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Deformation at minimum elastic strain energy
Scripta
METALLURGICA
Vol.
DEFORMATION
20, p p . 1 5 9 1 - 1 5 9 6 , Printed in t h e U . S . A .
AT
MINIMUM
ELASTIC
1986
STRAIN
Pergamon Journals Ltd All rights reserved
ENERGI
David J. Q u e s n e l * Werkstoffwissenschaften I University of Erlangen-NGrnberg D-8520 Erlangen Federal Republic of G e r m a n y (Received (Revised
July August
28, 1 9 8 6 ) 25, 1 9 8 6 )
0beervatiens of t,he dislocation structures of numerous defonmed metals Sqow them to be dominantly beterogeneous, oonsistirg of cegions of ~ dislocation density interspersed with regions of relatively low dislocation density. It has bean suggested that such configurations, often e~alled cell structures, form to mir~ize the stored energy of the system (I~ Not all deformed metals show such behavior, however. Notably, during creep and low strain-rate deformation, certa/n solid solution alloys (class A type) whose properties are discussed in terms of the viscous motion of dislocations, show remarkably hemogensous dislocation structures p ~ s t i n g ove~ relatively large strains(2). 1he description of inhomogeneous dislocation structure as a two ~hase structure has long been present in the literature from the time of the first direct observations of dislocation structure~ Tne two ~ng_~es are, by implication, homogeneous, so teat regions identified ~s the same p~se h~ve the same characteristic properties, if heterogeneous deformation st~,/ctu~'es consist of mixtures of homogeneous c ~ t e r subdomains, t~ml some criteria must exist whicn g~)vern the_ behavior of each of the homogeneous suedom-~0_ns. It is lilly that similar criteria govern the homcgeneocs behavior in class A alloys where only one p~ase domirates the str~cture. Consequantly, in t/lis paper, the behavior of regions homogensous in stress told dislocation structure is examined from the point of view of minimizing the total elastic strain energy density during deformation at a specific shear strain rate, ~. We begin by considering t2m total ~lastic strain energy aensity, E, of a homogeneous domain with applied stress T and total dislocation density p. 'Ibe fundamental expression is z 2 I I - i~: E : -- ~ dV (I) v J 2G w~are OV is the volume element, V the total volume, T ~ the local stress and G the appropriate elastic modulus. Writing the local stress as the externally applied stress, ~, plus the s~m of the stress contributions, ~i ' of each of the N defects (here. understood to be dislocations), we have N E =-V
i=I Ti)" T ~ N 1 2G dV = - - + ~ ~. dV + - 2G 2GV J i~I 1 2GV
N ~ ~. T. dV i=i j=! I 3
(2).
stresses, Ti, p~oduoed by the dislocations are in~ernal to the body and hqus must average to ~ Hence the second te.rm vanishes after integration, in 5he third term, those elements with i=j give rise to the line energy of the dislocations while tnoso with i ~ represent tce dislocation-dislocation interactions. Postulating the structure to be homogeneous means that the volume averaged interaction energy is only a Oanction of the dislocation density and arrangement and as such can be included as a dislocation density dependent line tension. If we adop~ the stress screening approach (I), S~slnsai (3) ~as s~ggested that the. logarithmic aependence of ~ne line energy on interdisl~tion spacing can be well represented by a power law approximation over the range of dislocation densities of interest, n~mely I (3) ~,Ln - - ~ k,oO" 86
b4 10n acaaemic leave f ~ m 14627, U.S.A.
Dep~rt:aent of ~4ecklanical Engineering, dniversity of ~ochester, Noehester, New Yor~
1591 0036-9748/86 $3.00 Copyright
(c)
1986
Pergamon
+
.00
Journals
Ltd.
1592
DEPOSITION
AND
STRAIN
ENERGY
Vol.
20,
No.
where b is the magnitude of the Burgers vector and k a constant r e q ~ for dimensional consistency, lhls simple a l ~ form will allow us to eetimate the influence of the screening effect in the oalculatices whiah follow but as will be seen, the screening efTeet is not the origin of the results we ~ obtain. As such, we proceed with a ecnstant value of [. the effieetive ener8~ of dislocations per unit len~ch and for a specimen umder an applied shear stress i we write 2 T P = __ + ~p (~) 2G as the total elastic strain energy density. Here G is the sYmar modulus and p the total dislocation density. Since T and p m-e independent, eq. 4 has only the trivial minimum T : p = (~ Sq. 4 defines a surface which rept~ants the elastic strain energy density that wou/d exist for all possible combinations of ~ and p. Nature, however, does not allow all ~nc~_~ible values; they must be consistent with 0rowan's strain rate eq~tio~ For a specific stress, the dislocations will have a specific average velocity. In class A alloys, the velocity behavic~ of dislocations is found to be viscous b~ca,_~e of the d ~ of solute atoms by the stress field of the moving dislocations. Taking B as the drag coefficient (or I/B as the mobility) we have b v -- - ~
(5)
B as the form in which the dislocation velocity-stress law is usually cast (4% Other plausible dislocation velocity laws are e~amlned later in the present paper, in the above, b is the ~urgers vector and v the average dislocation velocity when all dislocations are considered to he mobile+ Using eq. ~ the Orowan rate equation becomes
b2
"~
= O --
(6)
T
B
where ~ is the plastic shear strain rate. Since stress is generally thought to be the response to applied strain rate ~ at a specific dislocation density p, eq. 6 shows us that an increase in dislocation density decreases the required flow stress t • This unusual "oac~war~' t~havior is only observable experimentally, however, when p can be ocntrolled as an independent variable ~ as in yield strength measurements of single crystals of very low dislocation density (5) and in studies of yield point phenomena (6). Eq. 6 is an equation of oonstralnt embodying the deformation mechanism, the specific solution to eq. 6 that we desire is that which minimizes the energy of the system~ The location of the minimum can be seen schematically in Fig. I where a typical shape of eq. 6 has been drawn on the T - 0 plane and then projected onto the energy surface, as representad by eq. ~. The solid llne t~mm obtained represents the elastin energy of the system when the doformation mechanism reprasented by eq. 6 is followed. It contains a well aefined minimum Similar curves exist for other strain rate~ To locate this minimum analytically, we proceed fonm~lly by writing eq. 4 with eq. 6 used to eliminate "C (or p ) in favor of p (or T % This is equivalent to projecting the energy curve in Fig, I onto the E vs p (or E vs ) plane. ~ e resulting eY40ressions for E in terms of T and p are: 2 T #a E =-+ ~ 2G V
T
I yB 2 E =---( ) p-2 + ~.p 2G V
-I
(7a, "to).
To minimize R, we set the derivatives with respect to T and p equal to zero and solve for T and p :
Tm
=
#B 1/3 (C.~ --~ ) b+
Pm
=
~B 2/3 (Gr')-l/3 ( "-~ ) b-
(Sa, ~).
DMfe the subscript m has been added to Tand p to a i ~ that they minimize the elastic strain energy density.pm also represents the mobile dislocation density in the present case since all dislocatiorm are considered to be mobile. -___qeeond_ derivatives are everywhere positive indicating a unique global minimum~ ~s. 8a and 8b z~m~msent a set of values both dependent on ~ and as soch imply a relation between t m and Pln" This relation, which mir~m~Tes the strain energy for all ~ can be found by eliminating ~( from eq~ ~ and 8b as
II
Vol.
20,
No.
11
DEFOPJ{ATION AND STRAIN ENERGY
1593
we note t ~ t the stress w~ich minimizes the total elastic strain energy is an increasirg ~Iction of P m is identical in form to the f ~ e n t a l flow stress equatio~
= c~e>~
(~0),
w~ it is recalled t~t ~ is proportional to G b 2 and ~ is a constant of order one. The experimental validity o f e ~ 10 is of course alrcedy well establis~d. Fq. 8a cs~ be ccmbimd with eq. 5 to give ~ specific average dislocation velocity which minimizes the total elsstic strain energy density, ~/3
B2
(11)
wb_~ch t ~ t ~ with ec~ ~ represents a solutic~ to the Orowan equatio~ ~ m s e r~sults apply to homogeneous subd~ where both stress and dislocation density are sufficiently uniform to be cheracterized by single para,~ters. F4. ha is ~ t new to the literature on cree~ Becegnizing thet the line energy of a dislocation m s the ~enerel form Gb , eg. 8a can be written as _3 -
(12)
which is identical m the equation developed by 1 ~ u c h i and Argon (7) and has the s ~ e form as the earlier eq~ticn of Weertmas (8~ ~ validity of these e~tio~L~ in re~presenting c i ~ A steady state creep behavior has been well e s t a b l ~ ~he above results do not include dislocation interaction effect~ To include the effect of stress screening cn line energy, we use t ~ approximation described in eq. 3, arriving at • proportional to p 0.43 which Basinski (3) hes noted is acre clceely in agreement with careful experimental observations. With this apprcxi~tion, o t ~ are s ~ slightiy as well but t ~ f ~ e n t a l relationships just developed still domir~te the behevin~. ~ specific dislocation interactions w~c'n .~ffect the velocity of dislocations are mc6t easily irked indirectly in the form of the a v e ~ dislocation velocity-stress law to be discussed s~eque~tly. Also, the ~ t i a n of a specific density of d~s~ocations during deformation lowers ~be total elastic ~ r ~ hence dislocations are t ~ y r ~ y stable as long as the deformation cont£nuss. When the deformation stops, dtq!ocaticr~ are no longer stable defect8 "and the interaction energy t e r ~ are the w i ~ driving force for static st~Jcttral r ~ e m e ~ The metastable structures thus for~ed will influence subsequent dynamic dislocation motion end as such the structures we observe are likely to be_ the result of a combination of Dotn static effect~
in ~ ~ e ~ e v e l ~ t , we here ~ taoit~ ~ ~ ~ t i ~ wer~ ~ _ ~ . ~ ~ ~ dlelcc~tlc~s are in the form of non-inte~actirg loops of radius r with N loops in each 1 of volume as suggested by Fig. 2, then the dislocation denaity is just 2~Nr P =
13
(13)
The back stress ~rcduced by the curvature of the segments, equivalent in its effect to a ~ f o r m cs~ by line tension, is
internal stress
2Gb TLT. .-
(14) r
where the subscript LT represents line ter~c~ Since the internal stress (or back stress) postulated in this way is a uniform effect t/-zDu~hout the volume, no aver-~hn E procedures are needed to determine an effective internal stees~ S~tlal varlaticns in internal stress , on the other hand, would r~ui~e an averaging procedure similar to ttat discussed by Li (9) in order to determine tie effective internal str~s ~ are not considered ~-xplicitly in the present paper. Rather their influence is considered indirectly as a contribution to the form of the more general ran-linear a v e ~ dislocation velnoity-etress law discussed later. intrcducinE ~ as the ev~-ywhere oansr.-~fcstress drivlnE dislocation motion we have, in analc~ to eq. 5, b v
=-
:.: ~
B
(15)
1594
DEFORHATION AND STRAIN ENERGY
Vol.
20,
No.
ThuS the Orowan rate equation becem~s 2~Nr
4~NG
b ~,: • b ---~
Y--
13
b3
. ~:~
= .
B
~
13
(lea, lob)
•
B
"t-LT
which has the solution
Cloc)
4~NGb3--------~ TLT
which is formally similar to the Cottrell-Stokes law. In other woras, it suggests t~at the thermal (w:'~ and athermal ( ~ stress components are proportional to one another. Writing the total strain energy density as in eq. 4, we use eqs. 13 and 14 to eliminate ~ and subsequently r so that 2 2~N 2C~b E =-- + ~ ( ) (--) (17) 2G 7 ~LT Recognizing T : • ~ + ~L~' from eq. 16c we arrive at 13B-~ T=
(
4~N~ 3
(18) + I )~LT
Thus, 13By E=
+ I )2
(
4~N(~3
T2LT 4~NGb ~ + ~ (__~) 2G
.1 ~
(19)
~LT
differentiating, we have dE = •dTLT
(
13B~ " 3 +1) 4~N(~D
2
~LT --G
4~NC~ ~ C~')( i'~
I I " )( ) TLT TLT
(2O)
which can be rewritten using ~qs. 13 and 14 as I~B@
~LT
: ~
JY/(1
(2~)
+
4~NGb ~ Using eq. 18, el. 21 becomes T =
,/~ ,~
(22)
which is the same result we obtained earlier but now wit~ athermal line tension effects included in U e derivation. Thus an expression like eq. 12 holds for the ~ of glide resistance stemming from llne t~nsion effects .as well. Cottrell-Stokes-like expression, eq. 16c, s.hows which fraction of T is effective stress and whion fraction is internal ather~l back stress cauSed by line tensiorL From -~q. Itc, it is clear t~t T'~ in.-reases with increasing strain rate and incrsasing ,iTag coefficiant B as we would e~pect. in principle, t~e above treatment Sqould apply to dislocation strect,~es deforming by loop bow-out as well. At constant stress, all pinned segments should .have the s ~ e radius of curvature, ignoring differences in e ~ and screw mobility, and thus an array of strongly phzn~d segments anould ~ v e dynamic b~%-mvior similar to an ~rray of loops of appropriate radius of curvature ~md vol,,me density. '~hus the Co~trell-Sto~es-li~e benavi~ and the dependence is justifiable in these structures from minimum en~'KY ~,~umants. in situations where b ~ w a y from pinning sites is rate controlling the anove tre'~tmant can not De ~pplied directly. We now turn our attention to the problem of the m c ~ general ~ vs T law which m'~isht he observed in metals which eventually form cells. Certainly at T =0, we expect the average vclocity of disloc~tio[m to De zerc~ If all the dislocations are mobile, the effective average velocity wc seek should increase with s t r ~ approac/qing an upper limit of the sonic velocity as the suress beex~mes very large. As t~m s~_as decreases from t ~ t needed to produce sonic velocities, we expect a lin~r dependance as m~sured in pulse experiments (10) because the driving forces for such motions are e~treaely large in comparison with obstacle st~"engt~ As t~e stress is further redoced, the velocity will decreasc more rapidly th~n linearly ~.ause of the interaction of the dislocations with fixed obstacles and with c~e another. To fulfill the zero velocity-mero stress condition and yet remain mathematically continuous mud smocth, a property expect;~d of such :m :~verage, tnc velocity-st ~ress dependence must revise
11
Vol.
20,
No.
II
DEFOR}tATION
AND
STRAIN
ENERGY
1595
its curvatuz~_ ~hus, at least qualitatively, we expect 9 vs T to look similar to the ~ketch shown in Fi~ 3~ Compared to a linear velocity dependence, dislocation interactions cause a reduction in the average velocities at intermediate stresse~ ~ i s meems that to maintain a specific strain rate, the dislocation density must rise compared to that required for a linear-velocity stress law. As a result, P vs • develops a "hUmp" (sse Fi& 3b) ~qich, when observed cn the energy surface, causes a bifurcation into high ~ low stress and high and low dislocation density regions in order to minimize the total ener~. The forms of these energy curves are quite similar to those which are responsible for spinodal behavior in chemical demixing. < Once such a separation begins, h~wever, since initially the stresses will be the same in each phase, the relative strain rates will differ until sufficient iorg range internal stresses develop bstween the Bhases to equilibrate the strain rate. Urgar et. sl. (11) have recently shown by X-ray measurements that si~flcsnt long range stresses exist in the cell structures of deformed copper, consistent with a recent t~eory by M u ~ i that attributes their origin to such inhomogenenus deformation (12% The large differences in dlslocatien density also suggest to the present aurar the possibility of a bifurcation in the mechanism controlling glide resistance in the two phase~ In such a case, two separate energy curves would exist, one for each m e n ~ , each with one or more minim~. Further work is needed to address this issue in detail. Nevertheless, the curve shown in Fig, 4 suggests that an appropriate application of the double t a n ~ t rule would define the ec~,lllbrium condition that would ~miTe the total strain ane~y density. A b i r d i e d stuct~e is t ~ expected based on dislocation dynamics without reeourse to screening effects. Since screenirg also suggests bifurcation, the two effects should reinforce ~x]e anot~. Before summarizing, we point out that the dislocation density behavior shown in Fig, 3b is .not ~ysically observable. Rather it is to be considered as a visual variation in the ~ , m sense t ~ t ~ysically unre~l~hle displacements are used in virtual work cal~l~tior.~ ~bm particular displacement (or dislocation density in the present case) observed is that which minimizes the total energy of the syste~ While the detailed dislocationdislocation (or dislocation-defect) interactions have not been directly included in the present treatment, the form of the aver~g~ velocity vs stress curve is a direct result of these interactions. In this sense, it is indeed these interactions which produce the driving force for the development of either homogeneous or two ~hase dislocation structures during deformaticr~
This treatment shows that if the average velocity vs stress dependence is at all si~%lar to that shown in Fi~ 3a, at constant strain rate, one expects a bifurcation into regions of high and low dislocation density based soley on energy consideration~ ~rther, it is shown that dislocatior.s lower the ener~ of deforming systems and thus are ener~etically favorable defects durirg deformatio~ ~he quantitative treatment of viscous (linear velocity-stress law) dislocation motion of non-lnberacting dislocations, simi]~ to behavior expected and observed in c l ~ A alloy deformation, shows a single minimum in strain energy su~esting a driving force for homogeneous bahavior exlst~ [hder such conditions, the 4 ~ flow stress relation is found for both straight and curved dislocations , s ~ being t~e limit of large radii of curvature. ~ e internal athermal b~ck stress associated with llne tension is found to be p~oportional to the effective stress at oonstant strain rate, just_eying the Cottrell-~tokes law. Lastly, the TaM~uehl and Argon equation for behavior of ~ ! ~ A alloys is a direct result of the energy based approach to dislocation dynamics. Additional work is needed to explore firther the possibility of applying such ener~ b~__ approaches to inhomogeneous ~i~]ocation structug~s such as those occurring in fatigue and class M creep be~vior. Ackncwle~ The ;Llexand~ yon Humboldt Stifbu~ provided finar~isl a s s i s ~ e for this work t h r o ~ a fellowship to the ~uthor. Comments and discussion with Prof. K M u ~ i , Prof. W. Blum, Prof. P~ N e u h ~ , Dr. LL Essmann, Dr. ~bx>m~son ~ Dr. in~ ~ Christ are greatfully acknowledged. R e f ~ I. ~ Kuhlmmqn-Wilsdorf and & ~ van der Merwe, Mat. ScL Fag,, voL 55 (1982), p~ 79 2. ~% & Mills, J. C. Gibe.ling, and W. ~ Nix, Acta Met., vol. 33 (1985), pp. 1503-151~ 3. ~ & Basinski and '7- S Basinski, in Dislocatiors in Solids, ed. by F. ~. ~ Nabarro, ~ K~I l~nd Co. (1979), pp. 263-362. 4. ~t & Mills, P n ~ ~hesis, Stanford University, Dec., 1 9 ~ 5. & ~ Fatal and ~. ~. C ~ i , i Appl. ~hysics, voL 34, (1963), p~ 278~ 6. P. Faasen, Ph~ikal A e t a l ~ , Springer-Verlag, BerlinYPaidelberg, 1974, pp. 271. 7. S. Takeuchi and A. S. Argon, Phil. Mag~ A, vol. 40, (1975), pp. 65~ 6. J. Weertm~_n, Appl. ~y~, vol. 25, (1957), pp. 1 1 ~ 9. J.C.M. hi, in Dislocation Dynamics, ed. by A. F~ Rosenfield, G. T. Hahn, A. L. Bement, and R. i. Jaffee, ~,~
1596
DEFO~TION
AND S T P A I N E N E R G Y
Vol.
20, No.
FIG. 1 Schematic i l l u s t r a t i o n of a p o r t i o n of the e n e r g y s u r f a c e . A t y p i c a l s h a p e of the e q u a t i o n of c o n s t r a i n t is d r a w n on the T - p plane and p r o j e c t e d onto the energy surface. UJ
%
....
FIG. 2 Cube of m a t e r i a l s h o w - I ing N d i s l o c a t i o n loops in a v o l u m e 1
.~_~_ __ve_L~_,_T.Y_. . . . . . . . . . . . . . .
:
4~J'~1 ~ "@ "
1
UePER LMT
HIGHSTRESS LN IEAR REGION
." ' TRANSITION TOO{~TACLE .-~ CONTROLLE VELOCI D TY WITH [T~REAS~NG STRESS
,.L"-~'9"0_/ POWERLAWRECKON
FIG. 3a S c h e m a t i c of the a v e r a g e d i s l o c a tion velocity as a f u n c t i o n of a p p l i e d s t r e s s w h e n all d i s l o c a t i o n s are c o n s i d e r e d m o b i l e .
I
~'\' , ~~
~ \
%,/'
W
byi~*roctiomzwith
fix~ ~lacle.s;eendot~rd~locotioreJ
,
, ......
FIG. 3b S c h e m a t i c of the d i s l o c a t i o n d e n s i t y r e q u i r e d to k e e p a c o n s t a n t shear strain rate w h e n the d i s l o c a tion velocity-stress l a w s h o w n in Fig. 3a is o b e y a d .
FIG. 4 Projection of t h e v a r i a t i o n of dislocation density as a f u n c t i o n of s t r e s s n e e d e d to p r o d u c e c o n s t a n t s t r a i n r a t e o n t o the e n e r g y surface. The d o u b l e m i n i m u m s u g g e s t s use of the d o u b l e t a n g e n t rule to m i n i m i z e the total energy.
ii
METALLURGICA
Vol.
DEFORMATION
20, p p . 1 5 9 1 - 1 5 9 6 , Printed in t h e U . S . A .
AT
MINIMUM
ELASTIC
1986
STRAIN
Pergamon Journals Ltd All rights reserved
ENERGI
David J. Q u e s n e l * Werkstoffwissenschaften I University of Erlangen-NGrnberg D-8520 Erlangen Federal Republic of G e r m a n y (Received (Revised
July August
28, 1 9 8 6 ) 25, 1 9 8 6 )
0beervatiens of t,he dislocation structures of numerous defonmed metals Sqow them to be dominantly beterogeneous, oonsistirg of cegions of ~ dislocation density interspersed with regions of relatively low dislocation density. It has bean suggested that such configurations, often e~alled cell structures, form to mir~ize the stored energy of the system (I~ Not all deformed metals show such behavior, however. Notably, during creep and low strain-rate deformation, certa/n solid solution alloys (class A type) whose properties are discussed in terms of the viscous motion of dislocations, show remarkably hemogensous dislocation structures p ~ s t i n g ove~ relatively large strains(2). 1he description of inhomogeneous dislocation structure as a two ~hase structure has long been present in the literature from the time of the first direct observations of dislocation structure~ Tne two ~ng_~es are, by implication, homogeneous, so teat regions identified ~s the same p~se h~ve the same characteristic properties, if heterogeneous deformation st~,/ctu~'es consist of mixtures of homogeneous c ~ t e r subdomains, t~ml some criteria must exist whicn g~)vern the_ behavior of each of the homogeneous suedom-~0_ns. It is lilly that similar criteria govern the homcgeneocs behavior in class A alloys where only one p~ase domirates the str~cture. Consequantly, in t/lis paper, the behavior of regions homogensous in stress told dislocation structure is examined from the point of view of minimizing the total elastic strain energy density during deformation at a specific shear strain rate, ~. We begin by considering t2m total ~lastic strain energy aensity, E, of a homogeneous domain with applied stress T and total dislocation density p. 'Ibe fundamental expression is z 2 I I - i~: E : -- ~ dV (I) v J 2G w~are OV is the volume element, V the total volume, T ~ the local stress and G the appropriate elastic modulus. Writing the local stress as the externally applied stress, ~, plus the s~m of the stress contributions, ~i ' of each of the N defects (here. understood to be dislocations), we have N E =-V
i=I Ti)" T ~ N 1 2G dV = - - + ~ ~. dV + - 2G 2GV J i~I 1 2GV
N ~ ~. T. dV i=i j=! I 3
(2).
stresses, Ti, p~oduoed by the dislocations are in~ernal to the body and hqus must average to ~ Hence the second te.rm vanishes after integration, in 5he third term, those elements with i=j give rise to the line energy of the dislocations while tnoso with i ~ represent tce dislocation-dislocation interactions. Postulating the structure to be homogeneous means that the volume averaged interaction energy is only a Oanction of the dislocation density and arrangement and as such can be included as a dislocation density dependent line tension. If we adop~ the stress screening approach (I), S~slnsai (3) ~as s~ggested that the. logarithmic aependence of ~ne line energy on interdisl~tion spacing can be well represented by a power law approximation over the range of dislocation densities of interest, n~mely I (3) ~,Ln - - ~ k,oO" 86
b4 10n acaaemic leave f ~ m 14627, U.S.A.
Dep~rt:aent of ~4ecklanical Engineering, dniversity of ~ochester, Noehester, New Yor~
1591 0036-9748/86 $3.00 Copyright
(c)
1986
Pergamon
+
.00
Journals
Ltd.
1592
DEPOSITION
AND
STRAIN
ENERGY
Vol.
20,
No.
where b is the magnitude of the Burgers vector and k a constant r e q ~ for dimensional consistency, lhls simple a l ~ form will allow us to eetimate the influence of the screening effect in the oalculatices whiah follow but as will be seen, the screening efTeet is not the origin of the results we ~ obtain. As such, we proceed with a ecnstant value of [. the effieetive ener8~ of dislocations per unit len~ch and for a specimen umder an applied shear stress i we write 2 T P = __ + ~p (~) 2G as the total elastic strain energy density. Here G is the sYmar modulus and p the total dislocation density. Since T and p m-e independent, eq. 4 has only the trivial minimum T : p = (~ Sq. 4 defines a surface which rept~ants the elastic strain energy density that wou/d exist for all possible combinations of ~ and p. Nature, however, does not allow all ~nc~_~ible values; they must be consistent with 0rowan's strain rate eq~tio~ For a specific stress, the dislocations will have a specific average velocity. In class A alloys, the velocity behavic~ of dislocations is found to be viscous b~ca,_~e of the d ~ of solute atoms by the stress field of the moving dislocations. Taking B as the drag coefficient (or I/B as the mobility) we have b v -- - ~
(5)
B as the form in which the dislocation velocity-stress law is usually cast (4% Other plausible dislocation velocity laws are e~amlned later in the present paper, in the above, b is the ~urgers vector and v the average dislocation velocity when all dislocations are considered to he mobile+ Using eq. ~ the Orowan rate equation becomes
b2
"~
= O --
(6)
T
B
where ~ is the plastic shear strain rate. Since stress is generally thought to be the response to applied strain rate ~ at a specific dislocation density p, eq. 6 shows us that an increase in dislocation density decreases the required flow stress t • This unusual "oac~war~' t~havior is only observable experimentally, however, when p can be ocntrolled as an independent variable ~ as in yield strength measurements of single crystals of very low dislocation density (5) and in studies of yield point phenomena (6). Eq. 6 is an equation of oonstralnt embodying the deformation mechanism, the specific solution to eq. 6 that we desire is that which minimizes the energy of the system~ The location of the minimum can be seen schematically in Fig. I where a typical shape of eq. 6 has been drawn on the T - 0 plane and then projected onto the energy surface, as representad by eq. ~. The solid llne t~mm obtained represents the elastin energy of the system when the doformation mechanism reprasented by eq. 6 is followed. It contains a well aefined minimum Similar curves exist for other strain rate~ To locate this minimum analytically, we proceed fonm~lly by writing eq. 4 with eq. 6 used to eliminate "C (or p ) in favor of p (or T % This is equivalent to projecting the energy curve in Fig, I onto the E vs p (or E vs ) plane. ~ e resulting eY40ressions for E in terms of T and p are: 2 T #a E =-+ ~ 2G V
T
I yB 2 E =---( ) p-2 + ~.p 2G V
-I
(7a, "to).
To minimize R, we set the derivatives with respect to T and p equal to zero and solve for T and p :
Tm
=
#B 1/3 (C.~ --~ ) b+
Pm
=
~B 2/3 (Gr')-l/3 ( "-~ ) b-
(Sa, ~).
DMfe the subscript m has been added to Tand p to a i ~ that they minimize the elastic strain energy density.pm also represents the mobile dislocation density in the present case since all dislocatiorm are considered to be mobile. -___qeeond_ derivatives are everywhere positive indicating a unique global minimum~ ~s. 8a and 8b z~m~msent a set of values both dependent on ~ and as soch imply a relation between t m and Pln" This relation, which mir~m~Tes the strain energy for all ~ can be found by eliminating ~( from eq~ ~ and 8b as
II
Vol.
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DEFOPJ{ATION AND STRAIN ENERGY
1593
we note t ~ t the stress w~ich minimizes the total elastic strain energy is an increasirg ~Iction of P m is identical in form to the f ~ e n t a l flow stress equatio~
= c~e>~
(~0),
w~ it is recalled t~t ~ is proportional to G b 2 and ~ is a constant of order one. The experimental validity o f e ~ 10 is of course alrcedy well establis~d. Fq. 8a cs~ be ccmbimd with eq. 5 to give ~ specific average dislocation velocity which minimizes the total elsstic strain energy density, ~/3
B2
(11)
wb_~ch t ~ t ~ with ec~ ~ represents a solutic~ to the Orowan equatio~ ~ m s e r~sults apply to homogeneous subd~ where both stress and dislocation density are sufficiently uniform to be cheracterized by single para,~ters. F4. ha is ~ t new to the literature on cree~ Becegnizing thet the line energy of a dislocation m s the ~enerel form Gb , eg. 8a can be written as _3 -
(12)
which is identical m the equation developed by 1 ~ u c h i and Argon (7) and has the s ~ e form as the earlier eq~ticn of Weertmas (8~ ~ validity of these e~tio~L~ in re~presenting c i ~ A steady state creep behavior has been well e s t a b l ~ ~he above results do not include dislocation interaction effect~ To include the effect of stress screening cn line energy, we use t ~ approximation described in eq. 3, arriving at • proportional to p 0.43 which Basinski (3) hes noted is acre clceely in agreement with careful experimental observations. With this apprcxi~tion, o t ~ are s ~ slightiy as well but t ~ f ~ e n t a l relationships just developed still domir~te the behevin~. ~ specific dislocation interactions w~c'n .~ffect the velocity of dislocations are mc6t easily irked indirectly in the form of the a v e ~ dislocation velocity-stress law to be discussed s~eque~tly. Also, the ~ t i a n of a specific density of d~s~ocations during deformation lowers ~be total elastic ~ r ~ hence dislocations are t ~ y r ~ y stable as long as the deformation cont£nuss. When the deformation stops, dtq!ocaticr~ are no longer stable defect8 "and the interaction energy t e r ~ are the w i ~ driving force for static st~Jcttral r ~ e m e ~ The metastable structures thus for~ed will influence subsequent dynamic dislocation motion end as such the structures we observe are likely to be_ the result of a combination of Dotn static effect~
in ~ ~ e ~ e v e l ~ t , we here ~ taoit~ ~ ~ ~ t i ~ wer~ ~ _ ~ . ~ ~ ~ dlelcc~tlc~s are in the form of non-inte~actirg loops of radius r with N loops in each 1 of volume as suggested by Fig. 2, then the dislocation denaity is just 2~Nr P =
13
(13)
The back stress ~rcduced by the curvature of the segments, equivalent in its effect to a ~ f o r m cs~ by line tension, is
internal stress
2Gb TLT. .-
(14) r
where the subscript LT represents line ter~c~ Since the internal stress (or back stress) postulated in this way is a uniform effect t/-zDu~hout the volume, no aver-~hn E procedures are needed to determine an effective internal stees~ S~tlal varlaticns in internal stress , on the other hand, would r~ui~e an averaging procedure similar to ttat discussed by Li (9) in order to determine tie effective internal str~s ~ are not considered ~-xplicitly in the present paper. Rather their influence is considered indirectly as a contribution to the form of the more general ran-linear a v e ~ dislocation velnoity-etress law discussed later. intrcducinE ~ as the ev~-ywhere oansr.-~fcstress drivlnE dislocation motion we have, in analc~ to eq. 5, b v
=-
:.: ~
B
(15)
1594
DEFORHATION AND STRAIN ENERGY
Vol.
20,
No.
ThuS the Orowan rate equation becem~s 2~Nr
4~NG
b ~,: • b ---~
Y--
13
b3
. ~:~
= .
B
~
13
(lea, lob)
•
B
"t-LT
which has the solution
Cloc)
4~NGb3--------~ TLT
which is formally similar to the Cottrell-Stokes law. In other woras, it suggests t~at the thermal (w:'~ and athermal ( ~ stress components are proportional to one another. Writing the total strain energy density as in eq. 4, we use eqs. 13 and 14 to eliminate ~ and subsequently r so that 2 2~N 2C~b E =-- + ~ ( ) (--) (17) 2G 7 ~LT Recognizing T : • ~ + ~L~' from eq. 16c we arrive at 13B-~ T=
(
4~N~ 3
(18) + I )~LT
Thus, 13By E=
+ I )2
(
4~N(~3
T2LT 4~NGb ~ + ~ (__~) 2G
.1 ~
(19)
~LT
differentiating, we have dE = •dTLT
(
13B~ " 3 +1) 4~N(~D
2
~LT --G
4~NC~ ~ C~')( i'~
I I " )( ) TLT TLT
(2O)
which can be rewritten using ~qs. 13 and 14 as I~B@
~LT
: ~
JY/(1
(2~)
+
4~NGb ~ Using eq. 18, el. 21 becomes T =
,/~ ,~
(22)
which is the same result we obtained earlier but now wit~ athermal line tension effects included in U e derivation. Thus an expression like eq. 12 holds for the ~ of glide resistance stemming from llne t~nsion effects .as well. Cottrell-Stokes-like expression, eq. 16c, s.hows which fraction of T is effective stress and whion fraction is internal ather~l back stress cauSed by line tensiorL From -~q. Itc, it is clear t~t T'~ in.-reases with increasing strain rate and incrsasing ,iTag coefficiant B as we would e~pect. in principle, t~e above treatment Sqould apply to dislocation strect,~es deforming by loop bow-out as well. At constant stress, all pinned segments should .have the s ~ e radius of curvature, ignoring differences in e ~ and screw mobility, and thus an array of strongly phzn~d segments anould ~ v e dynamic b~%-mvior similar to an ~rray of loops of appropriate radius of curvature ~md vol,,me density. '~hus the Co~trell-Sto~es-li~e benavi~ and the dependence is justifiable in these structures from minimum en~'KY ~,~umants. in situations where b ~ w a y from pinning sites is rate controlling the anove tre'~tmant can not De ~pplied directly. We now turn our attention to the problem of the m c ~ general ~ vs T law which m'~isht he observed in metals which eventually form cells. Certainly at T =0, we expect the average vclocity of disloc~tio[m to De zerc~ If all the dislocations are mobile, the effective average velocity wc seek should increase with s t r ~ approac/qing an upper limit of the sonic velocity as the suress beex~mes very large. As t~m s~_as decreases from t ~ t needed to produce sonic velocities, we expect a lin~r dependance as m~sured in pulse experiments (10) because the driving forces for such motions are e~treaely large in comparison with obstacle st~"engt~ As t~e stress is further redoced, the velocity will decreasc more rapidly th~n linearly ~.ause of the interaction of the dislocations with fixed obstacles and with c~e another. To fulfill the zero velocity-mero stress condition and yet remain mathematically continuous mud smocth, a property expect;~d of such :m :~verage, tnc velocity-st ~ress dependence must revise
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20,
No.
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DEFOR}tATION
AND
STRAIN
ENERGY
1595
its curvatuz~_ ~hus, at least qualitatively, we expect 9 vs T to look similar to the ~ketch shown in Fi~ 3~ Compared to a linear velocity dependence, dislocation interactions cause a reduction in the average velocities at intermediate stresse~ ~ i s meems that to maintain a specific strain rate, the dislocation density must rise compared to that required for a linear-velocity stress law. As a result, P vs • develops a "hUmp" (sse Fi& 3b) ~qich, when observed cn the energy surface, causes a bifurcation into high ~ low stress and high and low dislocation density regions in order to minimize the total ener~. The forms of these energy curves are quite similar to those which are responsible for spinodal behavior in chemical demixing. < Once such a separation begins, h~wever, since initially the stresses will be the same in each phase, the relative strain rates will differ until sufficient iorg range internal stresses develop bstween the Bhases to equilibrate the strain rate. Urgar et. sl. (11) have recently shown by X-ray measurements that si~flcsnt long range stresses exist in the cell structures of deformed copper, consistent with a recent t~eory by M u ~ i that attributes their origin to such inhomogenenus deformation (12% The large differences in dlslocatien density also suggest to the present aurar the possibility of a bifurcation in the mechanism controlling glide resistance in the two phase~ In such a case, two separate energy curves would exist, one for each m e n ~ , each with one or more minim~. Further work is needed to address this issue in detail. Nevertheless, the curve shown in Fig, 4 suggests that an appropriate application of the double t a n ~ t rule would define the ec~,lllbrium condition that would ~miTe the total strain ane~y density. A b i r d i e d stuct~e is t ~ expected based on dislocation dynamics without reeourse to screening effects. Since screenirg also suggests bifurcation, the two effects should reinforce ~x]e anot~. Before summarizing, we point out that the dislocation density behavior shown in Fig, 3b is .not ~ysically observable. Rather it is to be considered as a visual variation in the ~ , m sense t ~ t ~ysically unre~l~hle displacements are used in virtual work cal~l~tior.~ ~bm particular displacement (or dislocation density in the present case) observed is that which minimizes the total energy of the syste~ While the detailed dislocationdislocation (or dislocation-defect) interactions have not been directly included in the present treatment, the form of the aver~g~ velocity vs stress curve is a direct result of these interactions. In this sense, it is indeed these interactions which produce the driving force for the development of either homogeneous or two ~hase dislocation structures during deformaticr~
This treatment shows that if the average velocity vs stress dependence is at all si~%lar to that shown in Fi~ 3a, at constant strain rate, one expects a bifurcation into regions of high and low dislocation density based soley on energy consideration~ ~rther, it is shown that dislocatior.s lower the ener~ of deforming systems and thus are ener~etically favorable defects durirg deformatio~ ~he quantitative treatment of viscous (linear velocity-stress law) dislocation motion of non-lnberacting dislocations, simi]~ to behavior expected and observed in c l ~ A alloy deformation, shows a single minimum in strain energy su~esting a driving force for homogeneous bahavior exlst~ [hder such conditions, the 4 ~ flow stress relation is found for both straight and curved dislocations , s ~ being t~e limit of large radii of curvature. ~ e internal athermal b~ck stress associated with llne tension is found to be p~oportional to the effective stress at oonstant strain rate, just_eying the Cottrell-~tokes law. Lastly, the TaM~uehl and Argon equation for behavior of ~ ! ~ A alloys is a direct result of the energy based approach to dislocation dynamics. Additional work is needed to explore firther the possibility of applying such ener~ b~__ approaches to inhomogeneous ~i~]ocation structug~s such as those occurring in fatigue and class M creep be~vior. Ackncwle~ The ;Llexand~ yon Humboldt Stifbu~ provided finar~isl a s s i s ~ e for this work t h r o ~ a fellowship to the ~uthor. Comments and discussion with Prof. K M u ~ i , Prof. W. Blum, Prof. P~ N e u h ~ , Dr. LL Essmann, Dr. ~bx>m~son ~ Dr. in~ ~ Christ are greatfully acknowledged. R e f ~ I. ~ Kuhlmmqn-Wilsdorf and & ~ van der Merwe, Mat. ScL Fag,, voL 55 (1982), p~ 79 2. ~% & Mills, J. C. Gibe.ling, and W. ~ Nix, Acta Met., vol. 33 (1985), pp. 1503-151~ 3. ~ & Basinski and '7- S Basinski, in Dislocatiors in Solids, ed. by F. ~. ~ Nabarro, ~ K~I l~nd Co. (1979), pp. 263-362. 4. ~t & Mills, P n ~ ~hesis, Stanford University, Dec., 1 9 ~ 5. & ~ Fatal and ~. ~. C ~ i , i Appl. ~hysics, voL 34, (1963), p~ 278~ 6. P. Faasen, Ph~ikal A e t a l ~ , Springer-Verlag, BerlinYPaidelberg, 1974, pp. 271. 7. S. Takeuchi and A. S. Argon, Phil. Mag~ A, vol. 40, (1975), pp. 65~ 6. J. Weertm~_n, Appl. ~y~, vol. 25, (1957), pp. 1 1 ~ 9. J.C.M. hi, in Dislocation Dynamics, ed. by A. F~ Rosenfield, G. T. Hahn, A. L. Bement, and R. i. Jaffee, ~,~
1596
DEFO~TION
AND S T P A I N E N E R G Y
Vol.
20, No.
FIG. 1 Schematic i l l u s t r a t i o n of a p o r t i o n of the e n e r g y s u r f a c e . A t y p i c a l s h a p e of the e q u a t i o n of c o n s t r a i n t is d r a w n on the T - p plane and p r o j e c t e d onto the energy surface. UJ
%
....
FIG. 2 Cube of m a t e r i a l s h o w - I ing N d i s l o c a t i o n loops in a v o l u m e 1
.~_~_ __ve_L~_,_T.Y_. . . . . . . . . . . . . . .
:
4~J'~1 ~ "@ "
1
UePER LMT
HIGHSTRESS LN IEAR REGION
." ' TRANSITION TOO{~TACLE .-~ CONTROLLE VELOCI D TY WITH [T~REAS~NG STRESS
,.L"-~'9"0_/ POWERLAWRECKON
FIG. 3a S c h e m a t i c of the a v e r a g e d i s l o c a tion velocity as a f u n c t i o n of a p p l i e d s t r e s s w h e n all d i s l o c a t i o n s are c o n s i d e r e d m o b i l e .
I
~'\' , ~~
~ \
%,/'
W
byi~*roctiomzwith
fix~ ~lacle.s;eendot~rd~locotioreJ
,
, ......
FIG. 3b S c h e m a t i c of the d i s l o c a t i o n d e n s i t y r e q u i r e d to k e e p a c o n s t a n t shear strain rate w h e n the d i s l o c a tion velocity-stress l a w s h o w n in Fig. 3a is o b e y a d .
FIG. 4 Projection of t h e v a r i a t i o n of dislocation density as a f u n c t i o n of s t r e s s n e e d e d to p r o d u c e c o n s t a n t s t r a i n r a t e o n t o the e n e r g y surface. The d o u b l e m i n i m u m s u g g e s t s use of the d o u b l e t a n g e n t rule to m i n i m i z e the total energy.
ii