Role of Peierls stress in power law dislocation creep

MATERIALS SCIENCE & ENGINEERING

A

Materials Science and Engineering A202 (1995) 52-56

ELSEVIER

Role of Peierls stress in power law dislocation creep J.N. Wang 1, T.G. Nieh Chemistry and Material Science, L-370, Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94551, USA Received 7 November 1994

Abstract

Based on the features of Harper-Dorn creep and the transition stress from this creep to power law creep, a relationship between the Peierls stress and the empirical constants in the Dorn equation for power law creep is derived, and is shown to agree very well with experimental data on metals, ceramics, and silicates. It is therefore inferred that the Peierls stress may play an important role in power law creep as in Harper-Dorn creep. Keywords: Peierls stress; Power law creep; Harper-Dorn creep

I. Introduction

It is well k n o w n that steady-state power law (PL) dislocation creep in crystalline materials at relatively high stresses and high temperatures can be described by the D o r n equation [1]: k-----~

(1)

where ~ is the axial strain rate, a is the axial stress, G is the shear modulus, b is the length o f the Burgers vector, k is B o l t z m a n n ' s constant, T is the absolute temperature, DL is the lattice diffusion coefficient o f the ratecontrolling species, n is the stress exponent which varies f r o m a b o u t 3 to 7, and ApE is a dimensionless constant which has values f r o m o f the order o f unity to as large as l016. A l t h o u g h m a n y theoretical models (for reviews see Ref. [2]) have been p r o p o s e d for this creep, none o f them can satisfactorily explain the observed values o f n and ApL. The large values o f ApL strongly suggest that some i m p o r t a n t physical quantity is not included in Eq. (1) [3]. A c c o r d i n g to Stocker and A s h b y [4] and B r o w n and A s h b y [5], ApL and n are not independent, but can be related to each other by the following equation logl0ApL = ~(n - fl)

where ~ and fl are constants. It was shown that alt h o u g h fl was usually close to 3, ~ t o o k different values depending on the class o f material (i.e. whether b.c.c. metals and alloys, f.c.c, and h.c.p, metals and alloys, or oxides and alkali halides). In the analysis o f Brown and A s h b y [5], a reference stress a0 was introduced into Eq. (1), and an alternative creep equation o f the form

~=

A' D L a ° b I a ) "

~i~ kT

~

(3)

was suggested with ao = G/IO ~ and A~,L : 10 "(] /~). ao was considered [5] to be a characteristic o f a material class, but A~L to be equal to approximately 1.6 x l0 -6 for all material classes so that it was essentially a universal constant [5]. The purpose o f this study is to illustrate the role o f the Peierls stress o f a crystal Zp in P L creep. Specifically, it will be demonstrated that in Eq. (2), a is a function o f Zp, and fl is a b o u t equal to 3, and that in Eq. (3), ao m a y be represented by Zp with A~L being a function o f Zp.

2. Derivation of a relationship between ApL and n

(2)

Formerly with Earthquake Research Institute, The University of Tokyo, Bunkyo-ku, Tokyo, ll3, Japan.

A relationship between ApL and n m a y be derived by considering the features o f H a r p e r D o r n ( l i D ) creep [6], a newtonian dislocation creep, and the stress level 0921-5093/95/$09.50 © 1995 - - Elsevier Science S.A. All rights reserved SSD1 0921-5093(95)09789-9

J.N. Wang, T.G. Nieh / Materials Science and Engineering A202 (1995) 52-56

at which the transition from this creep to PL creep takes place. Since most experiments have shown that H D creep, as PL creep, has an activation energy equal to that lattice self-diffusion, the general rate equation for H D creep is of the form

= A.D ~

(4)

in which AHD is a dimensionless constant having values from 10 12 to 10 6. It has been shown recently that the dislocation density in H D creep PUD is related to Vp by [7,8]

bpn D = 1.4(zp/G)

(5)

Moreover, it was suggested [9,10] previously for A1 that the rate-controlling flow process H D creep may be dislocation climb under saturated conditions. With the observed relationship between PHD and "gp, AHD in Eq. (4) for H D creep controlled by this flow process has been derived as [11]

AnD- 21r(vP/G)2

(6)

-- ln(zp/G) for all crystalline materials. It is shown that the strain rate in H D creep predicted by Eqs. (4)-(6) is within one order of magnitude of the experimental observations, suggesting that the previous conclusion on the rate-controlling flow process in H D creep for A1 probably also applies to all other materials (metals, ceramics and silicates) [11]. When the grain size of a material is relatively large, PL creep transits to H D creep rather than to diffusional creep as the applied stress decreases. The stress marking this transition at has been found to be close to the stress level of the Peierls stress. That is, O"t ~ X/3Z'p [12,13]. Thus, under the condition of a ~ xf3Tp, Eq. (1) m a y be equated with Eq. (4). Substituting in Eq. (6) and making appropriate arrangements, an expression describing the dependencies of ApL on n and Tp/G is obtained as the following:

ln(zp/G) range of "cp/G of

ApL -- --

(7)

The interest is approximately 4 × 1 0 - 6 - 5 × 10 3, which corresponds to - l n ( r p / G ) = 12.4 to 5.3, and that of n is generally about 3 - 7 , which g i v e s 3 ° n)/2= 0 . 3 3 to 0.04. By taking the average values of - l n ( z p / G ) ~ 8.85 and 3"-n)/2 ~ 0.18, Eq. (7) m a y be approximated by

AvL ~ 0.1 X

(8)

It can be noted from Eq. (8) that ApE is a constant (approximately 0.1) in the case of n = 3, but appears dependent o n ~p/G in all cases of n > 3. In the follow-

53

ing, these predictions are compared with experimental data.

3. Comparison with experimental data To compare the observed ApL value for a given n value with the predicted ApL value, it is necessary to know the magnitude of "(p/G of the material. Although experimental data on ~p are not available for most materials, they can be estimated approximately from the theoretical equation [14] 2"P= 1 - - V

where r/ is a material constant given by (3 -- 2v) 1 / - 4(1 - v)

(10)

in which L is the distance between the atomic planes, and v is Poisson's ratio. In estimating rp for a crystal, L is taken as the distance between the most closely packed atomic planes in the lattice, b as the spacing between atoms in the most closely packed direction on the most closely packed plane, and v as the average value of the dynamic, unrelaxed Poisson's ratio, as in previous analyses [8,11,13]. This procedure leads to values of Zp which represent the lower limits of the true Peierls stresses of crystals at T = 0 K and which are in reasonable agreement with most reported experimental measurements. F o r example, the estimated "~p/G for A1 at T = 0 K is 5.12 × 10 5, which agrees well with the experimentally determined value, 3 × 10 -5 [15], at T = 11 K. Owing to thermal vibration of the atoms, Dietze [16] estimated that the value of "gp/G would decrease by a factor of about 13 if the temperature increase from 0 K to the melting point Tin. By assuming that rp/G is inversely proportional to temperature, the value of rp/G for any material can be calculated. Such calculated values for some materials are listed in the left portion of Table 1. Creep data on PL creep are vast in the literature. The experimental values of n and AVL are listed in the right portion of Table 1 for the materials which rp/G data are available. In addition, the predicted value of ApL by Eq. (7) are also included for direct comparison. Inspection of the observed and the predicted ApL (Table 1 and Fig. 1) shows that the differences between these values are less than a factor for 20 for all the materials included. To illustrate further this agreement, ApL determined from Eq. (8) is plotted against "cp/G at n = 1, 3 - 7 in Fig. 2. Overplotted are the experimental data listed in Table 1 with n > 1 (PL creep) and those included in Ref. [1 l] with n = 1 ( H D creep). It is shown that in H D creep the dimensionless constant AHD de-

J.N. Wang, T.G. Nieh / Materials Science and Engineering A202 (1995) 52 56

54

Table 1 Comparison between the observed and the predicted ApE Material

v

AI Pb Sn ~-Fe e-Ti Si NaCI UO2 MgO A1203 MgAI204 (Mg, Fe)2SiO 4 fl-SiO 2 SiC

L/b

Structure

0.34 0.33 0.33 0.29 0.33 0.22 0.26 0.33 0.25 0.24 0.27 0.24 0.25 0.17

[17] [17] [18] [18] as ~-Zr [18] [19] [19] [19] [19] [19] [19] [19] [20]

Cubic (Lc.c) Cubic (f.c.c) B.c.t. Cubic (b.c.c) C.p.h. Cubic Cubic (f.c.c.) Tetragonal Cubic Hexagonal Cubic Orthorhombic Hexagonal Hexagonal

2 2 1.86 1.63 1.59 xf2 2 1.44 x/2 1.36 x~ 1.27 1.10 0.82

~Sp/G

"Cp/G

Observed

Observed

Predicted

at 0 K

at experimental T

n

ApL

ApL

4.00 4.00 8.71 7.20 6.90 1.66 6.10 1.73 7.34 2.05 1.44 2.93 1.18 4.93

4.4 5.0 6.6 6.9 4.3 5.0 3.6 4.0 3.3 3.0 2.7 3.0 3.0 5.7

3.4 × 106 [3] 2.5 X 108 [3] 5.5 x 1015 [21] 7.0 x 1013 [3] 7.7 x 104 [3] 2.5 × 106 [3] 6.6 x 102 [3] 3.9 x 103 [3] 20.0 [3] 3.38 [3] 0.16 [3] 0.45 [3] 0.011 [22]~ 4•03 × 104 [24]b

1.97 × 2.44 X 8.68 x 2.56 x 3.34 x 2.03 × 3.54 x 5.66 x 6.82 0.17 0.014 0.18 0.22 1.06 x

5.12 5.12 1.11 4.49 4.93 1.75 7.23 1.15 6.11 2.21 1.54 3.35 8.41 3.91

× × x × x x × x x x x x x ×

10 5 l0 5 10 4 10 4 10 4 10 3 10 5 10 3 10 4 10 3 10 3 10 -3 10 -3 10 -2

× × × × × × x x x x x x x ×

10 . 6 (0.98Tin) 10 -6 (0.98Tin) 10 -6 (0.98Tin) 10 4 (0.48Tin) 10 -5 (0.55Tin) l0 4 (0.81Trn ) 10 6 (0.91Tm) 10 -4 (0.51Tm) 10 -5 (0.63Tm) 10 -4 (0.83Tin) 10 . 4 (0.82Tm) 10 4 (0.88T~) 10 -3 (0.55Tm) 10 . 4 (0.61Tm)

~Taking DL=2.1 × 10 -11 exp(--138000/RT) m2s i [23]. bTaking D L = 8 . 3 6 × 103exp(-912OOO/RT)m2s pends on the Peierls stress proportionally

whereas in PL

ship between

the material

1 [25].

constant

AvL a n d t h e s t r e s s

c r e e p t h e d i m e n s i o n l e s s c o n s t a n t ApE is a l m o s t a c o n stant at n = 3 but has an inversely proportional depen-

e x p o n e n t n c h a r a c t e r i z i n g P L c r e e p is d e r i v e d . It is shown that this relationship agrees very well with ex-

d e n c e o n t h e P e i e r l s s t r e s s a t all o t h e r l a r g e r n v a l u e s . The prediction agrees well with experimental data for a n y g i v e n n. I t is n o t e w o r t h y t h a t t h i s a g r e e m e n t a p plies not only to the materials for which HD creep has been observed but also to the materials for which HD c r e e p h a s n o t b e e n o b s e r v e d (e.g. Si a n d M g A 1 2 0 4 ) .

perimental data. Therefore, such an agreement suggests that the Peierls stress may play a significant role in PL dislocation creep, and may be the missing physical q u a n t i t y i n t h e D o r n r a t e e q u a t i o n ( E q . (1)).

1023.

1o2]

.0

Power Law creep(n = 3 ~ 7)

! 1019 ~. =

4. Discussion

In the preceding sections, based on the transition stress from PL creep to HD creep and an experimentally confirmed rate equation for HD creep, a relation1018

,

~ ,

,

,

,

,

~ ,

,

,

I

t

I

,

l

,

I

,

I

/

1010 AI

108 8

102

,/@

Ct T i ( 4 " 4 ~

106

104

/\\

UO 2 (4-'3)J

, (4.0) ~ ~ r~aCl ~ a,1203 (3.6i J M g O

~-Fe (6.9)

Pb (5.0)

(3.0) , ~ ~O"~--

observed n value

.

0 •

~

Pb

107 105

II =

a-Ti

-4o

10 3

•

101 "~.~

~

M~O(3.3)

o.6)

n=

•

10"1 "3.0 10"3

"---.~5.7)',L,,, -

AJ20~(3)

-

~"

• olivine (3)

MgAI204 (2.7)

•

~-SiO2 (3)

~ Si ( 5 . 0 )

SiC (5.7) ........ (3.3)

Observed ApE ~ ..... "---- Predicted ApL

1 I t

10o

material

(6.9)~

•

10 I1 ~n = 5 109 n . =

A eL

1012

Sr

1013. "s . 5 ~

J

1014

.,~

1015. 6"0~ • n=

Sn(~66')QO

10 ]6

/

olivine (3.0) feldspar (3.2)

10"2

//" @~"'---- o o,~ .... MgAl204 (2.7) P'~tu2 ~a.u) 10 -4 . , , , . , . , , J . , , , . , , , . , . 1 0 - 4 10-2 I 0 0 102 104 106 I 0 8 I 0 I0 1012 1014 1016 1018

Predicted ApL Fig. I. Comparison between the observed and the predicted Ape.

10 -5 " 10"7-

" 10"9. 10-11 .

a-Fe •

!5-Co ~

Ct.SiO2

,,,,,"l~-Zr ~ , ~

•

Al~b~ n= ~ S n AI 10-13 [ ........ J 10-6 10-5

feldspar M olivine • gO • I ~ CaO

"~

a-Ti

~ Harper-Dorn c r e e p (n = 1)

........

i 10-4

........

, 1{)-3

........ 10-2

Fig. 2. Plot of the predicted (lines) and observed (solid dots) dependences of ApL on zp/G at various stress exponents• Data for n = I are from Ref. [5].

106 109 10 TM 10 TM 104 l06 102 102

105

55

J.N. Wang, T.G. Nieh / Materials Science and Engineering A202 (1995) 52-56

Comparison between Eq. (2) and Eq. (8) shows that o~= --loglo[zp/G ] and fl = 3 - {loglo[rp/G]} 1. Both and fl are noted to depend on the Peierls stress. Since rp << G, fl is relatively constant. For example, over the range of "rp/G of interest (about 4 × 10 6-5 × 10-3), changes from 5.3 to 2.3, and fl from 3.19 to 3.43. The dependence of ~ o n zp/G may explain why the ~ values determined by Brown and Ashby [5] are different for different classes of materials. If the reference stress ao in Eq. (3) is replaced by {p then ~.= A'eL

kT

(11)

with

kT

(13)

with \Zp/

(14)

The origin of the effect of the Peierls stress will now be examined even though underlying theories are still not understood. Assume that dislocation creep can be described by the well known Orowan equation: ~= p m b V

16,

where ~, is a dimensionless factor. The expression for V varies with the rate-controlling process. Consider that an edge dislocation, after being emitted from a source, glides a distance d and is stopped by an obstacle. Then, the dislocation passes a path h in a direction normal to the glide plane by climb, before meeting a dislocation of opposite sign and annihilating. Suppose this process is controlled by dislocation climb, the average dislocation velocity V can be expressed as [271 (17)

(12)

In this PL creep equation (Eq. (11)), the only unknown is the stress exponent n. It is different from Eq. (3) in that in Eq. (11) the physical meaning of the reference stress is assumed as the Peierls stress, and the Dorn constant varies with the square of ~p/G and thus with class of material. The value of Ave ranges from 10 -12 to 10 6 for materials from metals to ceramics and silicates. This observation is different from the demonstration of Brown and Ashby [5] that A~,t was a constant, having a value of approximately 10 6 for all materials. One of the most important results of this study is that for the case of n = 3, AvL is a genuine constant (approximately 0.1), but for all cases of n > 3 , AvL becomes dependent o n "¢p/G. Therefore, one may conclude that the third power stress dependence is an inherent and universal characteristic of PL creep in all crystalline materials. The additional stress dependence, giving a stress exponent n > 3, arises from the participation of the Peierls stress during deformation. Following the procedure of Horita and Langdon [26], Eq. (1) may be rewritten in the following form

A~L = 0.I

2

d V = Vc~

Apt = 0.1

i = A~,L

dislocation density p and the applied stress [1,2] by

(15)

where Pm is the density of mobile dislocations, and V is the dislocation velocity. Pm may be related to the total

with the climb velocity Vc Vc o c - ~

-

(18)

Combining Eqs. (15)-(18) gives rise to a rate equation of the form ~ : ~ hdDkGb ( G ) 3

(19)

Comparison of this equation with Eqs. (13) and (14) indicates that × -\ "/'p/t

(20)

Eq. (20) expresses the relationship between several micromechanistic variables, particularly rp, d, h, and ~b. At the present time, it is very difficult to separate further these variables and to find the individual functional interdependences among them.

5. Conclusions Using an extensive data base for metals, ceramics and silicates, it is shown that the Peierls stress of a crystal may play a significant role in power law creep as in H a r p e r - D o r n creep. Power law creep may have a "natural" law with the third power stress dependence. A higher stress dependence may be caused by some micromechanistic variable(s) being dependent on the ratio of applied stress to the Peierls stress.

Acknowledgements J.N.W. would like to acknowledge post-doctoral support from the Japan Society for the Promotion of Science and useful discussions with Professor F.R.N. Nabarro. This work was, in part, performed under the

56

J.N. Wang, T.G. Nieh / Materials Science and Eng#teering A202 (1995) 52-56

auspices of the US Department of Energy by LLNL under contract No. W-7405-Eng-48.

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