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Models for constrained cavity growth in polycrystals
M)o1-6160~x0.0?01-0143~0? oo:O
MODELS
FOR
CONSTRAINED CAVITY IN POLYCRYSTALS
Berkeley
Nuclear
GROWTH
W. BEERE C.E.G.B.,
Laboratories,
(Receiced
6 Auyust
Berkeley,
Gloucester,
England
1979)
Abstract-Cavities uniformly distributed on grain boundaries in polycrystals can grow without deformation in neighbouring grains. However cavities may be distributed non uniformly in which case a creep ~lccomm~ation process is necessary. Diffusion or dislocation creep accommodation can control cavity growth in certain specialised circumstances. The paper derives rate equations for the control processes and assesses the regimes in which they operate. R&sum&Des cavitks rbparties uniformtment sur les joints de grains d’un polycristal peuvent croitre sans d&formation, dans des grains voisins. Toutefois, lorsyue les cavitCs ne sont pas r&parties uniform& ment. un processus d’accommodation par fluage est nicessaire. Dans certains cas particuliers, I’accommodation par diffusion ou par Auage de dislocations peut contrbler la croissance des cavitts. Nous dicrivons dans cet article les equations qui donnent la vitesse des processus de contrble, et nous prL:cisons dans quels domaines its operent. 2usammenfa~ung--Hohlr~ume, die gleichmtil3ig iiber die Korngrenzen in Polykristallen verteiit sind, kannen ohne ohne Verformung benachbarter KGrner wachsen. Allerdings kijnnen HohlrLume ungleichrn$Jig verteilt sein; in diesem Fall ist ein Anpassungsprozcss durch Kriechen notwendig. Anpassung durch Diffusions- oder Versetzungskriechen kann das Hohlraumwachstum bei bestimmten Bedingungen kontrollieren. In dieser Arbeit werden Ratengleichungen fiir die Kontrollprozesse und die Bereiche ihres Auftretens abgeleitet.
1. INTRODU~ION
The failure of polycrystalline materials can occur by a variety of mechanisms depending on the material, the temperature and the stress. One type of failure, which can be relevent to the service conditions encountered in steam pipes and turbine blades, is caused by the nucleation and growth of small cavities confined to the grain boundary plane. It has been proposed that cavities can grow by vacancy diffusion between the boundary and the pore surface [I]. The mechanism assumes that a tensile stress acting across a boundary creates an excess of vacancies which diffuse to the pore surface. If the grain displacements caused by vacancy diffusion can take place without the restriction of an ~lccommod~~tion mechanism, then the cavity growth rate is proportional to the applied stress. The development of porosity in polycrystals is dependent both on nucleation and growth of cavities. Several experiments have been published in which the efrect of nucleation has been much reduced by either observing the growth rate of the largest cavities or nucleating all the cavities during or before the initial stages of creep [Z-.5]. Experimental observations of cavity growth rates occasionally show a linear stress dependence but it is more common for the observed growth rate to be proportional to the creep rate or a high power of
stress. The original diffusion theory can be modified to include the effect of grain plasticity[6]. At high stresses the theoretical growth rate then becomes proportional to creep rate. In this situation the vacancies are considered to be created non uniformly over the boundary. Diffusion then takes place predominently in the immediate vicinity of the cavity and the uneven collapse of the boundary is accommodated by grain plasticity. The combined diffusive~plastic process behaves similarly to a diffusive mechanism and a plastic growth mechanism acting independently in parallel. At low stresses the growth rate is predicted to be proportional to stress whilst at higher stresses the growth rate is proportional to creep rate. A good approximation to the combined process is obtained from summing the purely diffusion growth and plastic growth mechanisms. It follows that the growth rate will be proportional to the creep rate at growth rates crbave those predicted from diffusion theory. Whilst the combined diffusivc,‘plastic growth theory may be useful in discussing cavitation in some pure materials, it does not help in explaining why growth rates in some precipitate hardened materials can he related to creep strain rate at absolute rates several orders of magnitude ~YIOLVthe rate predicted from diffusion alone [4]. One possible explanation is that cavities are distributed non unifirmly in such a way that creep has to accommodate the displacements resulting from cavity growth 17.81. If for instance a 143
144
BEERE:
MODELS FOR CAVITY GROWTH
Fig. 1. Regions of cavitated grains in a polycrystal may be isolated (a) or connected
cluster of grains contain cavities but are surrounded by cavity free grains, the subsequent cavity growth can only take place at a rate in accordance with the creep of the surrounding mantle. 2. THE EFFECT OF CAVITY DISTRIBUTION The distribution of cavitated grains can have a considerable effect on the strength of the aggregate. Briefly three schemes may be considered. Firstly the cavitated regions may be isolated by uncavitated material, Fig. la. If we assume that a rapid cavity growth mechanism is operative and all the boundaries are cavitated, then the normal boundary stresses in the cavitated region will be relaxed to some reduced value. Strength is provided by the uncavitated grains. Assuming a rapid growth mechanism implies that the cavitated zones can easily follow the displacements of the surrounding material without any significant load being present in the cavitated zone. In this respect the cavitated zone can be removed and treated as one large hole. The overall cavity growth rate is then given by the plastic growth rate of the large closely spaced interacting holes. A second scheme is to have isolated volumes of uncavitated material surrounded by cavitated grains Fig. lb. In this case it is untenable to have ail grain faces cavitated with relaxed stresses, if the aggregate is to have creep resistance. Only a proportion of the boundaries can be cavitated and the remaining boundaries must transfer stress between grains. The rate of cavity growth will then depend on the details of the grain geometry and deformation mechanism. The last scheme is to have both the cavitated and uncavitated regions multiply connected and continuous Fig. 2. Such geometries are found between the pore phase and solid phase of powder compacts. Each phase is interconnected over large distances by branched connections. The properties of this aggregate will be intermediate between schemes 1 and 2. As yet there do not appear to be experimental observations on the topology of the cavitated fraction. However the creep properties of grains not having all faces cavitated are relevent to the schemes discussed
and the paper now develops this aspect. In the analysis it is assumed that a rapid cavity growth mechanism can operate. When the overall growth rate is limited by the accomm~ation mechanism the normal boundary stress is relaxed on cavitated boundaries. The relaxed value will be equal to the surface tension restraint at the cavity surface (27/p, where y is the surface tension and p the cavity radius) but for convenience will be put equal to zero. It has been suggested that the deformation of the grains to accommodate porosity may be controlied by dislocation creep in the matrix or diffusion creep between grain boundaries [7]. These mechanisms will now be evaluated. 3. ACCOMMODATION BY DIFFUSION CREEP The important feature of diffusive accommodation is that the grains remain essentially rigid and accommodation takes place by sliding along grain boundaries and the transfer of material between boundaries [9]. This type of deformation can usefully be modelled by representing the grains with geometrical shapes. In two dimensions it is appropriate to use hexagons since 3 boundaries then meet at angles of 2x/3. Figure 3 shows such a scheme in which 3 boundaries have been labelled a, b and c. The specimen
Fig. 2. When regions of cavitated and fully dense grains are both continuous the structure formed is interconnected.
BEERE:
MODELS
FOR
CAVITY
GROWTH
145
Diffusion creep requires that sliding takes place easily and diffusion between boundaries is rate controlling. The sliding mechanism which accommodates the normal displacements will in this case reduce the shear stresses to a vanishingly small value. Hence all the shear stresses in equations (1) and (2) can be put equal to zero. If one of the boundaries, say the c boundary is cavitated then a rapid cavitation mechanism reduces, the normal stress to some low value. (2) requires that Putting or = 0 in equation
Fig. 3. Displacements between grains resulting from porosity and diffusion creep can be analysed in hexagonal grains. Sliding and normal motion across boundaries can be related to strain in the specimen axes shown dashed (a). The stresses operative can also be analysed by isolating segments of the grains and balancing forces (b).
axes are dashed and are orientated at an arbitrary angle 0 to the hexagon axes. When this arrangement is deformed a sliding displacement s, and a normal displacement n, take place in the directions indicated in Fig. 3a on the ‘0’ boundary. The deformation generates a shear stress rb and a normal stress go acting on the boundary. The boundary stresses must balance with the externally applied stresses. This can be achieved by removing segments from the hexagons and balancing forces, Fig. 3b. For instance when the array is acted on by an external uniaxial stress cr in the 2’ direction balancing forces gives the following set of equations.
aces*,= :(ab +
u,) -
oa + ob = 60 sin*8 = 20 cos*@ Hence the grains can only support a uniaxial stress when oa = ob = g = 0 or a unique value of Q is chosen. The conclusion is that relaxing the shear stresses and one normal stress to zero leaves the grains unable to support a uniaxial load. In practice the grains will slide and the boundaries will cavitate until a stable stress system develops. The new system will be different from the externally applied stresses, will be localised and will depend on the orientation 0. The analysis may proceed if we make the assumption common to this type of problem that the centre of each grain follows the same path as the same point in a homogeneous body. That is for a specimen deforming uniaxially the strain between grain centres is uniaxial. Given the rate of deformation on each boundary the local stress can be found from the constitutive law for deformation. Finally the local stresses are summed over the distribution of grain orientations and equated to the applied stresses. The normal displacements on the boundary, n,. nb and n, are related to a uniaxial strain E in the 2’2’ direction Fig. 3a. The porosity area fraction AA/A is related by E,, + lZ2 = AA/A. Since E = E~.~. the strain in the transverse direction is E~.~. = (AA/A - e). The normal displacement across a boundary is equal to the product of the dilationary strain along the line between the adjacent grain crntres and $1, where I is the grain edge length. The strains between grain centres can be found in the usual way for tensors giving the following set of equations.
>I _
= +(cos’O - sin’ fl - 2,‘3 sin 0 cos 0)
+g+(g)
(3)
JT, -
= &(cos20 - sin*f3 + 2,‘3 sin Wcos 1))
2+tT;- 7:) \’
cr sin* Q = +a, + &ab + a,) -
2+(~A
t AM282
"
-
rb)
(2)
:;I
-=(sin’U--cos”O)(r-gJ+ji:--).
(5)
146
BEERE:
MODELS
FOR
Summing the three displacements gives the equation
CAVITY
GROWTH
bi.i. = +{o;, + ab)(cos2 8/3 + sir?@ + (b, - ab) sin 6 cos 6@
3,/% AA n* + nb + n, = ~ 2 A
(ri.2, = (0, + a&sin ecos 8/3
as required from conservation of area The stress on a particular boundary is related to the deformation rate. Differentiating the normal displacements with respect to time and representing this with a dot gives
+ (a, -
cr,)(cos2e
sin28)/2fi
-
Substituting for (a, - (Tb) and equations (9) and (10) gives 62j2.
=
P
c0s2e
+ g
(a, + era) from
-
(sin*@ -
c0S2e)
(7) sin28
(8) where K, is the coefficient relevant for the diffusive cavity growth on only, ‘c’ and K, refers to the diffusion creep processes on all boundaries. The stresses in brackets are the ‘deviatoric’ type stresses driving diffusion between boundaries. The diffusion creep and diffusive cavity growth theories embody identical assumptions regarding the production and flux of vacancies. An important difference is that vacancies flow a shorter distance between pore and boundary compared to the interboundary distance. It follows that for a distribution of many small closely spaced cavities on a boundary the displacement rates due to cavity growth greatly exceed those due to diffusion creep. The coefficient K i, equation (8), is correspondingly larger than Kz. Since .ri, is of the order rib N ti, the stress on ‘c’ is small. In the limit as K, -+ cc so Go--+0 and the calculation proceeds with ge = 0. Equations (3) and (4) can be differentiated with respect to time and the displacement rates substituted from equations (6) and (7). Rearrangement gives the normal stress on ‘a’ and ‘b’ for a given orientation.
Ak -5%
c0s2e
(--+
3sin2@
3
,2, = :sinBcosB*+
(11)
The average transverse stress isl,it must be zero. If cavitated boundaries are orientated randomly then the average transverse stress is zero when X = SAA/A. It follows that the transverse strain is - 0.66’. From equation (11) the average shear stress is also zero. In the direction of the applied stress the average is non zero 82.2.
=
-
1
27c
s 2a
62,2.
de
0
=$-d Putting ir,*,, equal to the applied stress g and rewritting in the more usual form gives 2.5 A1 = 6 = v’%~ A
1
(12)
The creep rate when one boundary is cavitated can be compared with the creep rate of un~avitated grains by putting AA/A = 0 in the strain equations. Hence from equation (5) -j$ The local state of stress in the undashed grain axes, Fig. 3a, can be found by balancing forces in the horizontal and vertical directions, Fig. 3b. 60 ,I
=
0,
+
bb
2022 = go + d,, 2J&
= cr, - 06.
= (sin2 6 - cos’ Ok?
(13)
The boundary stresses have been calculated previously [IO] and substitution in equation (8) gives the normal velocity across ‘c’. r& = 3&(+ -
COS’e)Cr
(14)
From (13) and (14) the creep rate is given by
The local stress in the dashed specimen axes is then found by the usual tensor method. 62’2’
=
&(a,
+
gb)(sin’8/3 + cos’ 0)
- (0, - cb) sin 6 cos t?/fi
Comparison of equations (12) and (15) shows that the creep rate is doubled in the cavitated grains.
BEERE:
MODELS FOR CAVITY GROWTH
147
cavitated then a random array of grains can support a general load. The grains are assumed to deform by a power law equation of the type iii = 2K35”-‘si,
(17)
where Eij is the symmetrical strain tensor, n the stress exponent, Ti, the deviatoric stress and z is given by
I
jTZ = 5:r + T.t2 + 2& Putting oZ2 = 0 and strain rate
Fig. 4. Displacements between grains resulting from porosity and dislocation creep are more easily analysed in square grains. Here porosity formation causes an extra displacement in the 2 direction.
G,~ = 0 gives
the
uniaxial
The strain between grain centres, et, depends on the strain in the gram, eij, and the cavitation on the boundary AA/A
et = eii The calculation considered a cavitated boundary which was orientated randomly with respect to the tensile direction. At low stresses the cavitated boundaries are more likely to lie perpendicular to the tensile direction [I I]. When cavitated boundaries are considered to lie only in the range 7-c/4< 0 < 37~14the analysis predicts that the ratio of cavitation to creep is c = 1.66Ak/A. The transverse strain is thus -0.4& Placing all the cavitated boundaries perpendicular to the stress axis also has the effect of increasing the creep rate to over six times the rate in uncavitated grains
1.18)
i#2
j*2
es2 = ez2 + AA/A. Volume is conserved within the grain but not between grain centres. In the specimen axes let a force be applied in the 2’2’ direction and let the strain in this direction be e, eZjZ. = e, then in the I’f’ direction the stram is, er.r. = AA/A - P. The strain between grriin centres and the specimen strain are related in the usual way for tensors. Then for uniaxial strain in the specimen the grain centre strain is given by
cos2 0 + e sin2 0 AA
(19)
AA - e sin2U + ecosZn i -x!
A+e22=
mt
elz = 4. PORE
GROWTH DISLOCATION
CONTROLLED
BY
CREEP
When the applied stress becomes sufficiently high the interior of the grains can no longer be considered rigid. Dislocation creep is more stress sensitive than diffusion creep or gain boundary sliding and above a certain stress the former mechanism dominates. Dislocation creep does not require any other mechanism for accommodation, Hexagons do not lend themselves easily to the analysis of deformation in the grain interior. An approximation can be made to the complex situation in real grains by considering cubes[12] or for the analysis to follow, square grains. The model grains are illustr~~ted in Fig, 4. When cavity growth is controlled by a much slower accommodation mechanism the normal boundary stress will be vanishingly small. When all the boundaries are cavitated only shear stresses appear on the boundaries. As with hexagons the grains cannot support a general external load and this situation will not be considered further. If say only the 22 faces are
Similarly the stresses appearing on the boundaries must be related to the stress system in the specimen axes local to grains at the orientation 0. When crj7 is zero the local stresses in the specimen axes are ffr.,. = o,,cos2fI
+ 2~~~sin~cos~
(21)
2
u2.2. =
ffrr sin fI - 2a,2sinOcost?
01.2. =
-cl,
sin0cos0
+ o,,(cos20
(27)
- sin’@) (23)
Initially it is easy to proceed if a linear relationship is chosen between strain rate and stress, r~ = 1. In this case, from equations (I 7) and (18) the strain rate is err = K3Qrr elz = 2K3tzrt Substituting
(24) (P,, = @zr)
(25)
(24) and (25) into (21) to (23) gives
K,o,,r,
= C;,rcos2Q + @,,sinOcosfI
K,cr,.,.
= P, ,sin’fI - e,,sin
K,o,.~,
= -PI,
0~0s 0
sin Ocos 0 + c;,,(cos’fl
- sin’(l)!:!
148
BEERE:
MODELS
FOR
CAVITY
But e,, and P,* can be obtained by differentiating equations (19) and (20). Hence the specimen stresses are
GROWTH
6 Y \
‘ool-----7 L
m
.uJ
1.
K3cr2.2. = f?sin*0 A/i K3els2, = -AsmtIcosO
/
I
I /-
The average stress in the transverse direction, ?ri,i,, is required to be zero and this is in fact the case for any f3 and any distribution of cavitated boundaries when A&A
= e. 11..
Similarly the average shear stress is required to be zero. The function sin ~9cos 8 is antisymmetric and integrates to zero for any symmetric distribution of cavitated boundaries. Hence the average specimen shear force is zero as required. The average stress in the 2’2’ direction must equal the applied stress. Putting a’2,2, = 0 the applied stress, and taking the average value of sin*@ gives
.
sin 0
el2
cos
STRESS
#“a
(26) (27)
From equations (21), (23) and, (19) (20) and (26) and (27) the stress in the 1’1’ and 1’2’ directions can be written in terms of K,, P, n and geometrical terms only. In the case of the transverse stress, oi,,,, is found to be always zero and 01,2, is an antisymmetrical function which integrates to zero for symmetrical distributions of cavitated boundaries about the tensile direction. The stress in the 2’2’ direction is non zero. From equations (21) and (22)
s e7
-
sin8’“+1)“d0
n/2
6 = 13K305
(28)
Hence cavitated grains are expected to creep much faster than fully dense grains. The numerical constant in equation (28) has been calculated for stress exponents between 1 and 10, Fig. 5. In general the creep rate takes on the form + = 2Aa” exp[(n - 1) In(lO)/S].
The analysis calculated the rate of cavitation and creep rate controlled by accommodation processes. A rapid cavity growth mechanism was assumed to operate which reduced the normal stress to near zero on cavitated boundaries. It is now necessary to find the conditions for rapid cavity growth by comparing the unrestricted growth rates with the accommodation limited rates. Comparison of cavity growth models is convenient if all models express the growth rate of a single cavity i. If the cavities are spaced a distance 2E. apart in the boundary then the growth rate of one cavity is given by in - 41*d AP/F where d is the grain size. The growth rate controlled by diffusion creep can be found by equating Ak/A with Ap/V. In uncavitated grains the creep rate is given by (see review
Cl63
1 xi* 011 do
for (~ii from equations
n
5. DISCUSSION
ii2 = K,2”(1 + sin*O/co~*O)(“-~)~*a~~.
Substituting
EXPONENT,
when n = 5 the creep rate is given by
0
dii = Ks(1 + cos20/sin2e)(“-1)‘2~~1
71s -n/Z
= a
The strain rates from equation (17) can now be written in terms of either c1 1 or o1 2
Q=-
I
.
IO
&l/n
ell -_=
.
Fig. 5. The creep rate in a randomised array of square grains having half the boundaries completely cavitated is sensitive to the creep exponent n. Increasing values of n increase the creep rate above the value for fully dense grains.
AA = 6 = 2K,o. A The strain rate is double the value for uncavitated grains as before but the transverse strain is predicted to be zero. The creep resistance when the stress exponent is greater than unity can also be found. If it is assumed initially that AA/A = e then it can be shown that the average transverse and shear stresses in the specimen axes are both zero. From equations (19) and (20) the ratio of the strain rates is
.
1
(26) and (19) gives
~2g5(l+!E&c)
BEERE:
MODELS
I49
FOR CAVITY GROWTH
where B is a constant, 12 the atomic volume, 6 the grain boundary width, D, the volume self diffusion coefficient, D, the grain boundary self diffusion coefficient and krthe thermai energy. From equations (15) and (16) the growth rate of a single cavity, when cavitated boundaries are predominantly across the stress axis. is given by ir Y 16B;$D,.(*
+ 12)
At higher stresses the accommodation mechanism changes to dislocation creep control. From equation (28) the cavity growth rate for cavitated boundaries randomfy orientated is given by L
G - 52K 3;r%G
(29)
where K3 is the creep constant in the creep rate expression for uncavitated grains Z = K$. A creep exponent of 5 has been taken although any value can be substituted provided the numerical constant in equation (29) is altered accordingly. The pore growth rate unrestricted by accommodation when vacancy diffusion in the grain boundary is rate controlling is given by [I 73 2nQD,6(a
- 2y/p)
’ - kT(ln(A/p) - (1 - p’/,?)(3--
p2/%‘)/4)
(30)
where y is the s&ace energy and p the pore radius. At high stresses the diffusive growth process is sfow compared to plastic growth. Above a transition stress the growth rate of non interacting pores is given
by l331 p - fp< where p is the pore radius. Replacing the strain rate by the applied stress the rate of change in volume is given by f31) The interaction of the mechanisms discussed can be demonstrated graphically. Figure 6 compares the rate of growth with the applied stress. The diagram illustrates the unconstrained growth rate For a pore growing by a plastic process at high stress, equation (31). and a diffusive process at low stress, equation (30). The constrained growth rate for didocation creep, equation (29)+ is faster than the unconstrained growth rate, equation (3 I), since i. and d are both much larger than p. This is true provided the pores are not closely interacting in which case equation (31) underestimates the growth rate. At low stresses both constrained and unconstrained growth processes can be controlled by diffusion. The change in mechanism is clear on a pfot of iog growth rate versus log stress. since the gradient changes from n to unity. In the diffusion controlkd region the accommodation mechanism is slower than the unconstrained mechanism and it is possible for the former to control pore growth. It must be empha-
LOG STRESS Fig. 6. The growth rate of porosity can be controlled by the causative growth mechanism or by geometrical creep constraint in surrounding grains. The slower mechanism controls growth. A comparison of the rate equations reveals that at high stresses growth is controlled by the causative plastic creep mechanism. At low stresses geometrical creep constraint can control cavity growth provided diffusion is not inhibited.
sised that when pore growth is controlled by the accommodation mechanism the specimen is deforming by diffusion creep. The particular unconstrained cavity growth mechanism considered, equation (3Of, is not the only diffusion mechanism which may operate. The cavity may develop as a long thin crack ClS] and there may be elastic effects to be considered [19]. These considerations do not alter the canclusion that growth can be controlled by accommodation provided not all boundaries are cavitated and provided the material deforms by diffusion creep. Diffusion creep can be inhibited in dispersion hardened materials [14] and there is reason to suspect that diffusive cavity growth is also inhibited. The scheme of accommodation control appears satisfactory for single phase polycrystals. However cavity growth rates orders of magnitude below those predicted from diffusive growth can be observed particulariy in precipitate hardened alloys. Care must be taken to ensure that a diffusive cavity growth mechanism can now operate. Consider for instance a cavity growing plastically above the trans~~on stress to diffusive growth, Fig. 6. Diffusion in the grain boundary is still present but is limited to a disc of boundary immediately adjacent to the cavity. Reducing the stress increases the area of this disc until ultimately discs from neighbouring cavities touch and growth becomes dominated by the diffusion process. When refractory precipitates are present on the boundary evidence suggests that vacancies are not created as easily at the particfematrix interface [2O]. In this case reducing the stress allows the disc of boundary creating vacancies only to approach the precipitate. The normal boundary displacement resulting from
1.50
BEERE:
MODELS
FOR
vacancy creation has to be accommodated by plastic deformation above the precipitate. Many complications now enter the situation as the stress is lowered. Stress becomes concentrated over the particle to drive the plastic deformation. This in turn means the second derivative of the stress with respect to distance from the cavity changes sign in the vicinity of the cavity. Diffusion theory now requires that this region of boundary becomes a vacancy sink. The plastic process now has to accommodate the displacements due to cavity growth and the local displacements resulting from the stress concentration at the particle. The latter displacements increases very rapidly with decreasing applied stress and a threshold stress can be calculated above which the diffusion process, as it is presently understood, cannot operate [20]. On lowering the stress the threshold stress is reached when the disc of vacancy creation encounters the refractory particle. For many alloys the precipitate spacings will be equal to or closer than cavity spacings and the threshold stress will be at or above the transition stress to diffusive cavity growth. It follows that the unrestrained diffusive cavity growth mechanism depicted in, Fig. 6, may not operate but may be inhibited by precipitates. In this case a much slower cavity growth rate wilt operate controlled by plastic accommodation around the grain boundary precipitates. Depending on the rate of the latter process the growth may be controlled by macroscopic accommodation of uncavitated grains or microscopic accommodation around refractory grain boundary precipitates. In the macroscopic process reduced cavity growth can be achieved by optimising the fine dispersion of precipitates in the matrix to reduce the creep rate. In the microscopic process the growth rate will be controlled by both the dispersion in the matrix and in the grain boundary. Returning to the growth rate at high stresses controlled by dislocation creep, the growth rate is considerably increased when cavities are close and strongly interacting. Equation (31) is then an underestimate and creep accommodation can be rate controlling. Typically in the later stages of growth just before failure of the specimen the cavities can link and some boundaries become completely cavitated. Further growth of the slab of porosity can then only be restricted by shear transmitted across uncavitated boundaries. The order of magnitude increase in the creep rate predicted for the power law creep accommodation process, equation (28), may describe the increased creep rate observed during tertiary creep. 6. CONCLUSIONS 6.1 At relatively high stresses cavity growth is controlled by the unconstrained dislocation creep process. C - 2np3K,a”
CAVITY
GROWTH
6.2 If diffusion is not inhibited by precipitates and not all boundaries are cavitated then diffusion creep
accommodation
can control cavity growth
6.3 When the cavitated boundaries lie predominently across the tensile axis the transverse strain is about 0.4 times the strain in the tensile direction. The above figure assumes that one in three boundaries are cavitated. In general the ratio of transverse to axial strain is dependent on the fraction of cavitated boundaries, their angular distribution with respect to the tensile axis and the creep mechanism. 6.4 When pores coalesce and cause complete separation of the boundary pore growth is limited by dislocation creep accommodation. For a material with half the boundaries cavitated creeping by power law creep with a stress exponent of five the cavitation rate and creep rate are given by
-AP z i + 13K,a5. V
6.5 Precipate hardened materials may not exhibit diffusion creep or diffusion controlled cavity growth. In this case cavity growth may be controlled by deformation around the grain boundary precipitates. ~c~~ff~~e~~~~~~~-This paper is published by permission of the Central Electricity Generating Board.
REFERENCES 1. D. Hull and R. Rimmer, Phil. Mug. 4, 673 (1959). 2. S. H. Goods and W. D. Nix, Actu Meta~l, 26, 739 (1978).
3. B. F. Dyson and M. J. Rodgers. ICF4 2, 621 (1977). 4. W. Pavinich and R. Raj, Mm/l. Trans. 8A, 1917 (1977). 5. 3. J. Cane and G. W. Greenwood. Metal Sci. 9, 55 (1975). W. Beere and M. V. Speight, Merat Sci. It. 172 (1978). B. F. Dyson, Metal. Sci. 10, 349 (1976). B. F. Dyson, Can. Metall. Q. to be published (1979). L. M. Gfshitz, Sou. Phys. JkTP, 17, 909 (1963). 10. W. Beere, Metal Sci. 10, 133 (1976). 11. A. G&ins and H. D. Williams. Phil. Msg. 16, 849 (1967). 12. M. V. Speight, Acta Mefall. 24, 727 (1976). 13. J. W. Hancock, Metal Sci. 10, 319 (1976). 14. B. Burton, Meral Sci. 5, ii (1971). 15. M. F. Ashbv. ScriDta Metal/. 3. 843 (1969). Materials. 16. B. Burton, i$$sion Creep q/.dolycr&~llhw Trans. Tech. Pub. (1977). 17. M. V. Speight and W. Beere, Metal Sci. 9, 190 (1975). 18. W. Beere and M. V. Speight, Metal Sci. 12,593 (1978). 19. M. V. Speight, W. Beere and G. Roberts, Metull. Sci. Engng 36, 155(1978). 20. W. Beere, to be published.
6. 7. 8. 9.
MODELS
FOR
CONSTRAINED CAVITY IN POLYCRYSTALS
Berkeley
Nuclear
GROWTH
W. BEERE C.E.G.B.,
Laboratories,
(Receiced
6 Auyust
Berkeley,
Gloucester,
England
1979)
Abstract-Cavities uniformly distributed on grain boundaries in polycrystals can grow without deformation in neighbouring grains. However cavities may be distributed non uniformly in which case a creep ~lccomm~ation process is necessary. Diffusion or dislocation creep accommodation can control cavity growth in certain specialised circumstances. The paper derives rate equations for the control processes and assesses the regimes in which they operate. R&sum&Des cavitks rbparties uniformtment sur les joints de grains d’un polycristal peuvent croitre sans d&formation, dans des grains voisins. Toutefois, lorsyue les cavitCs ne sont pas r&parties uniform& ment. un processus d’accommodation par fluage est nicessaire. Dans certains cas particuliers, I’accommodation par diffusion ou par Auage de dislocations peut contrbler la croissance des cavitts. Nous dicrivons dans cet article les equations qui donnent la vitesse des processus de contrble, et nous prL:cisons dans quels domaines its operent. 2usammenfa~ung--Hohlr~ume, die gleichmtil3ig iiber die Korngrenzen in Polykristallen verteiit sind, kannen ohne ohne Verformung benachbarter KGrner wachsen. Allerdings kijnnen HohlrLume ungleichrn$Jig verteilt sein; in diesem Fall ist ein Anpassungsprozcss durch Kriechen notwendig. Anpassung durch Diffusions- oder Versetzungskriechen kann das Hohlraumwachstum bei bestimmten Bedingungen kontrollieren. In dieser Arbeit werden Ratengleichungen fiir die Kontrollprozesse und die Bereiche ihres Auftretens abgeleitet.
1. INTRODU~ION
The failure of polycrystalline materials can occur by a variety of mechanisms depending on the material, the temperature and the stress. One type of failure, which can be relevent to the service conditions encountered in steam pipes and turbine blades, is caused by the nucleation and growth of small cavities confined to the grain boundary plane. It has been proposed that cavities can grow by vacancy diffusion between the boundary and the pore surface [I]. The mechanism assumes that a tensile stress acting across a boundary creates an excess of vacancies which diffuse to the pore surface. If the grain displacements caused by vacancy diffusion can take place without the restriction of an ~lccommod~~tion mechanism, then the cavity growth rate is proportional to the applied stress. The development of porosity in polycrystals is dependent both on nucleation and growth of cavities. Several experiments have been published in which the efrect of nucleation has been much reduced by either observing the growth rate of the largest cavities or nucleating all the cavities during or before the initial stages of creep [Z-.5]. Experimental observations of cavity growth rates occasionally show a linear stress dependence but it is more common for the observed growth rate to be proportional to the creep rate or a high power of
stress. The original diffusion theory can be modified to include the effect of grain plasticity[6]. At high stresses the theoretical growth rate then becomes proportional to creep rate. In this situation the vacancies are considered to be created non uniformly over the boundary. Diffusion then takes place predominently in the immediate vicinity of the cavity and the uneven collapse of the boundary is accommodated by grain plasticity. The combined diffusive~plastic process behaves similarly to a diffusive mechanism and a plastic growth mechanism acting independently in parallel. At low stresses the growth rate is predicted to be proportional to stress whilst at higher stresses the growth rate is proportional to creep rate. A good approximation to the combined process is obtained from summing the purely diffusion growth and plastic growth mechanisms. It follows that the growth rate will be proportional to the creep rate at growth rates crbave those predicted from diffusion theory. Whilst the combined diffusivc,‘plastic growth theory may be useful in discussing cavitation in some pure materials, it does not help in explaining why growth rates in some precipitate hardened materials can he related to creep strain rate at absolute rates several orders of magnitude ~YIOLVthe rate predicted from diffusion alone [4]. One possible explanation is that cavities are distributed non unifirmly in such a way that creep has to accommodate the displacements resulting from cavity growth 17.81. If for instance a 143
144
BEERE:
MODELS FOR CAVITY GROWTH
Fig. 1. Regions of cavitated grains in a polycrystal may be isolated (a) or connected
cluster of grains contain cavities but are surrounded by cavity free grains, the subsequent cavity growth can only take place at a rate in accordance with the creep of the surrounding mantle. 2. THE EFFECT OF CAVITY DISTRIBUTION The distribution of cavitated grains can have a considerable effect on the strength of the aggregate. Briefly three schemes may be considered. Firstly the cavitated regions may be isolated by uncavitated material, Fig. la. If we assume that a rapid cavity growth mechanism is operative and all the boundaries are cavitated, then the normal boundary stresses in the cavitated region will be relaxed to some reduced value. Strength is provided by the uncavitated grains. Assuming a rapid growth mechanism implies that the cavitated zones can easily follow the displacements of the surrounding material without any significant load being present in the cavitated zone. In this respect the cavitated zone can be removed and treated as one large hole. The overall cavity growth rate is then given by the plastic growth rate of the large closely spaced interacting holes. A second scheme is to have isolated volumes of uncavitated material surrounded by cavitated grains Fig. lb. In this case it is untenable to have ail grain faces cavitated with relaxed stresses, if the aggregate is to have creep resistance. Only a proportion of the boundaries can be cavitated and the remaining boundaries must transfer stress between grains. The rate of cavity growth will then depend on the details of the grain geometry and deformation mechanism. The last scheme is to have both the cavitated and uncavitated regions multiply connected and continuous Fig. 2. Such geometries are found between the pore phase and solid phase of powder compacts. Each phase is interconnected over large distances by branched connections. The properties of this aggregate will be intermediate between schemes 1 and 2. As yet there do not appear to be experimental observations on the topology of the cavitated fraction. However the creep properties of grains not having all faces cavitated are relevent to the schemes discussed
and the paper now develops this aspect. In the analysis it is assumed that a rapid cavity growth mechanism can operate. When the overall growth rate is limited by the accomm~ation mechanism the normal boundary stress is relaxed on cavitated boundaries. The relaxed value will be equal to the surface tension restraint at the cavity surface (27/p, where y is the surface tension and p the cavity radius) but for convenience will be put equal to zero. It has been suggested that the deformation of the grains to accommodate porosity may be controlied by dislocation creep in the matrix or diffusion creep between grain boundaries [7]. These mechanisms will now be evaluated. 3. ACCOMMODATION BY DIFFUSION CREEP The important feature of diffusive accommodation is that the grains remain essentially rigid and accommodation takes place by sliding along grain boundaries and the transfer of material between boundaries [9]. This type of deformation can usefully be modelled by representing the grains with geometrical shapes. In two dimensions it is appropriate to use hexagons since 3 boundaries then meet at angles of 2x/3. Figure 3 shows such a scheme in which 3 boundaries have been labelled a, b and c. The specimen
Fig. 2. When regions of cavitated and fully dense grains are both continuous the structure formed is interconnected.
BEERE:
MODELS
FOR
CAVITY
GROWTH
145
Diffusion creep requires that sliding takes place easily and diffusion between boundaries is rate controlling. The sliding mechanism which accommodates the normal displacements will in this case reduce the shear stresses to a vanishingly small value. Hence all the shear stresses in equations (1) and (2) can be put equal to zero. If one of the boundaries, say the c boundary is cavitated then a rapid cavitation mechanism reduces, the normal stress to some low value. (2) requires that Putting or = 0 in equation
Fig. 3. Displacements between grains resulting from porosity and diffusion creep can be analysed in hexagonal grains. Sliding and normal motion across boundaries can be related to strain in the specimen axes shown dashed (a). The stresses operative can also be analysed by isolating segments of the grains and balancing forces (b).
axes are dashed and are orientated at an arbitrary angle 0 to the hexagon axes. When this arrangement is deformed a sliding displacement s, and a normal displacement n, take place in the directions indicated in Fig. 3a on the ‘0’ boundary. The deformation generates a shear stress rb and a normal stress go acting on the boundary. The boundary stresses must balance with the externally applied stresses. This can be achieved by removing segments from the hexagons and balancing forces, Fig. 3b. For instance when the array is acted on by an external uniaxial stress cr in the 2’ direction balancing forces gives the following set of equations.
aces*,= :(ab +
u,) -
oa + ob = 60 sin*8 = 20 cos*@ Hence the grains can only support a uniaxial stress when oa = ob = g = 0 or a unique value of Q is chosen. The conclusion is that relaxing the shear stresses and one normal stress to zero leaves the grains unable to support a uniaxial load. In practice the grains will slide and the boundaries will cavitate until a stable stress system develops. The new system will be different from the externally applied stresses, will be localised and will depend on the orientation 0. The analysis may proceed if we make the assumption common to this type of problem that the centre of each grain follows the same path as the same point in a homogeneous body. That is for a specimen deforming uniaxially the strain between grain centres is uniaxial. Given the rate of deformation on each boundary the local stress can be found from the constitutive law for deformation. Finally the local stresses are summed over the distribution of grain orientations and equated to the applied stresses. The normal displacements on the boundary, n,. nb and n, are related to a uniaxial strain E in the 2’2’ direction Fig. 3a. The porosity area fraction AA/A is related by E,, + lZ2 = AA/A. Since E = E~.~. the strain in the transverse direction is E~.~. = (AA/A - e). The normal displacement across a boundary is equal to the product of the dilationary strain along the line between the adjacent grain crntres and $1, where I is the grain edge length. The strains between grain centres can be found in the usual way for tensors giving the following set of equations.
>I _
= +(cos’O - sin’ fl - 2,‘3 sin 0 cos 0)
+g+(g)
(3)
JT, -
= &(cos20 - sin*f3 + 2,‘3 sin Wcos 1))
2+tT;- 7:) \’
cr sin* Q = +a, + &ab + a,) -
2+(~A
t AM282
"
-
rb)
(2)
:;I
-=(sin’U--cos”O)(r-gJ+ji:--).
(5)
146
BEERE:
MODELS
FOR
Summing the three displacements gives the equation
CAVITY
GROWTH
bi.i. = +{o;, + ab)(cos2 8/3 + sir?@ + (b, - ab) sin 6 cos 6@
3,/% AA n* + nb + n, = ~ 2 A
(ri.2, = (0, + a&sin ecos 8/3
as required from conservation of area The stress on a particular boundary is related to the deformation rate. Differentiating the normal displacements with respect to time and representing this with a dot gives
+ (a, -
cr,)(cos2e
sin28)/2fi
-
Substituting for (a, - (Tb) and equations (9) and (10) gives 62j2.
=
P
c0s2e
+ g
(a, + era) from
-
(sin*@ -
c0S2e)
(7) sin28
(8) where K, is the coefficient relevant for the diffusive cavity growth on only, ‘c’ and K, refers to the diffusion creep processes on all boundaries. The stresses in brackets are the ‘deviatoric’ type stresses driving diffusion between boundaries. The diffusion creep and diffusive cavity growth theories embody identical assumptions regarding the production and flux of vacancies. An important difference is that vacancies flow a shorter distance between pore and boundary compared to the interboundary distance. It follows that for a distribution of many small closely spaced cavities on a boundary the displacement rates due to cavity growth greatly exceed those due to diffusion creep. The coefficient K i, equation (8), is correspondingly larger than Kz. Since .ri, is of the order rib N ti, the stress on ‘c’ is small. In the limit as K, -+ cc so Go--+0 and the calculation proceeds with ge = 0. Equations (3) and (4) can be differentiated with respect to time and the displacement rates substituted from equations (6) and (7). Rearrangement gives the normal stress on ‘a’ and ‘b’ for a given orientation.
Ak -5%
c0s2e
(--+
3sin2@
3
,2, = :sinBcosB*+
(11)
The average transverse stress isl,it must be zero. If cavitated boundaries are orientated randomly then the average transverse stress is zero when X = SAA/A. It follows that the transverse strain is - 0.66’. From equation (11) the average shear stress is also zero. In the direction of the applied stress the average is non zero 82.2.
=
-
1
27c
s 2a
62,2.
de
0
=$-d Putting ir,*,, equal to the applied stress g and rewritting in the more usual form gives 2.5 A1 = 6 = v’%~ A
1
(12)
The creep rate when one boundary is cavitated can be compared with the creep rate of un~avitated grains by putting AA/A = 0 in the strain equations. Hence from equation (5) -j$ The local state of stress in the undashed grain axes, Fig. 3a, can be found by balancing forces in the horizontal and vertical directions, Fig. 3b. 60 ,I
=
0,
+
bb
2022 = go + d,, 2J&
= cr, - 06.
= (sin2 6 - cos’ Ok?
(13)
The boundary stresses have been calculated previously [IO] and substitution in equation (8) gives the normal velocity across ‘c’. r& = 3&(+ -
COS’e)Cr
(14)
From (13) and (14) the creep rate is given by
The local stress in the dashed specimen axes is then found by the usual tensor method. 62’2’
=
&(a,
+
gb)(sin’8/3 + cos’ 0)
- (0, - cb) sin 6 cos t?/fi
Comparison of equations (12) and (15) shows that the creep rate is doubled in the cavitated grains.
BEERE:
MODELS FOR CAVITY GROWTH
147
cavitated then a random array of grains can support a general load. The grains are assumed to deform by a power law equation of the type iii = 2K35”-‘si,
(17)
where Eij is the symmetrical strain tensor, n the stress exponent, Ti, the deviatoric stress and z is given by
I
jTZ = 5:r + T.t2 + 2& Putting oZ2 = 0 and strain rate
Fig. 4. Displacements between grains resulting from porosity and dislocation creep are more easily analysed in square grains. Here porosity formation causes an extra displacement in the 2 direction.
G,~ = 0 gives
the
uniaxial
The strain between grain centres, et, depends on the strain in the gram, eij, and the cavitation on the boundary AA/A
et = eii The calculation considered a cavitated boundary which was orientated randomly with respect to the tensile direction. At low stresses the cavitated boundaries are more likely to lie perpendicular to the tensile direction [I I]. When cavitated boundaries are considered to lie only in the range 7-c/4< 0 < 37~14the analysis predicts that the ratio of cavitation to creep is c = 1.66Ak/A. The transverse strain is thus -0.4& Placing all the cavitated boundaries perpendicular to the stress axis also has the effect of increasing the creep rate to over six times the rate in uncavitated grains
1.18)
i#2
j*2
es2 = ez2 + AA/A. Volume is conserved within the grain but not between grain centres. In the specimen axes let a force be applied in the 2’2’ direction and let the strain in this direction be e, eZjZ. = e, then in the I’f’ direction the stram is, er.r. = AA/A - P. The strain between grriin centres and the specimen strain are related in the usual way for tensors. Then for uniaxial strain in the specimen the grain centre strain is given by
cos2 0 + e sin2 0 AA
(19)
AA - e sin2U + ecosZn i -x!
A+e22=
mt
elz = 4. PORE
GROWTH DISLOCATION
CONTROLLED
BY
CREEP
When the applied stress becomes sufficiently high the interior of the grains can no longer be considered rigid. Dislocation creep is more stress sensitive than diffusion creep or gain boundary sliding and above a certain stress the former mechanism dominates. Dislocation creep does not require any other mechanism for accommodation, Hexagons do not lend themselves easily to the analysis of deformation in the grain interior. An approximation can be made to the complex situation in real grains by considering cubes[12] or for the analysis to follow, square grains. The model grains are illustr~~ted in Fig, 4. When cavity growth is controlled by a much slower accommodation mechanism the normal boundary stress will be vanishingly small. When all the boundaries are cavitated only shear stresses appear on the boundaries. As with hexagons the grains cannot support a general external load and this situation will not be considered further. If say only the 22 faces are
Similarly the stresses appearing on the boundaries must be related to the stress system in the specimen axes local to grains at the orientation 0. When crj7 is zero the local stresses in the specimen axes are ffr.,. = o,,cos2fI
+ 2~~~sin~cos~
(21)
2
u2.2. =
ffrr sin fI - 2a,2sinOcost?
01.2. =
-cl,
sin0cos0
+ o,,(cos20
(27)
- sin’@) (23)
Initially it is easy to proceed if a linear relationship is chosen between strain rate and stress, r~ = 1. In this case, from equations (I 7) and (18) the strain rate is err = K3Qrr elz = 2K3tzrt Substituting
(24) (P,, = @zr)
(25)
(24) and (25) into (21) to (23) gives
K,o,,r,
= C;,rcos2Q + @,,sinOcosfI
K,cr,.,.
= P, ,sin’fI - e,,sin
K,o,.~,
= -PI,
0~0s 0
sin Ocos 0 + c;,,(cos’fl
- sin’(l)!:!
148
BEERE:
MODELS
FOR
CAVITY
But e,, and P,* can be obtained by differentiating equations (19) and (20). Hence the specimen stresses are
GROWTH
6 Y \
‘ool-----7 L
m
.uJ
1.
K3cr2.2. = f?sin*0 A/i K3els2, = -AsmtIcosO
/
I
I /-
The average stress in the transverse direction, ?ri,i,, is required to be zero and this is in fact the case for any f3 and any distribution of cavitated boundaries when A&A
= e. 11..
Similarly the average shear stress is required to be zero. The function sin ~9cos 8 is antisymmetric and integrates to zero for any symmetric distribution of cavitated boundaries. Hence the average specimen shear force is zero as required. The average stress in the 2’2’ direction must equal the applied stress. Putting a’2,2, = 0 the applied stress, and taking the average value of sin*@ gives
.
sin 0
el2
cos
STRESS
#“a
(26) (27)
From equations (21), (23) and, (19) (20) and (26) and (27) the stress in the 1’1’ and 1’2’ directions can be written in terms of K,, P, n and geometrical terms only. In the case of the transverse stress, oi,,,, is found to be always zero and 01,2, is an antisymmetrical function which integrates to zero for symmetrical distributions of cavitated boundaries about the tensile direction. The stress in the 2’2’ direction is non zero. From equations (21) and (22)
s e7
-
sin8’“+1)“d0
n/2
6 = 13K305
(28)
Hence cavitated grains are expected to creep much faster than fully dense grains. The numerical constant in equation (28) has been calculated for stress exponents between 1 and 10, Fig. 5. In general the creep rate takes on the form + = 2Aa” exp[(n - 1) In(lO)/S].
The analysis calculated the rate of cavitation and creep rate controlled by accommodation processes. A rapid cavity growth mechanism was assumed to operate which reduced the normal stress to near zero on cavitated boundaries. It is now necessary to find the conditions for rapid cavity growth by comparing the unrestricted growth rates with the accommodation limited rates. Comparison of cavity growth models is convenient if all models express the growth rate of a single cavity i. If the cavities are spaced a distance 2E. apart in the boundary then the growth rate of one cavity is given by in - 41*d AP/F where d is the grain size. The growth rate controlled by diffusion creep can be found by equating Ak/A with Ap/V. In uncavitated grains the creep rate is given by (see review
Cl63
1 xi* 011 do
for (~ii from equations
n
5. DISCUSSION
ii2 = K,2”(1 + sin*O/co~*O)(“-~)~*a~~.
Substituting
EXPONENT,
when n = 5 the creep rate is given by
0
dii = Ks(1 + cos20/sin2e)(“-1)‘2~~1
71s -n/Z
= a
The strain rates from equation (17) can now be written in terms of either c1 1 or o1 2
Q=-
I
.
IO
&l/n
ell -_=
.
Fig. 5. The creep rate in a randomised array of square grains having half the boundaries completely cavitated is sensitive to the creep exponent n. Increasing values of n increase the creep rate above the value for fully dense grains.
AA = 6 = 2K,o. A The strain rate is double the value for uncavitated grains as before but the transverse strain is predicted to be zero. The creep resistance when the stress exponent is greater than unity can also be found. If it is assumed initially that AA/A = e then it can be shown that the average transverse and shear stresses in the specimen axes are both zero. From equations (19) and (20) the ratio of the strain rates is
.
1
(26) and (19) gives
~2g5(l+!E&c)
BEERE:
MODELS
I49
FOR CAVITY GROWTH
where B is a constant, 12 the atomic volume, 6 the grain boundary width, D, the volume self diffusion coefficient, D, the grain boundary self diffusion coefficient and krthe thermai energy. From equations (15) and (16) the growth rate of a single cavity, when cavitated boundaries are predominantly across the stress axis. is given by ir Y 16B;$D,.(*
+ 12)
At higher stresses the accommodation mechanism changes to dislocation creep control. From equation (28) the cavity growth rate for cavitated boundaries randomfy orientated is given by L
G - 52K 3;r%G
(29)
where K3 is the creep constant in the creep rate expression for uncavitated grains Z = K$. A creep exponent of 5 has been taken although any value can be substituted provided the numerical constant in equation (29) is altered accordingly. The pore growth rate unrestricted by accommodation when vacancy diffusion in the grain boundary is rate controlling is given by [I 73 2nQD,6(a
- 2y/p)
’ - kT(ln(A/p) - (1 - p’/,?)(3--
p2/%‘)/4)
(30)
where y is the s&ace energy and p the pore radius. At high stresses the diffusive growth process is sfow compared to plastic growth. Above a transition stress the growth rate of non interacting pores is given
by l331 p - fp< where p is the pore radius. Replacing the strain rate by the applied stress the rate of change in volume is given by f31) The interaction of the mechanisms discussed can be demonstrated graphically. Figure 6 compares the rate of growth with the applied stress. The diagram illustrates the unconstrained growth rate For a pore growing by a plastic process at high stress, equation (31). and a diffusive process at low stress, equation (30). The constrained growth rate for didocation creep, equation (29)+ is faster than the unconstrained growth rate, equation (3 I), since i. and d are both much larger than p. This is true provided the pores are not closely interacting in which case equation (31) underestimates the growth rate. At low stresses both constrained and unconstrained growth processes can be controlled by diffusion. The change in mechanism is clear on a pfot of iog growth rate versus log stress. since the gradient changes from n to unity. In the diffusion controlkd region the accommodation mechanism is slower than the unconstrained mechanism and it is possible for the former to control pore growth. It must be empha-
LOG STRESS Fig. 6. The growth rate of porosity can be controlled by the causative growth mechanism or by geometrical creep constraint in surrounding grains. The slower mechanism controls growth. A comparison of the rate equations reveals that at high stresses growth is controlled by the causative plastic creep mechanism. At low stresses geometrical creep constraint can control cavity growth provided diffusion is not inhibited.
sised that when pore growth is controlled by the accommodation mechanism the specimen is deforming by diffusion creep. The particular unconstrained cavity growth mechanism considered, equation (3Of, is not the only diffusion mechanism which may operate. The cavity may develop as a long thin crack ClS] and there may be elastic effects to be considered [19]. These considerations do not alter the canclusion that growth can be controlled by accommodation provided not all boundaries are cavitated and provided the material deforms by diffusion creep. Diffusion creep can be inhibited in dispersion hardened materials [14] and there is reason to suspect that diffusive cavity growth is also inhibited. The scheme of accommodation control appears satisfactory for single phase polycrystals. However cavity growth rates orders of magnitude below those predicted from diffusive growth can be observed particulariy in precipitate hardened alloys. Care must be taken to ensure that a diffusive cavity growth mechanism can now operate. Consider for instance a cavity growing plastically above the trans~~on stress to diffusive growth, Fig. 6. Diffusion in the grain boundary is still present but is limited to a disc of boundary immediately adjacent to the cavity. Reducing the stress increases the area of this disc until ultimately discs from neighbouring cavities touch and growth becomes dominated by the diffusion process. When refractory precipitates are present on the boundary evidence suggests that vacancies are not created as easily at the particfematrix interface [2O]. In this case reducing the stress allows the disc of boundary creating vacancies only to approach the precipitate. The normal boundary displacement resulting from
1.50
BEERE:
MODELS
FOR
vacancy creation has to be accommodated by plastic deformation above the precipitate. Many complications now enter the situation as the stress is lowered. Stress becomes concentrated over the particle to drive the plastic deformation. This in turn means the second derivative of the stress with respect to distance from the cavity changes sign in the vicinity of the cavity. Diffusion theory now requires that this region of boundary becomes a vacancy sink. The plastic process now has to accommodate the displacements due to cavity growth and the local displacements resulting from the stress concentration at the particle. The latter displacements increases very rapidly with decreasing applied stress and a threshold stress can be calculated above which the diffusion process, as it is presently understood, cannot operate [20]. On lowering the stress the threshold stress is reached when the disc of vacancy creation encounters the refractory particle. For many alloys the precipitate spacings will be equal to or closer than cavity spacings and the threshold stress will be at or above the transition stress to diffusive cavity growth. It follows that the unrestrained diffusive cavity growth mechanism depicted in, Fig. 6, may not operate but may be inhibited by precipitates. In this case a much slower cavity growth rate wilt operate controlled by plastic accommodation around the grain boundary precipitates. Depending on the rate of the latter process the growth may be controlled by macroscopic accommodation of uncavitated grains or microscopic accommodation around refractory grain boundary precipitates. In the macroscopic process reduced cavity growth can be achieved by optimising the fine dispersion of precipitates in the matrix to reduce the creep rate. In the microscopic process the growth rate will be controlled by both the dispersion in the matrix and in the grain boundary. Returning to the growth rate at high stresses controlled by dislocation creep, the growth rate is considerably increased when cavities are close and strongly interacting. Equation (31) is then an underestimate and creep accommodation can be rate controlling. Typically in the later stages of growth just before failure of the specimen the cavities can link and some boundaries become completely cavitated. Further growth of the slab of porosity can then only be restricted by shear transmitted across uncavitated boundaries. The order of magnitude increase in the creep rate predicted for the power law creep accommodation process, equation (28), may describe the increased creep rate observed during tertiary creep. 6. CONCLUSIONS 6.1 At relatively high stresses cavity growth is controlled by the unconstrained dislocation creep process. C - 2np3K,a”
CAVITY
GROWTH
6.2 If diffusion is not inhibited by precipitates and not all boundaries are cavitated then diffusion creep
accommodation
can control cavity growth
6.3 When the cavitated boundaries lie predominently across the tensile axis the transverse strain is about 0.4 times the strain in the tensile direction. The above figure assumes that one in three boundaries are cavitated. In general the ratio of transverse to axial strain is dependent on the fraction of cavitated boundaries, their angular distribution with respect to the tensile axis and the creep mechanism. 6.4 When pores coalesce and cause complete separation of the boundary pore growth is limited by dislocation creep accommodation. For a material with half the boundaries cavitated creeping by power law creep with a stress exponent of five the cavitation rate and creep rate are given by
-AP z i + 13K,a5. V
6.5 Precipate hardened materials may not exhibit diffusion creep or diffusion controlled cavity growth. In this case cavity growth may be controlled by deformation around the grain boundary precipitates. ~c~~ff~~e~~~~~~~-This paper is published by permission of the Central Electricity Generating Board.
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6. 7. 8. 9.