird. Im station/iren Zustand ist D A ~ a -"2, wobei a die Spannung und m E ~ 1 ist. Potenzgestzkriechen liegt vor, i~ ~ a~, wobei m langsam mit der Spannung ansteigt, beginnend bei 3 bei niedriger Spannung. Die Dehnungsrate im station/iren Zustand h/ingt yon der Stapelfehlerenergie ab; das Potenzgesetz gilt nicht mehr. Die mittlere Geschwindigkeit der wandernden Subkorngrenzen folgt (v) ~ a "4 mit m 4 ~ m -- 1.

1. INTRODUCTION

This article introduces a micromechanical model for unifying two, distinct creep phenomenologies. The first, typified in Refs [1-13], emphasizes structure and properties at and near the steady state. The second [14-26] emphasizes flow properties at constant "hardness", or a*, and evolution of a* far from steady state. Each phenomenology maintains an internal consistency, separate from any microscopic interpretation:

Existence of the steady state has been demonstrated, based not on the strain rate minimum in a creep test [22], but rather an approach of average subgrain diameter (DA) , subgrain misorientation (0), forest dislocation density (pr), and strain rate (d) to steady values depending solely on present temperature (T) and stress (a). All of the steady state values are independent of previous history of deformation. Elevated temperature creep of A1 is an example for which the history-independent criteria are often met [1, 6, 8, 10]. Studies of AgC1 indicate the criteria are not always satisfied, however, even when an inverse proportionality can be found to exist between a and D A during secondary creep [14].

An equation of state among stress, strain rate, and hardness (a*) was first demonstrated by Hart and Solomon in high purity AI at room temperature [25]. The demonstration was rigorous but left open the significance of a*. (One controversy is seen in the claims by Bradley et al. [27] that A1 permanently softens during room temperature load relaxation.) Studies of Rhode et al. [29] showed D A remaining nearly constant during room temperature load relaxation of A1 while pf decreased. Alexopoulos et al. [20] demonstrated an inverse relationship between a* and the average distance between low angle grain boundaries (sub-boundaries) in 316 stainless steel. A reasonable conclusion is that a* is closely related to D A while anelastic and microplastic transients may be attributed to changes in pf. Other authors have drawn similar conclusions [16, 19, 20].

Below, a micromechanical model is proposed, that is consistent with both phenomenologies. Specifically, the model details a formal relationship between a* and D A and proposes a unified description of scaling laws comprising the two phenomenologies. The model primarily concerns high purity metals for which a steady state has been unambiguously demonstrated to exist. It also concerns the regime of stress and temperature for which the high temperature branch of Hart 's model describes material behavior [24]. These conditions confine our interests primarily

to the high temperature end of the athermal plateau in flow stress [16, 30] and the regime of stress and temperature known as power law dislocation creep.

Relevant scaling laws are listed below. They address flow properties at constant structure (a* or DA) , evolution of the structure, and structure and properties at steady state. Each law involves a scaling exponent. The exponents include/~, B, m, m2, m3, m4, N and P.

(1) At elevated homologous temperatures, stress relaxation data frequently show a high strain rate limit in the flow stress, a*, and a characteristic strain rate, ~*. It is possible to generate all possible stress relaxation curves, corresponding to different a*, from a single set of stress relaxation data by rigidly translating the data (plotted in log stress-log strain rate form) along a line of constant slope. This means that ~* is not an independent parameter, but instead depends on a* [23]:

~* oc (a*/G) l/u e x p ( - Q / R T ) (a)

where G is the shear modulus, Q the activation energy for creep, and # = 0 In ale9 ln~ Iv where v is the isostructural (constant a*) strain rate sensitivity.

(2) Workhardening is unambiguously defined in terms of the absolute hardening rate, F -= d In a*/de (where de refers to an increment in plastic strain). F is a function of a and a* only, and has a limit F* at high strain rates (a /a* ~ 1) with [17]:

F* ~ a *-8. (b)

(3) Steady state, power law creep exists, in which the following power laws hold (cf. [2]):

~,s ~c o-% (c)

DA oC a - ' : , (d)

~s~'~/.3, (e)

where D^ is average subgrain diameter, ~, is stacking fault energy in f.c.c, metals, m ~ 3-5, and m2 ~ 1. The exponent m3 is about 3 [31]. Steady state strain rate can be expressed as a product of powers of stress and subgrain diameter [8],

es~ oc D f f a N, (f)

where N is large ( ~ 8) and varies slowly with stress [16, 32]. Due to m z ~ 1, the difference between N and P is approx, m [8]. The subgrain boundaries migrate, and the average velocity of migration depends on stress as

d ln(v ) /d In a I ss = m4, (g)

where m4 ~ m - 1 [33].

STONE: SCALING LAWS IN DISLOCATION CREEP 601

The model, developed in Section 2, relies heavily on a principle of self-similarity like the one discussed by Weertman and Weertman [34]. The model defines a "structure" relevant to mechanical properties, defines a* in terms of this structure derives an expression for the flow properties at constant a*, introduces laws of evolution of tr*, and identifies structure and properties at steady state. The model is compared with those by Derby and Ashby [35] and Gottstein and Argon [36] in Section 3.1. It is applied to Hart 's phenomenology in section 3.2 and the phenomenology of steady state in Section 3.3.

2. D E F O R M A T I O N M O D E L

2.1. Structure

In this model, the average subgrain diameter D A uniquely specifies the state of internal structure relevant to macroplastic deformation. The density of dislocations at the centers of subgrains, Pf, will not be used, although pf is undoubtedly involved in some of the underlying physics. A justification for the choice of D A instead of pf is that the latter adjusts ~rapidly to changes in applied stress while both D A and ~ evolve more slowly [4, 5].

A second assertion is that the bulk properties derive from a statistical basis. With the apparent significance of DA and the ease of constructing flow laws in terms of local subgrain diameter, D, the present choice of the parameter giving rise to these statistics is D. (The subject of a future article will be to explore other physical parameters, related to D, giving rise to similar statistics.)

Let f ( D ) d D denote the volume fraction of subgrains with diameters between D and D + dD. Then

o ~ f ( D ) dD = I. (1)

All possible distributions f are confined to obey a law of self-similarity: if two specimens are viewed at magnifications such that the D g appear identical, then the subgrain distributions are indistinguishable in terms of size and shape [34]. Equivalently, if (D i) is the ith moment o f f , then (Di) /DA i is a constant (depending only on i), independent of all possible histories of work hardening and dynamic recovery that might result in D g.

As used here, self-similarity is to be distinguished from the principle of similitude [37]: only if the misorientation between subgrains were to vary inversely with D would self-similarity coincide with similitude. The possibility that self-similarity is broken is discussed briefly in Section 4.

The self-similarity property assures that the internal structure and macroplastic flow behavior can be uniquely defined in terms of a single internal state variable. Any of the moments, (Di ) , of f will serve. Self similarity implies

f ( D ) dD = g(r/) dr/, (2)

where r / = D/DA and g is some function. The magnitudes of the scaling exponents in (a)-(g) depend only weakly on the shape of g. (We need not be concerned about the shape of g, other than it be reasonably well-behaved.)

2.2. Flow properties at constant structure

Two, kinetically parallel deformation mechanisms can operate within each subgrain. This means that the strain rates add, and the mechanism having the lowest flow stress governs the local flow stress. The athermal mechanism is associated with glide of dislocations through subgrains; whereas the diffusional mechanism is associated with diffusion-controlled migration of subgrain walls.

The local stress a I required to operate the athermal mechanism depends only on D. Once this threshold stress is reached, the strain rate becomes indetermi- nate, governed solely by the condition that strain compatibility be maintained with surrounding subgrains. At locations in the crystal where the athermal mechanism governs flow stress, the flow stress is denoted cr~ and obeys

a~ = A D - ' . (3)

where A is a constant. For the purpose of clarity the quantity r is chosen as a parameter to allow the dependencies of the scaling law exponents on r to be tracked. When calculating the actual values of the scaling exponents we shall utilize r = 1 although other choices, like r = 1/2 corresponding to a true Hall-Petch effect [38, 39], can not be ruled out completely.

If the local stress level o- 1 is lower than the threshold value in equation 3, the athermal mechanism contributes negligibly to the local strain rate. In contrast, as long as the local stress is nonzero, the diffusional mechanism will always contribute a component, £t, to the local strain rate:

it = flal" D -q, (4)

where fl is a "constant" having roughly the same temperature dependence as the coefficient of self- diffusion. Both n and q are greater than 0. A candi- date expression for equation (4) is derived in the Appendix. At temperatures where lattice diffusion is rapid, n = 1 and q = 2 and the expression is similar to that for Nabarro-Herring creep. At low temperatures, n increases to a value of 3 due either to pipe diffusion of thermal vacancies or vacancies generated during deformation [30].

A subgrain boundary migrates at velocity v proportional to it:

v = itD/O, (5)

where 0 is the tilt misorientation between subgrains. In this work, 0 is assumed to be a constant. (There is some controversy whether or not 0 = constant is actually the case. Exell and Warrington [33] found that 0 is independent of stress level during


creep, whereas Blum et al. [4] report that the distance between dislocation in the subgrain boundary decreases as the inverse square root of creep stress. To introduce a dependence of 0 on D is possible, but does not radically change the present results. It affects the scaling exponents B and m4.)

Strain compatibility requires ~ to be uniform among subgralns, meaning that a~ is uniquely specified by i and D. For illustration, the local flow stress is plotted in Fig. 1 as a function of D for two strain rates (~2>gl). The flow stress approaches zero for both very large and very small D and is highest at D o . Temperature and strain rate uniquely define Do, the subgrain diameter for which al is just sufficient to operate the athermal mechanism:

D O = ~-l/(q+nOfll/(q+nr)An/(q+nO. (6)

The function f ( D ) is shown in Fig. 2 to illustrate how Do (or () partitions the crystal volume. The crystal contains small subgrains (D < Do) for which the diffusional mechanism controls the flow stress, and large subgrains (D > Do) for which the athermal mechanism controls. When ~ is very low Do is large, and if D0>>DA the flow stress is strongly rate-sensitive. At high (, Do is small. If D0<<DA, the flow stress is insensitive to strain rate. The degrees to which workhardening and dynamic recovery operate also depend on the ratio Do/DA; this ratio is obviously very important.

The athermal contribution to local strain rate, (~, is the difference between the total plastic strain rate and diffusional component of strain rate:

where

~ = ((1 - R); (7a)

R ~--- Et/E

= 1 for D < Do

= (D o/D)q ÷"r for D > D 0. (7b)

The bulk flow stress is the volume average of the local flow stress:

;°° +fo a = e l f (D) dD a~f (D) dD. (8) 0 o

Substitutions from equations (4) and (5) for ar and a 1 lead to

[;:° a = [a*/<n - ' ) ] tl;'(~/~o) q/" g(q) dq

+ f .7 q - ' g ( t / ) dr/] (9)

where qo = Do/DA and

Xo a * = A D - r f ( D ) dD = A Q / - ' > DA'. (10)

Due to self-similarity, (r /-r> is independent of D A. The high strain rate limit of flow stress at constant structure (DA) is o'*.

o-i (D)

~(2) > ~(I)

\ /

~(2) ~ /----~(i}

Do(2) Do(I)

D

Fig. l. Flow stress as a function of subgrain diameter. The model assumes two, parallel kinetic processes. The athermal process governs the flow stress of"large" subgrains, whereas the diffusional process governs the flow stress of "small" subgrains. Do is the diameter that distinguishes between

"large" and "small", and is a function of strain rate.

Expression 9 is a mechanical equation of state among stress, strain rate, and the state variable a*. a* is sufficient to uniquely specify the internal state of the crystalline solid relevant to macroplastic deformation.

The isostructural strain rate sensitivity is:

v =--t~ lna/O ln~ 1~.

I "° = (a*/ (q-r )na) q;-r(tl/q0)q/"g(q)dr/. (11) j0

This quantity decreases continuously from + 1/n at low strain rates to + 0 at high strain rates. From (9) and (11), v depends solely on rio (or equivalently, the ratio a/a*).

Curves of log strain rate vs log stress, corresponding to various DA (or a*), are shown in Fig. 3. The curves cross because q > 0, and they can be over- lapped by rigidly translating them along lines of slope 1/~, where

# = _ O l n a / O l n i l =r/(nr +q) . (12)

Flow stress governed by=

~ Athermal mechanism

Dif fusional mechanism

increases decreases

I f(O) i

dynamic I recovery

Do DA

D

Fig. 2. Partitioning of the crystal volume into subgrains in which the athermal mechanism governs the flow stress and those in which the diffusional mechanism governs. The relative volume fractions are governed by the strain rate (which determines Do) and average subgrain diameter, D A.

In a stress-controlled experiment D O (or the strain rate) adjusts so that the specimen can support the applied stress.


The scaling property (12) exists due to self-similarity. Lines of scaling slope 1/# correspond to constant Do/DA or, equivalently, constant a/a*.

2.3. Work hardening and dynamic recovery

Evolution of a* is to be evaluated in terms of the parameter, F = d In a*/de [17]. An expression for F is now derived.

Let L denote total area of subgrain boundary per unit volume. Due to the self-similarity property,

L = COD;, 1, (13)

where Co is a dimensionless constant. Moreover,

F = d In a*/dL x dL/de

= r x (Dh/Co) x dL/de. (14)

dL/de is the sum of two components:

dL/de = L + + L - . (15)

L + reflects the rate at which dislocations condense to form new subgrain walls. L - is a negative quantity representing the rate of loss due to subgrain boundary migration, collision, and annihilation. First, an expression for L + is derived.

To strain a subgrain of size D by amount de requires d N ~ ( 1 - - R ) ( D / b ) d e dislocations, where b is the Burgers vector and 1 - R is the fraction of strain contributed by the athermal mechanism. These dislocations rapidly condense to form a wall of area h x (bD/O)dN, where 0 is the tilt misorientation between subgrains. The factor h accounts for the possibility that not all of the mobile dislocations become part of the subgrain wall network. A large fraction of dislocations (1 - h ) are lost due to cross slip and recombination, a stress-assisted process [30, 36, 40]. In locations where dislocations are generated, the local stress is solely a function of D because the athermal mechanism controls the flow stress; therefore, h may be regarded as a function of D:

h oc (D/Dr) k (16)

where Dr is a unit of length whose value independent of Do and DA. For f.c.c, metals Dr is the stacking fault energy, 7, divided by shear modulus, G:

Dr = ylG. (17)

In general, k /> 0 to allows the efficiency, h, of dislocation storage to increase as the stacking fault energy decreases and decrease as D decreases. Ther- mal activation should affect h and introduce effects of temperature and strain rate; however, these dependencies are presumed weak. (In contrast, Gottstein and Argon [36] incorporate an explicit temperature dependence into the frequency of cross-slip.)

The number of subgrains, per unit volume, with dimensions between D and D + dD is D-~ f (D ) dD. The rate at which wall area is generated (per unit

volume) is therefore

L + =ClD~ -k ( 1 - R ) D k - l f ( D ) d D (18)

where Cl is a dimensionless constant. The diffusional component of strain rate intro-

duces dynamic recovery: From equation (5) the migration velocity of boundaries surrounding a cell is it(D/O). The rate at which volume is swept out is proportional to DEit(D/O). The rate at which boundary area is consumed by a single cell is proportional to [D2gt(D/O)][Co/DA]. These con- siderations lead to

L - = - - C 2 D ~ , 1 Rf(D) dD, (19) dO

where C: is another dimensionless constant. From equations (14), (15), (18), and (19):

If0 F = F * C l (1 -- R) t /k- lg( t / ) dr/

;o ]/ - C2(Dr/DA) k Rg(~/)d~/ C 3 (20)

where Cs is a third dimensionless constant, and

F* oc (DA/Dr) k. (21)

From equation (20), the absolute hardening rate is the product of an athermal or rate independent component, F*, and a temperature and strain rate- dependent component, the expression in brackets in (21). F* is the high strain rate limit of F. Behavior like this is observed experimentally [17].

2.4. Steady state

Stable, steady state creep only occurs under conditions where F = 0 and gF/gDAI{i or ~) > 0. From (20), F = 0 implies:

(DA/Dr) k = . (22a)

C1 (1 -- R)g(r#) d~l

An equivalent expression that demonstrates the relevant parameters is

F* = F(~/0 ) (22b)

where F is some function depending only on the shape of g.

The model predicts that, to every value of D A there corresponds a steady state stress and strain rate. To see this, note that both the numerator and denominator on the right-hand side of (22a) depend solely upon the ratio ~/0 = Do/DA. Compared to the denominator, the numerator in (22a) can be made arbitrarily large or small through adjustment of ~/0 (i.e. strain rate). This means that for every value of DA, there exists a strain rate such that (22) is satisfied. Under conditions where (22) is satisfied,

AM 39/~-L


log

OA(2I > OA(I) o~*, DA(~)

C > c~ z) ).. o-L D,(

log cr

Fig. 3. Isostructural log stress-log strain rate curves corresponding to average subgrain sizes DA(1 ) and DA(2). For every value of DA, there is a high strain limit of the flow stress, tr*. All other members of this family of curves can be generated through rigid translation of a single curve along a scaling line of slope 1/#. This property is due to self-similarity. The magnitude of the scaling slope is determined by

the deformation mechanisms.

t~F/t~DA[ o > 0. Therefore, a stable steady state exists for every value of D A.

Power law relations among stress, strain rate, average subgrain diameter, and stacking fault energy (Dr) can be derived through manipulat ion of equations (9) and (22a). Taking the logarithm of both sides of equation (9), then differentiating with respect to In ~ under conditions of steady state produces

d ln a /d ln ~ l~ = d ln a*/d ln ~ l~

- v ( n r + q ) d l n r / 0 / d l n g ] , . (23)

The last derivative is evaluated based on equation (22a) as:

d In r/0/d In d I

= -k / [r6 (nr + q)] d In a*/d In ~ [ , (24a)

where

+ . (24b)

Equations (23) and (24a), along with the definition of a* [equation (10)1 lead to the relation:

m 2 - - d l n D A / d l n a l ~ = ~ / ( r ~ +kv) . (25)

Other identities are:

m -- d l n ~ / d l n a [ ~

= [k + (nr + q)6]/[kv + r~] (26)

m s ~ d In d/d In Dr[ss,~/G =k[v(nr +q)+r]/[tSr +kv] (27)

N = ~ In ~/8 In a IDA = 1/v (28)

P = ~ In ~/8 In AAIo = r/v -- (nr + q). (29)

The average migration velocity is calculated as:

( v ) = ~O-1DA Rg(r/) dr/. (30)

The stress dependence of ( v ) at steady state is:

m4 = 8 l n ( v ) / 8 In a Iss

= m - [k8 ' + 8 ] / [ k v + rS] (31) where m is the steady state creep exponent [equation (26)] and

6' [ff Rg(r/)dr/]. (32)

Since the values of 6 and 8 ' help to determine the magnitudes of the scaling exponents, estimates of these two parameters are necessary. Both depend upon r/0. They also depend weakly on the shape of g. It is reasonable to assume that for small r/, g approaches 0 more rapidly than r/3. In this case, fi and 6 ' look somewhat like the curves shown in Fig. 4. The quanti ty v is also depicted.

Curves of ~ vs a at steady state are compared in Fig. 5 with curves of ~ vs a at constant DA. If k = 0, the steady state locus corresponds to a line of slope l/p, which is also the scaling slope for curves of constant D A. If k > 0, the locus of steady state points is not a straight line. Instead, its slope increases slowly from a value near 1/# at low stresses (we assume 6 >> 1 at steady state at low stresses) to a value of approximately [k/r + 1/#] at high stresses (6 ~ 1, v ~0 ) .

3. DISCUSSION

Major assumptions of the model are summarized in Table I.

3.1. Relationships with other models

Derby and Ashby [35] published a similar model of dislocation creep where properties derive from a discrete distribution of subgrain diameters. A fraction f of small subgrains flow by Nabarro-Herr ing creep (n = 1; q = 2) to accommodate deformation of a fraction ( 1 - f ) of large subgrains governed by the Orowan relationship (r = 1). Subgrain boundary migration leads to dynamic recovery, but dynamic recovery involving cross-slip is not considered: k = 0. In our model, these assumptions lead directly to the steady state relationships D A oc a-m2, ~ oc a m, and ( v ) oc a m4 with m 2 = 1, m = 3, and m4 = 2. Note that m - - m 4 : l , consistent with Exell and Warrington 's observations [33]. Due to the discreteness of the

STONE:

8,8',~

8 - 8 ' - - - 1/ . . . . . . . . . .

SCALING LAWS IN DISLOCATION CREEP 605

\ . . . ' " " " " "'" . . . . . . . . . . .

\ ..'" \ .."

.................... • ----..q ~ . . . .

"%

Fig. 4. The parameters 6, 6' and v as functions of rio = Do/D A .

Isostructial~

f IA/- S,eo0,__

. k

l og O"

Fig. 5. Log stress-log strain rate curves illustrating the relationship between isostructural and steady state flow data for the cases k = 0 and k > 0. The high stress limit of m, the

steady state power law exponent, is k / r + 1/~. distribution, the isostructural strain rate sensitivity also varies discretely with strain rate.

In the model of Gottstein and Argon [36], the strain rate is comprised o f two components, one due to the mot ion of mobile dislocations and the other from the migration of boundaries, much as is the case here. Steady state becomes established not through a dynamic equilibrium between subgrain boundary formation and annihilation, but instead by a balance between creation and annihilation of mobile dislocations. This feature is unlike the present model. Dynamic recovery in the model by Gottstein and Argon is dominated by diffusional processes at high temperatures and cross-slip at low temperatures, similar to the present model.

3.2. H a r t ' s p h e n o m e n o l o g y

Elevated temperature stress relaxation data frequently demonstrate a high strain rate limit in the flow stress, tr*. They also demonstrate scaling, with /~ ~ 0 . 2 frequently being quoted. The archetypal material obeying this type of behavior is A1, for which at room temperature # ~ 1/4.6. This is to be compared with the model value of /~ at room temperature, 1/5, corresponding to n = 3 and q = 2. At high temperatures, the predicted value o f /~ is 1/3, corresponding to n = 1 and q = 2. Indeed, an inspec- tion of Hart ' s 270°C load relaxation data of A1 [22, 30] reveals 1/3 provides a better fit than the low temperature value of 1/4.6 [25]. Another prediction is that load relaxation curves obtained from various

levels of workhardening will cross at sufficiently low strain rates, a phenomenon suggested by Hart 's data [22, 30].

Nabar ro [30] recently speculated whether, in A1, thermal vacancies are presented in sufficient concen- trations at room temperature to cause the drop in stress during load relaxation witnessed by Har t and Solomon [25]. His question can be addressed by estimating D o and DA, and then comparing them. If D0<<DA, then it is unlikely that thermal concen- trations of vacancies will be responsible for the drop in stress that occurs during relaxation.

To estimate D o and DA, explicit forms of the athermal and diffusional laws are required. For the athermal law, the correlation:

a~ = 10 G b / D (33)

will suffice based on the well-known correlation between subgrain diameter and yield stress at low temperatures [38, 39]. Fo r the diffusional mechanism, an expression derived in the Appendix is used:

40"l~D s dt k T D 2 0 2 . (34)

where f~ is the atomic volume and D s the coefficient of self-diffusion. At room temperature, pipe diffusion will dominate, so that D s ~ lO( t r /G)ZDc where De is the coefficient of self diffusion along dislocation

Table 1. Model assumptions Phenomenology Governing equation Significance

Self-similarity f ( D ) = g ( D / D A )/1) A

Athermal mechanism cq = AD -"

Diffusional mechanism ~ = Ba'~D -q

Dislocation cross-slip h oc (D/D,) k

Responsible for scaling laws in mechanical properties; allows the properties to be uniquely specified by a single internal state variable, DA. g gives rise to a strain rate sensitivity that decreases with increasing

Responsible for athermal plateau; associated with work hardening and dynamic recovery of the type involving dislocation cross-slip (r = 1)

Responsible for introducing strain rate and temperature effects; associated with subgrain boundary migration and the type of dynamic recovery associated with subgrain boundary annihilation (n = 1 at high temperatures, n = 3 at low temperatures, and q = 2)

Responsible for steady state stress exponent rn being higher than 3; responsible for stacking fault energy dependence of the strain rate, parabolic hardening at lower temperatures, and power law breakdown (D~ = y/G; k = 2)


cores. Literature values the parameters in equations (33) and (34) may be found in Ref. [41]. At a (tensile) strain rate of 10-9/s, the parameters give D o = 1.6#m. In Fig. 6 of Hart and Solomon's article [25], the level of stress prior to load relaxation is about 100 MPa (tension), which from equation (33) corresponds to a subgrain diameter of roughly 1.3#m (~DA). The stress drops by about 40% between 10-3/s and 10-9/s. Considering that Do/D A ~ 1 at 10-a/s, to propose that 40% drop in stress might be caused by diffusional migration of subgrain boundaries seems reasonable.

3.3. Phenomenology o f the steady state

The parameter F* can be viewed as the highest rate of workhardening achievable for a given level of tr*. While F* can only be reached at high strain rates (a /a* ~ 1), it still has an effect on the steady state, which is observed at comparatively low strain rates. The model suggests that m > 3 is caused by the dependence of F* on tr*. According to (22b), the condition for steady state can be expressed in terms of F* and the parameters rlo = D o IDA . Suppose F* were independent of a* (i.e. k = 0). Then the condition for steady state would be ~/0 = const., and steady state creep would obey the "natural" power law [34] ~ oc a m with m = 3. The "natural" values of other scaling exponents would be m E = 1, m 3 = 0, and m4 = m - 1. N would differ from P by 3. The values of the scaling exponents based on k = 0 are shown in Table 2 under the heading "Natural Values."

Experimentally, F* is not observed to be independent of tr*. Instead, F* is found to decrease with increasing tr*, a phenomenon that at lower temperatures gives rise to parabolic hardening. The reason, proposed here, that F* decreases with increasing tr* is the increased incidence of cross-slip and annihilation of dislocations at high stresses. This is also the reason why m > 3, Fig. 5. Because k > 0 the efficiency of workhardening decreases at high stresses. The reduced efficiency of workhardening at high stresses means that a greater fraction of the crystal must be devoted to the process of workhardening for steady

state to be maintained: thus, the ratio a/a* must increase with increasing stress for steady state to be maintained at higher stresses. This result is equivalent to m > 1/#.

In the model, the cross-slip form of dynamic recovery not only causes m > 3; it also causes the steady state strain rate in f.c.c, metals to depend on stacking fault energy. Additionally, cross-slip causes the other scaling exponents to deviate somewhat from their "natural" values. The values of the steady state creep scaling exponents estimated based on the model are shown in Table 2. The values are calculated based on t~ = 1, tS' = 0.1, and v = 1/8, which correspond to r/0 ~ 1. The calculated values of the scaling exponents are consistent with the experimental values. An important aspect of the scaling exponents, not represented in the table, is that they slowly change with creep stress.

The contribution of subgrain boundary migration to the specimen strain rate may be estimated based on the present model, then compared with experimental data. Exell and Warrington [33] reported grain boundary migration contributes about 25% to the total specimen strain rate. At 500°C and an imposed strain rate of 2.4 x 10-5/s, the level of stress was approx. 10MPa and D A was 150/~m. By using the model in the Appendix for diffusion-controlled migration, letting 0 ~ 1 ° and setting D ~ D A , w e

estimate the contribution of the diffusional mechanism to the strain rate of an "average" subgrain to be 1.3 x 10-6/s. This represents about 5% of the total strain rate, or a factor of 5 lower than what is estimated by Exell and Warrington. Considering the inaccuracy inherent to our estimate of the rate of subgrain boundary migration, a factor of 5 disrepancy is reasonable.

Caillard [9] measured the rate of dislocation climb in migrating subgrain boundaries and found it to be 50 times slower than the velocity of boundary migration. Considering that the boundaries migrate at a rate about 1/0 times the rate of dislocation climb, the observation by Caillard is entirely consistent with diffusion-controlled migration of the boundary.

Table 2. Summar y of results

Model Literature " N a t u r a l " Parameter Fo rmu la value value valuer Ref.

Scaling laws for constant structure # = t~ In a/d In ~ I, r/(nr + q) 0.2 0.22 0.2 25

(Low T) # = t3 l n a / d ln~l~ r / ( n r + q ) 0.33 ~0 .3 0.2 22

(High T ) Workhardening and dynamic recovery

B = - d l n F * / d l n a * k/r 2.0 1.6 0 17 Steady state

m = d l n ~ / d l n a l ~ [k+(nr+q)6] / [ kv+rr] 4.0 4.5 3 32 m 2 = - d l n D A / d l n a [ ~ t~/[kv +rt$ l 0.8 1 1 8 m 3 = d l n i / d l n D r l ~ k[v(nr +q)+r]/[Icv + r r ] 2.4 3 0 31 m 4 = d In ( v ) / d In o~ m - [ k r ' + t$]/[kv + r r ] 2.9 3.5 2 33 N = d In d/d In a IOA I/V 8 8 8 8 P = d In d/t3 In DAI . r/v -- (nr + q ) 5 3 5 8

Model values based on: n = 1 (3 at low temp.), q = 2, r = 1, k = 2, N = 8, 6 = 1, 6 ' = 0.1. (6, v, ~ ' correspond to t/0 ,~ 1.)

t " N a t u r a l " values based on k = 0.


(In contrast, Caillard cited the slow rate of dislocation climb as an argument to support that climb of dislocations in subgrain walls is not rate-controlling for migration.)

The model provides two insights into the origins of power law breakdown, both having to do with k > 0 (F* decreasing with increasing tr*). First, since m increases with increasing ~r, power law breakdown might simply be a continuation of this trend to high stresses. Stated in terms of mechanisms, power law breakdown might simply be due to the replacement of subgrain boundary migration by dislocation cross- slip as the dominant form of dynamic recovery. Second, according to equations (20) and (21), the strain required to establish steady state increases as the steady state value of D A decreases (as a increases). The strain should be roughly proportional to DA k, in which case a true steady state might not be achieved in many experiments performed at high stress levels because of the large strains required. Therefore, the "steady state" exponents will be arti- ficially high at high stresses, leading to a power law breakdown.

4. CONCLUDING REMARKS

(1) Scaling in macroscopic properties suggest an underlying scaling (e.g. self-similarity) in the micro- structure. Weertman and Weertman have already noted this in the case of steady state creep [34]. They propose that m > 3 might be caused by a breakdown of self-similarity or similitude [this proposition, incidently, is very reasonable--see item (2) below].

(2) Most of the results presented here are not specific to the details of the model. All models with similar physical basis, constructed in terms of parameters having meanings similar to DA, Do, and Dr, will provide results similar to those shown in Table 3. For instance, the result DA oc a- l i t at steady state does not depend on details of the model for workhardening and dynamic recovery; it simply follows from the formulation of the conditions for steady state in terms of the ratio DA/Do. Intro- duction of a third scale of length, Dr, means that DA oC a -~/'+~ where ~t represents a perturbation caused by the physics associated with Dr. As another example, the possibility that self-similarity is broken means that the distribution f is not only a function of D A but another parameter, say D~, that does not depend on Da in a simple fashion. If D" is the width of the distribution, and D~/DA decreases with decreasing DA, then the steady state creep exponent will be greater than 3.

(3) To interpret strain rate sensitivity based solely on specific mechanisms (e.g. activation area types of arguments) might be to overlook a significant statistical basis for the strain rate sensitivity. In the present model, the strain rate sensitivity v derives entirely from the statistics of two, relatively simple, mechanisms partitioned among large numbers of subgrains.

The dependence of the partitioning upon strain rate determines the strain rate sensitivity.

(4) A detailed micromechanical model has been provided, that is consistent with Hart 's phenomenology and links it with the phenomenology of steady state creep. Physical meanings have been attributed to tr*, F*, ~*, and the laws of scaling among them. Because the framework relies on a distribution of strengths, an internal stress exists. To extend the present model to include the Bauschinger effect and other microplastic phenomena is straight- forward.

Acknowledgements--The author wishes to acknowledge helpful discussions with R. F. Cooper, S. P. Hannula, M. Korhonen, C.-Y. Li, F. R. N. Nabarro, and W. D. Nix. This work has been supported by the Wisconsin Alumni Research Foundation. J. Jorgenson of Brooktondale, New York, performed the drafting.

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A P P E N D I X

Consider the migration of subgrain boundary walls, limited in rate by long-range diffusion of atoms. The walls are tilt with misorientation angle 0. We assume that for a subgrain boundary to migrate a distance Ax, the dislocations comprising it must climb a distance 0 ~ . The source for the vacancies will be a nearby subgrain boundary, which also migrates.

Climb of a distance OAx requires An =NbDOAx/~ vacancies, where b is the Burgers vector, D is the length of each dislocation, fl is the atomic volume, and N = DO/b is the number of dislocations in the wall. The work done by the stress a is AW = aNbAx.

The subgrain boundaries act as perfect sources and sinks for vacancies diffusioning across subgrains because the dislocation spacing in the walls is much smaller than the average subgrain diameter [42]. The work done when a vacancy diffuses from one wall to a nearby wall is roughly 2 x AW/An =2ag)/O, so the flux of vacancies toward a subgrain wall is J~4Dvctr/kT(D where c is the concentration of vacancies and Dv is the coefficient of diffusion for vacancies. The total number of vacancies impinging on the wall is ri = J x D 2 and the rate of migration is g =t i x (Ax/An). The migrating wall contributes to the strain rate by the amount i t ~ YcO/D, which means

Et ~' 4trD'Ds/k TD202, (A1)

where D~ = D¢/fl is the coefficient of self diffusion. This is essentially the formula for Nabarro-Herring creep but modified by a factor of 1/02.

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Scaling laws in dislocation creep