- Email: [email protected]
Void nucleation in astroloy: theory and experiments
VOID NUCLEATION IN ASTROLOY: EXPERIMENTS M. KIKUCHIf,
K. SHKOZAWA
THEORY AND
and J. R. WEERTMAN
Department of Materials Science and Engineering and the Materials Research Center, Northwestern University, Evanston, IL 60201, U.S.A.
Abstract-The process of nucleation of grain boundary voids in astroloy by room temperature deformation and annealing has been studied both experimentally and theoretically. Cavities form at the interface between matrix and grain boundary carbides, close to the ends of the carbides. Most cavities are associated with one or two slip bands. Slip bands tend to intersect grain boundaries close to the ends of the carbides. A calculation has been made of the stresses which are produced in the neighborhood of an elastic elliptical inclusion embedded in a deformable matrix when the matrix suffers a plastic strain. Plastic deformation causes shear stresses which peak near the ends of the inclusion. These concentrations of shear stress cause the slip bands observed near the carbide ends. Estimates of the interfacial stress produced by dislocation pile ups show that under optimum conditions, a plastic strain in the matrix of about 1% is sufficient to cause decohesion (i.e. void nucleation), Residual normal stresses also are produced which drive vacancy flow during annealing and thus cause cavity growth on certain boundaries. It is noted that the build up of compressive residual stresses on transverse boundaries during high temperature creep may affect void growth. R&sam&-Nous avons ttudie thbriquemcnt et exptrimentalement la germination de cavitts intergranulaires dans. Pastroloy, par d&formation g la temptrature ambiante et recuit. Les catit& se forment B l’interface entre la matrice et les carbures intergranulaires, pr&sde l’extrimitt des carbures. La plufant des cavitts sont associ&s B une ou deux bandes de glissement. Celles-ci ont tendance B confer les joints pr4s de l’extremite des carbures. Nous avons atIc& les contraintes qui se produisent au voisinage d‘une inclusion &stique elliptique noy& dans une matrice d&formable,lorsque cette matrlai est soumise B une d&fortna$on plastique. La d&formation plastique provoque des contra&es de cisaillement qui sont maximales p&s des extr&nit&sde l’irt&tsiott. Ces conamtrations de contrainte pro~~nt la formation des bandes de glissement que f’on observe au voisinage des extrimitts des carbures. Des estimations de la co&rain& produite 8 l’interface par un empilement de dislmtions montrent que dans des conditions optimales, une dfformation plastique de la matrice de 1% environ suffit pour provoquer une d&oh&ion (c’est g dire la germination de CavitcS).Des contraintes normales residuelIes sont tgalement produites; elles provoquent un Ccoulerncnt de lacunes au au tours du recuit et entrainent ainsi la croissance de cavitcP sur certains joints de grains. Nous avons remarquC que la formation de contraintes rtsiduelles de compression sur des joints transversaux au tours du Ruage B haute temp&ature pouvait modifier la croissance des cavitts. i%sammeafassang-Die Entstehung von Hohlrlumen an Korngrenxen in einer Superlegietung auf Nickelbasis ~Astroloy’) durch plastische Ve~ormung bei Raumtem~ratur und Auslagerung wurde experimentell und theoretisch untersucht. Hohlr~ume bilden sich an den Grenxtliichen zwischen Matrix und den in Korngrenzen enthaltenen Karbidtcikhen, und zwar in der Niihe deren Enden. Die meisten Hohlrliume htigen mit einem oder zwei Gleitbiindern zusammen, welche die Korngrenxen nahe den Enden der Karbidteilchen schneiden. Die Spannungen wurden berechnet, die in der N&he eines ellip soid&migen elastischen Einschlusses in ciner verformbaren Matrix erzeugt werden, wenn die Matrix plastisch verformt wird. Durch diese Verformung werden Schersparmungen etzeugt, die in der Nilhe der Ein~u~nden ihr Maximum em&hen. Diese Spannun~~~rh~hun~ wkderum erzeugen die be+ bachteten Gleitbtinder. Abscbftzungen der von den Ver~~n~uf~u~~n in der GrenxfBche vemrsac&n Spannungen zeigen, da0 unter optimaien J3edingungeneine Abgleitung von 1% in der Matrix ausreicht, eine AblaSung zu erxeugen (d.h. Keimbildung vori ‘Hohlriiumen). Restlich Normalspannungen werden such verursacht und fiihren wilhrend der Auslagerung zu einem LeerstellenfiuO,der an gewissen Korngrenzen Hohlriiume ausbildet. Restliche Drucks~nnunge~ die sich an querliegenden Komgrenzen wiihrend des H~htem~raturkriechens aufbauen, kijnnen das Hohlraumwac~tum beeinflussen.
NOMENCLATURE o,b ba
C
Semi-major and ~mi-minor axes of elliptical inelusion. Spacing between atoms.
t Present address: Dept. of Physics, Yokohama City University, Yokohama 236, Japan.
“’
ub u;
1747 A.M. 29,‘tO--F
Parameter in elliptical coordinates. Components of the external stress applied to matrix to return hok to original shape. Components of displacement of a pbint on the hole boundary caused by application of the external stress pv Components of displacement of a point on the hole boundary caused by the plastic strain e&.
KlKUCHl
er ul.:
VOID NUCLEATION
Coordinate axes, chosen to coincide with the axes of the elliptical inclusion. x + iy, x - iy. Distribution function of infinitesimal disfocations. Young’s modulus of inclusion and matrix (assumed to be the same). Dummy variable in X direction. Coordinate axes, as defined in Fig. 11. Angle between x axis and maximum principal strain (or external stress a”). Angle between uniaxial stress p and x axis (Fig. 11). Uniform plastic strain (principal value) sustained by matrix. Critical value of rf, needed to just nucleate voids. Components of uniform plastic strain. Variables in elliptical coordinates. Constant which defines an elliptical boundary, c = eo. Poisson’s ratio of inclusion and matrix (assumed to be the same). Shear modulus of inclusion and matrix (assumed to be the same). External stress applied to matrix. Stress field in matrix (with inclusion removed) produced by pv Final stress field in matrix/inclusion ensemble. Normal stress across matrix/inclusion interface. Critical value of oa to produce interracial decohesion. Peak value of UN, located near the end of the inclusion. Shear stress at X on slip plane Y = 0 driving edge dislocations in pile up. Shear stress at X, Y = 0, produced by the stress fields of all the dislocations in the pile up. Value of the residual shear stress when resolved on the plane which maximizes it. Peak value of 7,,,,,, located near the end of the inclusion. Rigid rotation of point on hole boundary caused by the plastic strain rb. Terms in complex stress function.
INTRODUCTION Prior cold working can have a serious deleterious effect on the subsequent high temperature service life of a component made from a nickel base superalloy [l J. Dyson and Henn [2] and Dyson and McLean [3] showed that the cold working nucleates many submicron size grain boundary voids, and that it is this by-passing of the usual slow cavity nucleation process which leads to premature intergranular failure. In a recent paper [4] we reported the results of a study of the influence of microstructure on cavitation in wrought astroloy induced by prior cold working. The size, spacing, and morphology of the grain boundary carbides (primarily M&J in this nickel base superalloy can be altered considerably by varying the heat treatment [SJ. It was found that the cavity spacing is proportional to carbide spacing, and TThese first experiments were carried out by Dr M. Uemura.
THEORY AND EXPERIMENTS
indeed shadowed two stage replicas of carefully prepared surfaces of strained and annealed samples showed that voids form at the ends of carbides where they intersect the grain boundary. As Dyson et al. [6] had previously observed, the cavities occur most frequently on boundaries parallel to the direction of maximum principal stress. We postulated that void growth occurs during annealing by the diffusion of vacancies driven by residual stresses caused by the difference in deformability between matrix and carbides. This qualitative model fits the observations on void growth in strained and annealed superalloys. In the present paper we examine the process whereby the voids are nucleated. A simple model is developed which accounts for the experimental results on nucleation and which puts the explanation for void growth on a more quantitative basis. A brief preliminary account of this research has been published recently [7]. EXPERIMENTAL
DETAILS
The experiments were carried out on samples of wrought astroloy, a nickel base superalloy. The composition of this ahoy is given in Table 1. Material to be made into samples was given the following heat treatment: solutionize at 1141°C for 4 h air cool; y‘ precipitation at 760°C for 8h, air cool; stabilization at 927°C for 8 h, air cool. The grain size in the astroloy so treated is about 100 micrometers. Samples were ground and shaped by an electric discharge machine into tensile specimens. Their surfaces were polished mechanically with silicon carbide paper, then diamond paste, and finally electropolished in a solution of 10% HClOh 90% CHsCOOH at 30 volts kept at a temperature below 10°C. It has been found [4] that electropolishing under these conditions produces smooth, pit-free surfaces in astroloy. The final gauge length, width, and thickness of the tensile specimens was 12 mm, 3.2 mm and 0.8 mm, respectively. The samples were pulled at room temperature to a strain of about IV/, at a strain rate of 10w4/s. They then were annealed at 810°C in vacuum for two hours. Long straight slip lines were very evident on the sample surfaces. In a first attempt to investigate the degree of correlation between cavitation and slip impingement at the boundary, two stage shadowed replicas were made of a portion of the surface of a strained and annealed samplet. A small amount of Table 1. Composition
of wrought astrofoy used in these experiments
[email protected].% Cr 14.00-16.00 MO 4.50-5.50
co
B Fe
0.02-0.03 wt.% max 0.50
Al
3.85-4.15
Mn Si
max 0.15 max 0.20
Ti c
3.35-3.65 0.030.09
Zr Cu
max 0.06 max 0.10
Ni
balance
S P
max 0.015 max 0.015
KIKIJCHI et al.:
VOID NUCLEATION THEORY AND EXPERIMENTS
1749
Fig, 1. Slip steps on the surface and voids just below it in a sample of astroloy pulled 14%and anncakd at 810°C for 3 h. Stressaxis horizontal.Boundariesmatchedby scratch marks.(Mi~o~aphs taken by M. Uemu~) material was removed from the sample by eiectropolishing and another set of replicas were made of the same area. (The extent of surface removal was sufEciently small that the original surface grains were still present but was deep enough to expose the region of bulk cavitation.) The number of slip steps impinging on a boundary were counted and compared with the number of voids seen on the same boundary in the corresponding seoond replica (Fig. 1). The nmber of steps was observed to be roughly equal to the number of voids. However this procedure does not permit a direct linkage to be made between slip implement and cavitation.
It was found that by carefully controlling the amount of electropolishing enough surface can be removed from a sample to expose the cavitated region without completely obliterating the stronger slip traces. in this manner slip traces and cavities can be observed on the same replica (Fig. 2). The replicas taken from the lightly polished specimen surfaces show that a strong correlation exists between slip impingement at the boundary and the presence of a grain boundary void there. The replicas also reveal that slip traces are most likely to intersect a boundary at a spot close to the end of a grain boundary carbide (see Figs 2 and 3). The strength of
Fig.2 Shadowedreplica of a sample of astroloy which has been pulled at room temperamreto a strain of about loo/, then annealed at 8WC for 2 h. The sample was lightiy ekctropolishcd. Stress axis paraW to grain boundary_ Note voids (disting~Iish~ by white shadow tails) and slip traes.
1750
KIKIJCHI et (I/.: VOID NUCLEATION
THEORY AND EXPERIMENTS
Fig. 3. Shadowed replica of a sample given the same treatment as that of Fig. 2. Stress axis perpendicufar to the grain boundary. Slip traces are very evident, but no voids are seen.
this tendency for slip impingement to be localized to the ends of carbides was estimatkd by counting the number of traces hitting the “end” of a carbide and comparing this number with the total number of trams seen to be hitting the boundary. As shown in Fig. 4 an end is defined as extending over lOo/,‘ofthe length of a carbide. The extent of slip localization was tabulated as a function of the angular orientation of the grain boundary with respect to the stress axk (Actually the tab~ation is given as a function of the apparent angular orientation of the grain boundary
with respect to the stress axis, i.e., the angular orientation of the grain boundary trace with respect to the stress axis. However, several investigations [g, 91 have shown that the difference between plots of data presented in terms of true angle and in terms of apparent angle is small.) The fraction of voids associated with one or two slip traces also was determined, and tabulated as a function of orientation of the grain boundary (Fig. 5).
REmILls The numbers in Fig. 4 clearly indicate that the slip bands tend to be localized to the regions near the
void ,
I M 23c6
I
a
I grain
I
%
Slip traces hitting carbide end
u
I
Fig. 4. Fraction of slip traces hitting the end of a grain boundary carbide as a function of the angle z between the stress axis and the grain boundary trace. The numerator in a fraction is the number of traces observed hitting an end; the denominator is the total number of traces observed.
Fig. 5. Fraction of voids associated with one or two slip traces as a function of the angle a between the stress axis and the grain boundary trace.
KIKUCHI
et al.:
VOID NUCLEATION THEORY AND EXPERIMENTS
The slip traces extend away from the boundaries in
Y
1 b
Fig. 6. Two dimensional elliptical inclusion with semimajor axis Aland semi-minor axis b embedded in a homogeneous matrix.
ends of the carbides. If slip band impingement were purely random the probability of hitting an end region would be WA. (Of the traces which do hit the boundary in the middle of a carbide the point of inter~ion for many comes at a protuberance in the carbide which appears rather simifar to a carbide end.) Figure 5 shows that the majority of the voids were associated with the presence of one or two slip traces. Almost W/, of the cavities were sited at the end of a slip trace. Since some of the traces must be polished away’during surface preparation the tabulated fractions are lower limits of void-slip band association. The data of Fii 5 are in agreement with previous reports [4,6’l on the angular dependence of cavitation in cold worked and annealed superalloy: voids are far more likely to occur on boundaries roughly parallel to the tensile stress axis than on high angle boundaries. Note that there are many slip traces which intersect a carbide without any void being formed (at least of a size discernible with our replica techniques, i.e., greater than about 2Onm in diameter). The absence of any cavitation at the end of the slip traces is almost universal on boundaries transverse to the stress axis (Fig. 3). However, as Fig. 5 shows, most voids which do occur are associated with a slip trace.
MODEL FOR ~AV~A~O~ WORKED
1751
IN COLD SUPRRALLOYS
A mode1of the process of nucleation and growth of cavities produced by cold ,working and heating must account for the following observations: 1. Voids nucleate primarily at the intersection of a slip band with a grain boundary carbide. 2. Many more slip bands intersect the grain boundary at a point near the end of a grain boundary carbide than would be predicted by the laws of chance. t Recently Tanaka and Mura[12] have derived the stress field around an elastic elliptical inclusion in a deformable matrix by making use of the equivalent inclusion method. Calculations based on their method give the same results as those presented in this paper.
straight lines over distances of many micrometers
(Figs. 1,2 and 3). (Sip in superalloys such as astroloy is highly planar in character[lOJ). Therefore, whatever causes the slip bands to intersect the boundaries near the ends of the carbides must be occurring in the grain boundary region. 3. Voids form primarily on boundaries oriented ap proximately parallei to the (tensile) stress axis. Little cavitation is seen on transverse boundari~. A considerably simplifiedpicture of a portion of the grain boundary region of an astroloy sample is depicted in Fig. 6. A two dimensional elliptical inclusion with semi-major axis a and semi-minor axis b is embedded in a homogeneous matrix. The x and y axes are chosen to coincide with the axes of the inclusion. For ease of calculation it is assumed that the inclusion and matrix have the same elastic constants (E, Young’smodulus, p, shear modulus and v, Poisson’s ratio) and that both are elastically isotropic. However only the matrix is able to sustain plastic deformation. The thickness of the inclusion-matrix ensemble is taken to be large so that plane strain conditions apply. it is imagined that an external uniaxial stress 8x is applied to the matrix at some large distance from the inclusion. So long as the external stress is below the yield stress of the matrix, the ensemble deforms as one material. However once the yield stress of the matrix is exceeded it deforms plastically while the inclusion continues to respond to 6” elastically. After a certain amount of plastic deformation has taken place the external stress is removed at which point the matrix and inclusion are left in a state of residual stress. It is the purpose of this section to find the value of the residual stress field in the neighborhood of the inclusion/matrix interface as a function of the permanent plastic strain E$ sustained by the matrix. It can be appreciated readily that the process of applying tiX and permanently deforming the material corresponds to the cold working phase of the astroloy experiment. The method of calcutation is based on a procedure used in a classic paper by Eshelby [l I]. In order to find the residual stress near the mat~~mclusion interface the deformation process is pictured as consisting of 5 steps (Fig. 7):t 1. The elliptical inclusion is removed from the matrix. 2. The matrix (which now contains an elliptically shaped hole) is given a uniform permanent plastic strain e& Once the defo~ing stress has been removed the matrix is stress free, but the shape of the hole has changed. 3. A uniform external stress is applied to the matrix whose components p$$have just the value needed to bring the hoIe back to its otiginal shape. This external stress gives rise to a spaoe-dependent stress field o&z, J) in the matrix.
1752
KIKUCHI et al.:
VOID NUCLEATION
THEORY AND EXPERIMENTS
(d)
(e)
Fig. 7. Steps in the calculation of the residual stress ibid mar the inclusion/matrix interface after the matrix has been given a permanent plastic strain cg. Figures 7(a) through (e) correspond to steps l-5 described in the text.
4. The inclusion is reinserted. At t&isstage the inclusion still is stress and strain fkee. 5. The external stress is relaxed, which is equivalent to imposing additional external stresses equal to -pfj to the matrix.
At the conchision of step 5 the residual stress u& y) left in the material is: u&y)
= $,(x, y) - plj
Q(XPY)= --Pij
in the matrix in the inclusion
(1)
ul: = &x
-I-
u; = (&
(E$ - ca)y
+ 0)x + E'ny
12)
where w denotes a rigid rotation and the plastic strain components l6 are given by: & = 4
m=6gCQS2U
E~y=EgsinZa
(3)
In order to return the hole to its original shape uniform external stresses pi) are applied to the matrix [Fig 7(c)]. The value of the components pu are chosen so that the (elastic) displacements (u, uY)they produce at the hoie boundary just of&et (tl$, u;).
Note that all deformation except that of step 2 is assumed to be elastic. (Later the shear stress which is concentrated around the ends of the inclusion wil1 be u$ + u, = 0 allowed to relax by plastic deformation.) We now outu; + UJ = 0 (4) line the calculation of the value of the residual stress field rr,, in the neighborhood of the inclusion/matrix These conditions on the displacements combined with interface. Details are given in Appendix A-l. equations (A-6) for II, and ur generalized from After the inclusion has been removed (step I, Ref. cl33 lead to the foIlo~ng equations for pii: Fig 7a) the matrix is given a uniform plastic strain. The principal values of the plastic strain are (~6, - c!$) (since the deformation is accomplished without volume change). The direction of the maximum printW)2 -_ 2i4 (5) cipal strain makes an angle a with respect to the x (1 b/& axis (Fig. 7b). As a result of the deformation of step 2 2c a point originalIy located at (x, y) on the hole bound(bia) Pxr = v)Gtsin24 ary suffers a displacement (I& a;).
pvr - v)~~@~~)(I + t1
_
f1
+
bja,z.
KIKUCHI er a!.:
VUlD NUCLEATION
Y
I
b
I
I Unit of
THEORY AND EXPERIMENTS
I753
that plane is chosen which gives the largest magnitude for the shear stress, T,,,~~.Calculations of r_ were carried out for three geometries: deforming stress 0~~ and plastic strain rg (a) parallel to the long axis of the inclusion; @) perpendicular to the long axis; (c) inclined at 45” to the long axis. It is assumed that an external stress a’” and the plastic strain EC;it produces lie in the same direction. The residual shear stress WBS evaluated for a (h/u) ratio of 1W a value which is representative of the grain boundary carbides in the astroloy samples. It ~8s found that the curves of T,, are the same for cases [a), (b), and {c) (Fig. 8Nt,
Fig. g_ Residual shear sfrcss T_ at the ~~~x~~us~n in&face as a function of x, the distance along the long axis of the inclusion. At each point around the inierface the shear stress has bten resolved on the plane which maximizes its value. The shear stress distribution of this figure corresponds to d”: (a) parallel to the long axis of the inclusion; (b) perpendicular to the long axis; or (c) inclined 45” to the long axis. c, has been calculated for the ratio b/a = l/4.
Since the traction stresses must be ~nt~uous acrass the ~n~~us~on/matrix interface the normal stress crN at the interface can be found by evaluating the component of the stress field -plr in the direction perpendicular to the surface of the inclusion. Figure 9 shows the value of uN as a function of distance x along the carbide for the three situations (a), (b) and fc) mentioned above. Again the ratio of (b/u) is taken to be l/4_ The magnitu~ of arp at y = 0 outside the inclusion has been evaluated and is shown in Fig. 9.
Once the pz;‘sare known, the spatially variable stress
field +(x., y) which is produoed in the matrix can be calculated ftom equations (A-2), (A-3), and (A-4). Finally the inclusion is reinserted in ttK: matrix and the external stresses pu relaxed to zero pig. 7(d),(e)]. Since the inclusion/matrix ensemble is elastically homogeneous, this stress relaxatian leads to a uniform change in the stress and strain state of the material. The final stress state is given by equations 0). While we now have general expressions for the residual stress field in the material, it is af special interest to evaluate three particular stresses: 1. The maximum residual shear stress in the matrix adjacent to the interface. 2. The normal stress across the interface. 3. +, at y = 0 (corresponding to the aormal stress acting across the grain boundary) in the vicinity of the inclusion.
r f
I
0 ------
i
f
IOj
I
1
J
_-_c__
!
0
The residual shear stress in the matrix at the interface which is produced by the plastic deformation is shown in Fig. 8. The plane upon which the shear stress of Fig. 8 has been resolved varies in going around the periphery of the inclusion. At each poinh t Figure 8 varies from Fig. 2 in Ref. f73 because the shear stress of the latter curve was resolved on a fixed plane, one which makes a 45” angle with the x axis. Note that the peak values of the shear- stress in the two cases are identical.
Fig. 9. Normal stress a~ at the rna~~x~~nc~usio~interface and ox3 at Ix] 1 u, jr = 0, as a function of x_ (a) Plastic strain 29 parallel to the long axis of the inclusion; (b) perpendicular to the long axis; (c) inclined at 45” to the long axis. The stresses uN and uyy have been calculated for the ratio of b/a = 114.
1754
KIKUCHI
et al.:
VOID NUCLEATION
DISCUSSION Void nucleation (decohesion)
A number of excellent models of cavity nucleation at inclusions have been proposed (e.g., Ashby [14], Argon et al. [15] and Fleck, et al. [16]). However none appears to completely explain the observations listed at the beginning of the previous section. The model developed in the previous section is inaccurate for several reasons. The calculations are based on a material behavior which is unrealistic: after the initial uniform plastic straining E{, all further deformation is elastic. Large residual shear stresses, highly localized, are allowed to build up to values which exceed the elastic limit of the material before they are relaxed by slip (see below). Probably a more serious deviation of the model from reality concerns its overlooking the close proximity of other inclusions. However, despite these shortcomings, it is expected that the general predictions of the model are applicable to our samples. In the type of experiment described in this paper a sample is pulled at room temperature in a tensile testing machine until a predetermined plastic strain is achieved. The external stress a’” required to produce the deformation then is released and later the sample is annealed. The largest stress experienced by the sample occurs at the end of the straining, before the external stress is released. It is at this point that microcracking would take place to its fullest extent. With the present model it has been shown (Fig. 8) that a plastic strain et leaves a large localized shear stress close to the interface of an elastic inclusion in a deformable matrix. In the case of the external stress 6” applied parallel or perpendicular to the major axis of the inclusion, the peak residual shear stress near the long ends of the inclusion acts in the same sense as the shear stress component of a”. Therefore at the moment of maximum stress reached at the end of the room temperature deformation operation, the total localized shear stress at the interface is even larger than that shown in Fig. 8. It is not surprising that most of the heavy slip traces in the plastically deformed astroloy are seen to intersect carbides close to their extremities when the carbides lie on grain boundaries oriented roughly parallel or perpendicular to the stress axis (Fig. 4). In the case of boundaries lying at 45” to the applied stress axis, the calculations show that the peak residual shear stress is opposed to the applied shear stress. In agreement with this conclusion, slip traces on inclined boundaries are found to impinge against the carbides in a somewhat more nearly random fashion (Fig. 4). (It may be noted in passing that another assumption of the model, namely, that the inclusion is twodimensional, is not so far removed from fact. Extraction replicas of the grain boundary carbides [4] reveal a dendritic shape. The carbides shown in section in micrographs such as Figs 2 and 3 actually are cuts through long fingers.)
THEORY
AND EXPERIMENTS
The shear stresses calculated with the model on the basis of elastic response (after the initial straining ~6) are so large that stress relaxation through further plastic deformation must occur. It is this relaxation which produces the observed slip lines which are largely localized near the ends of the carbides. The dislocation pile ups created by relaxation of the shear stress of Fig. 8 produce high interfacial normal stresses near the ends of the carbides. An estimate of the magnitude of these normal stresses has been made from the model by assuming that the (known) maximum residual shear stress r,,,_(O) at the interface falls off with a relaxation distance equal to the semi-minor axis b of the inclusion, r,,,(X) P r,,,(O) e-“I*, where X is distance from the inclusion measured on the plane of the pile up (Fig. 11). The calculations, which are given in Appendix A-2, show that the greatest normal tensile stress CN,one atom distance, bo, away from the interface is:
where i,(O) is the peak value of r,, at the interface. Since ?,,,,(O) is directly proportional to the original plastic strain ~6, once a criterion has been set for the minimum normal stress needed to produce decohesion at the particle/matrix interface it will be possible to estimate the critical value of the strain, a, required to nucleate voids. A stress criterion for deco hesion may be obtained by setting On equal to the theoretical strength of the material, taken to be ap proximately equal to E/10. This condition leads to the following equation for the critical plastic strain:
If b = 2.5
x
lo-’ m, b. = 2.5 x 10-i” m, and v = l/3, r&c = 0.7%.
A critical plastic strain of approximately 1% is consistent with our observation that microcracking can be seen after 1.5% plastic strain (the smallest strain used in the tests). These estimates of the critical plastic strain needed for microcracking through decohesion do not include the contribution of the external stress, which acts in concert with the residual stress in the case of pulling parallel or perpendicular to the long axis of the inclusion or against the residual stress in the case of an inclusion inclined at 45” to a’“. It also should be pointed out that in a real crystal it is unlikely that the plane of highest shear stress will coincide with a crystallographic slip plane. In summary, both experiments and model indicate that in the course of room temperature deformation voids are nucleated in the alloy by microcracking caused by caused by decohesion at the particle/matrix interface. This decohesion is caused by the high nor-
KIKUCHI
et nf.:
VOID NUCLEATtON
ma1 tensile stresses associated with dislocations piled up against the inclusion. It should be emphasized that the slip bands associated with decohesion do not originate with some event far from the grain boundaries. Rather they result from the incompatibilities under deformation between carbide and matrix and are created by stress concentrations in the close vicinity of the boundaries. It could be argued that the microcracking occurs first and that slip bands then are produced by stress con~nt~tions around the cracks. However, as has been noted, the majority of slip lines are not ~sociated with any cracks or voids. On the other hand few voids are seen which are not sited at the intersection of a slip line with a carbide. Therefore it seems logical to conclude that slip leads to microcracking, not that voids produce the observed slip. Void growth The model shows how cold working an alloy containing hard grain boundary particles can lead to decohesion at the particle ends. Voids do not become apparent, however, until the material has been heated. It appears likely that at an elevated tem~rature the cavities grow by vacancy diision driven by residual tensile stresses [4,6]. Voids are observed primarily on those boundaries which lie parallel to the stress axis although deco&ion should take place equally well on parallel and perpendicular boundaries. Figure 9 shows that the normal stresses at the particle/matrix interface are large and tensile at the ends of carbides on boundaries parallel to the stress axis. Of course in the immediate vicinity of the decohesion the normal stress drops to zero, but a tensile stress persists elsewhere. It is the stress gradient surrounding the decohesed region which drives the vacancy flow toward the void [lq. On perpendicular boundaries the residual stresses are compressive, so no voids grow even though microcracking has taken place. Indeed, the direction of the vacancy flow would be such that these microcracks would tend to heal. Application of the model to creep cavitation
While the experiments and model described in this report refer to voids produced by cold working and annealing the application to cavitation in creep is obvious. It has been observed by several workers that the number of cavities is pro~rtion~ to plastic strain. Usually a small incubation strain is required before cavitation begins [16,18,19]. In a study of creep cavitation in 347 austenitic steel, Needham and Gladman [t8] found that the number of grain boundary voids in a sample is a linear function of the steady state creep strain, but is essentially independent of temperature or stress. The commonly observed facts of cavity nucleation during creep in material containing hard grain boundary particles are consistent with the predictions of our model. When the creep strain reaches the critical value E&., particle decohesion commences on
THEORY
AND EXPERIME~S
1755
those boundaries which are oriented parallel or perpendicular to the stress axis and which are adjacent to grains with slip planes oriented favorably with respect to the localized shear stress T,, arising from incompatibility of deformation between carbide and matrix, As creep proceeds void nucleation can occur on boundaries with less optimum conditions for decohesion. Thus nucleation is continuous during creep. Unlike the case of cavitation produced by cold work and annealing, voids which occur during creep are most likely to be found on boundaries transverse to the stress axis. Since the external &fo~ing stress a’” (i.e., the creep stress) is applied to the sample throughout the test, it is the transverse boundaries which are likely to experience the greatest tensile stress. It should be noted, however, that the compressive residual stresses continue to grow (unless relieved by some recovery process) as creep proceeds and thus they reduce the effective stress driving the vacancy flow. In view of the magnitude of the residual stresses it is perhaps not surprising that cavitation growth is a slower process than calculated on the basis of applied stress alone [ZO,211. SUMMARY
AND CONCLUSIONS
The process of nucleation of grain boundary voids produced in astroloy by room temperature deformation and subsequent annealing has been studied both experimentally and theoretically. Cavities form at the interface between grain boundary carbides and matrix close to the ends of the carbides. There is a very strong correlation between the presence of a void on the end of a particular carbide and the impingement of a coarse slip band with the carbide at just that spot. While slip bands are observed to hit ali the grain boundaries, cavities are seen most frequently on those boundaries which are roughly parallel to the deformation axis (tentrife stress axis). Coarse slip bands are most likely to intersect the boundaries close to the ends of the carbides. A calculation has been made of the residual stresses which exist around an elastic elliptical inclusion embedded in a deformable matrix when the matrix suffers a plastic strain. It was found that residual shear stresses are left around the inch&on in the wake of the plastic defo~atio~ and that these shear stresses are sharply peaked in the region around the ends of the inclusion. Such concentrations of shear stress are responsible for the slip bands observed near the ends of the carbides. Estimates of the stress produced at an inclusion interface by the pile up of the dislocations in a slip band show that a plastic strain in the matrix of about 1% is sufficient to cause decohesion (i.e., void nucleation). Residual normal stresses also are produced around the inclusion when the matrix is deformed. If the long axis of the inclusion lies parallel to the deformation axis the residual stresses acting across the ends of the
1756
KIKUCHI et al.:
VOID NUCLEATION
inclusion at the interface are tensile. During the annealing period these tensile stresses drive vacancy flow and thus are responsible for cavity growth. The residual normal stresses around an inclusion with long axis perpendicular to the strain axis are compressive. The model predicts cavitation on parallel boundaries, none on transverse boundaries. It is noted that the mechanism of void nucleation proposed to explain cavitation resulting from room temperature deformation and annealing also can be responsible for cavity nucleation during high temperature creep. Once a critical plastic strain has been reached nucleation becomes continuous. The presence of large compressive residual stresses on transverse boundaries is likely to have a considerable effect on the rate of void growth. Acknowledgements-We are greatly indebted to Mr M. VanWanderham of Pratt and Whitney Aircraft for the astroloy. We are grateful to Dr M. Uemura, formerly a Postdoctoral Fellow at Northwestern University and now an Assistant Professor at Osaka University, Osaka, Japan, for supplying Fig. 1. In this research, extensive use was made of the facilities of Northwestern University’s Materials Research Center, funded by the NSF-MRL program, grant number DMR76-80847. This work was soonsored bv the United States Air Force Office of Scientitickesearch &ant 77-3251, under the direo tion of Dr Alan Rosenstein.
REFERENcE!3 1. B. F. Dyson and M. J. Rodgers, Merol Sci. 8, 261 (1974). 2 B. F. Dyson and D. E. Henn, J. Microsc. 97, 165 (1973). 3. B. F. Dyson and D. McLean, Metal Sci. 6, 220 (1972). 4. T. Saegusa, M. Uemura and J. R. Weertman, Metall. Trans. A l&l453 (1980). 5. J. C. Runkle, Sc.D. Thesis, M.I.T., Cambridge, Massachusetts (1978). 6. B. F. Dyson, M. S. Loveday, and M. J. Rodgers, Proc. R. Sot. Land. A 349,245 (1976). 7. M. Kikuchi and J. R. Weertman, Scripta metoll. 14, 797 (1980). 8. P. W. Davies, K. R. Williams and B. Wilshire, Phil. Msg. 18, 197 (1968). 9. T. Saegusa, Ph.D. Thesis, Northwestern University, Evanston, Illinois (1978). 10. R. F. Decker and C. T. Sims, The Superalloys (edited bv C. T. Sims and W. C. Haael) op. 33-77, Wiley, New _ ._York (1972). 11. J. D. Eshelby, Proc. R. Sot. Land. A 241, 376 (1957). 12. K. Tanaka and T. Mum, J. appl. Mech. in press. 13. A. C. Stevenson, Proc. R. Sot. Lond. A 184, 128, 218 (1945). 14. M. F. Ashby, Phil. Msg. 14, 1157 (1966). 15. A. S. Argon, J. Im, and R. Safoglu, Metall. Truns. A 6, 825 (1975). 16. R. G. FIeck, D. M. R. Taplin, and C. J. Beevers, Acta metall. 23,415 (1975). 17. D. Hull and D. E. Rimmer, Phi/. Mag. 4, 673 (1959). 18. N. G. Needham and T. Gladman. Metal Sci. 14. 64 (198tl). 19. T. Johannesson and A. ThSlen. J. Il~sr. Merals 97, 243 ( 1969).
THEORY AND EXPERIMENTS 20. W. Pavinich and R. Raj, Metall. Trans. A 8, 1917 (1977). 21. B. F. Dyson, Metal Sci. 10, 349 (1976). 22 S. P. Timoshenko and J. N. Goodier. Theorv of Elasticity, 3rd edn, McGraw-Hill Book’ Co., New York (1970). 23. J. Weertman, Bull. Seism. Sot. Am. 54, 1035 (1964).
A-l
APPENDIX
We consider the problem of determining the uniform external stresses p,, which must be applied to the inclusion/ matrix ensemble in order to produce the (elastic) displacements (uI, uI) specified by equations (4). Stevenson [13] has solved the problem of a thick plate containing an elliptical hole which is loaded under uniaxial tension p acting at an angle /3 with respect to the major axis of the hole (Fig 10). It is assumed that the stresses are applied at a position far removed from the hole. He obtained the following equations for the stress function [22] Q, = Re[TY(z) + _Y(z)],where z=x+iyandi=x-iy: 4Y(z) = pc[e2Cocos2fic&hC
+ (1 - e24*2’“)sinh[] (A-l)
4x(z) = -pcq(cosli
2&J - cos Z/!I)C+ +?” x cash 2([ - Co - is)]
Here use has been made of elliptical coordinates:
x = ccosh{cosq,y
= csinh
The elliptical boundary of the hole is given by C:= cc. The semi-major and semi-minor axes can be expressed in terms of Co and c: a = ccosh Co, b = c sinh Co. Stevenson’s solution for a uniaxial stress can be generalized to the case of an arbitrary stress p,, applied to the plate by superposition of the following special cases: B = 0.
Pf,
P-*Pn.
s-f,
P
/I__!!,
-p
Fig. 10. Uniaxial stress p applied to a thick plate containing an elliptical hole. The stress p makes an angle B with respect to the long axis of the elliptical hole.
KIKUCHI
et al.:
VOID NUCLEATION
TWEORY
AND EXPERIMENTS
1757
distributed infinitesimal dislocations, and define the distribution function B(X)dX as equal to the sum of the Burgers vectors of all the infinitesimal dislocations located between X and X + dX. The shear stress ox&, 0) on the slip plane caused by the stress fields of the dislocations in the pile up is:?
a&X. 0) =
- B(S)dS P 2x(1 - v) f 0 (x).
(A-7
Under equilibrium conditions, the total shear stress on the slip plane in the region 0 s X e co must vanish: o*r(X, 0) + a&(X, 0) = 0.
I
Equation (A-7) can be inverted [UJ and combined with (A-8) to yield an expression for B(X) in terms of a&:
Fig. Il. Dislocation pile up at end of inclusion.
4Y’(zf = c&,
4x(2) = --c’
+ pry) sinh I + e’“((p,,
iI
HP,, + p&=h%
(A-8)
(A_9)
- p&z-’ - 2ip,, sinh < )]
- (P, - P,X I
(A-2)
The normal stress uxx(X, Y) produced by the stress field of the continuous dislocations is: o&X,
- ip*, sinh 2(5 - to)
Y) = --
(o ~(S)Y~3(X - S)* + Y*; dS 2a(l
:(x - S)2 + Y2)2
11
The calculation of the ~spla~ments and stress components in the plate is a straight forward process once the stress function is known (22). o:, + a,: = 2[vr’(z) + @(‘(i)J
(A-3)
uzr - o:~ + Ziu.$ = 2[gY “(2) + x”(z)] 2&,
(A-4)
+ iu,) = (3 - 4v)Y’(z)- @‘(Z) - f’(i)
(A-5)
where the primes denote differentiation and the superscript bars the complex conjugate. From equations (A-2) and (A-S), with the substitution that x = a cos q, y = b sin r), the following equations are found for the displacements u, and ur on the edge of the hole as a function of pi,:
We wish to evaluate plane from below the tensile normal stress. involving Dirac delta
:z
a&X, Y) as Y approaches the slip X axis, i.e., in the region of maximum Use is made of the following limits functions, a(X):
Y = &nRa(X - S) (X - s)* 4 Y2
Y(X - s)* lim =+5(X-S). r-0 ((X - s)* + Y2f2 The sign to be taken is that of the direction of approach to zero, in this case the minus sign. (A-10) Equations (A-9) and (A-f0) are combined to give:
Equations (A-6) are combined obtain equations (5):
APPENDIX
with Q), (3), and (4) to
A-2
In this section we first calculate the distribution of dislocations in a pile up which is driven by a known shear stress, a&, acting on the slip plane. Once the distribution function is known it is possible to determine the normal stress at the barrier caused by the pile up. A stress criterion for decohesion can be obtained by setting this stress equal to the theoretical strength of the material. Consider a pile up of edge dislocations on the slip plane Y = 0 (Fig. 1I). The barrier to dislocation motion is located at X = 0. We make use of the concept of continuously t It is assumed that the Cauchy principal taken for this, and the following integrals.
values are
2 1 @xX(X*Y) = - -2 r-o-
dDu&(S, 0)S”2 dS f0
(S - X)
. (A-11)
The driving stress u&(X,0) can be identified with the residual shear stress r,,,&X) if it is assumed that the Y = 0 plane coincides with the plane on which r,, a&s. In the case of cf” parallel or perpendicular to the long axis of the inclusion, the sense of r,, is such that ~~r(X,O) = -r,,(X). In order to estimate a value for uxx(X,O-) as X approaches the inclusion interface it is assumed that ?MI falls off from its value close to the interface, ‘c,.(O), as: T&X)
= r,,(0)e-x’b
where b is the semi-minor axis of the inclusion. With this substitution equation (A-i 1) becomes:
KIKUCHI et al.:
1758
For values of X approaching evaluated directly:
X-O
uxxw,Y) -
$
VOID NUCLEATION
zero the integral can be
L(OlwX)“‘.
At a distance of one atom spacing, b,,, from the inclusion interface, oxx(X. O-) becomes:
udbo. 0-l -
-&~,.(0)Wbo)“2.
0412)
THEORY AND EXPERIMENTS
point along the inclusion/matrix interface where T,,(O) has its peak value&,,(O)). Therefore axx(bo. O-) at T,,,_(O)can be identified with Z,,(b,,) of equation (6) in the Discussion Section. From Fig. 8 it can be seen that the peak value of TV_ near an elliptical inclusion with shape b/a = l/4 is
L,(O) -
&
=g$).
Dccohcsion at the intcrfaa should be insured when the normal stress u,(b,) reaches a critical value a,,,: UN.<* E/10.
It can be shown that, for d” and rg parallel or perpendicular to the long axis of the inclusion the outward normal to the interfaa coincides with the X axis at the
(A-13)
(A-14)
The combination of (A-12), (A-13), and (A-14) leads to equation (7) for the critical plastic strain lJvc needed to nucleate voids.
K. SHKOZAWA
THEORY AND
and J. R. WEERTMAN
Department of Materials Science and Engineering and the Materials Research Center, Northwestern University, Evanston, IL 60201, U.S.A.
Abstract-The process of nucleation of grain boundary voids in astroloy by room temperature deformation and annealing has been studied both experimentally and theoretically. Cavities form at the interface between matrix and grain boundary carbides, close to the ends of the carbides. Most cavities are associated with one or two slip bands. Slip bands tend to intersect grain boundaries close to the ends of the carbides. A calculation has been made of the stresses which are produced in the neighborhood of an elastic elliptical inclusion embedded in a deformable matrix when the matrix suffers a plastic strain. Plastic deformation causes shear stresses which peak near the ends of the inclusion. These concentrations of shear stress cause the slip bands observed near the carbide ends. Estimates of the interfacial stress produced by dislocation pile ups show that under optimum conditions, a plastic strain in the matrix of about 1% is sufficient to cause decohesion (i.e. void nucleation), Residual normal stresses also are produced which drive vacancy flow during annealing and thus cause cavity growth on certain boundaries. It is noted that the build up of compressive residual stresses on transverse boundaries during high temperature creep may affect void growth. R&sam&-Nous avons ttudie thbriquemcnt et exptrimentalement la germination de cavitts intergranulaires dans. Pastroloy, par d&formation g la temptrature ambiante et recuit. Les catit& se forment B l’interface entre la matrice et les carbures intergranulaires, pr&sde l’extrimitt des carbures. La plufant des cavitts sont associ&s B une ou deux bandes de glissement. Celles-ci ont tendance B confer les joints pr4s de l’extremite des carbures. Nous avons atIc& les contraintes qui se produisent au voisinage d‘une inclusion &stique elliptique noy& dans une matrice d&formable,lorsque cette matrlai est soumise B une d&fortna$on plastique. La d&formation plastique provoque des contra&es de cisaillement qui sont maximales p&s des extr&nit&sde l’irt&tsiott. Ces conamtrations de contrainte pro~~nt la formation des bandes de glissement que f’on observe au voisinage des extrimitts des carbures. Des estimations de la co&rain& produite 8 l’interface par un empilement de dislmtions montrent que dans des conditions optimales, une dfformation plastique de la matrice de 1% environ suffit pour provoquer une d&oh&ion (c’est g dire la germination de CavitcS).Des contraintes normales residuelIes sont tgalement produites; elles provoquent un Ccoulerncnt de lacunes au au tours du recuit et entrainent ainsi la croissance de cavitcP sur certains joints de grains. Nous avons remarquC que la formation de contraintes rtsiduelles de compression sur des joints transversaux au tours du Ruage B haute temp&ature pouvait modifier la croissance des cavitts. i%sammeafassang-Die Entstehung von Hohlrlumen an Korngrenxen in einer Superlegietung auf Nickelbasis ~Astroloy’) durch plastische Ve~ormung bei Raumtem~ratur und Auslagerung wurde experimentell und theoretisch untersucht. Hohlr~ume bilden sich an den Grenxtliichen zwischen Matrix und den in Korngrenzen enthaltenen Karbidtcikhen, und zwar in der Niihe deren Enden. Die meisten Hohlrliume htigen mit einem oder zwei Gleitbiindern zusammen, welche die Korngrenxen nahe den Enden der Karbidteilchen schneiden. Die Spannungen wurden berechnet, die in der N&he eines ellip soid&migen elastischen Einschlusses in ciner verformbaren Matrix erzeugt werden, wenn die Matrix plastisch verformt wird. Durch diese Verformung werden Schersparmungen etzeugt, die in der Nilhe der Ein~u~nden ihr Maximum em&hen. Diese Spannun~~~rh~hun~ wkderum erzeugen die be+ bachteten Gleitbtinder. Abscbftzungen der von den Ver~~n~uf~u~~n in der GrenxfBche vemrsac&n Spannungen zeigen, da0 unter optimaien J3edingungeneine Abgleitung von 1% in der Matrix ausreicht, eine AblaSung zu erxeugen (d.h. Keimbildung vori ‘Hohlriiumen). Restlich Normalspannungen werden such verursacht und fiihren wilhrend der Auslagerung zu einem LeerstellenfiuO,der an gewissen Korngrenzen Hohlriiume ausbildet. Restliche Drucks~nnunge~ die sich an querliegenden Komgrenzen wiihrend des H~htem~raturkriechens aufbauen, kijnnen das Hohlraumwac~tum beeinflussen.
NOMENCLATURE o,b ba
C
Semi-major and ~mi-minor axes of elliptical inelusion. Spacing between atoms.
t Present address: Dept. of Physics, Yokohama City University, Yokohama 236, Japan.
“’
ub u;
1747 A.M. 29,‘tO--F
Parameter in elliptical coordinates. Components of the external stress applied to matrix to return hok to original shape. Components of displacement of a pbint on the hole boundary caused by application of the external stress pv Components of displacement of a point on the hole boundary caused by the plastic strain e&.
KlKUCHl
er ul.:
VOID NUCLEATION
Coordinate axes, chosen to coincide with the axes of the elliptical inclusion. x + iy, x - iy. Distribution function of infinitesimal disfocations. Young’s modulus of inclusion and matrix (assumed to be the same). Dummy variable in X direction. Coordinate axes, as defined in Fig. 11. Angle between x axis and maximum principal strain (or external stress a”). Angle between uniaxial stress p and x axis (Fig. 11). Uniform plastic strain (principal value) sustained by matrix. Critical value of rf, needed to just nucleate voids. Components of uniform plastic strain. Variables in elliptical coordinates. Constant which defines an elliptical boundary, c = eo. Poisson’s ratio of inclusion and matrix (assumed to be the same). Shear modulus of inclusion and matrix (assumed to be the same). External stress applied to matrix. Stress field in matrix (with inclusion removed) produced by pv Final stress field in matrix/inclusion ensemble. Normal stress across matrix/inclusion interface. Critical value of oa to produce interracial decohesion. Peak value of UN, located near the end of the inclusion. Shear stress at X on slip plane Y = 0 driving edge dislocations in pile up. Shear stress at X, Y = 0, produced by the stress fields of all the dislocations in the pile up. Value of the residual shear stress when resolved on the plane which maximizes it. Peak value of 7,,,,,, located near the end of the inclusion. Rigid rotation of point on hole boundary caused by the plastic strain rb. Terms in complex stress function.
INTRODUCTION Prior cold working can have a serious deleterious effect on the subsequent high temperature service life of a component made from a nickel base superalloy [l J. Dyson and Henn [2] and Dyson and McLean [3] showed that the cold working nucleates many submicron size grain boundary voids, and that it is this by-passing of the usual slow cavity nucleation process which leads to premature intergranular failure. In a recent paper [4] we reported the results of a study of the influence of microstructure on cavitation in wrought astroloy induced by prior cold working. The size, spacing, and morphology of the grain boundary carbides (primarily M&J in this nickel base superalloy can be altered considerably by varying the heat treatment [SJ. It was found that the cavity spacing is proportional to carbide spacing, and TThese first experiments were carried out by Dr M. Uemura.
THEORY AND EXPERIMENTS
indeed shadowed two stage replicas of carefully prepared surfaces of strained and annealed samples showed that voids form at the ends of carbides where they intersect the grain boundary. As Dyson et al. [6] had previously observed, the cavities occur most frequently on boundaries parallel to the direction of maximum principal stress. We postulated that void growth occurs during annealing by the diffusion of vacancies driven by residual stresses caused by the difference in deformability between matrix and carbides. This qualitative model fits the observations on void growth in strained and annealed superalloys. In the present paper we examine the process whereby the voids are nucleated. A simple model is developed which accounts for the experimental results on nucleation and which puts the explanation for void growth on a more quantitative basis. A brief preliminary account of this research has been published recently [7]. EXPERIMENTAL
DETAILS
The experiments were carried out on samples of wrought astroloy, a nickel base superalloy. The composition of this ahoy is given in Table 1. Material to be made into samples was given the following heat treatment: solutionize at 1141°C for 4 h air cool; y‘ precipitation at 760°C for 8h, air cool; stabilization at 927°C for 8 h, air cool. The grain size in the astroloy so treated is about 100 micrometers. Samples were ground and shaped by an electric discharge machine into tensile specimens. Their surfaces were polished mechanically with silicon carbide paper, then diamond paste, and finally electropolished in a solution of 10% HClOh 90% CHsCOOH at 30 volts kept at a temperature below 10°C. It has been found [4] that electropolishing under these conditions produces smooth, pit-free surfaces in astroloy. The final gauge length, width, and thickness of the tensile specimens was 12 mm, 3.2 mm and 0.8 mm, respectively. The samples were pulled at room temperature to a strain of about IV/, at a strain rate of 10w4/s. They then were annealed at 810°C in vacuum for two hours. Long straight slip lines were very evident on the sample surfaces. In a first attempt to investigate the degree of correlation between cavitation and slip impingement at the boundary, two stage shadowed replicas were made of a portion of the surface of a strained and annealed samplet. A small amount of Table 1. Composition
of wrought astrofoy used in these experiments
[email protected].% Cr 14.00-16.00 MO 4.50-5.50
co
B Fe
0.02-0.03 wt.% max 0.50
Al
3.85-4.15
Mn Si
max 0.15 max 0.20
Ti c
3.35-3.65 0.030.09
Zr Cu
max 0.06 max 0.10
Ni
balance
S P
max 0.015 max 0.015
KIKIJCHI et al.:
VOID NUCLEATION THEORY AND EXPERIMENTS
1749
Fig, 1. Slip steps on the surface and voids just below it in a sample of astroloy pulled 14%and anncakd at 810°C for 3 h. Stressaxis horizontal.Boundariesmatchedby scratch marks.(Mi~o~aphs taken by M. Uemu~) material was removed from the sample by eiectropolishing and another set of replicas were made of the same area. (The extent of surface removal was sufEciently small that the original surface grains were still present but was deep enough to expose the region of bulk cavitation.) The number of slip steps impinging on a boundary were counted and compared with the number of voids seen on the same boundary in the corresponding seoond replica (Fig. 1). The nmber of steps was observed to be roughly equal to the number of voids. However this procedure does not permit a direct linkage to be made between slip implement and cavitation.
It was found that by carefully controlling the amount of electropolishing enough surface can be removed from a sample to expose the cavitated region without completely obliterating the stronger slip traces. in this manner slip traces and cavities can be observed on the same replica (Fig. 2). The replicas taken from the lightly polished specimen surfaces show that a strong correlation exists between slip impingement at the boundary and the presence of a grain boundary void there. The replicas also reveal that slip traces are most likely to intersect a boundary at a spot close to the end of a grain boundary carbide (see Figs 2 and 3). The strength of
Fig.2 Shadowedreplica of a sample of astroloy which has been pulled at room temperamreto a strain of about loo/, then annealed at 8WC for 2 h. The sample was lightiy ekctropolishcd. Stress axis paraW to grain boundary_ Note voids (disting~Iish~ by white shadow tails) and slip traes.
1750
KIKIJCHI et (I/.: VOID NUCLEATION
THEORY AND EXPERIMENTS
Fig. 3. Shadowed replica of a sample given the same treatment as that of Fig. 2. Stress axis perpendicufar to the grain boundary. Slip traces are very evident, but no voids are seen.
this tendency for slip impingement to be localized to the ends of carbides was estimatkd by counting the number of traces hitting the “end” of a carbide and comparing this number with the total number of trams seen to be hitting the boundary. As shown in Fig. 4 an end is defined as extending over lOo/,‘ofthe length of a carbide. The extent of slip localization was tabulated as a function of the angular orientation of the grain boundary with respect to the stress axk (Actually the tab~ation is given as a function of the apparent angular orientation of the grain boundary
with respect to the stress axis, i.e., the angular orientation of the grain boundary trace with respect to the stress axis. However, several investigations [g, 91 have shown that the difference between plots of data presented in terms of true angle and in terms of apparent angle is small.) The fraction of voids associated with one or two slip traces also was determined, and tabulated as a function of orientation of the grain boundary (Fig. 5).
REmILls The numbers in Fig. 4 clearly indicate that the slip bands tend to be localized to the regions near the
void ,
I M 23c6
I
a
I grain
I
%
Slip traces hitting carbide end
u
I
Fig. 4. Fraction of slip traces hitting the end of a grain boundary carbide as a function of the angle z between the stress axis and the grain boundary trace. The numerator in a fraction is the number of traces observed hitting an end; the denominator is the total number of traces observed.
Fig. 5. Fraction of voids associated with one or two slip traces as a function of the angle a between the stress axis and the grain boundary trace.
KIKUCHI
et al.:
VOID NUCLEATION THEORY AND EXPERIMENTS
The slip traces extend away from the boundaries in
Y
1 b
Fig. 6. Two dimensional elliptical inclusion with semimajor axis Aland semi-minor axis b embedded in a homogeneous matrix.
ends of the carbides. If slip band impingement were purely random the probability of hitting an end region would be WA. (Of the traces which do hit the boundary in the middle of a carbide the point of inter~ion for many comes at a protuberance in the carbide which appears rather simifar to a carbide end.) Figure 5 shows that the majority of the voids were associated with the presence of one or two slip traces. Almost W/, of the cavities were sited at the end of a slip trace. Since some of the traces must be polished away’during surface preparation the tabulated fractions are lower limits of void-slip band association. The data of Fii 5 are in agreement with previous reports [4,6’l on the angular dependence of cavitation in cold worked and annealed superalloy: voids are far more likely to occur on boundaries roughly parallel to the tensile stress axis than on high angle boundaries. Note that there are many slip traces which intersect a carbide without any void being formed (at least of a size discernible with our replica techniques, i.e., greater than about 2Onm in diameter). The absence of any cavitation at the end of the slip traces is almost universal on boundaries transverse to the stress axis (Fig. 3). However, as Fig. 5 shows, most voids which do occur are associated with a slip trace.
MODEL FOR ~AV~A~O~ WORKED
1751
IN COLD SUPRRALLOYS
A mode1of the process of nucleation and growth of cavities produced by cold ,working and heating must account for the following observations: 1. Voids nucleate primarily at the intersection of a slip band with a grain boundary carbide. 2. Many more slip bands intersect the grain boundary at a point near the end of a grain boundary carbide than would be predicted by the laws of chance. t Recently Tanaka and Mura[12] have derived the stress field around an elastic elliptical inclusion in a deformable matrix by making use of the equivalent inclusion method. Calculations based on their method give the same results as those presented in this paper.
straight lines over distances of many micrometers
(Figs. 1,2 and 3). (Sip in superalloys such as astroloy is highly planar in character[lOJ). Therefore, whatever causes the slip bands to intersect the boundaries near the ends of the carbides must be occurring in the grain boundary region. 3. Voids form primarily on boundaries oriented ap proximately parallei to the (tensile) stress axis. Little cavitation is seen on transverse boundari~. A considerably simplifiedpicture of a portion of the grain boundary region of an astroloy sample is depicted in Fig. 6. A two dimensional elliptical inclusion with semi-major axis a and semi-minor axis b is embedded in a homogeneous matrix. The x and y axes are chosen to coincide with the axes of the inclusion. For ease of calculation it is assumed that the inclusion and matrix have the same elastic constants (E, Young’smodulus, p, shear modulus and v, Poisson’s ratio) and that both are elastically isotropic. However only the matrix is able to sustain plastic deformation. The thickness of the inclusion-matrix ensemble is taken to be large so that plane strain conditions apply. it is imagined that an external uniaxial stress 8x is applied to the matrix at some large distance from the inclusion. So long as the external stress is below the yield stress of the matrix, the ensemble deforms as one material. However once the yield stress of the matrix is exceeded it deforms plastically while the inclusion continues to respond to 6” elastically. After a certain amount of plastic deformation has taken place the external stress is removed at which point the matrix and inclusion are left in a state of residual stress. It is the purpose of this section to find the value of the residual stress field in the neighborhood of the inclusion/matrix interface as a function of the permanent plastic strain E$ sustained by the matrix. It can be appreciated readily that the process of applying tiX and permanently deforming the material corresponds to the cold working phase of the astroloy experiment. The method of calcutation is based on a procedure used in a classic paper by Eshelby [l I]. In order to find the residual stress near the mat~~mclusion interface the deformation process is pictured as consisting of 5 steps (Fig. 7):t 1. The elliptical inclusion is removed from the matrix. 2. The matrix (which now contains an elliptically shaped hole) is given a uniform permanent plastic strain e& Once the defo~ing stress has been removed the matrix is stress free, but the shape of the hole has changed. 3. A uniform external stress is applied to the matrix whose components p$$have just the value needed to bring the hoIe back to its otiginal shape. This external stress gives rise to a spaoe-dependent stress field o&z, J) in the matrix.
1752
KIKUCHI et al.:
VOID NUCLEATION
THEORY AND EXPERIMENTS
(d)
(e)
Fig. 7. Steps in the calculation of the residual stress ibid mar the inclusion/matrix interface after the matrix has been given a permanent plastic strain cg. Figures 7(a) through (e) correspond to steps l-5 described in the text.
4. The inclusion is reinserted. At t&isstage the inclusion still is stress and strain fkee. 5. The external stress is relaxed, which is equivalent to imposing additional external stresses equal to -pfj to the matrix.
At the conchision of step 5 the residual stress u& y) left in the material is: u&y)
= $,(x, y) - plj
Q(XPY)= --Pij
in the matrix in the inclusion
(1)
ul: = &x
-I-
u; = (&
(E$ - ca)y
+ 0)x + E'ny
12)
where w denotes a rigid rotation and the plastic strain components l6 are given by: & = 4
m=6gCQS2U
E~y=EgsinZa
(3)
In order to return the hole to its original shape uniform external stresses pi) are applied to the matrix [Fig 7(c)]. The value of the components pu are chosen so that the (elastic) displacements (u, uY)they produce at the hoie boundary just of&et (tl$, u;).
Note that all deformation except that of step 2 is assumed to be elastic. (Later the shear stress which is concentrated around the ends of the inclusion wil1 be u$ + u, = 0 allowed to relax by plastic deformation.) We now outu; + UJ = 0 (4) line the calculation of the value of the residual stress field rr,, in the neighborhood of the inclusion/matrix These conditions on the displacements combined with interface. Details are given in Appendix A-l. equations (A-6) for II, and ur generalized from After the inclusion has been removed (step I, Ref. cl33 lead to the foIlo~ng equations for pii: Fig 7a) the matrix is given a uniform plastic strain. The principal values of the plastic strain are (~6, - c!$) (since the deformation is accomplished without volume change). The direction of the maximum printW)2 -_ 2i4 (5) cipal strain makes an angle a with respect to the x (1 b/& axis (Fig. 7b). As a result of the deformation of step 2 2c a point originalIy located at (x, y) on the hole bound(bia) Pxr = v)Gtsin24 ary suffers a displacement (I& a;).
pvr - v)~~@~~)(I + t1
_
f1
+
bja,z.
KIKUCHI er a!.:
VUlD NUCLEATION
Y
I
b
I
I Unit of
THEORY AND EXPERIMENTS
I753
that plane is chosen which gives the largest magnitude for the shear stress, T,,,~~.Calculations of r_ were carried out for three geometries: deforming stress 0~~ and plastic strain rg (a) parallel to the long axis of the inclusion; @) perpendicular to the long axis; (c) inclined at 45” to the long axis. It is assumed that an external stress a’” and the plastic strain EC;it produces lie in the same direction. The residual shear stress WBS evaluated for a (h/u) ratio of 1W a value which is representative of the grain boundary carbides in the astroloy samples. It ~8s found that the curves of T,, are the same for cases [a), (b), and {c) (Fig. 8Nt,
Fig. g_ Residual shear sfrcss T_ at the ~~~x~~us~n in&face as a function of x, the distance along the long axis of the inclusion. At each point around the inierface the shear stress has bten resolved on the plane which maximizes its value. The shear stress distribution of this figure corresponds to d”: (a) parallel to the long axis of the inclusion; (b) perpendicular to the long axis; or (c) inclined 45” to the long axis. c, has been calculated for the ratio b/a = l/4.
Since the traction stresses must be ~nt~uous acrass the ~n~~us~on/matrix interface the normal stress crN at the interface can be found by evaluating the component of the stress field -plr in the direction perpendicular to the surface of the inclusion. Figure 9 shows the value of uN as a function of distance x along the carbide for the three situations (a), (b) and fc) mentioned above. Again the ratio of (b/u) is taken to be l/4_ The magnitu~ of arp at y = 0 outside the inclusion has been evaluated and is shown in Fig. 9.
Once the pz;‘sare known, the spatially variable stress
field +(x., y) which is produoed in the matrix can be calculated ftom equations (A-2), (A-3), and (A-4). Finally the inclusion is reinserted in ttK: matrix and the external stresses pu relaxed to zero pig. 7(d),(e)]. Since the inclusion/matrix ensemble is elastically homogeneous, this stress relaxatian leads to a uniform change in the stress and strain state of the material. The final stress state is given by equations 0). While we now have general expressions for the residual stress field in the material, it is af special interest to evaluate three particular stresses: 1. The maximum residual shear stress in the matrix adjacent to the interface. 2. The normal stress across the interface. 3. +, at y = 0 (corresponding to the aormal stress acting across the grain boundary) in the vicinity of the inclusion.
r f
I
0 ------
i
f
IOj
I
1
J
_-_c__
!
0
The residual shear stress in the matrix at the interface which is produced by the plastic deformation is shown in Fig. 8. The plane upon which the shear stress of Fig. 8 has been resolved varies in going around the periphery of the inclusion. At each poinh t Figure 8 varies from Fig. 2 in Ref. f73 because the shear stress of the latter curve was resolved on a fixed plane, one which makes a 45” angle with the x axis. Note that the peak values of the shear- stress in the two cases are identical.
Fig. 9. Normal stress a~ at the rna~~x~~nc~usio~interface and ox3 at Ix] 1 u, jr = 0, as a function of x_ (a) Plastic strain 29 parallel to the long axis of the inclusion; (b) perpendicular to the long axis; (c) inclined at 45” to the long axis. The stresses uN and uyy have been calculated for the ratio of b/a = 114.
1754
KIKUCHI
et al.:
VOID NUCLEATION
DISCUSSION Void nucleation (decohesion)
A number of excellent models of cavity nucleation at inclusions have been proposed (e.g., Ashby [14], Argon et al. [15] and Fleck, et al. [16]). However none appears to completely explain the observations listed at the beginning of the previous section. The model developed in the previous section is inaccurate for several reasons. The calculations are based on a material behavior which is unrealistic: after the initial uniform plastic straining E{, all further deformation is elastic. Large residual shear stresses, highly localized, are allowed to build up to values which exceed the elastic limit of the material before they are relaxed by slip (see below). Probably a more serious deviation of the model from reality concerns its overlooking the close proximity of other inclusions. However, despite these shortcomings, it is expected that the general predictions of the model are applicable to our samples. In the type of experiment described in this paper a sample is pulled at room temperature in a tensile testing machine until a predetermined plastic strain is achieved. The external stress a’” required to produce the deformation then is released and later the sample is annealed. The largest stress experienced by the sample occurs at the end of the straining, before the external stress is released. It is at this point that microcracking would take place to its fullest extent. With the present model it has been shown (Fig. 8) that a plastic strain et leaves a large localized shear stress close to the interface of an elastic inclusion in a deformable matrix. In the case of the external stress 6” applied parallel or perpendicular to the major axis of the inclusion, the peak residual shear stress near the long ends of the inclusion acts in the same sense as the shear stress component of a”. Therefore at the moment of maximum stress reached at the end of the room temperature deformation operation, the total localized shear stress at the interface is even larger than that shown in Fig. 8. It is not surprising that most of the heavy slip traces in the plastically deformed astroloy are seen to intersect carbides close to their extremities when the carbides lie on grain boundaries oriented roughly parallel or perpendicular to the stress axis (Fig. 4). In the case of boundaries lying at 45” to the applied stress axis, the calculations show that the peak residual shear stress is opposed to the applied shear stress. In agreement with this conclusion, slip traces on inclined boundaries are found to impinge against the carbides in a somewhat more nearly random fashion (Fig. 4). (It may be noted in passing that another assumption of the model, namely, that the inclusion is twodimensional, is not so far removed from fact. Extraction replicas of the grain boundary carbides [4] reveal a dendritic shape. The carbides shown in section in micrographs such as Figs 2 and 3 actually are cuts through long fingers.)
THEORY
AND EXPERIMENTS
The shear stresses calculated with the model on the basis of elastic response (after the initial straining ~6) are so large that stress relaxation through further plastic deformation must occur. It is this relaxation which produces the observed slip lines which are largely localized near the ends of the carbides. The dislocation pile ups created by relaxation of the shear stress of Fig. 8 produce high interfacial normal stresses near the ends of the carbides. An estimate of the magnitude of these normal stresses has been made from the model by assuming that the (known) maximum residual shear stress r,,,_(O) at the interface falls off with a relaxation distance equal to the semi-minor axis b of the inclusion, r,,,(X) P r,,,(O) e-“I*, where X is distance from the inclusion measured on the plane of the pile up (Fig. 11). The calculations, which are given in Appendix A-2, show that the greatest normal tensile stress CN,one atom distance, bo, away from the interface is:
where i,(O) is the peak value of r,, at the interface. Since ?,,,,(O) is directly proportional to the original plastic strain ~6, once a criterion has been set for the minimum normal stress needed to produce decohesion at the particle/matrix interface it will be possible to estimate the critical value of the strain, a, required to nucleate voids. A stress criterion for deco hesion may be obtained by setting On equal to the theoretical strength of the material, taken to be ap proximately equal to E/10. This condition leads to the following equation for the critical plastic strain:
If b = 2.5
x
lo-’ m, b. = 2.5 x 10-i” m, and v = l/3, r&c = 0.7%.
A critical plastic strain of approximately 1% is consistent with our observation that microcracking can be seen after 1.5% plastic strain (the smallest strain used in the tests). These estimates of the critical plastic strain needed for microcracking through decohesion do not include the contribution of the external stress, which acts in concert with the residual stress in the case of pulling parallel or perpendicular to the long axis of the inclusion or against the residual stress in the case of an inclusion inclined at 45” to a’“. It also should be pointed out that in a real crystal it is unlikely that the plane of highest shear stress will coincide with a crystallographic slip plane. In summary, both experiments and model indicate that in the course of room temperature deformation voids are nucleated in the alloy by microcracking caused by caused by decohesion at the particle/matrix interface. This decohesion is caused by the high nor-
KIKUCHI
et nf.:
VOID NUCLEATtON
ma1 tensile stresses associated with dislocations piled up against the inclusion. It should be emphasized that the slip bands associated with decohesion do not originate with some event far from the grain boundaries. Rather they result from the incompatibilities under deformation between carbide and matrix and are created by stress concentrations in the close vicinity of the boundaries. It could be argued that the microcracking occurs first and that slip bands then are produced by stress con~nt~tions around the cracks. However, as has been noted, the majority of slip lines are not ~sociated with any cracks or voids. On the other hand few voids are seen which are not sited at the intersection of a slip line with a carbide. Therefore it seems logical to conclude that slip leads to microcracking, not that voids produce the observed slip. Void growth The model shows how cold working an alloy containing hard grain boundary particles can lead to decohesion at the particle ends. Voids do not become apparent, however, until the material has been heated. It appears likely that at an elevated tem~rature the cavities grow by vacancy diision driven by residual tensile stresses [4,6]. Voids are observed primarily on those boundaries which lie parallel to the stress axis although deco&ion should take place equally well on parallel and perpendicular boundaries. Figure 9 shows that the normal stresses at the particle/matrix interface are large and tensile at the ends of carbides on boundaries parallel to the stress axis. Of course in the immediate vicinity of the decohesion the normal stress drops to zero, but a tensile stress persists elsewhere. It is the stress gradient surrounding the decohesed region which drives the vacancy flow toward the void [lq. On perpendicular boundaries the residual stresses are compressive, so no voids grow even though microcracking has taken place. Indeed, the direction of the vacancy flow would be such that these microcracks would tend to heal. Application of the model to creep cavitation
While the experiments and model described in this report refer to voids produced by cold working and annealing the application to cavitation in creep is obvious. It has been observed by several workers that the number of cavities is pro~rtion~ to plastic strain. Usually a small incubation strain is required before cavitation begins [16,18,19]. In a study of creep cavitation in 347 austenitic steel, Needham and Gladman [t8] found that the number of grain boundary voids in a sample is a linear function of the steady state creep strain, but is essentially independent of temperature or stress. The commonly observed facts of cavity nucleation during creep in material containing hard grain boundary particles are consistent with the predictions of our model. When the creep strain reaches the critical value E&., particle decohesion commences on
THEORY
AND EXPERIME~S
1755
those boundaries which are oriented parallel or perpendicular to the stress axis and which are adjacent to grains with slip planes oriented favorably with respect to the localized shear stress T,, arising from incompatibility of deformation between carbide and matrix, As creep proceeds void nucleation can occur on boundaries with less optimum conditions for decohesion. Thus nucleation is continuous during creep. Unlike the case of cavitation produced by cold work and annealing, voids which occur during creep are most likely to be found on boundaries transverse to the stress axis. Since the external &fo~ing stress a’” (i.e., the creep stress) is applied to the sample throughout the test, it is the transverse boundaries which are likely to experience the greatest tensile stress. It should be noted, however, that the compressive residual stresses continue to grow (unless relieved by some recovery process) as creep proceeds and thus they reduce the effective stress driving the vacancy flow. In view of the magnitude of the residual stresses it is perhaps not surprising that cavitation growth is a slower process than calculated on the basis of applied stress alone [ZO,211. SUMMARY
AND CONCLUSIONS
The process of nucleation of grain boundary voids produced in astroloy by room temperature deformation and subsequent annealing has been studied both experimentally and theoretically. Cavities form at the interface between grain boundary carbides and matrix close to the ends of the carbides. There is a very strong correlation between the presence of a void on the end of a particular carbide and the impingement of a coarse slip band with the carbide at just that spot. While slip bands are observed to hit ali the grain boundaries, cavities are seen most frequently on those boundaries which are roughly parallel to the deformation axis (tentrife stress axis). Coarse slip bands are most likely to intersect the boundaries close to the ends of the carbides. A calculation has been made of the residual stresses which exist around an elastic elliptical inclusion embedded in a deformable matrix when the matrix suffers a plastic strain. It was found that residual shear stresses are left around the inch&on in the wake of the plastic defo~atio~ and that these shear stresses are sharply peaked in the region around the ends of the inclusion. Such concentrations of shear stress are responsible for the slip bands observed near the ends of the carbides. Estimates of the stress produced at an inclusion interface by the pile up of the dislocations in a slip band show that a plastic strain in the matrix of about 1% is sufficient to cause decohesion (i.e., void nucleation). Residual normal stresses also are produced around the inclusion when the matrix is deformed. If the long axis of the inclusion lies parallel to the deformation axis the residual stresses acting across the ends of the
1756
KIKUCHI et al.:
VOID NUCLEATION
inclusion at the interface are tensile. During the annealing period these tensile stresses drive vacancy flow and thus are responsible for cavity growth. The residual normal stresses around an inclusion with long axis perpendicular to the strain axis are compressive. The model predicts cavitation on parallel boundaries, none on transverse boundaries. It is noted that the mechanism of void nucleation proposed to explain cavitation resulting from room temperature deformation and annealing also can be responsible for cavity nucleation during high temperature creep. Once a critical plastic strain has been reached nucleation becomes continuous. The presence of large compressive residual stresses on transverse boundaries is likely to have a considerable effect on the rate of void growth. Acknowledgements-We are greatly indebted to Mr M. VanWanderham of Pratt and Whitney Aircraft for the astroloy. We are grateful to Dr M. Uemura, formerly a Postdoctoral Fellow at Northwestern University and now an Assistant Professor at Osaka University, Osaka, Japan, for supplying Fig. 1. In this research, extensive use was made of the facilities of Northwestern University’s Materials Research Center, funded by the NSF-MRL program, grant number DMR76-80847. This work was soonsored bv the United States Air Force Office of Scientitickesearch &ant 77-3251, under the direo tion of Dr Alan Rosenstein.
REFERENcE!3 1. B. F. Dyson and M. J. Rodgers, Merol Sci. 8, 261 (1974). 2 B. F. Dyson and D. E. Henn, J. Microsc. 97, 165 (1973). 3. B. F. Dyson and D. McLean, Metal Sci. 6, 220 (1972). 4. T. Saegusa, M. Uemura and J. R. Weertman, Metall. Trans. A l&l453 (1980). 5. J. C. Runkle, Sc.D. Thesis, M.I.T., Cambridge, Massachusetts (1978). 6. B. F. Dyson, M. S. Loveday, and M. J. Rodgers, Proc. R. Sot. Land. A 349,245 (1976). 7. M. Kikuchi and J. R. Weertman, Scripta metoll. 14, 797 (1980). 8. P. W. Davies, K. R. Williams and B. Wilshire, Phil. Msg. 18, 197 (1968). 9. T. Saegusa, Ph.D. Thesis, Northwestern University, Evanston, Illinois (1978). 10. R. F. Decker and C. T. Sims, The Superalloys (edited bv C. T. Sims and W. C. Haael) op. 33-77, Wiley, New _ ._York (1972). 11. J. D. Eshelby, Proc. R. Sot. Land. A 241, 376 (1957). 12. K. Tanaka and T. Mum, J. appl. Mech. in press. 13. A. C. Stevenson, Proc. R. Sot. Lond. A 184, 128, 218 (1945). 14. M. F. Ashby, Phil. Msg. 14, 1157 (1966). 15. A. S. Argon, J. Im, and R. Safoglu, Metall. Truns. A 6, 825 (1975). 16. R. G. FIeck, D. M. R. Taplin, and C. J. Beevers, Acta metall. 23,415 (1975). 17. D. Hull and D. E. Rimmer, Phi/. Mag. 4, 673 (1959). 18. N. G. Needham and T. Gladman. Metal Sci. 14. 64 (198tl). 19. T. Johannesson and A. ThSlen. J. Il~sr. Merals 97, 243 ( 1969).
THEORY AND EXPERIMENTS 20. W. Pavinich and R. Raj, Metall. Trans. A 8, 1917 (1977). 21. B. F. Dyson, Metal Sci. 10, 349 (1976). 22 S. P. Timoshenko and J. N. Goodier. Theorv of Elasticity, 3rd edn, McGraw-Hill Book’ Co., New York (1970). 23. J. Weertman, Bull. Seism. Sot. Am. 54, 1035 (1964).
A-l
APPENDIX
We consider the problem of determining the uniform external stresses p,, which must be applied to the inclusion/ matrix ensemble in order to produce the (elastic) displacements (uI, uI) specified by equations (4). Stevenson [13] has solved the problem of a thick plate containing an elliptical hole which is loaded under uniaxial tension p acting at an angle /3 with respect to the major axis of the hole (Fig 10). It is assumed that the stresses are applied at a position far removed from the hole. He obtained the following equations for the stress function [22] Q, = Re[TY(z) + _Y(z)],where z=x+iyandi=x-iy: 4Y(z) = pc[e2Cocos2fic&hC
+ (1 - e24*2’“)sinh[] (A-l)
4x(z) = -pcq(cosli
2&J - cos Z/!I)C+ +?” x cash 2([ - Co - is)]
Here use has been made of elliptical coordinates:
x = ccosh{cosq,y
= csinh
The elliptical boundary of the hole is given by C:= cc. The semi-major and semi-minor axes can be expressed in terms of Co and c: a = ccosh Co, b = c sinh Co. Stevenson’s solution for a uniaxial stress can be generalized to the case of an arbitrary stress p,, applied to the plate by superposition of the following special cases: B = 0.
Pf,
P-*Pn.
s-f,
P
/I__!!,
-p
Fig. 10. Uniaxial stress p applied to a thick plate containing an elliptical hole. The stress p makes an angle B with respect to the long axis of the elliptical hole.
KIKUCHI
et al.:
VOID NUCLEATION
TWEORY
AND EXPERIMENTS
1757
distributed infinitesimal dislocations, and define the distribution function B(X)dX as equal to the sum of the Burgers vectors of all the infinitesimal dislocations located between X and X + dX. The shear stress ox&, 0) on the slip plane caused by the stress fields of the dislocations in the pile up is:?
a&X. 0) =
- B(S)dS P 2x(1 - v) f 0 (x).
(A-7
Under equilibrium conditions, the total shear stress on the slip plane in the region 0 s X e co must vanish: o*r(X, 0) + a&(X, 0) = 0.
I
Equation (A-7) can be inverted [UJ and combined with (A-8) to yield an expression for B(X) in terms of a&:
Fig. Il. Dislocation pile up at end of inclusion.
4Y’(zf = c&,
4x(2) = --c’
+ pry) sinh I + e’“((p,,
iI
HP,, + p&=h%
(A-8)
(A_9)
- p&z-’ - 2ip,, sinh < )]
- (P, - P,X I
(A-2)
The normal stress uxx(X, Y) produced by the stress field of the continuous dislocations is: o&X,
- ip*, sinh 2(5 - to)
Y) = --
(o ~(S)Y~3(X - S)* + Y*; dS 2a(l
:(x - S)2 + Y2)2
11
The calculation of the ~spla~ments and stress components in the plate is a straight forward process once the stress function is known (22). o:, + a,: = 2[vr’(z) + @(‘(i)J
(A-3)
uzr - o:~ + Ziu.$ = 2[gY “(2) + x”(z)] 2&,
(A-4)
+ iu,) = (3 - 4v)Y’(z)- @‘(Z) - f’(i)
(A-5)
where the primes denote differentiation and the superscript bars the complex conjugate. From equations (A-2) and (A-S), with the substitution that x = a cos q, y = b sin r), the following equations are found for the displacements u, and ur on the edge of the hole as a function of pi,:
We wish to evaluate plane from below the tensile normal stress. involving Dirac delta
:z
a&X, Y) as Y approaches the slip X axis, i.e., in the region of maximum Use is made of the following limits functions, a(X):
Y = &nRa(X - S) (X - s)* 4 Y2
Y(X - s)* lim =+5(X-S). r-0 ((X - s)* + Y2f2 The sign to be taken is that of the direction of approach to zero, in this case the minus sign. (A-10) Equations (A-9) and (A-f0) are combined to give:
Equations (A-6) are combined obtain equations (5):
APPENDIX
with Q), (3), and (4) to
A-2
In this section we first calculate the distribution of dislocations in a pile up which is driven by a known shear stress, a&, acting on the slip plane. Once the distribution function is known it is possible to determine the normal stress at the barrier caused by the pile up. A stress criterion for decohesion can be obtained by setting this stress equal to the theoretical strength of the material. Consider a pile up of edge dislocations on the slip plane Y = 0 (Fig. 1I). The barrier to dislocation motion is located at X = 0. We make use of the concept of continuously t It is assumed that the Cauchy principal taken for this, and the following integrals.
values are
2 1 @xX(X*Y) = - -2 r-o-
dDu&(S, 0)S”2 dS f0
(S - X)
. (A-11)
The driving stress u&(X,0) can be identified with the residual shear stress r,,,&X) if it is assumed that the Y = 0 plane coincides with the plane on which r,, a&s. In the case of cf” parallel or perpendicular to the long axis of the inclusion, the sense of r,, is such that ~~r(X,O) = -r,,(X). In order to estimate a value for uxx(X,O-) as X approaches the inclusion interface it is assumed that ?MI falls off from its value close to the interface, ‘c,.(O), as: T&X)
= r,,(0)e-x’b
where b is the semi-minor axis of the inclusion. With this substitution equation (A-i 1) becomes:
KIKUCHI et al.:
1758
For values of X approaching evaluated directly:
X-O
uxxw,Y) -
$
VOID NUCLEATION
zero the integral can be
L(OlwX)“‘.
At a distance of one atom spacing, b,,, from the inclusion interface, oxx(X. O-) becomes:
udbo. 0-l -
-&~,.(0)Wbo)“2.
0412)
THEORY AND EXPERIMENTS
point along the inclusion/matrix interface where T,,(O) has its peak value&,,(O)). Therefore axx(bo. O-) at T,,,_(O)can be identified with Z,,(b,,) of equation (6) in the Discussion Section. From Fig. 8 it can be seen that the peak value of TV_ near an elliptical inclusion with shape b/a = l/4 is
L,(O) -
&
=g$).
Dccohcsion at the intcrfaa should be insured when the normal stress u,(b,) reaches a critical value a,,,: UN.<* E/10.
It can be shown that, for d” and rg parallel or perpendicular to the long axis of the inclusion the outward normal to the interfaa coincides with the X axis at the
(A-13)
(A-14)
The combination of (A-12), (A-13), and (A-14) leads to equation (7) for the critical plastic strain lJvc needed to nucleate voids.