- Email: [email protected]
Back-stresses, image stresses, and work-hardening
BACK-STRESSES,
IMAGE
STRESSES, L.
M.
AND
WORK-HARDENING*
BROWN?:
d comparison is made between recently published theories on the plastic behaviour of dispersionhardened alloys in which no plastic relaxation occura. It is shown thet Tanaka and Mori’s ‘1’ calculation Some new derivations are given for thr is a rigorous lower bound for the hardening to be expected. mean strains in matrix and inclusions. A general formula for the image strain is presented. The theories are compared with experiment, and it is concluded that for spherical obstacles considerable plastic relaxation occurs. CONTRAISTES
REFLECHIES,
COSTR.UNTES
IMAGES
ET
ECROUISSAGE
L’auteur effectue uno companriaon entre 10s theories, r&emment publi&es, du comportement plautiquc des ailiages durcis par dispersion dans lesquels aucune relaxation plaatique ne se produit. Le calcul de Tanaka et Mori”’ apparait comme constituant, une borne infbrieure pour l’apperition du durcisaement. L’auteur donne quelques r&ndtat.s pour les dCform8tiOns moyennes dens 18 metrice et les inclusions. et pr6sent.e une formule generale pour la contrainte image. Aprils comparaiaon des theories et des resultsts exp&imentaux, l’auteur conclut que, pour des obstacles nphbriques, il se produit unc relaxation plastique importante. RCCKSPANNUSGES,
BILDRXFTE
UXD
VERFESTIGUSG
Kiinlich ver6ffentlichte Theorien zum plastischen Verhalten dispersionsverfeetigter Legierungen, in denen keine plaatische Relaxation stattfindet, werden verglichen. Ea wird gezeigt, de0 die Rechnungen von Tanaka und Mori”’ eine strenge untere Grenze fiir die zu erwartende Verfeetigung liefem. Fiir die mittlere Abgleitung in der Matrix und in den Einschliiasen werden einige neue Ableitungen angegeben. Fiir die BildkrHfte wird eine allgemeine Formel abgeleitet. Die Theorien werden mit dem Experiment verglichen; dieter Vergleich zeigt, da0 bei kugelf6rmigen Hindernissen eine betrtihtliche plastieehe Helaxetion Rtattfindet.
In view of the recent papers published in Acta by Tanaka and Mori,(l) Hart(z) and Tanaka et uZ.‘~), on the work-hardening of dispersionhardened crystals, it seems worthwhile to publish a short note comparing these papers with the papers of Brown and Stobb.@) on the same subject. Let us first consider an elastic-plastic matrix, yielding at a stre8s uO, and containing a volume fraction f of elastic inclusions. For simplicity, let the elastic constants of matrix and inclusions be identical. Then when the matrix has undergone a (symmetrical) plastic shear strain* E,, elastic strains of the order of E, are induced in the inclusions. The shear strain may be regarded as a transformation strain in Eshelby’s theory’b*7) of elastic inclusions, and Eshelby shows how to calculate the resulting state of stress and strain. Let us first consider a situation in which the strain in the inclusion is a uniform pure shear strain with the same components as the plastic strain. Eshelby defines an “accommodation factor” y such t.hat Met.
E1=
yEP
where E’ is the strain in the inclusion; inclusion is simply J = 2&
the stress in the
21, JULY
needle, in slip plane, parallel to slip vector
r=l
circular plate, in slip plane, with aspect ratio c/a < 1
y = 4(1 -. 11)n
1973
x(2 -
circular plate perpendicular to slip plane. parallel to slip direction
y=
1.
Y) c
l(c)
1((l)
Eshelby@) also proves that the elastic energy of an inclusion volume V, given by
E,,=
av,y,uED2.
(3
Now the basis of Tanaka and Mori’s(2’ calculation ir; to equate the work done by the applied stress to the elastic energy stored, plus the frictional energy dissipated against the yield stress of the matrix. Tanaka and Mori identify the yield stress of the matrix with the Orowan stress, u,. If d’ is the applied stress, de, = 4fy/4cp da, + (1 - f)2a, CIF”
u‘4 = 00 + 2Y1P&,/(1 - f). (3) Equation (3) is a simplified form of Tanaka and Mori’s result. The following points should be noted : (1) We have considered pure shear only, whereas Tanaka and Mori deal mostly with dilatation-free extension : in other words, in the case discussed here,
Received June 5, 1972; revised September 25. 1972. t Depertment of Metallurgy and Materials Science, McMester University, Hamilton, Ontario, Cened8. z On leave from Cevendish Laboratory, Cambridge. l E, ia equal to the fractional plestio tensile extension of the matrix, if the glide plane is inclined at 46’ to the tensile axis. VOL.
5v
whence
l
METALLURGICA,
i -
sphere
(1 - f) .2d’
and is a pure shear stress. For various cases Eshelby@)
ACTA
quotes values of y as follows:
879
8Nl
ACT.4
METALLURGICA,
slip occurs on a single glide plane, and the cross-section of a tensile specimen changes shape during deformation, whereas Tanaka and Mori quote formulae for multiple slip, in which the cross-section is unchanged in shape. For a spherical inclusion, Tanaka and Mori show that Yaingie-aiir, = +SYmu~tip~e-si~p. (2) The factor (1 - f) in the denominator of equation (3) should not be taken to imply that equation (3) is accurate to second-order in f. If (1 - f)-’ differs significantly from unity, then a more elaborate calculation must be done from first principles. An attempt is made to do this later in this paper. (3) Tanaka and Mori consider the general case when the inclusion is elastically inhomogeneous. Elastic inhomogeneit,y of the inclusion has three consequences: first, it modifies the average elast.ic constants of the material : second, it changes the value of the Orowan stress by modifying the energy of a dislocation in the neighbourhood of the inclusion; and third, it alters Brown and t.he effective transformation strain. Stobbst4) give a simple argument to show that if the elastic constants of the inclusion are p*, v*, then ep in equation (3) must be replaced as follows:
This replacement gives the alteration in transformation strain. For small volume fract.ions Tanaka and Mori’s result agrees with this. However for large volume fractions, Tanaka and Mori’s result is substantially different. The discrepancy appears to arise because Tanaka and Mori treat the elastic inhomogeneity effect in a formal manner which results in a change of elastic constants affecting all the terms on the right-hand side of equation (3). It seems unreasonable that the effect of the elastic inhomogeneity is properly accounted for in this wag; certainly the Orowan stress is not correctly modified in Tanaka and Mori’s treatment. However, for small volume fract.ions, the numerical discrepancy between Tanaka and Mori’s result and equation (3) is negligible. It is worth pointing out here that E, in equation (3) refers to plastic strain. It is assumed that in any comparison with experiment, the elastic strain has been subtracted from the total strain to calculate the plastic strain. It follows that it is not necessary to know the elastic response of the composite to be able to calculate the flow st.ress. Son Tanaka and Mori’s estimate of the applied stress required to continue plastic flow is clearly a loner bound to the true value. This is because in their treatment all the work done by the increment of the
VOL.
21,
1973
applied stress over the initial yield stress is stored as internal elastic energy; any additional heat production will increase the flow stress. Thus Tanaka and Mori’s calculation provide a rigorous lower bound for the strength of a model solid in which one component deforms plastically and another component deforms elastically by the same strain. This ideal solid retains a perfect elastic ‘memory’ of its shape before deformation, that is to say, measurement of the internal stress enables one to determine its original shape. Furthermore, viscous deformation of the matrix driven by the stored elastic stress tends to return the body to the shape it had before plastic deformation. so for brevit! let us call it the ‘PMS’ or perfect memory solid. It is instructive to derive equation (3) by another method. Once again, let us first consider an elasticall> homogeneous body, and treat the inhomogeneous case later. Eshelby(?) defines the total displacement u,’ caused by an elastic inclusion in such a finite body. This displacement is the sum of uic, the displacement. which would be observed in an infinite medium, and uiim, the ‘image displacement’ caused by the boundary conditions on the surface of t.he body. Xow Eshelby shows that the interaction energy between an applied stress and the elastic inclusion is given* by Eint = -
s
aijAuiF dSj = -
8,
s
(T. .&. .T dp ’ ,.l ‘J ‘J
(4)
In equation (4), the surface integral gives the work done by the applied tractions when the external surface moves through uiF. The volume integral is to be taken over the volume of the inclusion. But if the transformation strain ei, T is uniform (as it. is in the PMS), then the integral over the inclusion volume can be written as f times the integral over the specimen volume. Furthermore, the surface integral can be transformed to a volume integral by Gauss’ theorem: then we 6nd
s
aijAeijF dv = f u~,*~E~,T dr Ve I r-
(-5)
but the left-hand side of equation (5) is proportional to the mean strain in the body, so that
(EijF)= fEijT. Thus the mean value of the strain, as it would be calculated from average displacements on the extend
surface of the
body, is given by the volume fraction of inclusion times the transformation strain. Equation (6) appears to be
valid for all values off, because Eshelby’s proof of equation (4) does not assume that the volume fraction l In this and subsequent equations. we assume summation over repeated indices.
BROWS:
BACK
STRESSES,
IMAGE
STRESSES
_4SD
WORK-HARDESISG
881
stress is a lower bound for the flow stress of the perfect memory solid, and the calculation just outlined gives us another way of understanding this. Let us imagine a rigid dislocation passing through the matrix. It feels an average st,ress (/), = 2~(~~)~+~,and this will be be equal to the flow stress; the sense of this stress is of course, to prevent further deformation. Xow, if we allow the dislocation to have flexibility, it. will bend forwards and backwards and sample more adverse stress than the mean stress. The increment in adverse stress depends sensitively on the sampling process and (7) CEijFJ= (l -.f)(Ei,F>_*f + ftEijF)I is difficult to calculate, but as the flexible dislocation where the subscript M refers to an average over the moves the bending will dissipate energy in the form matrix, and the subscript I to an average over the in- of heat. Thus the flow stress calculated from the mean clusions. But in a given inclusion, the strain consists stress is a bower bound to the true flow stress of the of t.he constrained strain, Q,~; the image strain, E~~~“‘; model solid. The calculations of Brown and Stobbsf4) and of and the strain due to all the other inclusions. ProHartt3) differ from thdse of Tanaka and Mori”) in one vided the dispersion of inclusions is fine enough that chief respect-they attempt to calculate the true flow any one of them does not account for a large part of stress of the PMS, allowing for dislocation flexibility. the volume fraction, that is, provided the body conHowever, Brown and Stobbs conclude that the major (ains more than say ten inclusions, it will be a good component in the hardening at large strains is the approximation to put the strain due to all the other ‘image stress’, which, as luck would have it, coincides inclusions equal to the mean strain in the matrix: thus with the mean stress in the matrix provided the inCEijF)I e (&ij’)I + CEijtm)I + fEijF).ll* (8) clusions are spherical. It is clear from the arguments given above that for non-spherical inclusions no genSubstitution in equation (7) gives eral rigorous connection can be made between the tEijF)_41 =ftEijT C&i:)1 lEijim>I) tg) image stress and the flow stress of the PMS. However, it is worthwhile evaluating the image stress, for several Equation (9) appears to be valid for all values off, subject only to the condition t.hat the number of in- reasons: first, because it is a uniform component of clusions is large.* the strains in the matrix, it controls t.he peak shift in an X-ray pattern from the crystal-the local fluctuIf we restrict ourselves to ellipsoidal inclusions, for ating strains produce an asymmetrical line broadening; which ei, ’ is uniform inside the inclusion, and for which Eshelby@‘” writes second, because it is independent of the shape of the inclusion, it depends only on the transformation strain, so that simple formulae for it can be found ; third, the mean image stress appears in equation (9), so that it must be evaluated if an expression accurat.e If we further restrict ourselves to the case of pure to second-order in f for the mean strain in the matrix shear, with ~~~~= eSIT = E,, and, following Eshelby, is to be found. In the appendix, a general formula is we call y = I-%,,,,, then equation (10) gives a pure derived for the mean value of the image stress caused shear strain by an inclusion of arbitrary shape in the centre of a spherical body. For an elastically homogeneous body, (E13 F\ /*If = YfE, -f'\%31m,, (11) subject to pure shear but giflm is itself proportional tof, so that to first order T T El3 = E31 = El,, in f the stress derived from equation (11) agrees with the increment in stress (6” - aO) in equation (3). If. the result is
of inclusion be small. Equation (6).is obviously true forf = 1. Equation (6) is of particular interest in discussing the dimensional stability of work-hardened materials, because it gives directly the change in overall shape of the body resulting from a given transformation strain. Sow the quantity needed to calculate the flow stress of the plastic matrix is the mean value of the elastic strain in the matrix. It can be found from
ia see?&,then, that Tanaka and Mori’s”’
calculationgives
the mean &errs in the matrix, including the contribution from the image stress. As previously mentioned, this
* Translated
into stresses, equation (9) simply Rtaten the mean value of any stress component is zero. for the righthand side of equation (9) is simply the negative of the strain which causes stress in the inclusions.
(Ei:‘“l) =
f&,
ACTA
882
METALLUR~~CA.
VOL.
21.
1973
The connection between equation (12) and the previous work can be appreciated if it is compared with equation (ll), taking y from equation l(a). Then it is easily seen that for a spherical inclusion,(.ziji*>_,, = '.~EijF):Ua
An important application of equation (12) is to insert the mean value of the image strain into equation rll), to find
!&,sF>.u = Yf%-
7 - 5v 15(1 -
v)
f2%
(13)
which gives the mean strain in the matrix, valid to second-order in f, provided the number of inclusions is large.* If the elastic constants of the inclusion and the matrix are not the same, equation (3a) can be used to modify equation (6), equation (lo), and equation (13). It should be pointed out that for inclusions whose geometry is such that the strain in the inclusion is not parallel to the strain in t.he matrix-i.e. for which Sirkahas more than one non-zero component-a more elaborate ~lculation must be done, following the methods of Eshelbp.c7) The comparison between all published theories for the PMS and experiment is shown in Fig. 1. Considering first the forward curves, i.e. t.he ‘work-hardening’, it is clear that because the theoretical estimates of Hart(*) and of Brown and Stobb+) take account of the s&as required to bow dislocations between particles, these theories give larger estimates of the stress The than the lower-bound limit of Tanaka and Nori. two former theories are in substantial agreement: both predict that the hardening curves will be parabolic at low strains, and become linear at larger strains. Both predict the same dependence on volume fraction; Hart’s calculation predicts parabolic hardening about 30 per cent greater than that predicted by Brown and Stobbs and this is the only discrepancy between the two theories. It is probable that Hart’s calculation overestimates the true hardening This is because Hart takes no account of the stresses due to particles lying outside the slip plane. These particles act to reduce the stresses on elements of the bowing dislocation, Tt is central to Brown and Stobbs’ arguments that the mean local stress around a particle is zero. Hart’s expression (his equation (1)) does not give the stresses correctly in three dimensions: his stresses are everywhere of the same sign, tending to promote negative slip in the matrix. Thus it is impossible from his treatment to derive the contribution of the image stress to the hardening, and more important, imposl Equation (13) can also be derived by Tanaka and Mori’s method, by altering the esprr.wions for the elastic energy.
FIG. 1. Comparison
of theories for the P31S with csperi. mental obserl-at ion
The experimental stress strain curse is taken from Brown and Stobbs.“’ For this metcrial (Cu SiO,). f = 0.54:;; 8’ = 0.4;
[C = 4.x ‘.: 103 rip mu-‘; I”+ -
j” 7(/‘*
- ,8’)
= 0.8:
and mean volumetric particle radius - tX0 .i. The curve labelled Tanaka and Yori is constructc*cl from equation (3). The unloading behaviour is determined by the assumption that the Otowan stre-CS3acts as a friction stress, RO that no reverse flop occura until thch stress drops by 2a,. The curve labelled Hart is taken fnml reference (2). equation (22) as corrected in t.he apprndis of that paper In Hart’s notation, a* is the unrelaxed nlide strain and is equal to 2EP. The unloading curve is c&tructed accord. inn to Hart’s mvscriution. ?J?he curve iabelleh Brown and Stobbs is constructed from reference (4). The unloading behaviour haq been oonstructod aa follows: BRsumo that the fluctuating stresses with zero mean value act es*txntially as a friotion stress, so that they play the rolt! of uO in Tanaka and Miori’s theory. Then tho stress must be lowered bv twice the amount by which Brown and Stobbs’ curve &oeed$ Tanaka and Mori’e curve before reverse flow starts. and thereafter the reverse flow follows a curve which is simply the forward work-hardening curve, displaced downwards by a constant amount. This figure replaces and improves upon Fig. 5 of refer. once (4). which is erroneously tirnwn.
sible to see that plastic relaxation can take place by positive slip on secondary systems. However, prosided the image stress is included, Hart’s treatment of the bowing dislocation is not greatly affected by his assumed form of stress-field. On thr other hand, because Brown and Stubbs make a vrry crude approximation for the shape of the bowing dislocation, it is difficult to assess the reliability of their treatment.
BROWS:
BACK
STRESSES,
IMAGE
The close agreement between the two theories leads one to believe that they may be valid to within (say) the extent of the difference between them. Figure 1showsthat the experimentalwork-hardening curve lies between the lower-limit estimate of Tanaka and Mori and the two other estimates. Thus it might be argued that the forward work-hardening of Cu-SiO, can be accounted for by assuming that it deforms as a PMS. In the present author’s view, this is not so : the curves of Hart and of Brown and Stobbs lie significantly higher than experiment, and furthermore they show the wrong dependence on volume fraction. However uncertainty in the theory, and of how to average over particle size, perhaps make this conclusion controversial. It is known that the experimental curve can be accounted for equally well by a pure forest theory, or at least by theories which take no account of long-range elastic stresses (Ashby( Hirsch and Humphreys@)): thus one has a particulyrly clear demonstration that models whose basic premises are opposite extremes can give an ‘adequate’ account of stress-strain curves. These models, although they can all be faulted on points of detail, are much more convincingly worked out than their counterparts in single-phase materials: namely Seeger’s mainly back-stress theory, and Hirsch’s mainly forest theory.* It is not surprising, therefore, that attempts to distinguish between one or other theory on the basis of work-hardening curves alone have been unsuccessful. However, as soon as one admits other data, progress can be made. Figure 1 shows that the behaviour of copper-silica on reverse straining cannot be accounted for if it deforms like a perfect memory solid. Clearly a large part of the hardening is “frictional” and not elastic. This point was recognised by Hart’*’ and earlier by Fisher et ccl.“l) It is also clear, that a purely forest theory cannot account for the existence of the Bauschinger effect, unless special assumptions are made about the asymmetry of forest intersections. However, the X-ray work of Wilson(l*’ shows that internal stresses are linearly related to the magnitude of the Bauschinger effect,, and very probably account for all of it. Furthermore, the extensive electron-microscope observations (Humphreys and Hirsch,f13) Brown and Stobbsc5)) show that plastic strain causes dislocation structures to be generated around inclusions, and that these structures are of a t,ype which reduce greatly the long-range elastic stresses; further it shows that they are of a type which present symmetrical obstacles to forward and backward flow. There is additional less direct data on the l
For a review, se0 Naberro, Basinski and Holt .(I01
STRESSES
AXD
WORK-HARDESISC
883
dimensional stability of cold-worked dispersion-hardened materials, and on plastic cavitation in them, all of which is consistent with the picture that an appreciable part of the hardening is ‘forest’ hardening, but that a by no means negligible part is ‘back-stress hardening’, which causes the Bauschinger effect. For the data in Fig. 1, the forest hardening accounts for about 85 per cent of the total hardening increment. Further experimental work on this problem is in progress in an attempt to reach a quantitative theory. The results will be published elsewhere. Brown and Stobbs(5) have already discussed the principles of such a theory. Finally, it is worth pointing out that in fibre-reinforced materials, and in materials with plate-like inclusions, back-stress hardening can become the major component. For these cases, Tanaka and Mori’s treatment may well be adequate. ACKNOWLEDGEMENTS
The author would like to acknowledge numerous discussions with his colleagues in the Metal Physics Group, Cambridge; and to thank especially Professor J. D. Embury and the group at McMaster, who contributed much metallurgical good sense to the development of these ideas, and whose hospitality is great 1) appreciated. Drs. K. Tanaka and T. Mori pointed out an error of a factor of 2 in Fig. 5 of reference (4). which is corrected in this paper; the author is grateful for discussions held with them and with E. Hart. REFERENCES 1. K. TANAKA and T. MORI, Ada nfek 18, 931 (1970). 2. E. W. tiT, AC&.Met. 20, 275 (1972). 3. K. TASAKA, K. NARITA and T. Mom, _&la. .lief. 20, 297 (1972). 4. L. M. BBOWN and W. (1971). 5. L. M. BROWN and W.
M. STOBBS.
Phil. Nag. 23,
1185
M. STOBBS,
Phil. _Vag. 23,
1201
(1971). 6. i. 8. 9.
J. D. ESHELBY, Pm. R. Sot. A241, 376 (1957). J. D. ERHELBY. Prog. Solid Mech. 2. 69 (1961). M. F. ASHBY, Phil. Msg. 14. 1157 (1966). I’. B. HIRSCH and F. J. HL.MPHRETX, I’roc. H. SOC. A318. 4.5 (1970). 10. F. R. N. NABARRO, Z. S. BARINYKI and D. B. HOLT. .-1111*.
I’hya. 13. 193 (1964).
11. J. C. FISHER, E. W. &RT
and R. H. PRY, Aclo iVef. 1,
336 (1953).
12. D. V. Wmsoa, Acfa. Mel. 1s. 807 (1965). 13. F. J. HIJMPRREYB, and P. B. HIRSCH, Pm. 73 (1970). APPENDIX
Evaluation
R. SOC.A318,
of the mean image strain caused by an
ellipsoidal in&&n
a4 the center of a large spherical body
If in equation (9) of the text, eijF is split into its components ~~~~and &ijim,then we find
ic.. ’ 11I’“)
=f(EijT
-
(EL,~),)- (&,jC)JI-J(&ij’m)L,,. A.1
ACTA
881
JIETALLCRGICA.
Thus to find (~,,‘~)_,r to first order in j, we need to find (sijC)_,, ; the last term on the right-hand side of equation A.1 is proportional to j2. Consider an ellipsoidal inclusion whose surface is S, and whose volume is VI sitting at the center of a large, elastically uniform body with surface S,, and volume Vc. Then
l&ij9.11
1
\-OL.
tribution
21.
19i3
comes from integrals
x2 IdSI
‘i1 = (Ej,’ t (~iIc) with similar expressions b2 and .r2/c2 respectively.
for I,, and Ii3 containing If we define p so that
then IdSI can be expressed ?(I
-
[/syzc
j)r’,
-aspic
dSj
+IsfjC
]dSj = hp3JdRI dSj
(-4.2)
-~?djcO!Si).
i(L:i’dSj+krj’ds.) =- VI~,kTQ$)
in terms of solid angle:
dSi
For the integrals over the ozlter surface, we use Eshelby’s expression for the distant displacement field due to an inclusion: see Ref. 7, equation 2.23, or ref. 6, equation 2.18. The integrals are readily evaluated, and the result is
2(1 -
y2/
s matrix
(l - f)r’s
=
(-4.5)
i n’h ..sI
Ei jc dv
=
’
of the type
[6,,6,, + 6i,6,j -
and
(A.6) Thus, substituting A.6, A.5 and A.3 into A.2, we can 6nd an expression for the mean constrained strain in the matrix. The result is then substituted into equation A.l, to find
6,,6ij]
V) -2 gbi&
For t,he integrals write
I
sirsr j t
6ijslk] . (A.3) I over the surface of the inclusion, n-e UiC = (QrC + cQ2.k
where the (symmetrical) strain and (antisymmetrical) rotation are those appropriate to the solut.ion inside the inclusion. This is permissible, because the displacements are cont.inuous across the interface. The integral of interest is now of the type Ii j = (saC + Girt) Sow for an ellipsoidal vector
inclusion,
xk dS,. I SI the normal
(A.4)
+ +[di,a,j
I
dik6,, 2 b,,dij] .
-
This expression depends only on j and the transformation strain, and is valid only to first-order in j. It can easily be modified for inhomogeneous inclusions by replacing .Q,~ by Ed,** as described by Eshelby.“) When applied to single-phase materials, it is best to think of the image strain as being caused by dislocation loops. Thus if an array of shear loops, of total area A, with Burgers vector b in the X-direction, and lying on the z-y plane are trapped in a crystal, there will bc a long-range strain
to S, is a
0
0
7 -
5v
15(1 -
(E,(m)=
$
0
0
0
o
0
0 7-5v
where
15(1 )$ =
Furthermore, tains squares
the integral A.4 vanishes unless it conof co-ordinates, so that the only con-
(A.71
This easily derived plate-like inclusions and cIgT = b/26. In a single-phase
v)
v)
.1 (A.8a)
by imagining that the loops are of thickness b, so that VI = A6 crystal,
the mean
stress
in the
BROWX:
BACK
STRESSES,
IMAGE
matrix of course, depends on the shape of the tangles of dislocations which prevent the shear loops from collapsing, and prevent moving dislocations from sampling all the volume of the crystal. It is probably not a bad approximation to use equation A.8a to give an estimate of the back stress. In cases of radiation damage, it is of interest to calculate the effect of an array of interstitial prismatic loops, with Burgers’ vectors parallel to the z-direction lying on the plane. Then
STRESSES
AND
WORK-HARDENIXG
885
(A.8b) The specimen is subjected to a uniform, approximately tensile, strain.
IMAGE
STRESSES, L.
M.
AND
WORK-HARDENING*
BROWN?:
d comparison is made between recently published theories on the plastic behaviour of dispersionhardened alloys in which no plastic relaxation occura. It is shown thet Tanaka and Mori’s ‘1’ calculation Some new derivations are given for thr is a rigorous lower bound for the hardening to be expected. mean strains in matrix and inclusions. A general formula for the image strain is presented. The theories are compared with experiment, and it is concluded that for spherical obstacles considerable plastic relaxation occurs. CONTRAISTES
REFLECHIES,
COSTR.UNTES
IMAGES
ET
ECROUISSAGE
L’auteur effectue uno companriaon entre 10s theories, r&emment publi&es, du comportement plautiquc des ailiages durcis par dispersion dans lesquels aucune relaxation plaatique ne se produit. Le calcul de Tanaka et Mori”’ apparait comme constituant, une borne infbrieure pour l’apperition du durcisaement. L’auteur donne quelques r&ndtat.s pour les dCform8tiOns moyennes dens 18 metrice et les inclusions. et pr6sent.e une formule generale pour la contrainte image. Aprils comparaiaon des theories et des resultsts exp&imentaux, l’auteur conclut que, pour des obstacles nphbriques, il se produit unc relaxation plastique importante. RCCKSPANNUSGES,
BILDRXFTE
UXD
VERFESTIGUSG
Kiinlich ver6ffentlichte Theorien zum plastischen Verhalten dispersionsverfeetigter Legierungen, in denen keine plaatische Relaxation stattfindet, werden verglichen. Ea wird gezeigt, de0 die Rechnungen von Tanaka und Mori”’ eine strenge untere Grenze fiir die zu erwartende Verfeetigung liefem. Fiir die mittlere Abgleitung in der Matrix und in den Einschliiasen werden einige neue Ableitungen angegeben. Fiir die BildkrHfte wird eine allgemeine Formel abgeleitet. Die Theorien werden mit dem Experiment verglichen; dieter Vergleich zeigt, da0 bei kugelf6rmigen Hindernissen eine betrtihtliche plastieehe Helaxetion Rtattfindet.
In view of the recent papers published in Acta by Tanaka and Mori,(l) Hart(z) and Tanaka et uZ.‘~), on the work-hardening of dispersionhardened crystals, it seems worthwhile to publish a short note comparing these papers with the papers of Brown and Stobb.@) on the same subject. Let us first consider an elastic-plastic matrix, yielding at a stre8s uO, and containing a volume fraction f of elastic inclusions. For simplicity, let the elastic constants of matrix and inclusions be identical. Then when the matrix has undergone a (symmetrical) plastic shear strain* E,, elastic strains of the order of E, are induced in the inclusions. The shear strain may be regarded as a transformation strain in Eshelby’s theory’b*7) of elastic inclusions, and Eshelby shows how to calculate the resulting state of stress and strain. Let us first consider a situation in which the strain in the inclusion is a uniform pure shear strain with the same components as the plastic strain. Eshelby defines an “accommodation factor” y such t.hat Met.
E1=
yEP
where E’ is the strain in the inclusion; inclusion is simply J = 2&
the stress in the
21, JULY
needle, in slip plane, parallel to slip vector
r=l
circular plate, in slip plane, with aspect ratio c/a < 1
y = 4(1 -. 11)n
1973
x(2 -
circular plate perpendicular to slip plane. parallel to slip direction
y=
1.
Y) c
l(c)
1((l)
Eshelby@) also proves that the elastic energy of an inclusion volume V, given by
E,,=
av,y,uED2.
(3
Now the basis of Tanaka and Mori’s(2’ calculation ir; to equate the work done by the applied stress to the elastic energy stored, plus the frictional energy dissipated against the yield stress of the matrix. Tanaka and Mori identify the yield stress of the matrix with the Orowan stress, u,. If d’ is the applied stress, de, = 4fy/4cp da, + (1 - f)2a, CIF”
u‘4 = 00 + 2Y1P&,/(1 - f). (3) Equation (3) is a simplified form of Tanaka and Mori’s result. The following points should be noted : (1) We have considered pure shear only, whereas Tanaka and Mori deal mostly with dilatation-free extension : in other words, in the case discussed here,
Received June 5, 1972; revised September 25. 1972. t Depertment of Metallurgy and Materials Science, McMester University, Hamilton, Ontario, Cened8. z On leave from Cevendish Laboratory, Cambridge. l E, ia equal to the fractional plestio tensile extension of the matrix, if the glide plane is inclined at 46’ to the tensile axis. VOL.
5v
whence
l
METALLURGICA,
i -
sphere
(1 - f) .2d’
and is a pure shear stress. For various cases Eshelby@)
ACTA
quotes values of y as follows:
879
8Nl
ACT.4
METALLURGICA,
slip occurs on a single glide plane, and the cross-section of a tensile specimen changes shape during deformation, whereas Tanaka and Mori quote formulae for multiple slip, in which the cross-section is unchanged in shape. For a spherical inclusion, Tanaka and Mori show that Yaingie-aiir, = +SYmu~tip~e-si~p. (2) The factor (1 - f) in the denominator of equation (3) should not be taken to imply that equation (3) is accurate to second-order in f. If (1 - f)-’ differs significantly from unity, then a more elaborate calculation must be done from first principles. An attempt is made to do this later in this paper. (3) Tanaka and Mori consider the general case when the inclusion is elastically inhomogeneous. Elastic inhomogeneit,y of the inclusion has three consequences: first, it modifies the average elast.ic constants of the material : second, it changes the value of the Orowan stress by modifying the energy of a dislocation in the neighbourhood of the inclusion; and third, it alters Brown and t.he effective transformation strain. Stobbst4) give a simple argument to show that if the elastic constants of the inclusion are p*, v*, then ep in equation (3) must be replaced as follows:
This replacement gives the alteration in transformation strain. For small volume fract.ions Tanaka and Mori’s result agrees with this. However for large volume fractions, Tanaka and Mori’s result is substantially different. The discrepancy appears to arise because Tanaka and Mori treat the elastic inhomogeneity effect in a formal manner which results in a change of elastic constants affecting all the terms on the right-hand side of equation (3). It seems unreasonable that the effect of the elastic inhomogeneity is properly accounted for in this wag; certainly the Orowan stress is not correctly modified in Tanaka and Mori’s treatment. However, for small volume fract.ions, the numerical discrepancy between Tanaka and Mori’s result and equation (3) is negligible. It is worth pointing out here that E, in equation (3) refers to plastic strain. It is assumed that in any comparison with experiment, the elastic strain has been subtracted from the total strain to calculate the plastic strain. It follows that it is not necessary to know the elastic response of the composite to be able to calculate the flow st.ress. Son Tanaka and Mori’s estimate of the applied stress required to continue plastic flow is clearly a loner bound to the true value. This is because in their treatment all the work done by the increment of the
VOL.
21,
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applied stress over the initial yield stress is stored as internal elastic energy; any additional heat production will increase the flow stress. Thus Tanaka and Mori’s calculation provide a rigorous lower bound for the strength of a model solid in which one component deforms plastically and another component deforms elastically by the same strain. This ideal solid retains a perfect elastic ‘memory’ of its shape before deformation, that is to say, measurement of the internal stress enables one to determine its original shape. Furthermore, viscous deformation of the matrix driven by the stored elastic stress tends to return the body to the shape it had before plastic deformation. so for brevit! let us call it the ‘PMS’ or perfect memory solid. It is instructive to derive equation (3) by another method. Once again, let us first consider an elasticall> homogeneous body, and treat the inhomogeneous case later. Eshelby(?) defines the total displacement u,’ caused by an elastic inclusion in such a finite body. This displacement is the sum of uic, the displacement. which would be observed in an infinite medium, and uiim, the ‘image displacement’ caused by the boundary conditions on the surface of t.he body. Xow Eshelby shows that the interaction energy between an applied stress and the elastic inclusion is given* by Eint = -
s
aijAuiF dSj = -
8,
s
(T. .&. .T dp ’ ,.l ‘J ‘J
(4)
In equation (4), the surface integral gives the work done by the applied tractions when the external surface moves through uiF. The volume integral is to be taken over the volume of the inclusion. But if the transformation strain ei, T is uniform (as it. is in the PMS), then the integral over the inclusion volume can be written as f times the integral over the specimen volume. Furthermore, the surface integral can be transformed to a volume integral by Gauss’ theorem: then we 6nd
s
aijAeijF dv = f u~,*~E~,T dr Ve I r-
(-5)
but the left-hand side of equation (5) is proportional to the mean strain in the body, so that
(EijF)= fEijT. Thus the mean value of the strain, as it would be calculated from average displacements on the extend
surface of the
body, is given by the volume fraction of inclusion times the transformation strain. Equation (6) appears to be
valid for all values off, because Eshelby’s proof of equation (4) does not assume that the volume fraction l In this and subsequent equations. we assume summation over repeated indices.
BROWS:
BACK
STRESSES,
IMAGE
STRESSES
_4SD
WORK-HARDESISG
881
stress is a lower bound for the flow stress of the perfect memory solid, and the calculation just outlined gives us another way of understanding this. Let us imagine a rigid dislocation passing through the matrix. It feels an average st,ress (/), = 2~(~~)~+~,and this will be be equal to the flow stress; the sense of this stress is of course, to prevent further deformation. Xow, if we allow the dislocation to have flexibility, it. will bend forwards and backwards and sample more adverse stress than the mean stress. The increment in adverse stress depends sensitively on the sampling process and (7) CEijFJ= (l -.f)(Ei,F>_*f + ftEijF)I is difficult to calculate, but as the flexible dislocation where the subscript M refers to an average over the moves the bending will dissipate energy in the form matrix, and the subscript I to an average over the in- of heat. Thus the flow stress calculated from the mean clusions. But in a given inclusion, the strain consists stress is a bower bound to the true flow stress of the of t.he constrained strain, Q,~; the image strain, E~~~“‘; model solid. The calculations of Brown and Stobbsf4) and of and the strain due to all the other inclusions. ProHartt3) differ from thdse of Tanaka and Mori”) in one vided the dispersion of inclusions is fine enough that chief respect-they attempt to calculate the true flow any one of them does not account for a large part of stress of the PMS, allowing for dislocation flexibility. the volume fraction, that is, provided the body conHowever, Brown and Stobbs conclude that the major (ains more than say ten inclusions, it will be a good component in the hardening at large strains is the approximation to put the strain due to all the other ‘image stress’, which, as luck would have it, coincides inclusions equal to the mean strain in the matrix: thus with the mean stress in the matrix provided the inCEijF)I e (&ij’)I + CEijtm)I + fEijF).ll* (8) clusions are spherical. It is clear from the arguments given above that for non-spherical inclusions no genSubstitution in equation (7) gives eral rigorous connection can be made between the tEijF)_41 =ftEijT C&i:)1 lEijim>I) tg) image stress and the flow stress of the PMS. However, it is worthwhile evaluating the image stress, for several Equation (9) appears to be valid for all values off, subject only to the condition t.hat the number of in- reasons: first, because it is a uniform component of clusions is large.* the strains in the matrix, it controls t.he peak shift in an X-ray pattern from the crystal-the local fluctuIf we restrict ourselves to ellipsoidal inclusions, for ating strains produce an asymmetrical line broadening; which ei, ’ is uniform inside the inclusion, and for which Eshelby@‘” writes second, because it is independent of the shape of the inclusion, it depends only on the transformation strain, so that simple formulae for it can be found ; third, the mean image stress appears in equation (9), so that it must be evaluated if an expression accurat.e If we further restrict ourselves to the case of pure to second-order in f for the mean strain in the matrix shear, with ~~~~= eSIT = E,, and, following Eshelby, is to be found. In the appendix, a general formula is we call y = I-%,,,,, then equation (10) gives a pure derived for the mean value of the image stress caused shear strain by an inclusion of arbitrary shape in the centre of a spherical body. For an elastically homogeneous body, (E13 F\ /*If = YfE, -f'\%31m,, (11) subject to pure shear but giflm is itself proportional tof, so that to first order T T El3 = E31 = El,, in f the stress derived from equation (11) agrees with the increment in stress (6” - aO) in equation (3). If. the result is
of inclusion be small. Equation (6).is obviously true forf = 1. Equation (6) is of particular interest in discussing the dimensional stability of work-hardened materials, because it gives directly the change in overall shape of the body resulting from a given transformation strain. Sow the quantity needed to calculate the flow stress of the plastic matrix is the mean value of the elastic strain in the matrix. It can be found from
ia see?&,then, that Tanaka and Mori’s”’
calculationgives
the mean &errs in the matrix, including the contribution from the image stress. As previously mentioned, this
* Translated
into stresses, equation (9) simply Rtaten the mean value of any stress component is zero. for the righthand side of equation (9) is simply the negative of the strain which causes stress in the inclusions.
(Ei:‘“l) =
f&,
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882
METALLUR~~CA.
VOL.
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1973
The connection between equation (12) and the previous work can be appreciated if it is compared with equation (ll), taking y from equation l(a). Then it is easily seen that for a spherical inclusion,(.ziji*>_,, = '.~EijF):Ua
An important application of equation (12) is to insert the mean value of the image strain into equation rll), to find
!&,sF>.u = Yf%-
7 - 5v 15(1 -
v)
f2%
(13)
which gives the mean strain in the matrix, valid to second-order in f, provided the number of inclusions is large.* If the elastic constants of the inclusion and the matrix are not the same, equation (3a) can be used to modify equation (6), equation (lo), and equation (13). It should be pointed out that for inclusions whose geometry is such that the strain in the inclusion is not parallel to the strain in t.he matrix-i.e. for which Sirkahas more than one non-zero component-a more elaborate ~lculation must be done, following the methods of Eshelbp.c7) The comparison between all published theories for the PMS and experiment is shown in Fig. 1. Considering first the forward curves, i.e. t.he ‘work-hardening’, it is clear that because the theoretical estimates of Hart(*) and of Brown and Stobb+) take account of the s&as required to bow dislocations between particles, these theories give larger estimates of the stress The than the lower-bound limit of Tanaka and Nori. two former theories are in substantial agreement: both predict that the hardening curves will be parabolic at low strains, and become linear at larger strains. Both predict the same dependence on volume fraction; Hart’s calculation predicts parabolic hardening about 30 per cent greater than that predicted by Brown and Stobbs and this is the only discrepancy between the two theories. It is probable that Hart’s calculation overestimates the true hardening This is because Hart takes no account of the stresses due to particles lying outside the slip plane. These particles act to reduce the stresses on elements of the bowing dislocation, Tt is central to Brown and Stobbs’ arguments that the mean local stress around a particle is zero. Hart’s expression (his equation (1)) does not give the stresses correctly in three dimensions: his stresses are everywhere of the same sign, tending to promote negative slip in the matrix. Thus it is impossible from his treatment to derive the contribution of the image stress to the hardening, and more important, imposl Equation (13) can also be derived by Tanaka and Mori’s method, by altering the esprr.wions for the elastic energy.
FIG. 1. Comparison
of theories for the P31S with csperi. mental obserl-at ion
The experimental stress strain curse is taken from Brown and Stobbs.“’ For this metcrial (Cu SiO,). f = 0.54:;; 8’ = 0.4;
[C = 4.x ‘.: 103 rip mu-‘; I”+ -
j” 7(/‘*
- ,8’)
= 0.8:
and mean volumetric particle radius - tX0 .i. The curve labelled Tanaka and Yori is constructc*cl from equation (3). The unloading behaviour is determined by the assumption that the Otowan stre-CS3acts as a friction stress, RO that no reverse flop occura until thch stress drops by 2a,. The curve labelled Hart is taken fnml reference (2). equation (22) as corrected in t.he apprndis of that paper In Hart’s notation, a* is the unrelaxed nlide strain and is equal to 2EP. The unloading curve is c&tructed accord. inn to Hart’s mvscriution. ?J?he curve iabelleh Brown and Stobbs is constructed from reference (4). The unloading behaviour haq been oonstructod aa follows: BRsumo that the fluctuating stresses with zero mean value act es*txntially as a friotion stress, so that they play the rolt! of uO in Tanaka and Miori’s theory. Then tho stress must be lowered bv twice the amount by which Brown and Stobbs’ curve &oeed$ Tanaka and Mori’e curve before reverse flow starts. and thereafter the reverse flow follows a curve which is simply the forward work-hardening curve, displaced downwards by a constant amount. This figure replaces and improves upon Fig. 5 of refer. once (4). which is erroneously tirnwn.
sible to see that plastic relaxation can take place by positive slip on secondary systems. However, prosided the image stress is included, Hart’s treatment of the bowing dislocation is not greatly affected by his assumed form of stress-field. On thr other hand, because Brown and Stubbs make a vrry crude approximation for the shape of the bowing dislocation, it is difficult to assess the reliability of their treatment.
BROWS:
BACK
STRESSES,
IMAGE
The close agreement between the two theories leads one to believe that they may be valid to within (say) the extent of the difference between them. Figure 1showsthat the experimentalwork-hardening curve lies between the lower-limit estimate of Tanaka and Mori and the two other estimates. Thus it might be argued that the forward work-hardening of Cu-SiO, can be accounted for by assuming that it deforms as a PMS. In the present author’s view, this is not so : the curves of Hart and of Brown and Stobbs lie significantly higher than experiment, and furthermore they show the wrong dependence on volume fraction. However uncertainty in the theory, and of how to average over particle size, perhaps make this conclusion controversial. It is known that the experimental curve can be accounted for equally well by a pure forest theory, or at least by theories which take no account of long-range elastic stresses (Ashby( Hirsch and Humphreys@)): thus one has a particulyrly clear demonstration that models whose basic premises are opposite extremes can give an ‘adequate’ account of stress-strain curves. These models, although they can all be faulted on points of detail, are much more convincingly worked out than their counterparts in single-phase materials: namely Seeger’s mainly back-stress theory, and Hirsch’s mainly forest theory.* It is not surprising, therefore, that attempts to distinguish between one or other theory on the basis of work-hardening curves alone have been unsuccessful. However, as soon as one admits other data, progress can be made. Figure 1 shows that the behaviour of copper-silica on reverse straining cannot be accounted for if it deforms like a perfect memory solid. Clearly a large part of the hardening is “frictional” and not elastic. This point was recognised by Hart’*’ and earlier by Fisher et ccl.“l) It is also clear, that a purely forest theory cannot account for the existence of the Bauschinger effect, unless special assumptions are made about the asymmetry of forest intersections. However, the X-ray work of Wilson(l*’ shows that internal stresses are linearly related to the magnitude of the Bauschinger effect,, and very probably account for all of it. Furthermore, the extensive electron-microscope observations (Humphreys and Hirsch,f13) Brown and Stobbsc5)) show that plastic strain causes dislocation structures to be generated around inclusions, and that these structures are of a t,ype which reduce greatly the long-range elastic stresses; further it shows that they are of a type which present symmetrical obstacles to forward and backward flow. There is additional less direct data on the l
For a review, se0 Naberro, Basinski and Holt .(I01
STRESSES
AXD
WORK-HARDESISC
883
dimensional stability of cold-worked dispersion-hardened materials, and on plastic cavitation in them, all of which is consistent with the picture that an appreciable part of the hardening is ‘forest’ hardening, but that a by no means negligible part is ‘back-stress hardening’, which causes the Bauschinger effect. For the data in Fig. 1, the forest hardening accounts for about 85 per cent of the total hardening increment. Further experimental work on this problem is in progress in an attempt to reach a quantitative theory. The results will be published elsewhere. Brown and Stobbs(5) have already discussed the principles of such a theory. Finally, it is worth pointing out that in fibre-reinforced materials, and in materials with plate-like inclusions, back-stress hardening can become the major component. For these cases, Tanaka and Mori’s treatment may well be adequate. ACKNOWLEDGEMENTS
The author would like to acknowledge numerous discussions with his colleagues in the Metal Physics Group, Cambridge; and to thank especially Professor J. D. Embury and the group at McMaster, who contributed much metallurgical good sense to the development of these ideas, and whose hospitality is great 1) appreciated. Drs. K. Tanaka and T. Mori pointed out an error of a factor of 2 in Fig. 5 of reference (4). which is corrected in this paper; the author is grateful for discussions held with them and with E. Hart. REFERENCES 1. K. TANAKA and T. MORI, Ada nfek 18, 931 (1970). 2. E. W. tiT, AC&.Met. 20, 275 (1972). 3. K. TASAKA, K. NARITA and T. Mom, _&la. .lief. 20, 297 (1972). 4. L. M. BBOWN and W. (1971). 5. L. M. BROWN and W.
M. STOBBS.
Phil. Nag. 23,
1185
M. STOBBS,
Phil. _Vag. 23,
1201
(1971). 6. i. 8. 9.
J. D. ESHELBY, Pm. R. Sot. A241, 376 (1957). J. D. ERHELBY. Prog. Solid Mech. 2. 69 (1961). M. F. ASHBY, Phil. Msg. 14. 1157 (1966). I’. B. HIRSCH and F. J. HL.MPHRETX, I’roc. H. SOC. A318. 4.5 (1970). 10. F. R. N. NABARRO, Z. S. BARINYKI and D. B. HOLT. .-1111*.
I’hya. 13. 193 (1964).
11. J. C. FISHER, E. W. &RT
and R. H. PRY, Aclo iVef. 1,
336 (1953).
12. D. V. Wmsoa, Acfa. Mel. 1s. 807 (1965). 13. F. J. HIJMPRREYB, and P. B. HIRSCH, Pm. 73 (1970). APPENDIX
Evaluation
R. SOC.A318,
of the mean image strain caused by an
ellipsoidal in&&n
a4 the center of a large spherical body
If in equation (9) of the text, eijF is split into its components ~~~~and &ijim,then we find
ic.. ’ 11I’“)
=f(EijT
-
(EL,~),)- (&,jC)JI-J(&ij’m)L,,. A.1
ACTA
881
JIETALLCRGICA.
Thus to find (~,,‘~)_,r to first order in j, we need to find (sijC)_,, ; the last term on the right-hand side of equation A.1 is proportional to j2. Consider an ellipsoidal inclusion whose surface is S, and whose volume is VI sitting at the center of a large, elastically uniform body with surface S,, and volume Vc. Then
l&ij9.11
1
\-OL.
tribution
21.
19i3
comes from integrals
x2 IdSI
‘i1 = (Ej,’ t (~iIc) with similar expressions b2 and .r2/c2 respectively.
for I,, and Ii3 containing If we define p so that
then IdSI can be expressed ?(I
-
[/syzc
j)r’,
-aspic
dSj
+IsfjC
]dSj = hp3JdRI dSj
(-4.2)
-~?djcO!Si).
i(L:i’dSj+krj’ds.) =- VI~,kTQ$)
in terms of solid angle:
dSi
For the integrals over the ozlter surface, we use Eshelby’s expression for the distant displacement field due to an inclusion: see Ref. 7, equation 2.23, or ref. 6, equation 2.18. The integrals are readily evaluated, and the result is
2(1 -
y2/
s matrix
(l - f)r’s
=
(-4.5)
i n’h ..sI
Ei jc dv
=
’
of the type
[6,,6,, + 6i,6,j -
and
(A.6) Thus, substituting A.6, A.5 and A.3 into A.2, we can 6nd an expression for the mean constrained strain in the matrix. The result is then substituted into equation A.l, to find
6,,6ij]
V) -2 gbi&
For t,he integrals write
I
sirsr j t
6ijslk] . (A.3) I over the surface of the inclusion, n-e UiC = (QrC + cQ2.k
where the (symmetrical) strain and (antisymmetrical) rotation are those appropriate to the solut.ion inside the inclusion. This is permissible, because the displacements are cont.inuous across the interface. The integral of interest is now of the type Ii j = (saC + Girt) Sow for an ellipsoidal vector
inclusion,
xk dS,. I SI the normal
(A.4)
+ +[di,a,j
I
dik6,, 2 b,,dij] .
-
This expression depends only on j and the transformation strain, and is valid only to first-order in j. It can easily be modified for inhomogeneous inclusions by replacing .Q,~ by Ed,** as described by Eshelby.“) When applied to single-phase materials, it is best to think of the image strain as being caused by dislocation loops. Thus if an array of shear loops, of total area A, with Burgers vector b in the X-direction, and lying on the z-y plane are trapped in a crystal, there will bc a long-range strain
to S, is a
0
0
7 -
5v
15(1 -
(E,(m)=
$
0
0
0
o
0
0 7-5v
where
15(1 )$ =
Furthermore, tains squares
the integral A.4 vanishes unless it conof co-ordinates, so that the only con-
(A.71
This easily derived plate-like inclusions and cIgT = b/26. In a single-phase
v)
v)
.1 (A.8a)
by imagining that the loops are of thickness b, so that VI = A6 crystal,
the mean
stress
in the
BROWX:
BACK
STRESSES,
IMAGE
matrix of course, depends on the shape of the tangles of dislocations which prevent the shear loops from collapsing, and prevent moving dislocations from sampling all the volume of the crystal. It is probably not a bad approximation to use equation A.8a to give an estimate of the back stress. In cases of radiation damage, it is of interest to calculate the effect of an array of interstitial prismatic loops, with Burgers’ vectors parallel to the z-direction lying on the plane. Then
STRESSES
AND
WORK-HARDENIXG
885
(A.8b) The specimen is subjected to a uniform, approximately tensile, strain.