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On creep fracture by void growth
Progress in Materials Science Vol. 27, pp. 189 to 244, 1982
0079-6425/82/040189-56528.00/0 Copyright © 1982 Pergamon Press Ltd
Printed in Great Britain. All rights reserved
ON CREEP FRACTURE BY VOID GROWTH A. C. F. Cocks and M. F. Ashby
Cambridge University, Engineering Department, Trumpington Street, Cambridge CB2 1PZ, England ABSTRACT The growth of void-like creep damage is analysed. Voids can grow by mechanisms controlled by grain-boundary diffusion, by surface diffusion, by power-law creep, and by any combination of two of these Approximate analytical equations for the growth rate by each mechanism, under multiaxial stress states, are developed, and related to a number of published analyses. The growth equations are integrated to give times and strains to fracture under constant (multiaxial) stress, constant load and two simple load histories. Both quasi-uniform and non-uniform distributions of voids are considered. The formulation can be related to the continuum damage method of Kachanov and suggests modifications to this method. It leads to a new prescription for extrapolating creep data. A unified picture of void growth appears possible, embracing most aspects of previously published models.
CONTENTS LIST OF SYMBOLS I. INTRODUCTION: CREEP DAMAGE 2. MECHANISMS OF CREEP AND OF HOLE GROWTH 3. THE MODELLING OF VOID GROWTH
3.1. Void Growth Controlled by Boundary Diffusion Alone 3.2. Void Growth Controlled by Surface Diffusion 3.3. Void Growth Controlled by Power-Law Creep Alone 3.4. Void Growth by Coupled Boundary Diffusion and Power-Law Creep 3.5. Void Growth by Coupled Surface Diffusion and Power-Law Creep 3.6. Void Growth by Coupled Boundary and Surface Diffusion 3.7. Interface-Reaction Control of Void Growth 3.8. Transgranular Creep Fracture 3.9. Void Growth Maps 3.10. Importance of Grain-Boundary Sliding 4. THE TIME AND STRAIN TO FRACTURE
4.1. 4.2. 4.3. 4.4. 4.5. :4.6.
Boundary Diffusion Alone Surface Diffusion Alone Power-Law Creep Alone Coupled Boundary Diffusion and Power-Law Creep Coupled Surface Diffusion and Power-Law Creep Coupled Surface and Boundary Diffusion
5. NON-UNIFORM DISTRIBUTIONS OF VOIDS: CONSTRAINED CAVITY GROWTH
5.1. 5.2. 5.3. 5.4. 5.5.
Both Void Growth and Grain Deformation by Diffusion Alone Power-Law Creep Alone Coupled Growth Creep Rupture Plots Other Distributions of Voids
6. COMPARISON WITH THE CONTINUUM THEORY OF CREEP DAMAGE MECHANICS
6.1. The Continuum Theory of Kaehanov (1958) and Robotnov (1969) 6.2. Comparison of the Continuum and the Mechanistic Models: Power-Law Creep 6.3. Comparison of the Continuum with the Mechanistic Model: Diffusion Controlled Growth 6.4. Results of the Comparisons LP,'~.S. 27 3-4 ,~
189
190 191 192 194 195 196 199 201 202 204 204 205 206 211 213 213 214 214 215 216 216 217 218 220 221 222 223 223 223 224 225 225
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7. EXTRAPOLATION OF CREEP DATA
7. I. 7.2. 7.3. 7.4.
Present Methods for Extrapolating in Time Model-Based Procedure for Extrapolating in Time Extrapolation to Non-Steady Histories of Stress and Temperature Extrapolation in Stress Space
8. SUMMARY REFERENCES
APPENDIX A: APPENDIX B: APPENDIX C: APPENDIX D" APPENDIX E" APPENDIX E
Void Growth by Coupled Boundary Diffusion and Power-Law Creep Void Growth by Coupled Surface Diffusion and Power-Law Creep Void Growth by Coupled Surface and Boundary Diffusion Stress State Caused by Grain-Boundary Sliding Creep Fracture Resulting from Precipitate Coarsening
226 226 227 230 234 234 235 235 237 239 240 242 243
LIST OF SYMBOLS He ~1 ~2 U3 ad ~+
p ~c G~
E
ff n, ~YO,to t,
t[ tf rh
rd re 2! d
I. /, f; f~ A ~:, V A
6. DI, D,, D~
QJ, Q,, Q~ k R
T
Equivalent tensile stress (N m - 2) Principal stresses (N m-2) Remote principal stresses (N m - 2) Mean normal stress in diffusion zone (N m -2) Stress used in continuum theory of creep damage (N m -2) The hydrostatic pressure; p = - J (al + ~72 + 03)(N m -z) Mean stress in damaged region when voids are non-uniformly distributed (N m - 2) Mean stress in undamaged region when voids are non-uniformly distributed (N m - 2) Young's modulus (N m - 2) Shear modulus (N m -2) Steady state creep-rate in the absence of voids (s- 1) Principal strain-rates (s- 1) Axial strain-rate of a grain-boundary slab of material (s-1) Strain to nucleate voids Strain to failure Creep constants (dimensionless, N m-2, s-1) Time to nucleate voids (s) Time to failure (s) Time to failure at constant load (s) Radius of growing void (m) Radius of diffusion zone (m) Effective void radius as seen by power-law creep zone (m) Centre-to-centre void spacing (m) Grain-size (m) Area fraction of holes on grain-boundary;fh = r~/I 2 Area fraction of diffusion zones; fa = r~/I 2 Effective area fraction of holes;fe --- r~/I z Initial area fraction of holes Area fraction of holes at which linkage occurs, taken as 0.25 Area fraction of holes at transition from growth by boundary diffusion to power-law creep growth in approximate analysis Area fraction of voids at transition from growth by surface diffusion to power-law creep growth Area fraction of voids when stress is changed from a diffusion-controlled regime to a power-law creep one Volume (m 3) Constant in creep-law Atomic volume (m 3) Grain-boundary thickness (m) Thickness of layer of material in which surface diffusion takes place (m) Grain-boundary, surface and lattice diffusion coefficients (m~ s- i) Activation energies for boundary, surface and lattice diffusion (J mole- 1) Surface free energy (J m -2) Boltzmann's constant (1.38 x 10 -23 J K -1) Gas constant (8.31 J mol- ~ K - 1) Absolute temperature (K)
ON
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¢o #'0 Ao
O3
H,J
CREEP
FRACTURE
BY VOID
GROWTH
191
Reference temperature (K) Length of Burger's vector (m) Dimensionless quantity which appears when grain-boundary diffusion and power-law creep are coupled Dimensionless quantity which appears when surface diffusion and power-law creep are coupled Dimensionless quantity which appears when surface diffusion and grain-boundary diffusion are coupled Parameter that measures effect of stress state on void growth-rate Value of//corresponding to simple tension Measure of damage in continuum theory of creep damage Parameters used to plot creep rupture data.
1. INTRODUCTION: CREEP DAMAGE
When a solid deforms at high temperatures, its structure changes. Holes or cracks may nucleate and grow inside the solid; its grain size may increase (because of grain growth) or decrease (because of dynamic recrystallisation); precipitates or dispersoids within it may coarsen or dissolve; a substructure introduced by prior working may be destroyed; or the microstructure may, in some other sense, be altered. All these changes we refer to as "creep damage". This damage usually accelerates the creep-rate, which in turn accelerates the rate of appearance of damage, until the specimen or component fails. The tertiary part of the creep curve shown in Fig. 1 is a direct result of this coupling of creep-rate and damage-rate. The figure defines the other quantities which interest us here: the time-to-fracture t~- (s), the strain-to-fracture %,, and the steady-state creep-rate ~, (s- 1). This paper examines one class of creep damage: holes and cracks which will be collectively termed "creep voids". We shall not discuss their nucleation (for which see Goods and Brown, 1979; Argon et al., 1980), but include it merely as an additive nucleation time (t,) or strain (E~).We focus attention on void growth and the way in which stress, temperature and stress-state affect it. We combine results from two earlier papers (Cocks and Ashby, 1980, 1981) with newly derived solutions to formulate differential equations for creep-rate and for the rate of growth of void-damage. The results are presented as "void growth rate maps", and are compared and related to earlier microscopic models of Hull and Rimmer (1959), Beere and Speight (1978), Edward and Ashby (1979), Chuang et al. (1979), and Dyson (1976, 1978), to the finite-element calculations of Needleman and Rice (1980) and to the continuum-damage mechanics of Kachanov (1958).
STRAIN TERTIARY STRAIN SECONDARY
= Pccstf ELASTIC* PRIMARY = TIME ~-
tf
=
FIG. 1. A schematic creep curve.
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PROGRESS
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These damage growth equations are then integrated for conditions of constant stress, constant load and simple load histories, to give the time and strain to fracture. A modelbased procedure for extrapolating creep data is developed. The effect of stress state is discussed and illustrated by constructing isochronous fracture surfaces. Where possible, the results are compared with experiment and their implications for the extrapolation of creep data are indicated. It must be remembered throughout that creep fracture is a very complicated phenomenon, only one aspect of which is analysed fully here. The creep life of creep-resistant alloys like the Nimonics or the low-alloy ferritic steels, is influenced strongly by the ease or difficulty of void nucleation and by microstructural changes other than void growth, both of which are considered at only the most elementary level. But in other ways the conclusions reached here are encouraging: we believe a general consensus is emerging on how holes grow and link; a single set of equations is presented which, at an approximate level, describe power-law creep growth, growth controlled by boundary or surface diffusion and growth involving the coupling of pairs of these. Further, a direct connection appears between the microscopic theories and the continuum-mechanics of creep damage; and the method leads to a new, model-based, extrapolation procedure which looks very promising.
2. MECHANISMSOF CREEP AND OF HOLE GROWTH Above 0.35 TM (or thereabouts) a crystalline solid creeps. At high stresses, the solid suffers power-law creep, described by the constitutive law d,~, = do
(2.1)
where ae is the equivalent (tensile) stress, defined by tl tze = L:r[(al - a2) 2 + (a2 - a3) 2 + (a3 - al)2]} 1/2
(2.2)
and d,., is the equivalent (tensile) strain-rate d,~. = ~g[(e12 " - d 2 ) 2 + (d2 - d 3 ) 2 + (da - d l ) 2 ] } 1/2.
(2.3)
The quantity do depends exponentially on temperature do ~c exp
Qc
RT
where Qc is the activation energy for creep. It is discussed further in Section 7. The quantities n and ao(MN/m -2) are creep constants and ala2a3 and dld2d3 are the principal stresses and strain-rates. At lower stresses and in small-grained solids, deformation is by diffusionalflow, and the strain-rate is controlled instead by diffusion across or around grains (Nabarro, 1948; Herring, 1950; Coble, 1963; Raj and Ashby, 1971):
12oo~Ge( 1 + rc~Da~ dD~ /
ds~, '" ~
(2.4)
where Dv and 6Ds describe the rates of latticeand of grain boundary diffusion,and d is the grain size. These regimes of deformation are shown for copper (which typifiesmany f.c.c. metals) in Fig. 2, which also shows the conditions under which dynamic rccrystallizationis
ON
CREEP
FRACTURE
BY VOID
TEM PERATURE I0"
-200
0
200
,
•
,
ml mBIIDEAL I m qml!'HEAR el i
400 .
('C) 600
800
I000
I
i
I
STRENGTH I m II m m ~ I m ~
mD
PURE
COPPER
d -
KS~ w:}')',
193
GROWTH
PLASTICITY
O.I mm
79
,o'
DYN~R~CsTALUSATIO N
b
I \\\~",~13 EAK
U~ Ld n,
I-- J6 ~
tO n,< UJ "r"
(L.T. CRE"EP)~ x"
~
~)1~
o
POWER ILAW
C
LLI
W
n."
m_ ._1 <
0 z
DIFFUSIONAL
KS:
16
F LOK-_.
' D,.'}.,o., O
O. 2
0.4
0.6
0.8
1.0
HOMOLOGOUS TEMPERATURE, T/'rM FIG. 2. A deformation-mechanism map for copper, showing the regimes of plastic deformation.
observed, and the transition region (labelled "Breakdown") between power-law creep and classical (rate-independent) plasticity. Creep fracture is more complicated. Under creep conditions, fracture most commonly occurs by the growth and coalescence of voids which lie on grain-boundaries. The growth can be controlled by boundary diffusion (Hull and Rimmer, 1959; Speight and Harris, 1967; Raj and Ashby, 1975); by surface diffusion (Chuang et al., 1979); by power-law creep (Edward and Ashby, 1979; Cocks and Ashby, 1980); or by a coupling of diffusion and power-law creep (Beere and Speight, 1978; Edward and Ashby, 1979; Needleman and Rice, 1980; Cocks and Ashby, 1981). At any instant in time, a particular growth mechanism dominates, so that a map, analagous to Fig. 2, can be constructed for void growth (examples are given later). But the time to fracture reflects the entire history of the growth of voids, from nucleation to final linkage. Voids usually grow by diffusion when they are small, but as they become larger, power-law creep takes over as the dominant growth mechanism: this coupling of mechanisms must be taken into account in calculating the time to fracture. And the stress state matters too: creep fracture depends not only on the equivalent stress (tr,) but also on the magnitude of the maximum principal stress (al) and the hydrostatic pressure p = -~(trl + tr2 + a3).
(2.5)
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PROGRESS IN MATERIALS SCIENCE
The next section describes approximate ways of calculating damage rates b.y each of the void growth mechanisms, under multiaxial stresses. Section 4 shows how these can be combined and integrated to give times to fracture. 3. THE MODELLINGOF VOID GROWTH Voids may grow in a way controlled by boundary diffusion, or by surface diffusion, or by power-law creep or by combinations of these. When diffusion is involved, the voids grow on grain-boundaries, but when power-law creep is dominant they may grow within the grains as well. We now consider each growth mechanism in turn. For the purpose of analysis we isolate a cylindrical element of material of diameter 21 (the void spacing) and height d (the grain size) centred on a grain-boundary void of diameter 2rh, shown in Fig. 3. The surfaces of this unit are subjected to the stress field at, a2, a3 as shown. We calculate the void growth rates caused by this stress field, remembering that it may differ from the remote (applied) field cry, a~, a~ because of grain-boundary sliding. Finally, the two fields are related (Section 3.10) to give the rates of void growth caused by a given remote field. The calculations which follow are approximate. In solving diffusion problems, refinements which influence the results by less than a factor of two are dropped. In solving problems involving power-law creep, bound-methods are employed which also involve errors; but comparison of our results with the finite element calculations of Needleman and Rice (1980) shows agreement to within a factor of about two, giving confidence in the method. The value of the approach given here is that it is very simple, it deals with multiaxial stress states, and it gives analytical results which can easily be integrated to give times and strains to fracture for any given history of load and temperature. By far the largest source of error in using these results is the material data: power-law creep data can
ttftfil
---X
tt
" Y--Illlli
o',~
Ill
I
o-,
Fro. 3. Void growth on a grain-boundary, showing the unit of structure.
ON
CREEP
FRACTURE
BY VOID
GROWTH
195
,.Jim
~.= (~O(~o)nI ~-~-
RIGID GRAIN IOUNDARY
SURFACE DIFFUSION ,,.
(i
~n
%--..q arh
- ~ 2rh~-
- - - - - ~ 2,~ - - - - - - ~ RIGID GRAIN
RIGID GRAIN
..q-.
•
..i,-
)
)
(a)
)
t
)
;
;
t
.
o-n
)
t
....
¢=%1~o) )
(b)
;
(c)
FlG. 4. The three simple mechanisms which limit void growth.
generally be trusted to within a factor of about two, boundary diffusion coefficients, when known at all, are accurate only to a factor of about four; surface diffusion, to a factor of perhaps ten. 3.1. Void Growth Controlled by Boundary Diffusion Alone Hole growth controlled by boundary-diffusion alone is shown in Fig. 4a. Matter diffuses by boundary diffusion out of the growing void and plates onto the grain-boundary. The void remains spherical because surface diffusion rapidly redistributes matter within it. This diffusion problem can be solved with more than enough precision for our purposes (Hull and Rimmer, 1959; Speight and Harris, 1967; Raj and Ashby, 1975). There is general agreement that the volume of matter flowing out of the void per second is well approximated by
(
r23
dr _ 2~D,6Balfl \1 12"] dt kT ln(//rh)
(3.1)
where 2rh is the hole diameter and 21 their spacing. (The equation becomes inaccurate if the void size is near the "sintering limit" at which surface tension balances the driving force due to the applied stress. Then al/(1 - r2/l 2) must be replaced by [~1/(1 - r2/I 2) - 2?s/rs], where ?, is the surface energy of the solid.) The other components of local stress field (a2, a3) have no important influence on void growth by this mechanism. We now define the area fraction of holes, fh, by f~ _ r~
12
(3.2)
2De6Bft ao k Tl3 #-o
(3.3)
and we define a new material property, ~bo,by
C~o =
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PROGRESS
where So and 4o were defined both because atoms flow out boundary over the area 7r(I2 of the void caused by diffusion
IN
MATERIALS
SCIENCE
in eq. (2.1). We further note that the void grows in volume of it by diffusion and because these atoms plate onto the r 2) jacking the two half-crystals apart. The total growth-rate plus jacking is
dt
dt
and the growth-rate of the damage, dfh/dt, is easily shown to be
df. 1 dV d--t = 2zrrh~ dt" Substituting these into eq. (3.1) gives the final equation for the damage-rate; and straightforward manipulation of eq. (3.1) gives also the strain-rate caused by void growth.
ldf
4o dt
4,o
-
f]/2
ln(1/~)
(a)
(3.4)
l de 4o & = In(l/A) \d,]\0"o,]"
(3.5)
[To this must be added the strain-rate due to diffusional flow, eq. (2.4) which, when boundary diffusion is dominant, becomes: 1 de = 6n4~o ae 40 dt -d3 ~o "
(3.6)
Power-law creep does not contribute to pure diffusional growth, of course; 0-0 and 40 appear only because they are convenient normalizing parameters.] The dependence of damage-rate on the local stress state can be illustrated by plotting surfaces of constant dfh/dt in principal stress space. Such a surface for plane stress and plane strain are shown in Fig. 5a. The axes are normalized stress: 0-1.
0"2.
X3
-~- 0"..3_3
0-0
where 0-0 is the simple tensile stress that produces the given damage growth rate. The surfaces are those of constant maximum principal stress. When grain-boundary sliding is included, the surfaces become more complicated: they are illustrated, and discussed, in Section 3.10. Under certain circumstances, voids growing by boundary diffusion or by a coupling of boundary and surface diffusion (Section 3.6) can become unstable. The perifery of the void develops perturbations which grow into finger-like extensions. These fingers tunnel ahead of the main void, increasing the damage rate (dfh/dt). The instability and subsequent growth can be analysed (Fields and Ashby, 1976) but seems rare in practice; we shall ignore it here. 3.2. Void Growth Controlled by Surface Diffusion When surface diffusion within the void is slow, it ceases to grow as a sphere. Matter flows out of it at the equator, causing it to become flatter and more crack-like (Fig. 4b) until the curvature difference between the poles and the equator is sufficient to drive a surface flux
ON
"
CREEP
FRACTURE
,,,~,,4
ANO STRAIN
O.S
~
~I~o.s
20
197
GROWTH
F~WER-LAW CREEP
PLANE STRESS
El
BY VOID
tr
-,.o
-O.S
PLANE STRESS
~ o \o.,
o
-O.SX
2.C
2.0
"~l~°'s !,,
2.0
,.;3.0
- 1.0" ~
//~
f..o
•
r~3.o
- 1.01" ~"
FIG. 5. Fracture surfaces for void-growth by diffusion, and by power-law creep, ignoring the stressconcentrating effects of grain-boundary sliding. These are introduced later and shown in Fig. 18.
which matches that leaving, by boundary diffusion, at the equator. This problem has been analysed in detail by Chuang and Rice (1973) and Chuang et al. (1979). This section provides a simple, approximate, analysis of the limit in which surface diffusion completely controls void growth. The coupling of this with boundary diffusion and power-law creep are considered later. The shape of a crack-like void is one with continuously changing curvature, as shown by the broken line in Fig. 6. We idealize this as a penny-shaped crack of constant width and semicircular tip of radius r0 a s s h o w n by the full line. The governing equation for the surface-diffusion current is:. I, =
O-k--T~\d-S]
(3.7)
where I, is the number of atoms flowing across unit length of surface per second driven by the surface potential gradient (d#/dS), and D,h, is the surface diffusion coefficient times
198
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IN MATERIALS SCIENCE
o-,
I
DIFFUSION
i_ ~-
r h ________~
I I I
I
I_ If
1
II
1
Fro. 6. Void growth limited by surface diffusion.
its diffusion thickness. The potential gradient is given by: d#_ d dS dS [-7sf~(xt + x2)].
(3.8)
Here S is a coordinate measured along the surface, and xl and x2 are the principal curvatures at S. We now approximate the differential by the difference in curvature between the crack tip and the crack flank (where, in our approximation, the curvature is zero) divided by the distance between them, which we take to be (Tr/2)ro. This gives:
dS
rr ro
where 1
Xl = - ro
and
x2 = -
1
rh
are the principal curvatures at the crack tip and 7, is the surface free energy. If the crack advances with velocity dr,/dt and rh >> to, then conservation of matter requires that:
drh 2ro -~- = 2I, f/.
(3,9)
But if surface diffusion totally controls the growth, the potential gradient in the grain boundary is negligible; and for the potential at the void tip to be continuous we require that:
1
ro
-
al
(3.10)
~,(1 - A )
Combining these equations gives:
dra at
2 -
D,6,f~
7r kTy~,(1 - A P
a~.
(3.11)
ON CREEP FRACTURE BY VOID GROWTH
199
This is identical with the equation derived by Chuang et al. except for the constant n/2; they find 2 x/~ instead. This difference is slight, but we can amend eq. (3.11) to include it. We define a second material property ~ko, analagous to ~bo [eq. (3.3)], ¢,o =
(3.121
and using
dfh dt
2rh drh 12 dt
and
& 8xrorh drh a t = d12(1 - J h ) d t we obtain
1 dfh d/of~/2 (trl~ 3 ~o d--t-= (1 - ~ \ ao---/ 1 dl~ do
(3.13)
41~of I/2 ~s (o'_1_1~2
(3.14)
=
To this must be added the strain-rate due to diffusional flow [eq. (3.6)], if appropriate. As before, it should be realized that power-law creep makes no contribution here; ao and do appear as normalizing parameters only. The surfaces of constant growth-rate in stress space, for this mechanism, are identical with those for boundary-diffusion limited growth (Fig. 5a).
3.3. Void Growth Controlled by Power-Law Creep Alone A void can grow by the power-law creep of the surrounding matrix as shown in Fig. 4(c) (Hellan, 1975; Edward and Ashby, 1979; Cocks and Ashby, 1980). The growth-rate can be computed by using finite element methods (Needleman and Rice, 1980); or it can be calculated approximately (Cocks and Ashby, 1980) by using a bound theorem due to Martin (1966). This second method is approximate; but it leads to analytical expressions for the void growth-rate, it can be adapted for multiaxial loading, and it can be checked against the results for the finite element calculations. In simple tension, the result is particularly simple: the shaded slab containing the void in Fig. 4(c)extends at a rate determined by the net section stress a d o -fh), while the rest of the cylinder (unshaded) extends at a rate determined by tre alone. The result is that the shaded slab dilates, causing the holes to grow. The application of the continuity equation (Cocks and Ashby, 1980):
27r-ff-rh dt
d
)
- fhi:ss
(3.15)
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PROGRESS IN MATERIALS SCIENCE
leads immediately to expressions for the damage rate (dfddt) and the strain-rate (d~/dt): 1 dA
E
1
(1 - A )
(3.16)
~o Z = # (i _7~),
]a~
{
~o ~ =
1
+
2r._F 1 -z~L(1
t-l~(o2}"
---Ay'
(3.17)
jo,,oo/
where fl is a constant ( ~ 0.6) and ae is the von Mises equivalent stress (eq. 2.2), equal to the axial stress al in simple tension. The behaviour under combined axial and hydrostatic tension is less obvious, but can be treated by the bound method (Cocks and Ashby, 1980), and by finite element methods (Needleman and Rice, 1980). The results show that a hydrostatic tension -p/ae accelerates void growth. The damage-rate and the strain-rate are still given by eqs. (3.16) and (3.17) with fl--sinh-
2 ~ - ~
.
(3.18)
The result can be illustrated (as before) by plotting surfaces in stress space on which the hole growth-rate has a constant value. They are shown in Fig. 5b, c and d for plane stress, plane strain and axisymmetric loading. 102
~o =O.I n =5 t h - i O -2
I01 -
r_,,~
ool
COUPLED GROWTH
/
/
-
.,//
/ 7 Idt
//
/
~o-
/
I
I
/
/
//
/ OIFF'USIONAL
GROWTH
/
/"~-POWER- LAW CREEP GROWTH
I, %,
FIG. 7. The damage-rate as a function of stress, when voids grow by diffusion and power-law creep.
ON CREEP FRACTURE BY VOID GROWTH
201
3.4. Void Growth by Coupled Boundary Diffusion
and Power-Law Creep If the damage-rates given by eqs. (3.4) and (3.16) are plotted against stress, the behaviour shown in Fig. 7 is found. When the stress is low, the voids grow by diffusion; when high, the3; grow by power-law creep; in between, they grow by a combination or coupling of both (full line, the derivation of which is described below). If, instead, the damage-rates (dJ~/dt) are plotted against fh the behaviour is as shown in Fig. 8. When J~ is small, the voids grow by diffusion; when large, they grow by power-law creep; in between, they grow by a coupling of both (full line). The growth-rate has a minimum at the point at which the two curves intersect. This is important: the time-to-fracture is largely determined by the depth and position of this minimum, and therefore depends on the rates of both mechanisms. Hole growth by coupled boundary diffusion and power-law creep has been analysed in several recent studies (Beere and Speight, 1978; Edward and Ashby, 1979; Needleman and Rice, 1980; Cocks and Ashby, 1981). An analytical calculation of the rate is summarized in Appendix A. The equations given there have been solved and lead to the full lines shown on Figs 7 and 8. It is obvious from these two Figures that a reasonable approximation for the coupled rate of growth is simply given by adding the rates of the two simple mechanisms: boundary diffusion [eq. (3.4)] and power-law creep [eq. (3.16)]. This is such a convenient approximation that much of the following development will be based on it. The point at which the broken lines intersect on the two figures can be calculated by equating eqs (3.4) and (3.16). This is the point at which the dominant void
102
<~,., = 0.1 n
= 5
l dfh
/
COUPLED MECHANISM
DIFFUSIONAL GROWTH
\ \ ~
,o-'
.j/
/
//
/
/
/ ~ W E R - LAW CREEP / GROWTH /
/
/
JdK~e
I
Id s
I
Io-~
i
Id 3
/
i
Id 2
I
Io-'
fh FIG. 8. The damag¢-rate as a function of damage, wh¢n voids grow by diffusion and power-law creep.
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PROGRESS
IN MATERIALS
SCIENCE
growth mechanism changes. The result is
~b0 = #f~/21n
(I)[ ~
I
(1 - J ~ ) "
(I- J~)l(e'y(e°'~ d\eo/\ell"
(3.19)
(The result for simple tension is obtained by setting fl = 0.6 and ee = at.) This is the equation of the line, in stress/temperature space, separating the field in which boundary diffusion is dominant from that in which power-law creep is dominant. It can be solved to give the critical area fraction, f~, at which the change of mechanism occurs (Cocks and Ashby, 1981). It is
I f ~ = [a(ln a - 1)] a/2
(3.20)
where
4(n + 1)(ae~"(e0~//. a =
(3.21)
3cb~ \go/ \ell
3.5 Void Growth by Coupled Surface Diffusion and Power-Law Creep When the damage-rates given by eqs (3.13) and (3.16) are plotted against stress (Fig. 9) and againstfh (Fig. 10), they are seen to intersect. As before, diffusion controls the early part ~ro = O . I n = 5 fh = IO-2
/
/
/
/
/
/
/
/
GROWTH
/ SURFACE DIFFUSION CONTROLLED
j6 ~
16'
/•//11 I /
I I I
1.0
%
IO
FIG. 9. The damage-rate as a function of stress, when void growth is limited by surface diffusion,and
power-law creep.
ON
CREEP
FRACTURE
BY VOID
GROWTH
203
I01
,o=O, . io 0
/!
/I
=s
~'= I %
/ ,/ t
I0 i I
df h
io dt
,d
SU~rAC~
Ol~~ ///' /Y/l/it,
1111 I
i0 "l'
/
I
iO-e io-S io-~
I
I
9 3 io-Z
I I 0 -~
i0 °
fh FIG. 10. The damage-rate as a function of damage, when void growth is limited by surface diffusion and power-law creep.
of void growth, but power-law creep takes over as the voids become larger and the net section stress rises. The coupling of these two mechanisms is treated in Appendix B. For most practical purposes it is adequate to assume that the coupled rate is simply the sum of the rates of the two individual mechanisms: sur]'ace diffusion [eq.(3.13)] and power-law creep [eq. (3.16)]. The point at which the broken lines intersect on Figs 9 and 10 is important in calculating the fracture time. It is the point at which the dominant mechanism changes, and is obtained by equating eqs (3.13) and (3.16). The result is
+o
=
ill2
'
(1 - - A ) n + '
,l:°'Y(=ol _J\~oo./ \~tt]' "
(3.22)
(Simple tension is described by setting ~ = 0.6 and o, = or.) This is the equation of the line, in stress/temperature space, separating the field in which surface diffusion is dominant from that in which power-law creep is dominant. It can be solved to give the critical area fraction, f~, at which the change of mechanism occurs. Provided f~ << 1 (as it usually is) the result is
[ +o (ooi?o,71'
I: ~- (,,+ I)/~,,~i ,,~o: j
(3.23)
204
PROGRESS
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SCIENCE
3.6 Void Growth by Coupled Boundary and Surface D!ffusion When surface diffusion is very rapid, the growing void can maintain its equilibrium shape (two spherical caps that meet in the boundary plane). All the driving force is available to drive the boundary diffusion required to make the damage grow, which it does at a rate described by eq. (3.4). When, instead, surface diffusion is slow, the void becomes crack-like (Section 3.2). In the limit that all the driving force is used to drive surface diffusion, the damage-rate is given by eq. (3.13). Between these two limits, the void adopts a shape which reflects the relative rates of boundary and of surface diffusion, and the driving force is partitioned in such a way as to drive them at matching rates. This problem has been analysed in detail by Chuang et al. (1979). Appendix C contains a simple, approximate treatment which gives the void shape and damage-rates in this coupled regime. Obviously, both surface and boundary diffusion are necessary for void growth, so it is not simply the faster of the two which is dominant (as it was in the last two sections). We require some other criterion for the transition from surface to boundary diffusion controlled growth. A sensible criterion is that the void maintains its shape as it grows. This is examined in Appendix C, where we show that the transition occurs when a_l = 1.5 + 9A0 ln(1/fs) aCAV 2 X/2(1 -- A)
(3.24)
where acAP is the capilarity stress at the crack tip aCAP --
27,(1 - A)
(3.25)
rh
and Ao is a new dimensionless material property Ds~s A° = DatSs"
(3.26)
This is the equation of the line in stress/temperature space, separating the field of dominance of surface diffusion from that of boundary diffusion.
3.7 Interface-Reaction Control of Void Growth If grain boundaries are imperfect sources of vacancies (or if the void itself is an imperfect sink) then the boundary conditions of the diffusion problems are changed. Instead of equating the chemical potential of atoms (or the concentration of vacancies) to that in equilibrium with a grain-boundary on which a mean local stress at acts, the potential (or concentration) differs from the equilibrium value by the amount necessary to drive the "interface reaction"--in this case, the generation of vacancies. This explanation was proposed by Ashby (1969) to explain the observed inhibition of diffusional flow in dispersion-strengthened alloys. It applies equally to the inhibition of void growth. Since then other models (Harris, 1972; Burton, 1977) have been proposed which extend this sort of approach. All can lead to a threshold stress for void growth; and all can lead to a simple diminution of its rate. Interface-reaction control can be readily incorporated into the treatment of diffusional void growth (for the method, see Ashby, 1969) and is
ON
CREEP
FRACTURE
BY
VOID
205
GROWTH
0 C)
0
0
0
iI
C)
2rh0
0
0
FIG. 11. A h o m o g e n e o u s distribution of voids, growing by power-law creep.
likely to be important when voids are sharply facetted, or when alloys contain a fine dispersion of a stable second phase. It is not discussed further here.
3.8 Transgranular Creep Fracture Voids that lie within a grain grow only by power-law creep. It is true that lattice diffusion from the void to the free surfaces or to grain-boundaries will enlarge the void, but under all realistic circumstances, the rate of such diffusional growth is negligibly small. Figure 11 shows how a material containing a random array of voids within its grains can be divided into slabs, each containing a single void. The slabs are no longer separated by the grain size d but are stacked on top of each other. The growth equations of Section 3.3 can be adapted to describe the growth of this array. The damage-rate equation is unchanged; but the strain-rate equation is altered because the slab height 2rh replaces the distance d, giving the approximate results
1 dfs
1
(1 - fs)
1 d~ ~fl_ l ( a e ~ " Co ~ = L1 - fl + (1 -A)"_[\~oo] "
(3.27)
(3.28)
In simple tension fl -~ 0.6 and ~e = 31 as before. These results allow a direct comparison of the mechanistic approach of this paper with the continuous damage mechanics of Kachanov, discussed in detail in Section 6. J.PM.S.27 3-4 a
206
PROGRESS
IN MATERIALS SCIENCE
iO-'[
COPPER
~1 GROWTH BY LOWI0-'~- TEMPERATURE
0
Z
t
/
DATA
CONTROLLED GROWTH
--6[
~°o
POWER-LAW H
I
0.2
I
0.4
I
OF RAJ I
0.6 o.a TEMPERATURE (T/T M)
HOMOLOGOUS
~.o
FIG. 12. A void growth map for copper with 1 = 12#m andf~ = 10 -1.
3.9 Void Growth Maps The results developed in the previous sections are conveniently displayed as void growth maps (Figs 12-16)'I". The axes are the local tensile stress, normalized by Young's modulus, ~I/E, and temperature, T/Tu, (where Tu is the melting temperature). Each map is divided into fields in which a single mechanism is dominant. Superimposed on the fields are contours of constant damage-rate, (df~/dt). The damage-rate contours are calculated from the equations for simple power-law creep [eq. (3.16)] and simple boundary diffusion [eq. (3.4)], (taking the faster of the two) together with the full equation for crack-like growth by coupled surface and boundary diffusion (Appendix C)
I dfh
3~bo
( ~ "~3~'[1
2w/-~trliAof~/Zln(i/fh)]i/2 }a.
"o dt = 2 x/2fh(ln l/fh)3Aos \hTo// (k +
Y-~ ~-~)3"
..j
- 1
(3.29)
In the limit of surface-diffusion controlled growth, this reduces to eq. (3.13), which properly describes void growth provided
2 v/2f]/2 ln(1/J~) O'l/AO ~ 1. -
-
(3.30)
?A similar concept has been developed simultaneously by Svensson, L. E. and Dunlop, G. L. (1980) Proc. I U T A M Meeting, Leicester, England.
_
l1 Z-,~,~,h-,o' I POWER'LAW
Ul (Jr) LIJ rY
~,~
CREEP GROWTH
I
,o-~ Z
.
C
O
N
T
R
O
~
_u) <
Q:" O Z °
DATA OF RAJ
..t . . . . .
I
0.2
I
.............
0.4
1 ........
O,6
I
0.8
HOMOLOGOUS TEMPERATURE (T/T M)
1
1.0
Fro, 13. A void growth map for copper with I = 12#m and]~ = I0 -~.
~'
"U3 " "~
~
I CoPPER
............................
POWF.R..LAW TEMPERATURE PLASTICITY ~. .r...~ p GROWTH
~d~
m
"3 O"51 <1[ IZ I
DATA OF / NIEH & NIX
BOUNDARYDIFFUSION CONTROLLEDGROWTH
O Z
vo
0.2
0.4
o.~
o.s
HCMOLOGOUS TEMPERATURE (T/TM ) FIG. 14. A void&rowth map for cop_per with I = 1.6~um a n d f , = 0.2.
207
~.o
tot
I SILVER
-
I
?~ '~ J ~ A S T I C I T Y
Z
............
4
......!:~
,o
~to
,
BOUNDARY DIFFUSION
CONTROLLEDGROWTH
nf
O Z
i0 ~ O
i O. 2
1 O.4
I O.6
I 0.8
I.O
HOMOLOGOUS TEMPERATURE (T/TM ) FIG. 15. A void growth m a p for copper with 1 = 2 . 2 # m andfh = 0.2.
Io-'
S I LVE R .,Z =ll#mifh= O.8
,~, io-"
GROWTH BY LOWTEMPERATURE P L A S T I C I T Y
LLI -3 I~ I 0 ~
C R E E P GROWTH
""-
DATA OF
~
~
I
~ , \ "-,o-.~
,o i S U R F A C E DIFFUSION
iO'6 :;:;;~':/.:':!::::::!'~i:~'i/i~i~!~!i
O
0-2
O.4
0.6
O-8
HOMOLOGOUS TEMPERATURE (T/TM) FIG. 16. A void g r o w t h m a p for silver with I = 11 # m andfh = 0.8.
208
I.O
ON
CREEP
FRACTURE
BY
VOID
209
GROWTH
The field boundaries (heavy lines) separate the regions of dominance of each mechanism. The boundary between the two diffusion mechanisms is given by eq. (3.24). The boundaries between the diffusional mechanisms and power-law creep are given by equating the simple damage-rates [as described by eq. (3.19 and 3.22)], except when surface and boundary diffusion couple; then the damage-rate due to power-law creep (eq. 3.16) is best equated to eq. (3.29) giving an implicit equation of ~rl and T which can be solved by normal iterative methods. Equation (3.30) is shown on the maps as a broken line. To the left of this line. crack-like growth becomes dominated by surface diffusion so that eq (3.29) can be replaced by the simpler eq. (3.13). The shading on the maps shows the width of the transition between pairs of mechanisms. We have based the calculation of this width on the discussion given in the earlier sections. The transition from sphere-like to crack-like growth occurs over the range (Appendix C) 3 In(l/A) or1 6 In(l/A) 1 + (~ ~ - ~ Ao <~ aCAP <~ 2 + ~-f --~-j~ Ao
(3.31)
where aCAP is given by eq. (3.25). The transition between boundary diffusion and power-law creep has width (Appendix A): In(l/A) f qb0 "~o/,-1, a~ //~o ~,~/n-,, [1 + 2 In(l/A)] ~"/"- ~) \ f ~ / 2 ] <~ --ao <~ (1 - fh) \~37h3/~] .
(3.32)
That between surface diffusion and power-law creep has width (Appendix B): [(3)"(1 - A ) " - ' f ~ / 2 ~ o ] "/"-3~ ~< a--L ~ eo
f]/2( 1 - A)A
(1 -¼A).
(3.33)
The maps show three main fields: within the unshaded regions, simple power-law creep, simple boundary diffusion CHull-Rimmer growth"), and simple surface diffusion control the growth-rate. The area of boundary-diffusion control is relatively small. That of surfacediffusion is generally larger and dominant at low temperatures and slow growth-rates. Power-law creep tends to become dominant at high temperatures and, as one would expect, gives high growth-rates. These are separated by (shaded) regions in which the coupling of two mechanisms is important: they can be regarded as broad transitions between one "simple" mechanism and another. The bottom line on the figure shows the capilarity stress. When the local tensile stress is less than this, the voids sinter instead of growing. Interfacereaction controlled growth can be included on such a map, but it is not shown here because there is no evidence for it in pure copper or silver. The positions of the boundaries between the mechanisms depend on A (compare Figs 12 and 13). Because of this the boundaries sweep across the maps as the voids grow, and more than one mechanism has a term of dominance during the life of the component, even under conditions of constant stress and temperature. This is taken into account in Section 4 where times to failure are calculated. The growth maps of Figs 12-16 have been constructed for given values of void spacing, 21 and area fraction of voids, j~. In all of these experiments, voids were either present in the material at the start of the tests (Goods and Nix, 1978a, b; Nieh and Nix, 1980) or were nucleated early on in the tests (Raj, 1978). So in each situation the problem becomes one of void-growth controlled life. Figures 12 and 13 show the range of dominance of each mechanism for pure copper for I = 12 vm and two values of A. (The additional data used to construct these and subsequent
210
PROGRESS
IN
MATERIALS
SCIENCE
Table 1 Copper n A D o v ( m 3 s -1) Qv (KJ m o l e - 1) b (m) # o ( M N m -2) Tm (K) A#r(K -l) t2(m 3) D o n f s ( m a s -1) Qn (KJ m o l e - 1) y, (J m -2) Do,6,(mas -1) Q, (kJ m o l e - 1) ao (MN m -2)
4.8 2.1 x 10 -9 6.2 x 10 -5 207 2.56 x 10 -1° 4.21 x 104 1356 3.97 x 10 - a 1.18 x 10 -29 5.12 x l0 - i s 105 1,72 6 x 10 -1° 205 39.0
Silver 5.3 2.1 × 10 -9 4.4 x 10 - s 185 2.89 x 10 -1° 2.64 x 104 1234 4.36 x 10 - a 1.71 × 10 -29 6.94 x 10 -15 89.8 1.12 6 x 10-1°? 186? 61.0
?The values of Do,f, and Q, for silver were calculated by averaging over a n u m b e r of f.c.c, metals ( N e u m a n n and N e u m a n n , 1972).
maps is given in Table 1). The choice of I corresponds with that found by Raj (1978) in his experiments on copper bicrystals. The range of stress and temperature used in the experiments is illustrated on the maps. At low stresses, Raj (1978) showed that the times to fracture were consistent with the boundary diffusion model. At high stresses he observed that growth occurred by a plastic mechanism and failure times were found to be less than that predicted by the boundary diffusion model. These observations are borne out by the maps which show that the experimental range bridges the boundary diffusion and powerlaw creep mechanisms. The map for pure copper shown in Fig. 14 was constructed using l = 1.6/~m andA = 0.2, corresponding to the initial values found in the experiments of Nieh and Nix (1980). The map shows that experiments were conducted well within the regime where surface diffusion controls void growth. This is consistent with their finding that the times to failure agreed with the surface diffusion model. Figure 15 shows a void growth map for pure silver using I = 2.2/~m and f, = 0.2, together with the range of stress and temperature used in the experiments of Goods and Nix (1978a). Again, it is expected that surface diffusion should control void growth, as they observed. Goods and Nix (1978b) performed additional experiments on specimens containing large closely spaced voids. An appropriate map is shown in Fig. 16 on which the range of stress and temperature used in their experiments is drawn. Failure was found to be controlled b'y power-law creep. The experiments, however, seem to have been conducted in a regime where surface diffusion controls growth. This discrepancy is probably due to the large uncertainty in the data for surface diffusion coefficients. In constructing these maps the effects of stress concentration due to grain-boundary sliding was ignored; the stress shown here is the local stress. Taking grain-boundary sliding into account results in a translation in the experimental data upwards by an average factor of 1.3 on the figures, though local peaks in stress can be much larger. This shift would, in fact, move the experiments of Goods and Nix (1978b) towards the regime where power-law
ON CREEP FRACTURE BY VOID GROWTH
211
creep controls growth. The overall effects of grain-boundary sliding are the subject of the next section. 3.10
Importance of Grain-Boundary Sliding
The growth equations describe the growth of voids on a single, flat boundary like those shown in Fig. 4. Real engineering solids are made up of nesting polyhedral grains. At high temperatures the boundaries between these polyhedra slide, relaxing shear stresses there. Such sliding is essential to accommodate diffusional void growth; without it, the holes cannot grow. It has the effect of concentrating stress onto some boundaries and it also induces a locally triaxial stress field. The magnitudes of these effects can be calculated for an hexagonal array of grains (as in Fig. 17) subjected to a remote stress ~ in the 1-direction and a~ in the 3-direction (Appendix D). If these directions are coincident with the 1' and 3' directions then by considering equilibrium of the element ABCA'B'C' it can be shown that the traction normal to the boundary BC is
1,3o~ In practice the grain can be orientated at any angle to the applied stress system. By averaging this over all values of 0 the mean values of the maximum tractions, ol, is found to be =
s(~r2 + a~)
(3.34)
where a~, ~r~2 and tr~ are the applied principal stresses. Void growth by diffusion depends on the normal traction acting across the boundary on which the voids sit. Therefore, identical voids in specimens subjected to the same value of o~ will grow at the same rate. If the voids in a specimen subjected to a uniaxial stress ai grow
II~ I, O'L
~
3t
3,0" 3
FiG. 17. The unit used in the calculation of the stress concentrationdue to grain-boundarysliding.
212
PROGRESS
IN MATERIALS
SCIENCE
POWER-I.AW CREEP PLANE STRESS, n-5
DIFFUSION PLANE STRESS
z;
z °
~o
-21.O
z;"
-I.0
I
-,7..O
-IIO ~ ~" 11 -LO
-2.O to q
11 \
I.O POWER-LAW CRI~P PLANE STRAIN, n . 5 z.
DIFFUSION PLANE STRAIN
£**
I to -5.3
-2'.o -~o -I.O
-2(:
FIG. 18. Fracture surfaces for diffusionand power-lawcreep, incorporating the stress redistribution due to grain-boundarysliding.
dfh/dt,
at a rate then the voids in a specimen subjected to a stress state a ~1, aT, a~ will grow at the same rate, if ~I-Y-2 + Y~J:'I = 1 where E~, y.~ and Y.~ are the applied principal stresses normalized by ai. This result, plotted in stress space, describes a surface of constant growth-rate. It is shown in Fig. 18 for plane stress and plane strain loading conditions. The element ABCA'B'C' of Fig. 17 is constrained by the surrounding material to deform as it does. This constraint induces a local triaxial stress field within the grains. In the vicinity of the grain boundary an estimate of its magnitude can be found by equating the local and distant (applied) deviatoric components of stress (Appendix D). The result is
[Pl - ( 4La=~ + a ~ + a6a~ ')~
(3.35)
Void growth by power-law creep in a multiaxial stress field was discussed in Section 3.3. Growth is determined by the equivalent stress, a e, biased by the hydrostatic pressure, p. When grain-boundaries slide, the local value of is different from the applied value. By using eq. (3.35) to calculate the local value and substituting it into eq. (3.•8) for fl, the void growth-rate for this mechanism can be calculated. The result can again be plotted as a
p/ae ofp/ae
ON CREEP FRACTURE BY VOID GROWTH
213
surface of constant growth-rate in stress space. Surfaces for plane stress and plane strain loading conditions are shown in Fig. 18.
4. THE T I ~
AND STRAIN TO FRACTURE
We now integrate the differential equations of Section 3 to calculate times and strains to fracture. For simplicity we assume that the voids all nucleate at time t. (and strain c,); but if some other nucleation rate were known to apply, it could be combined with the growth equations (Raj and Ashby, 1975) and integrated to give times and strains to fracture for this case also. To evaluate the integrals we require a fracture criterion: a numerical value f¢ for fh at the instant of failure. Clearly, the sample has failed when J~ -- 1. But in reality it will fail sooner than this because, asfh approaches one, the true stress on the remaining ligaments increases rapidly and they will fail by ductile tearing, or by cleavage, or by some other fast-fracture mechanism. We have adopted the fracture criterion fc = 0.25, except (as explained in Section 5) when constrained growth is considered.
4.1. Boundary Diffusion Alone To find the time to fracture we integrate eq. (3.4) between the limits Jh=fi
at
t=t,
fh = f ~
at
t=
t:.
(4.1)
Here f~ is the initial area fraction of the holes, and t, the nucleation-time; fc is the value of fh at which the holes coalesce. The result is
ty = t, + ~
f3/2 In
+
-
•
In
+
--
(4.2)
when f~ is small ( < 10- z) as it almost always is, and f~ = 0.25, this simplifies to
(4.3)
tf ~ tn + ~
The shape of the creep curve, and the strain to failure, E.r, can be obtained by integrating both eqs (3.4) and (3.5) as a coupled set. But it is obvious that if diffusion acts alone, the strain is that due to the growth of the voids. If a hole of volume 4~ r h3 is inserted into a grain boundary of area (rtl2 - nr~), the relative displacement of the grains is 4 3
"srh
rtl2 and the strain at failure is u/d, giving 4-3/2 I
Ef ---- 3J c
(l"
(4.4)
This is the tertiary strain when voids remain roughly spherical. It depends on the area
214
PROGRESS
IN
MATERIALS
SCIENCE
fraction of voids at fracture, f~. Taking this to be 0.25 gives 1
Es ~ 0.2 ~.
To this must be added the nucleation strain, ~., and a contribution from diffusional creep, if appropriate. When boundary-diffusion provides the only growth mechanism, strains to fracture are usually small. Then the fracture times at a constant load will be almost equal to those at constant stress. 4.2. Surface Diffusion Alone The time to fracture is given by integrating eq. (3.13) between the limits of eq. (4.1). At small values ofJ~, eq. (3.13) is well approximated by ,
_
~-o d t
\aol
Integrating at constant stress gives
ts
=
t,
(4.5)
7o o
+
If f~ is small and f, is set equal to 0.25 this simplifies to
1 - 2~(ao] 3 t:
=
t.
+
~o~
\~,i
(4.6)
"
Because the void is flat and crack-like, it contributes almost no strain as it grows, Since ~f is small, we expect fracture times at constant load to be almost equal to those at constant stress. ~o is given by eq. (3.12).
4.3. Power-Law Creep Alone If eq. (3.16) is integrated between the limits given by eq. (4.1) at constant stress, we obtain (Cocks and Ashby, 1980)
,
tf = t. + fl(n + 1)do In
[1-(,-
(1 --f~)~+ ' J \ e , / I "
(4.7)
When f~ is small and f~ is large (of the order 0.25) this simplifies to tf = t, + [3(n
+,
Voo .
1)doIn (n + 1 ) A ] \ e J "
(4.8)
Fracture times at constant load are obtained by noting that (in simple tension) if the initial stress is ~ri, then after time t it has increased to (7 I
a = (1 -
n¢ j-'-,,t "1/"
(4.9)
ON
CREEP
FRACTURE
BY V O I D
GROWTH
215
giving (in this instance) t~, = [(n + l)J]]("/"+nt. + 1( - [(n t +rneo l)J~]("/"+') ~ "
(4.10)
which properly reduces to Hoff's result (Hoff, 1953) 1
t~ -- n~,,
(4.11)
when ~ goes to zero. The shape of the creep curve, and the strain to failure cf are given by integrating eqs (3.16) and (3,17) as a coupled set. But E~, can be obtained by inspection, It is the strain due to creep in the specimen as a whole (ts~,,) plus that due specifically to the voids [eq. (4.4)] ef - Q'Co where
+ 0.2
(4.12)
tf is given by eq. (4.8) and includes the nucleation time.
4.4. Coupled Boundary Diffusion and Power-Law Creep The time to fracture is obtained by integrating eq. (A4) (Appendix A) using the boundary conditions given by eq. (4.1). The results for both constant stress and load are given by Cocks and Ashby (1981), who compare them with the finite element computations of Needleman and Rice (1980): for the range of computed results available, agreement is good. The time to fracture obtained in this way depends on the quantity ~o and on the stress. In simple tension, for instance, Cocks and Ashby (1981) show that, at one limit (low 4)o, large ~r), it reduces to the value calculated for diffusion alone [eq. (4.3)]; at the other it reduces to the result for power-law creep alone [eq. (4.8)]. The entire behaviour is well approximated if each mechanism is integrated over the range for which it is dominant, and the two times are added to give Q.. Diffusion is dominant from f~ to f~ (Fig 8) and power-law creep fromf~ tof~, wheref~ is given by eq. (3.20). Then, from eqs (4.2) and (4.8) with f~ inserted in the obvious way, we obtain
tf = t. + ~ ~o~---~ In
+ 3J\~,]
+ B(. + I)~0 In (n + -l)f{ \ o r , ] "
(4.13)
This result has the proper limits at low and high stress and 4)o. Its value for simple tension, given by setting fl = 0.6 and (re = th is plotted in Fig. 19 (broken line) which shows that it closely approximates the more detailed calculations. The shape of the creep curve and the strain to fracture are obtained by integrating eqs (A4) and (AS) as a coupled set. But cs can be obtained, as before, by inspection. It is given by eq. (4.12) with tI given by eq. (4.13).
216
PROGRESS
IN MATERIALS SCIENCE
NE~OLEMAN
RICE
tf
• t : L : ~ .~ ..~. -
ANALYTICAL RESULT/
J
~.7..""--,
0
,o
n-5 ,
tOM.rED
ft = 10"3
RESULT I~.SULT
"Jd -N% \
,o"
,,, ~
~o-3
I
io-2
I
io-~
I0-:
I0 0
I I01
10 2
FIG. 19. The creep life, plotted against the parameter ~, comparing the approximate analysis with the numerical computations of Needleman and Rice (1980).
4.5. Coupled Surface Diffusion and Power-Law Creep The problem is analysed in Appendix B. But as in the last section, the time to fracture is well aiaproximated by adding the time taken to grow from fi to f~ [eq. (3.23)] by surface diffusion to the time taken to grow from f~ to f~ by power-law creep. These two times are given by substitutingf~ in the obvious way into eqs (4.5) and (4.8). The result (assumingf~ is small andf~ is of order 1) is
tf = tn +
2[(f7),/2 _ f~,2] (O.o~3 -3tIn F _ i I .l( oy C:O----~O \¢~1.] [3(n + 1)~o L(n + 1)fTd\aJ"
(4.14)
The result for simple tension, given by setting fl = 0.6 and ae = trl, is shown in Fig. 20. Since crack-like voids contribute almost nothing to the strain, the failure strain is close to E'f = ~ss tf. 4.6. Coupled Surface and Boundary Diffusion Surface and boundary diffusion are not alternative mechanisms (in the way that diffusion and power-law creep are); both must operate if the void is to grow by diffusion. But regimes exist in which one or the other totally controls void growth (Sections 3.1 and 3.2). Between these limits, the behaviour must be treated as a coupled diffusion problem. The results are given as eqs (C7) and (C8) of Appendix C. These equations are readily integrated numerically to give the time and strain to fracture
ON
CREEP
FRACTURE
BY
VOID
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f t ,. I o -3
n=S
0.4 n=4
i0-1
~'II.~
"NI
lot,.,
I
to 2
I
I°3
"~ t
I
i°~
iilfo t "
-0.4
-0.6
-0.8
FIG. 20. The time-to-failure as a function of stress for various creep exponents.
by this coupled mechanism, for given values of Ao, f~ etc. Strains to fracture are calculated in Sections 4.1 and 4.2. 5. N O N - U N I F O R M DISTRIBUTIONS OF VOIDS: CONSTRAINED CAVITY G R O W T H
It is sometimes observed that voids nucleate on some grain facets but not on others. In previous sections it was assumed that voids were uniformly distributed on all boundaries which lay normal to a given direction (Fig. 21a). But it may be that some boundaries have many voids on them while others have fewer, or none (Fig. 21b). Then a pair of grains which have voids on a shared boundary may be surrounded by a shell of grains which are unvoided. This introduces a new aspect to the problem. Dyson (1976) calls it "constrained cavity growth" because, if a void is to grow, the surrounding shell of grains must deform also, and this requirement may under certain circumstances restrict void growth. The problem is best tackled by introducing a new variable; the spacing 2L of the clusters of cavities (Fig. 21b); if the grain size is d the quantity ( 2 L / d - 1) measures the thickness of the constraining shell. A number of regimes of behaviour are now possible; but to an adequate approximation, the rate of strain and of void growth in each can be obta!ne d b y th e use_of equations already given. We shall illustrate-this by considering clustered voids which grow by boundary diffusion, while the surrounding cage of grains deforms either by diffusional creep or by power-law creep (an analysis including surface diffusion follows the same method). The quantitiesfh and 21 are redefined so that they now describe the area fraction and spacing of voids on a cavitated boundary. We consider growth in uniaxial tension only, although the analysis can readily be extended (by the methods already given) to multiaxial stress states.
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(al
<
(b) FIo. 21. (a) Voids distributed uniformly on one set of boundaries. (b) Voids on some boundaries only.
5.1. Both Void Growth and Grain Deformation by Diffusion Alone Consider first void growth and grain deformation by diffusion alone. In the damaged region the strain-rate is due mainly to the plating out of material from the growing voids, so that grains A and B move apart at a rate which is given by eq. (3.5). Allowing for grain-boundary sliding (which increases the local stress by a factor 4/3) we have
1 d ~ _ _ 8 ~bo a t l ~o dt 3 In 1/fk ao d
(5.1)
where a¢ is the mean stress in this cavitated region. The surrounding shell of grains deform by diffusional flow I-eq. (3.6)] at a rate 13 tr. 1 de = 18~bo d3 ~o dt ao
(5.2)
where a. is the mean stress in the unvoided grains. To satisfy equilibrium (2L - d)a. + dab = 2La ~.
(5.3)
ON CREEP FRACTURE BY VOID GROWTH
219
Equating the two strain-rates, and using eq. (5.3) gives
(2L/d)a ® 4d(2L - d)" 1+ 27• 2 In 1/.~
(5.4)
Substituting this into eqs (3.4) and (5.1) gives
do dt
4 dpo (a*~r (2L/d) 3 fl/2 In(I/A) k~oo,] [1 + 4dX(2L/d - 1) 2712 In 1/fh
]
1dE 8 ~bo ( l ~ ( a ~ ' ~ [ (2L/d) "~ -= 4d2(2L/d - 1)|" C:o dt 3 ln(1/fh) \ }k, ~o 1 + 27--~n-l-~fh j
(5.5)
(5.6)
When all boundaries are cavitated (2L = d), this pair of equations reduces identically to those given earlier for diffusional growth [eqs (3.4) and (3.5)], as they should. But when L >> d, the strain-rate [eq. (5.5)] becomes that due to Coble creep [eq. (3.6)] and the void growth-rate becomes controlled by the rate of this creep in the surrounding shell of grains. This regime of behaviour exists only when the rate of Coble creep in the undamaged region [eq. (5.2)] exceeds the rate of power-law creep there. For this range of stress
_
-
o.~:,
so that the boundary of the regime is defined by
(5.7) When all boundaries are cavitated (2L = d) no constraint occurs. But when 2L >> d, constraint is possible. We assume (for the purpose of further calculation) that fracture occurs when voids on cavitated boundaries link. Then the time to fracture is calculated (as before) by integrating eq. (5.5) between the limits given by eq. (4.1). Iff~ is small, and fc = 0.25t and assuming a constant distribution of voids (so that ! and L do not change with time) we find the remarkable result
tf = t, + 2-Ll_\~o4oJa--iJ +
1 - 2Ljk2712dpo~ ° ~-; .
(5.8)
?In evaluatingthe integral leading to the second term in eqn (5.8), we take f, = 1. This is because separation can take place in the voided region only if the strain in the surrounding,unvoided,region is sufficientto give the full tertiary strain. When calculatingtimes to failure,the cut-offf~is not critical, and our choice of 0.25 is as good as any; but when calculating strains to failure, the choice of cut-offfc is important and must match the physical situation as closelyas possible.
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The time to fracture is the sum of two terms. The first is the time boundaries to fracture by unconstrained diffusion under a stress a ~ fraction of boundaries which are cavitated. The second is the time to strain" Ieq. (4.4)] in the uncavitated region under a stress er~, weighted uncavitated boundaries.
for the cavitated weighted by t h e give the "tertiary by the fraction of
5.2. Power-Law Creep Alone At the other extreme, the voids grow by power-law creep alone. We use the same method as that of Section 5.1. In the cavitated region (grains A and B, Fig. 21b) the strain-rate is given by eq. (3.17) with the stress set equal to ac. In the surrounding grains it is given by eq. (2.1) with the stress set equal to au. Equating the two, and taking J~ << 1, gives ~.-~ac
1 +--j- BA .
Substituting this into the equilibrium equation [eq. (5.3)] gives 2(L/d)a ~
ac =
(5.9)
1 + (-~ - 1)(1 + ~21 flf~/2)" In the limit that all boundaries are cavitated (2L = d), ac = a~; for all other values of 2L > d, the equation states that a, is very close to a ~. So in this case, the rate of void growth and of strain are given by
E {
1 dA = # 1 ~o dt (1 -J~)" ldE oZ=
1+
(1 - A )
2r, F 1 -1 dl_I1-A)"
1(;o) ]}(°7
(5.1o)
"
(5.11)
Note that the growth-rate of the voids is independent of the degree of clustering, and depends only on the local area fraction, f,, on the boundary. The strain-rate, of course, decreases a little as the cavitated boundaries become fewer. The condition for this pure power-law behaviour has already been considered in Section 3.5. If the void growth is to be predominantly by such creep, then eq. (3.19) must be satisfied, When ~ << 1, this simplifies to
- - >/ ao
k(n + 1)f~ :z In(i/A)_]
(5.12)
The time to fracture, in this case, is given by the result calculated earlier [eq. (4.8)]. Remember that fh has been redefined in this section as the area fraction on the cavitated boandaries. The result shows that the growth-rate and time to fracture, when power-law creep only is involved, depend on the local volume fraction and are not changed if neighbouring boundaries are uncavitated.
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GROWTH
5.3. Coupled Growth The two sets of equations developed above describe two extremes of behaviour: diffusion alone, and power-law creep alone. Between these extremes, voids grow by various couplings of two processes. Of these, the most obvious and interesting is that brought about when voids between grains A and B grow by diffusion alone, but the surrounding cage of grains deforms by power-law creep. Then, proceeding as before, we first impose compatibility by equating the diffusive strain-rate in the cavitated region [eq. (5.1)] to that in the power-law creeping region [eq. (2.1) with tre --- tru]; and then use the equilibrium equation [eq. (5.3)] to calculate the stress tr, in the uncavitated region. This results in the implicit equation for tr,
tro/
8°0
,)o.]
3 In UA J
The solution is wall approximated by
= { tro
1_
1
\1 - d/2L ] ao
ac
(2L/d - 1) ao
where ~
(1 - d/2L)" \ Cro}
ao-[(1
- d/2L)"l
1dk](a~]"-'ao} + 3d In (1/fh)81'°
(5.13)
The void growth-rate and strain-rate in this coupled region are then given by 1 dfh 4*0 {a~'~ do -~ -- 3f~/2 ln(1/fh) \~oo]
(5.I4)
d~ d-t = 3 ln(1/J~) d
(5.15)
with trc given by eq. (5.13). Note, first, that if every boundary is cavitated (2L = d) then cc = or" and the equations become identical with those derived earlier [eqs (3.4) and (3.5)]; there is no constraint. Second, at low stresses or when *o is large, the second term on the bottom of eq. (5.13) is dominant and eq. (5.14) reduces to do d~- = 2f~/2(I -
d/2L)"
"
(5.15)
The damage-rate is then completely controlled by the deformation-rate of the surrounding cage of uncavitated grains.But when tr__.~~_= ( 8•*0 ')(1/,-,) ao 3d In 1/f•]
(5.16)
ac = a ~ and growth is unconstrained. If a ~ is greater than that given by eq. (5.16) the J.p.M.S.27 3-4--c
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rate of deformation of the damaged region is primarily due to the power-law creep of grains A and B, and void growth is unconstrained. Equation (5.16) therefore represents the transition to constrained growth. It can be expressed in the form 81 ~bo a ~ ~--~ ~< 3--dIn 1/fh a-~"
E,s
(5.17)
This is exactly the expression given by Dyson (1979, p. 36, with his 22 = nl2) to define the transition to constrained growth. The time to fracture is once again given by integrating eq. (5.14) between the limits given by eq. (4.1). Again, taking f~ = 0.25 we find t; = t, + [ 1 -
(1
- ~ ) l + ( 1 - d/2L), [3~-~o 41 \( aao'~"~. ~JJ d/2L)"][ ~ " ( aao
-
(5.18)
As with pure diffusional growth, the time to fracture is the sum of two terms. The first is the time to fracture by diffusion alone, under the stress a ~, weighted (in a non-linear way) by the fraction of cavitated grains. The second is the time to accumulate the "tertiary strain" [eq. (4.4] in the uncavitated grains by power-law creep under the stress a ~, weighted (non-linearly) by their fraction.t 5.4. Creep Rupture Plots These results are plotted in Fig. 22, for 2L/d = 2 (every second boundary is cavitated). The curve has an unusual shape: over a wide range of stress it is concave upwards.
%
qbo = 0.I fi
IO PURE
10-3
=
I00
21.=
POWER-LAW
I(32
=
d n
I0"
I\
I
I
I0
I02
i
I
I0 ~
2
=
5
L
I04
IOs
a L
Lt,
I06 ~ •
~..~-ON CON STRAI N ED ~BOUNDARY DIFFUSION
CONTROLLED FAILURE I0' -
•
~
/DIFFUSIVE GROWTH ~
,/
CONSTRAINED BY REEP
\
-
DIFFUSION
\
FiG. 22. The time-to-failure as a function of stress when constrained growth is important. tVery recently, Rice, J. R. (to appear in Acta Met., 1981) has also examined this problem, and reached a similar conclusion,
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5.5. Other Distributions of Voids The discussion so far relates to a particularly simple, constant, non-uniform distribution. More complicated constant distributions can be dealt with in the manner outlined by Cocks (1980). But it appears to us to be unlikely that a void distribution remains constant with time. The nucleation of voids, during the test, on the previously void-free boundaries, would appear to be of importance. When the number of voids increases with time, it is still possible to calculate the damage rates and times to fracture. An example of the method is given by Raj and Ashby (1975), but does not, in general, lead to analytical solutions for the times to fracture. 6. COMPARISONWITH THE CONTINUUMTHEORYOF CREEP DAMAGE MECHANICS In this Section we compare the rate-equations for void growth described in Section 3 with the corresponding equations of continuum damage mechanics. We find that the two are very closely related, but that there are, nonetheless, discrepancies, so that their predictions differ. 6.1. The Continuum Theory of Kachanov (1958) and
Robotnov (1969) Creep damage-mechanics (Kachanov, 1958; Robotnov, 1969; Leckie and Hayhurst, 1977) has been developed as a continuum approach to the analysis of creep fracture. The damage (which can include all the processes listed in the Introduction: loss of section by cavity growth, degradation of the microstructure by strain, etc.) is measured by the scalar parameter co which varies from zero (no damage) to one (failure). Both the damage and the strain are assumed to grow with time in a way which depends on stress and on temperature, and on the current extent of the damage, so that in simple tension dco rb = - - = g(~l, T, o)
dt
d~
(6.1)
= d t = f ( a l , T, co). The creep life, the strain to failure and the shape of the creep curve are obtained by integrating the equations between appropriate limits. This procedure exactly parallels that developed in Sections 3 and 4; thus far, the mechanistic model and the continuum model are identical. To proceed further, functions are assumed for f a n d g. The functions chosen by Kachanov and others who have extended his work, are generally
•
¢-O= COO
~1
\no/\l
1
- co/
(6.2) (6.3)
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Here d~o is a temperature-dependent rate constant (like ~o) and v is a constant exponent (like n); the quantities tro,~o and n have already been defined [eq: (2.1)]. When the stress state is triaxial, the continuum eqs (6.2) and (6.3) are modified. Leckie and Hayhurst (1977), following Kachanov (1958), do this by writing
•
~,., = ,Ot~oo) t ~
\" )
1
•
cb = 09ot~---o-o)t ~ 1
(6.4)
"r
)
(6.5)
where tre and 7~ are defined by eqs (2.2) and (2.3) and tr + is identified either with the equivalent stress, ae, or the maximum principal stress, 61, depending on the material. To use the equations, of course, it is necessary to know whether the component is made of an "equivalent stress" material or a "maximum principal stress" material. 6.2. Comparison of the Continuum and the Mechanistic Models: Power-Law Creep When a random array of voids grow by power-law creep alone, the damage-rate and strain-rate are given by the mechanistic model as eqs (3.27) and (3.28). If we identify the area fraction of voids, fh, with the damage, 09 and restrict ourselves to stress states near simple tension (for which fl -- 1), then the strain-rate equations of the continuum and the mechanistic treatment are identical. At the same level of approximation, the mechanistic model (eq. 3.27) gives the damagerate _dt_ =
_Tfh)
This equation resembles the continuum damage-rate equation [eq. (6.2)] with r = n and &0 = 4o, and becomes identical to it when fh is large (since the - ( 1 --fh) term can then be neglected). But there is an important difference: when fh is small, the term [1/(1 - 09)]" is replaced by [1/(1 -.~)" - 1]. Physically the continuum model states that the damage-rate is finite even when there is no damage (09 = 0); the mechanistic model says the damage-rate is zero under these conditions: non-existent holes do not grow. Because of this, results calculated from the continuum and the mechanistic equations differ. If the time to fracture, for example, is derived by integrating the continuum eq. (6.2) with v = n and ~'Oo= ~o, the result is
(1- J~) '--'['°°~l".
(6.7)
t: = t, + (n + 1)~o \ t r y / By contrast, the mechanistic model leads to the result [cf. eq. (4.8)]
I
tf
---- t n d-
[
I
l(ao'~" '
fl(n + 1)~o In (n + 1)f~lktrt/
(6.8)
These two times are almost identical when J~ is large; but when f~ is small, the mechanistic model predicts a much longer life because the holes (when small) are growing more slowly. There are corresponding differences in the strain to fracture.
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225
For multiaxial loading, the mechanistic and continuum models differ in additional ways. First, the mechanistic model shows that the strain-rate as well as the damage-rate are accelerated by hydrostatic tension, and which enters through the term fl [eq. (3.18)]; the continuum model does not have this feature. When voids grow by power-law creep on boundaries [eqs (3.16) and (3.17)] the comparison is made more complicated because the grain size now enters the equations. But the same close resemblance and the same fundamental differences are still found. 6.3. Comparison of the Continuum with the Mechanistic Model: Diffusion Controlled Growth The mechanistic equations for hole growth by boundary diffusion alone were given by eqs (3.4) and (3.5). When fh approaches 1, we may write f~/2 ln(1/fh)~ (1 --fh), and the damage-rate equation becomes dfh ~--
. trl 1 ~oEo(~00) (1--~fh).
(6.9)
If, as before, we identify fh with damage, to, this equation is identical with the first of the continuum equations (eq. 6.2), with t.bo = ~boE0 and v = 1. But when fh is much less than 1, the two equations differ significantly; in particular, eq. (3.4) leads to a damage rate which decreases as damage grows (see Fig. 8), whereas the continuum equation predicts a steadily increasing damage rate. The strain-rate equation, too, becomes identical with the second of the continuum equations when fh is close to 1. In this limit In 1/fh ~ (1 -J~), and if holes are randomly distributed, l = d, giving ~-~ - 2 ~ o ~ o
(1 - fh)"
(6.10)
But, as before, the two equations differ significantly when fh is small--and this is the regime in which diffusional growth is important. Exactly the same conclusions hold when voids grow in a way which is limited by surface diffusion [eqs (3.13) and (3.14)]. When fh "- 1, the mechanistic and the continuum equations are identical, but when fh is small, the two sets of equations diverge. In the regime in which we expect surface diffusion to be important, this divergence leads to large differences between the two approaches. 6.4. Results of the Comparisons In summary, all the mechanistic equations have a close resemblance to those of continuum damage mechanics, and become identical to them when fh (or co) approaches the value 1. But whenfh is small (as it is for most of the life) there are important differences. The comparison highlights what we believe to be certain fundamental weaknesses of the continuum equations: first, the prediction that the damage-rate is finite even when there is no damage; second, the prediction that the damage-rate always accelerates with damage; and finally the incorporation of triaxiality in a way which is not consistent with (approximate) physical models. It must be emphasized that the mechanistic equations, too, are approximate. But they are
226
PROGRESS IN MATERIALS SCIENCE
based on models and have limits, which are consistent with the physical processes which are thought to lead to fracture. They resemble the continuum equations, and can be thought of as a mechanistically-based extension of them to describe specific damage mechanisms. For these mechanisms we believe they are a better description than the simpler (though very flexible) equations of the continuum theory.
7. EXTRAPOLATIONOF CREEP DATA The engineer, when designing high-temperature structures, is often faced with the problem of extrapolating creep data. In particular, he may be given uniaxial creep data at constant load and for relatively short times, and find it necessary to extrapolate the data in three ways: (i) to longer times, (ii) to histories of non-steady load and temperature and (iii) to stress states other than simple tension. The models described in Section 3 of this paper give ways of doing this. Some of the results are limited (at present) to materials in which damage takes the form of voids and for which the void-growth time (rather than the time for voids to nucleate) controls the life of a component. But the procedure for extrapolating creep data to longer times is more general than this and appears to be valid for many sorts of creep damage.
7.1. Present Methods for Extrapolatin9 in Time Current extrapolation procedures for uniaxial creep data are empirical. In using the Graphical Method, data for log t.r are plotted against trl or log a x at a constant temperature and the resulting curve extended by eye. The method is formalized in the Reoression Analysis Method (e.g. Booker, 1976, 1977): the data are fitted to one of many possible empirical equations, of which a typical example is A2
log t s = At + T
A3
+ T log at
(7.1)
and the constants A1, A2, A3 are determined by a multivariable linear regression analysis (Draper and Smith, 1966; Daniel and Wood, 1971). The extrapolation by eye is now replaced by the use of the fitted equation, but no physical modelling is involved. The Time-Temperature Parameter Method (Larson and Miller, 1952; Conway, 1969; Penny and Marriott, 1971) has the merit that physical understanding of the fracture process can more readily be introduced. It breaks down the fitting procedure into two steps. A parametric combination of t.c and T is chosen and assumed to be a function of stress only
P(ts, T) = 9(al).
(7.2)
Some 20 or more such parameters have been suggested, among which the most widely used are the Larson-Miller Parameter (Larson and Miller, 1952)
P(tI, T) = T(C + log t r)
(7.3)
and the Dorn, or Fisher, parameter (Orr et al. 1954; Fisher, 1959)
P(ts, T) = log ts + C/T.
(7.4)
(This last equation is based on kinetic theory, but without any clear model for the fracture mechanism.) The data are plotted against the parameter P to give a master-plot, selecting
ON
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227
the constant C to minimize the spread. Then a polynomial (usually a quadratic in P) is fitted to the master plot, and this is extended to longer times and lower stresses, to give a stress corresponding to the design life. Finally a safety factor is applied to this stress. The parametric approach is made more general in the Minimum Commitment Method of Manson (1964 and 1968) although contact with physical models is thereby lost. The data are fitted to a basic equation of the form log ty + A P(T)log ts + P(T) = G(o.~)
(7.5)
where P(T) is to be determined from the data, and G(o.t), usually a polynomial in o.1, then describes the shape of the master curve. The equation obviously includes the two parameters described above as special cases. Numerical procedures have been devised to obtain P(T) and G(o.1), which then allow the equation to be used in design. All these methods must be regarded as completely empirical; where physical understanding is incorporated, it is at too vague a level to give the method itself any real physical basis. All assume that the physical mechanism which controls creep or fracture in the extrapolated region is fully represented in the available data, and that the presence in this data of information about other mechanisms exerts no influence on the extrapolation procedure. Further, none have the capacity to predict the creep life or creep strain if the load or temperature change during the life. To do this requires a differential method. The Kachanov-Rabotnov equations and the equations developed in this paper, are examples of differential equations for damage and strain. All have the general form (for simple tension) of eq. (6.1). If such equations are known for each damage-mechanism and for each mechanism of creep, then they can be integrated to give the strain at any time, and the time to fracture, for any history of loading. The great strength o f this method is that it allows the progress of damage-accumulation to" be followed throughout the life. Its weakness is a practical one: the data needed to determine the functions f and g of eq. (6.1) are particularly difficult to obtain. But if the models are good enough to give the forms of f and g, this difficulty is removed and the method leads to powerful extrapolation procedures. The remainder of this paper illustrates this approach. 7.2. Model-Based Procedure for Extrapolating in Time In Sections 4 we showed that the time to fracture in tension by any single mechanism or by the coupling of any pair of mechanisms of void growth gave a normalized fracture time to tI which depends on normalized stress (th/o.0), on a material property (q~o, ~ko or Ao), on f/ and (weakly) on n. Boundary diffusion coupled to power-law creep, for example, gives fracture times described by ~otj- =
o , - - , n,
.
(7.6)
O'o
For any given material, n a n d ~ are constant. Then to ts depends only on ~bo and o.1/tro. Fracture by this coupling of mgchanisms, or of either acting alone, at all stresses and temperatures, is described by a family of curves, each characterized by a value of ~bo [defined by eq. (3.3)]. At this point, it is necessary to introduce an explicit temperature-dependence for power-
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PROGRESS IN MATERIALS SCIENCE
law creep. We shall use an expression? which is standard in the materials science literature (e.g. Sherby and Burke, 1967)
where Dc is the creep diffusivity: Dc = Do~exp(-Q
i° - AD
(7.8)
and 2flDs~ ( ~ f f -I
dP° = AD
"
(7.9)
We may choose go and ~o in any way which is consistent with the constitutive law (eq. 2.1). We may therefore redefine a0 to give a~, and Eo to give ~,, such that ~bo is independent of temperature. Then if creep rupture data at different temperatures is normalized by plotting it on axes of log al/a* and log ~* t:, all points must fall on a single master curve. Eq. (7.9) can be rewritten
where Qc and QB are the activation energies for creep and grain-boundary diffusion respectively and B is a dimensionless material constant. We then define a , in such a way that t~0 is constant and equal to unity
I- Qc - Q~ -I :~F, B.,.-,, expLtn- r)~r~J
(7.11)
=
and, from eq. (7.8)
i~ - COo<.b kT
exp -
[ ~-
O.c]
(7.12)
where C is a dimensionless material constant and Doc is the pre-exponential of the diffusion coefficient. A single master curve should, therefore, be obtained when data is plotted on axes of P1 = loglo tri/a~j and P 2 --- loglo t/i*. From eqs (7.11) and (7.12)
,, =
-
.(1
. -
logio-
/~o
"" loglo ai
-
H
(7.13)
and P2 = logio tf
--
J
.
1
.
logto. To
. logio .
~o
"" loglo tf
J
To
.
(7.14)
i'The detailed form of this expression is unimportant. Exactly the same result is obtained if. for example, we choose instead i = A' exp - QJR T (tr/a,)" where A' is a constant and a, a referencestress.
ON CREEP FRACTURE BY VOID GROWTH
229
Here To is a reference temperature (that of the centre of the range of data, for instance) and #0 is the value of the modulus at that temperature. The quantities H and J are explicitly given by the model H =
Qc - Qs
2.3(n - 1)R j
_
"QB -
Qc
2 . 3 ( n - 1)R
(7.15)
though in practice they should be chosen to minimize the spread of data. The procedure for doing this is illustrated by Fig. 23; it is adapted from the method of Manson and Ensign (1978). The creep rupture data, plotted on axes of logto aa and logao t I are traced onto a transparent sheet containing no grid work. Smooth lines are drawn by eye through the data at each temperature. The sheet is placed over a fine grid and rotated
SUBSHEET (3ONTAINING GRIDWORK
_I
I / /
I,~. /i~
/
~
/ I
\\ "%
d
i
.
• m
m
m
. m
. m
~' m
m
m
m
m
m
m
m
ami==-__Immmmmmm mmmmmmmmmi==im ISOTHERMALS
BUT NO
GRIDWORK
"To
o TI
To>T, >T~>Ts
°T3 FIG. 23. Illustration of the method used to determinethe scaling numbers H and J.
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clockwise about the point A until the horizontal distances L o between two isothermals T = T~and T = Tj is as nearly constant as possible along their length, and the ratios of the lengths is consistent with 1
1
1
1
When these conditions are met, the angle of rotation 0, and the lengths Lo, are noted, and used to form the quantity
L =
Lij
The two scaling numbers, H and J, are then given by H = L sin0 J = Lcos0
(7.17)
The data are replotted on axes of PI and P2 (eqs 7.13 and 7.14) to give a master curve for the material. Finally, a curve of constant ~bo, is fitted through, and extended beyond, the data, giving a modelbased extrapolation. As an example of the use of the method, we apply it to Williams' (1968) data for the creep fracture of 304 stainless steel at a number of stresses and temperatures. The raw data is shown in Fig. 24a. The same data, replotted using the parameters P1 and P2 with H = 419 K and J = 15,000 K, (chosen by the fitting procedure just described) are shown in Fig. 24b. The solid line is that predicted from the theory for n = 9. Data for the creep-rupture of a steel, S-590 (Grant and Bucklin, 1950) are shown in Fig. 25a. The same data, replotted according to the method described above with H = 860 K and J = 14,000 K, are shown in Fig. 25b, together with the computed curve for failure resulting from precipitate coarsening (Appendix E). As a third example, data for O F H C copper (Goldhoff, 1974) are shown in Fig. 26a, The same data are replotted with H = 811 K and J = 3900 K in Fig. 26b. In all three examples the procedure works well. Further, the quantities H and J obtained in this way can be related back to fundamental quantities (n, Qc, Qn) through eq. (7.15). The merit of this procedure is its basis in a physical model: it is not an empirical procedure (like almost all other extrapolating methods), but one which fully incorporates a mechanistic understanding. It is readily seen that the same procedure is valid for any one, or any coupled pair of the three growth mechanisms discussed in this paper; and Appendix E shows that it is valid also for creep fracture caused by certain sorts of microstructural instability. It is also valid for simple instances of nucleation dominated creep fracture. For instance, if void nucleation occurs catastrophically after a strain En, where En is a material property, the failure time can still be written in terms of ~bo, a/tro and material properties and the extrapolation procedure holds. 7.3. Extrapolation to Non-Steady Histories of Stress
and Temperature When the stress or the temperature vary during a creep test, it is normal engineering practice to calculate the life by using the linear cumulative damage rule (or Robinson's rule;
ON
CREEP
FRACTURE
&A
0
A
BY
O
VOID
D o
_
O
F
o
A~
a A
i~
231
GROWTH
~O. o 5380(:: O 593~C; A 649oC 0 7040(: v 816oc 87toc o 927°(:; ":-
v
0
eC K~4
6
TIME-TO-FAILURE (hrs) 2.4
!~ q:~
4
STAINLESS
2.~
! -I~-
t-!
o To - $93 °C -866K g
Jr
t.~
O" ,, Ilvl~l tf ,, (hrs)
I
I 3
I
I
1 6
I 7
FIG. 24. The extrapolation procedure, applied to data for 304 Stainless Steel of Williams (1968); raw data above, normalised data below.
Robinson, 1952). If the time spent at stress tr~ is t~, and the failure time at that stress is tyi, the rule states that failure occurs in a time ty where
~
ti
C
tfi
or
f[~
dt
tf
C
(7.18)
with C = 1. It is easily shown (Appendix F) that each of the equations describing void growth by a single mechanism [eqs (3.4), (3.13) and (3.16)] satisfies the life-fraction rule with C = 1. But if the voids grow by a coupled mechanism, or if they grow by a single mechanism at a low stress and by another at a higher stress, then changing the stress changes the mechanism,
232
PROGRESS
IN
MATERIALS
SCIENCE
and C is no longer equal to unity. Consider a sample in which voids grow by diffusion alone at a low stress and that a change of stress (or temperature) occurs such that for the remaining part of their life they grow by power-law creep alone (Fig. 27). If the voids grow by boundary diffusion until their area fraction isf~, the elapsed time t") is given by setting f~ = f~ in eq. (4.2). At this point the stress (or temperature) changes so that they now grow by power-law creep. A further time elapses as the holes grow from f~ to f~ (when failure occurs) by power-law creep. This time, t(2), is given by settingf~ = f~ in eq. (4.7). We have formed the sum t(z)/t~ I) + t(2)/t~2), and plotted the result in Fig. 27a. The sum is less than 1 unless one mechanism controls the entire life; for the conditions of the plot it can be as low as 0.3, so that Robinson's Rule is unsafe. If, instead, we cause the holes to grow first by power-law creep and then by diffusion, the result shov, n in Fig. 27b is obtained. The life-sum is greater than 1, so that Robinson's Rule is conservative.
03
- ~S-590 ALLOY o
o
°° 0
ee
a.
o
o
v
•
0
u 0
•
~
0
3Eeo 2
•
,
o
o o V
eO
U~ uO UJ rY
•
I0
0
0
v
•
o 650"C o 73TC
A 816"C
0 871eC v 927"C • I038* C
I lo-~
I
~
IO ~
TIME-TO-FAILURE
s-sN
'
IO4
(hrs)
AL-C-C-
g g, e To-e7ve - 1144K
--
O'(MPo) t~(hrs)
0 -d
-2
0
t,-J oo (+
2
4
FIG. 25. The extrapolation procedure applied to data for an alloy steel, S590, of Grant and Bucklin
(1950); raw data above, normalised data below.
ON
CREEP
FRACTURE
BY VOID
233
GROWTH
[ OFHC COPPER • ;IlI'C • 316"C o 371"C :,4g~
_ ,d
. St)T C A Mg'C eTO~C • 760"C
+ 48TC ~ 871"C o S38"C x
UJ rY I-u~
0
o
O
I0
+
V
•
X
e
0
6
•
•
+
000
•
I)
x
Aa
13Q
O
O O
+
TIME-TO-FAILURE (hrs) 3
"[OFHC GOPPER To - S,I~ C
I
- 8IlK 0'lldPa)
v
o
-2
-I
I
o
2
,o%,,- 3.900
3
(+-+o)
4
$
FIG. 26. The extrapolation procedure applied to data for OFHC Copper of Ooldhoff (1974); raw data above, normalised data below.
POWER-LAW CREEP GRow'rI~
POWF-q-I.~W
(7 lorT~
CnEEP G~OwT.
DIFFUSIONAL GROWTH
l
|
I'
OtFFUSIONAL GROWTH
1'
fs
fh
fh
fs n-9
sI
0.5~ I.O
.
I I °o-"
.
.
I Io -3
.
.
n-'9 I Io -2
.
I lo-'
I<)
0.5
fs
-
0_4
1 IO-]
I IO'2
I iO ~
FIG. 27. An illustration of deviations from the linear cumulative damage rule.
fs
234
PROGRESS IN MATERIALS SCIENCE 7.4. Extrapolation in Stress Space
When a pressure vessel or pipe supports an internal pressure, or when a component with a change of section is loaded in tension, the stress state is a multiaxial one. Current design practice assumes either that the equivalent stress, ae, controls fracture ('Won Mises materials") or that it is controlled by the maximum principal stress at ("Maximum Principal Stress materials"). Explicit equations were given earlier for the growth-rate of voids, and the time and strain to fracture, under local multiaxial stress fields. When grain boundaries slide, the local field is related to the remote (applied) field by the equations given in Section 3.10. Combining the two sets of equations gives expressions which show how the time and strain to fracture vary with the triaxiality of the remote field. This was the approach used in constructing the second set of constant growth-rate surfaces (Fig. 18). If the nucleation time, tn, is short, then these figures become identical with the isochronous fracture surfaces for the material (surface in stress space on which the fracture time has a constant value) provided one mechanism is dominant throughout life. But if the nucleation process dominates the fracture time, it is necessary to know also how tn depends on stress state. This problem is outside the scope of the present paper.
8. SUMMARY
Void growth by boundary diffusion, surface diffusion and power-law creep, acting either singly or together, is analysed. Equations for rate of growth of void damage and for the strain-rate are developed in Section 3, for multiaxial stress states. The results arc illustrated by growth-mechanism maps by plotting surfaces of constant growth-rate in stress space. The time and the strain to fracture arc obtained by integrating the rate-equations. The results of doing so are given in Section 4. When voids are non-uniformly distributed, their growth may be constrained by the need to deform adjacent grains with fewer or no voids on them. Some simple results for fracture under these conditions are given in Section 5. The results of the mechanistic approach of this paper bear a close relationship to the equations of continuum-damage mechanics. This relationship is discussed in Section 6, where it is shown that, in spite of their similarities, the results of the two methods differ. Simple modifications to the continuum method are possible which would make it consistent with the mechanistic approach. An important aspect of an analysis like that given here is the help it gives in extrapolating creep fracture data. A new procedure for extrapolating failure times to longer times, and for producing creep-rupture master curves, is given in Section 7, which also includes a discussion of the validity of Robinson's Rule and of creep fracture under multiaxial stress states.
ACKNOWLEDGEMENTS We wish to acknowledge financial support from the S.R.C. (in the form of a studentship) and from the U.K.A.E.A., Harwell. We are particularly grateful to Dr. B. F. Dyson for his critical comments and suggestions.
ON
CREEP
FRACTURE
BY V O I D
GROWTH
235
REFERENCES ARGON,A. S., CHEN, L-W. and LAU, C. W. (1980) Creep and Fatigue Fracture, (ed. Pelloux, R. M. N. and Stoloff, N.) A.I.M.E., N.Y. ASHBY, M. F. (1969) Scripta Met. 3 837. BEERE,W. and SPEIGHT, M. V. (1978) Metal Sci. 4, 172. BOOKER, M. K. (1976) ORNL Report ORNL/TM-5329; (1977) ORNL Report ORNL/TM-5831. BROWN, A. M. and Astray, M. F. (1980) Acta Met. 28, 1085. BURTON, B. (1977) Diffusional Creep in Polycrystalline Materials, Trans. Tech. Publications No. 5, p. 62. CHUANG,T-J. and RICE, J. R. (1973) Acta Met. 21, 1625. CHUANG, T-J., KAGAWA,K. I., RICE, J. R. and SILLS, L. B. (1979) Acta Met. 27, 265. COBLE, R. L. (1963) J. Appl. Phys. 34, 1679. COCKS, A. C. F. (1980) Ph.D. Thesis, Cambridge University. COCKS, A. C. F. and ASHBY,M. F, (1980) Metal Sci. 14, 395. COCKS, A. C. F. and ASHBY,M. F. (1981), CUED Report No. CUED/C/MATS/TR 27; to be published. CONWAV, J. B. (1969) Stress-Rupture Parameters: Origin, Calculation and Use, Gordon and Breach, N.Y. DANIEL, C. and WOOD, F. S. (1971) Fitting Equations to Data, John Wiley, N.Y. DRAPER, N. R. and SMITH, H. (1966) Applied Regression Analysis, Wiley, N.Y. DvsoN, B. F. (1976) Metal Sci. 10, 349. DVSON, B. F. (1978) Canad. Met. Quarterly 18, 31. EDWARD, G. H. and ASHBY, M. F. (1979) Acta Met. 27, 1505. FIELDS, R. J. and AstraY, M. F. (1976) Phil. Mag. 33, 33. FISHER, W. A. P. (1959) RAE TN Structures, 270. GRANT, N. J. and BUCKLIN,A. G. (1950) Trans. Am. Soc. Metals 42, 720. GOLDHOFF, R. M. (1974) J. Testing Evaluation 2, 387. GOODS, S. H. and BROWN, L. M. (1979) Acta Met. 27, 1. GOODS S. H. and NIX, W. D. (1978a) Acta Met. 26, 739. GOODS, S. H. and Nix, W. D. (1978b) Acta Met, 26, 753. HARRIS, J. E. (1972) Met. Sci. J. 7, 1. HELLAN, K. (1975) Int. J. Mech. Sci. 17, 369. HERRING, C. (1950) J. Appl. Phys. 21,437. HOFF, N. J. (1953) J. Appl. Mech. 20, 105. HULL, D. and RIMMER,D. E. (1959) Phil. Mag. 4, 673. KACHANOV,L. M. (1958) lzv. Akad. Nauk. SSSR No. 8, p. 26. LARSON, F. R. and MILLER, J. (1952) Trans. Am. Soc. Mech. Engrs 74, 765. LECKIE, F. A. and HAYHURST,D. R. (1977) Acta Met. 25, 1059. MANSON, S. S. (1964) Proc. Inst. Mech. Eng. 178, Part 3A. MANSON, S. S. (1968) Am. Soc. Metals Publication D8-100, p. 1. MANSON, S. S. and ENSIGN,C. R. (1978) Am. Soc. Mech. Engrs Publication MPC-7. NARARRO,F. R. N. (1948) Strength of Solids, Phys. Soc. London, p. 75. NEEDLEMAN,A. and RICE, J. R. (1980) Acta Met. 28, 1315. NEUMANNG. and NEUMANN,G. M. (1972) Surface Self-Diffusion of Metals, Diffusion Inf. Centre. NIEH, T. G. and NIX, W. D. (1980) Acta Met. 28, 557. OOQWST, F. K. G. and HULT, J. (1961) Ark. Fys. 19, 379. ORR, R. L., SHERBY,O. D. and DORN, J. E. (1954) Trans. Am. Soc. Metals 46, 113. PENNY, R. K. and MARRIOTT,D. L. (1971) Design for Creep, McGraw-Hill, pp. 192-238. RAaOTNOV YU. N. (1969) Proc. X I I I U T A M Congress, Stanford (ed. M. H6tenyi and W. G. Vincenti); Springer, Berlin, p. 342. RAJ, R. (1978) Acta Met. 26, 341. RAJ, R. and ASHBY,M. F. (1971) Met. Trans. 2, 1113. RAJ, R. and AsI-mY, M. F. (1975) Acta Met. 23, 653. ROBINSON,E. L. (1952) Trans. Am. Inst. Min. Engrs 74, 777. SHERBY,O. D. and BURKE, P. M. (1967) Prog. Mat. Sci. 13, 325. SI-mWMON,P. G. (1969) Transformations in Metals, McGraw-Hill, pp. 299-302. SPEiGtrr, M. V. and HARRIS,J. E. (1967) Metal Sci. l, 83. SWlNKELS` F. B. and ASHBY,M. F. (1981) to appear in Acta Met. WILLIAMS,W. L. (1968) Am. Soc. Metals Publication D8-100, 351.
APPENDIXA : Void Growth by Coupled Boundary Diffusion and Power-Law Creep The method of dealing with coupled diffusion and power-law creep is illustrated by Fig. A1. All current treatments (Beere and Speight, 1978; Edward and Ashby, 1979; Cocks and Ashby, 1981) assume that voids grow, by boundary diffusion, on a grain-boundary across which a mean normal traction a~ is applied. The material
236
PROGRESS
i
#
IN
#
MATERIALS
SCIENCE
i
#
t
EFFECTIVE, x DIFFUSION HOLE SIZE) / t ZONE ~'x-it
'I
..\ ~/
-
-~
-- -'~/"_ . ~
-~
t
•
-
--
t
A~OWER - LAW CREEP ZON~
/
~'.~~~:.~'~:..~:~::..'.'~
~.:::!::..~:~:.:.~.:.;..::.:....::.?.:.;....~..:...:~.:..:.::::.....yv.~:::.::::::~ ~.'-"-:':'.."!:y!:.~:..~'..!..~-..? :'!:!':..:'..':'!'....:-: '.:..'._-.~:.~".'-":':
.~
~.:~.'"':':"': ": :').:!~'~.'#~:'":';':.:"""~;':':~':" r:-:~"~'.'~.:~...'::~-!~-'~:.: ~':~.~
I
.----- r e ---.~
rd
Ji
i
¢
i
i
i i
i I
t
t
i
i
i
i'
FIG. A1. A void growing by coupled boundary diffusion and power-law creep.
leaving the void plates out onto the boundary over a ring of outer radius r~ (occupying an area fraction fj). This diffusion zone is an important feature of the model; if its radius r4 extends to the hole half-spacing I, hole growth is purely by diffusion. But, if it shrinks until r~ = rk the surrounding power-law creep zone (I > r > rj) totally controls hole growth. When rj lies between these extremes, growth is by a coupling of two mechanisms. Within the diffusion zone, the stress is partly relaxed by the material which deposits on the boundary plane. This sheds load onto the surrounding power-law creep zone, which behaves as if the slab contained a hole of effective size r e (with corresponding area fraction re). The method of Cocks and Ashby (1981) is to show that the axial creep-rate in the power-law creep zone is bounded by ~, = ~o(a'~"
1
(A1)
\ a o / (1 - fe)"" This must match the axial strain-rate of the central part of the slab, caused by diffusive plating. A straightforward application of eq. (3.1) for the annular ring of radius rd shows this to be
~z =
2flDB3al 1 kTl 3 f~/zfa In(fdfh)
(A2)
where f~ = r~/I 2. Equating the two gives
¢o
(~t)'-'f~lZf,
=
ln(fdf,)
(A3)
(1 - f.r
where ~o has already been defined by eq. (3.3). If this is now optimized with respect to fa (to give the value of fd which maximizes the strain-rate), an expression for fd is obtained
,o(,Oy-', .
=
f,f~,'El + 21n(f;f~)]"
{. - sJ1 + (in L
t, JlJ
,n
(A4)
l,,J
For known values of $0, *t/ao and .~, the quantity f~ can be calculated. The quantity fe is then found using eq. (A3). In the diffusion limit, the diffusion zone extends halfway to the next void. Settingfd = 1 in eq. (A4) gives
ON
CREEP
FRACTURE
BY VOID
237
GROWTH
In the power-law creep limit the diffusion zone shrinks onto the void. Setting fj = fh gives _
\ai}
(1 - fhP- l"
These two equations are used to give the range of the coupling (shaded band) on the growth maps. By reasoning which exactly parallels that leading to eq. (3.16) and (3.17) the damage-rate and strain-rate are found to be
i~o dt -
(1
i~dt =
1+
f~)"
(l
(AS)
d L(1 -f~)~
dJ\ao/
This approach is the most refined of the recent analytical treatments of this problem, all of them approximate. It gives results which agree well with the finite element calculations of Needleman and Rice (1980), over the range their calculations cover. All these calculations have in common the feature that the hole growth-rate depends only on al/ao and on the quantity t~o, which appears in the work of Needleman and Rice (1980) as =
\ol/
(A7)
and in Edward and Ashby (1979) as the parameter P
1 V2 P = ~L
{ao~n-I~] 2In ~°~-lJ
~
.
(AS)
This has the important consequence, used in Section 7, that a parametric plotting method can be prescribed which leads to a single "creep-rupture master curve" when fracture is caused by the mechanism analysed here. The quantity q~o,then, is the important material property which controls void growth. APPENDIX B: Void Growth by Coupled Surface Diffusion and Power-Law Creep Fig. B1 shows a void growing by coupled diffusion and power-law creep. Surface diffusion limits void growth,
causing the void to become flat and penny-shaped. Matter is transported along the void surfaces to the tip, where it enters the boundary. Here it diffuses up to a distance rd from the centre of the void, and deposits as a layer of uniform thickness. The radius of the crack tip is r0 and the stress in the diffusion zone is aj. They are related by 7s ro = --. (B1) ad
t
t
t
#
I
~
t
l
DIFFUSION
L
ez
/ZON~
! #
--~.,~
- -
--
- r -i- - - - T
--P
. . . . .
~ "~-.
I
,
re
~_
' i
.
~
rd
,-
POWER-LAW
t /CREEP ZONE
_I
£z
-,
Z
i
l
i
iT|
l
i
i
i
FIG. B1. A void growing by coupled surface diffusion and power-law creep.
J.P•.s. 27 3-4--D
238
PROGRESS
IN
MATERIALS
SCIENCE
Matter flows out of the penny-like void at the rate dv dt
--
dr h 8nrorh - - . dt
(B2)
The velocity drh/dt is given by eq. (3.10) and the form of eq. (3.11) given by Chuang et al. (1979). Combining these with the last equation gives dv 2 v ~ n DdS,?,~rh dt = r~k T
(B3)
This is deposited uniformly over a disc of area n(r~ - r~) giving 1 dv 4, = 4nro(r~ -- r 2) ~ "
(B4)
Defining r~
A-
12
(B5)
and using the definition of ~'o [eq. (3.12)-I we find
~-z fl/2~O ( O'a'~3 ~o
(B6)
8(:,-A)~,}Too "
The power-law creep zone sees a void of effective radius re, and the strain-rate within the shaded zone is therefore ~o
(1 - re)""
(B7)
Equilibrium requires that °'a = (1
--
f~) \ f d - f h /
If we now equate G in the two zones, and substitute for a,, we find 8(fa--fh) 4
~bo = f~/2(1 _ f,,),,_3(fd _ f , ) 3
( O I ) n-3 ~o "
(B9)
This is a relationship between f, and f~. The optimum value off~ is that at which df,,/df4 = O. Differentiating the last equation w.r.t, fj, setting the result equal to zero and solving for f~ gives f, = ¼A + ~ .
(B10)
Inserting this into the equation for ~o gives ~o
.
(0"1~(n-3)
(fd -- ,~)
. . . 27 f~/2[1 - (¼f~ + ~fh)]"-3 \ a o /
(BI 1)
In the diffusion limit, the diffusion zone extends halfway to the next void, a distance I. Setting fa = 1 gives
~o =
f~/'(l - A t - " \ ~ o /
If ¢'o is greater than this, power-law creep is unnecessary, and the voids grow by diffusion alone, at a rate controlled by surface diffusion. In the power-law creep limit, the diffusion zone shrinks down onto the void. If fj is set equal to fh in the equation for ~b0, we merely obtain ~o = 0 meaning that a small diffusion zone exists. If, instead, we substitute fj = 2J~ we obtain the condition for substantial power-law creep control
(o,?-, ~,o ~< 2-~ (1 - ~ f , r - 3 \ a o }
(BI3)
Equations (BI2) and (B13) give the range of the coupling, shown as the shaded band on the void-growth maps. As was the case for the coupled mechanism described in Appendix A the void growth-rate, and the fracture time, by this coupled mechanism, are completely determined by stress and the value of ~o. This has the important consequence, used in Section 7, that a parametric plotting method can be prescribed which leads to a single "creep-rupture master curve" for fracture by this coupled mechanism also. In this case, the quantity ~#o is the
ON C R E E P
FRACTURE
BY V O I D
239
GROWTH
important material property controlling void growth and fracture, but otherwise the method of Section 7 applies unchanged. APPENDIXC: Void Growth by Coupled Surface and Boundary DiJJi*sion This problem is treated in some detail by Chuang et al. (1979). Here we present a simple, approximate analysis which gives the necessary results. Consider a periodic array of voids of spacing 21 and diameter, in the boundary plane, of 2rh (Fig. 6). Material flows by surface diffusion to the tip, where it enters and diffuses along the boundary, depositing uniformly. The driving force is distributed so as to drive these two processes at matching rates. We allow for this by introducing a tip radius, r* (to be determined) which adjusts itself so that the surface flux equals the boundary flux at the tip. Then the normal traction or* in the boundary where it meets the tip and the chemical potential ~* at the tip are given by ~*
=
- #*fl =
- 7~fl
+
""
r*
provided r s >> r*. The volume of matter arriving at the tip per second is dv - - = n~ r . rj, drs -~- = 4nr*l 2 dfh dt dt
(C2)
2x/-2nD,6,7,f~rh k Yr .2
(C3)
-
using eqs (3.10) and the Chuang et al. (1979) form of eq. (3.11). This must be matched by the boundary flux. The volume of matter flowing into the boundary is given by eq. (3.1), but since the tip radius is near the sintering limit, we must replace O'l/(l - J~) by [al/(l - fh) - ys/r*], giving
Equating the two equations and solving for r* gives r.
( 1 - f~)z
I~
r*
,,/-2 In ( ~ ) Ao
+ ~.(1
-
A)3
-I
(C5)
where Ao is a further material property
Ao
=
D,6, DI~e"
(C6)
The equations for coupled growth arc obtained from eqs (C2) and (C3).
q,3 ~'1/2
1 dA Co r.jh do d~- = ~o3,*~
(c7)
I d~ 4~00f.t/27~ do dt ffi (1 - J~) d ~ r .2
(C8)
where r* is given by eq. (C5), and Co by eq. (3.12). Equation (C7) is the same as that given by Chuang et aL for the crack velocity. When 2,J/2~.r. In ( ~ ) A o ~< 1 ~,(I - A ) s
(C9)
240
PROGRESS
IN
MATERIALS
SCIENCE
eq. (C5) reduces to 1
(71
r*
y,(1 -- A)
Then the two rate-equations (C7) and (C8) reduce identically to those given as eqs (3.13) and (3.14), as they should. .Equation (C-x)) is that of the broken line on the growth maps. When Ao is large, the void becomes more nearly spherical, and the calculation breaks down because r* is no longer small compared to r,. In the limit of very large Ao, the boundary diffusion limits the rate of diffusion, the damage-rate and strain-rate are given by eqs (3.4) and (3.5). We require a definition for the boundary separating crack-like growth from equilibrium void-growth. We adopt the criterion that r* is a fraction of the void radius r* rh
1 3
giving [from eq. (C5)] 9:ln(~) o_~_t = 1.5 + a~p
4
A° (CIO)
(1 - j~)2
where a~p -- 2y,(! - f,)/r, is the capilarity stress. This is the equation of the boundary between the regime of crack-like growth and boundary diffusion controlled growth. The transition from crack-like to equilibrium growth is, in fact, a gradual one. We assume that the void grows in a crack-like manner if r* 1 ~< (Cll) rh 4 and it maintains an equilibrium shape if r* ~< 1 rh 2"
(C12)
Substituting these limits into eq. (C5) gives stress ranges over which the transition occurs [eq. (3.31)].
APPE~qDIX D: Stress State Caused by Grain-Boundary Slidino Consider the hexagonal array of grains shown in Fig. 17, subjected to remote stresses a~: and a~: in the 1 and 3-directions, which are rotated through the angle 0 with respect to the 1' and 3' axes. Resolving forces in the 1' and 3' directions gives d~ =
a~
2
-
cos 20
+ ~
2
2
cos 20
~ 3 = o~ - o~ sin 20. 2
(D1)
Isolating the microscopic element of material ABCA'B'C' gives the average stresses shown in Fig. (DI). The boundaries AB, BB' and A'B' slide freely and cannot therefore support any shear stresses. Equilibrium requires that a~c = ½ ( 3 ~
-
~)
(D2)
O'bb = O'3 + N. 3r~ 3.
Substituting for o'~, a'3 and r~3 using eq. (D1) gives abo = ½ ( a ~ + O D + ( a ~ -- a ~ ) C O S
20
abb = ½(or + a]:) - (a~ - a]~)cos(20 + 60°).
(D3)
abe and Obb are the mean tractions on the grain-boundaries. As the stress system is rotated, the maximum traction shifts from the BB' boundary to the AB (and A'B') boundary. The magnitude of the maximum traction is repeated every 30 ° of rotation. In calculating the mean value of the maximum tractions, ax, only 30 ° of rotation
ON
CREEP
FRACTURE
BY V O I D
Crab
A-- ¢ab
241
GROWTH
C
l
B t
i
C
,rac ~
---%c b TOC
C*
'rob- ~
%c
O'ob
FIG. D1. The analysis of grain-boundary sliding.
need be considered. Therefore,
•[16
~b~ dO
~x
--
(D4)
nt6
do 00
which gives or, = 4~al~t - . a-3- . 3
(D5)
It is equally probable that the grain could lie in the 12 plane. For this orientation the mean of the maximum tractions, a r is 3 "
(D6)
Averaging over both orientations gives the overall mean of the maximum tractions, at, as a l = ~(r~: - ~(cr~ + a;:).
(D7)
The element ABCA'B'C' of Fig. DI is constrained by the surrounding material to deform as it does. This constraint can be satisfied if the deviatoric components of stress remain constant and equal to the applied components throughout the element. A measure of the level of triaxiality in the vicinity of the grain-boundary can be found by equating the principal deviatoric component there with the principal applied component, If S~: is the applied principal deviatoric component, then S; ~ = { a ; ~ - ~ ( ~
+ a~)
(D8)
equating this with the local principal component, $1, where ~ 1 = ~C~a~ 2 4 ~: -
~(e~ + ~ ) ]
- ~(~
+
~)
(D9)
a~ and a~ are the other two local principal stresses, gives a local level of triaxiality of tP/ L
where a [ is the applied effective stress.
= _ (4(;~ -i-(r~ q- (r~) 6e~:
(DIO)
242
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APPENDIX E: Creep Fracture Resulting from Precipitate Coarsening If do is taken as a reference strain-rate in the creep law [eq. (2.1)] then Oo can be interpreted as the stress that results in a strain-rate do. If the material ages during the course of a test then the stress needed to maintain a strain-rate do decreases, i.e. Oo decreases. At low temperatures the strength of a material is found to be inversely proportional to the precipitate spacing Ip. It therefore seems reasonable to assume that ~ro is inversely proportional to lp. Then ~o -- %0
(El)
where at the start of the test ¢0 -- aoo and lp = Io. The mean radius of a spherical precipitate after a time t is given by (Shewmon, 1969) r~
ro3 + 8DpypC°I'22
9RT
t
(E2)
where ~,p is the surface free energy of precipitate material, Dp is the diffusivity of precipitate material through the matrix, and Co is the equilibrium solubility of large precipitates, and since the volume fraction of precipitates must remain constant, i.e. fv = r~/l~ = r~/l~, we obtain 1~ = 1 + 0odot
(E3)
Oo - 8D#pC°D2 9R Tffl~d o "
(E4)
where
From eqs (El) and ( E 4 ) (E5)
Ooo
ao
=
(1 + Oodot)1/3"
Substituting this into the creep law [eq. (2.11] gives d =
(1 + OodotY'/3.
do
(E6)
Now, consider a uniaxial creep test conducted at constant load. During the course of the test, the volume of the specimen must remain constant.
Aoho = Ah = const
(E7)
where A is the cross.sectional area and h the height of the specimen. Differentiating eq. (E7) w.r.t, t gives 1 dA A dt
ldh hdt
. . . . . . .
d
(E8)
and if o~ -- P/Ao is the initial uniaxial stress then eq. (E6) can be written as 1 dA / o ' " / A \" A dt =do~-~ooo)~A)(1 + 0odotY '3.
(E9)
The time failure is found by integrating eq. (E9) between the limits A/Ao = 1 at t = 0; and A/Ao = 0 at t = ty, giving
dot:=
I + ~
\ - -o~ / 3
-
1 .
(EIO)
Equation (El01 can be plotted as a creep rupture curve using axes of al/aoo and dots. Curves for 0o = 1.0 and a range of values of n are shown in Fig. El. Equation (El0) can be written as o'l Oo is a material parameter. It can be defined in the same way as 4~o in Section 7.2 to give a model-based procedure for extrapolating creep data (Cocks, 19801. The result is again that data should be plotted using axes PI = loglo ol - H
-
ON
CREEP
FRACTURE
BY VOID
243
GROWTH
% % n=5
~
~o=1.0
n = 5
JO- 6
2
~.0
o.5
1I0l -l.
IC) I~
~
i
~
I0 0
102
-o._c
-I.O
-I.5 FIG. El. Creep rupture plots when fracture is a result of precipitate coarsening.
and
(, ,)
P2 = loglotf - d ~ -
Too
where H and J are now defined by H
-Q,+Q.+Qo 2.3nRT
j _ Qp + Qo 2.3RT
and Qc, Q~ and Qo are the activation energies of the exponentially varying quantities D, D r, Co. APPENDIX F Consider any mechanism which causes damage to accumulate in a way which can be written dfh dt
1 h(a)k(T)#(fh)
(F1)
that is, stress, temperature and damage are (mathematically) separable. The eqs (3.4), (3.13) and (3.16) are all of this type. Then the time-to-fracture for any history of loading is given (ignoring nucleation) by h[o(t)]k[T(t)]
-
o(fh) dfh.
(F2)
At constant stress and temperature, the time-to-fracture is t r = h(a)k(T)
fo*
g(fh) dfh.
(F31
244
PROGRESS
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SCIENCE
We now form the life-fraction [eq. (7.18)] giving
ft's ~ - . f l h(a)k(T)g(fk) dfh = 1. do •/o ty h(tr)k(T) g(fh)dfh
(F4)
Io'
A proof of this sort was first given by Odqvist and Hult (1961) who, however, limited themselves to a simple power-law for h(a) and g(A).
0079-6425/82/040189-56528.00/0 Copyright © 1982 Pergamon Press Ltd
Printed in Great Britain. All rights reserved
ON CREEP FRACTURE BY VOID GROWTH A. C. F. Cocks and M. F. Ashby
Cambridge University, Engineering Department, Trumpington Street, Cambridge CB2 1PZ, England ABSTRACT The growth of void-like creep damage is analysed. Voids can grow by mechanisms controlled by grain-boundary diffusion, by surface diffusion, by power-law creep, and by any combination of two of these Approximate analytical equations for the growth rate by each mechanism, under multiaxial stress states, are developed, and related to a number of published analyses. The growth equations are integrated to give times and strains to fracture under constant (multiaxial) stress, constant load and two simple load histories. Both quasi-uniform and non-uniform distributions of voids are considered. The formulation can be related to the continuum damage method of Kachanov and suggests modifications to this method. It leads to a new prescription for extrapolating creep data. A unified picture of void growth appears possible, embracing most aspects of previously published models.
CONTENTS LIST OF SYMBOLS I. INTRODUCTION: CREEP DAMAGE 2. MECHANISMS OF CREEP AND OF HOLE GROWTH 3. THE MODELLING OF VOID GROWTH
3.1. Void Growth Controlled by Boundary Diffusion Alone 3.2. Void Growth Controlled by Surface Diffusion 3.3. Void Growth Controlled by Power-Law Creep Alone 3.4. Void Growth by Coupled Boundary Diffusion and Power-Law Creep 3.5. Void Growth by Coupled Surface Diffusion and Power-Law Creep 3.6. Void Growth by Coupled Boundary and Surface Diffusion 3.7. Interface-Reaction Control of Void Growth 3.8. Transgranular Creep Fracture 3.9. Void Growth Maps 3.10. Importance of Grain-Boundary Sliding 4. THE TIME AND STRAIN TO FRACTURE
4.1. 4.2. 4.3. 4.4. 4.5. :4.6.
Boundary Diffusion Alone Surface Diffusion Alone Power-Law Creep Alone Coupled Boundary Diffusion and Power-Law Creep Coupled Surface Diffusion and Power-Law Creep Coupled Surface and Boundary Diffusion
5. NON-UNIFORM DISTRIBUTIONS OF VOIDS: CONSTRAINED CAVITY GROWTH
5.1. 5.2. 5.3. 5.4. 5.5.
Both Void Growth and Grain Deformation by Diffusion Alone Power-Law Creep Alone Coupled Growth Creep Rupture Plots Other Distributions of Voids
6. COMPARISON WITH THE CONTINUUM THEORY OF CREEP DAMAGE MECHANICS
6.1. The Continuum Theory of Kaehanov (1958) and Robotnov (1969) 6.2. Comparison of the Continuum and the Mechanistic Models: Power-Law Creep 6.3. Comparison of the Continuum with the Mechanistic Model: Diffusion Controlled Growth 6.4. Results of the Comparisons LP,'~.S. 27 3-4 ,~
189
190 191 192 194 195 196 199 201 202 204 204 205 206 211 213 213 214 214 215 216 216 217 218 220 221 222 223 223 223 224 225 225
190
PROGRESS
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7. EXTRAPOLATION OF CREEP DATA
7. I. 7.2. 7.3. 7.4.
Present Methods for Extrapolating in Time Model-Based Procedure for Extrapolating in Time Extrapolation to Non-Steady Histories of Stress and Temperature Extrapolation in Stress Space
8. SUMMARY REFERENCES
APPENDIX A: APPENDIX B: APPENDIX C: APPENDIX D" APPENDIX E" APPENDIX E
Void Growth by Coupled Boundary Diffusion and Power-Law Creep Void Growth by Coupled Surface Diffusion and Power-Law Creep Void Growth by Coupled Surface and Boundary Diffusion Stress State Caused by Grain-Boundary Sliding Creep Fracture Resulting from Precipitate Coarsening
226 226 227 230 234 234 235 235 237 239 240 242 243
LIST OF SYMBOLS He ~1 ~2 U3 ad ~+
p ~c G~
E
ff n, ~YO,to t,
t[ tf rh
rd re 2! d
I. /, f; f~ A ~:, V A
6. DI, D,, D~
QJ, Q,, Q~ k R
T
Equivalent tensile stress (N m - 2) Principal stresses (N m-2) Remote principal stresses (N m - 2) Mean normal stress in diffusion zone (N m -2) Stress used in continuum theory of creep damage (N m -2) The hydrostatic pressure; p = - J (al + ~72 + 03)(N m -z) Mean stress in damaged region when voids are non-uniformly distributed (N m - 2) Mean stress in undamaged region when voids are non-uniformly distributed (N m - 2) Young's modulus (N m - 2) Shear modulus (N m -2) Steady state creep-rate in the absence of voids (s- 1) Principal strain-rates (s- 1) Axial strain-rate of a grain-boundary slab of material (s-1) Strain to nucleate voids Strain to failure Creep constants (dimensionless, N m-2, s-1) Time to nucleate voids (s) Time to failure (s) Time to failure at constant load (s) Radius of growing void (m) Radius of diffusion zone (m) Effective void radius as seen by power-law creep zone (m) Centre-to-centre void spacing (m) Grain-size (m) Area fraction of holes on grain-boundary;fh = r~/I 2 Area fraction of diffusion zones; fa = r~/I 2 Effective area fraction of holes;fe --- r~/I z Initial area fraction of holes Area fraction of holes at which linkage occurs, taken as 0.25 Area fraction of holes at transition from growth by boundary diffusion to power-law creep growth in approximate analysis Area fraction of voids at transition from growth by surface diffusion to power-law creep growth Area fraction of voids when stress is changed from a diffusion-controlled regime to a power-law creep one Volume (m 3) Constant in creep-law Atomic volume (m 3) Grain-boundary thickness (m) Thickness of layer of material in which surface diffusion takes place (m) Grain-boundary, surface and lattice diffusion coefficients (m~ s- i) Activation energies for boundary, surface and lattice diffusion (J mole- 1) Surface free energy (J m -2) Boltzmann's constant (1.38 x 10 -23 J K -1) Gas constant (8.31 J mol- ~ K - 1) Absolute temperature (K)
ON
To b
¢o #'0 Ao
O3
H,J
CREEP
FRACTURE
BY VOID
GROWTH
191
Reference temperature (K) Length of Burger's vector (m) Dimensionless quantity which appears when grain-boundary diffusion and power-law creep are coupled Dimensionless quantity which appears when surface diffusion and power-law creep are coupled Dimensionless quantity which appears when surface diffusion and grain-boundary diffusion are coupled Parameter that measures effect of stress state on void growth-rate Value of//corresponding to simple tension Measure of damage in continuum theory of creep damage Parameters used to plot creep rupture data.
1. INTRODUCTION: CREEP DAMAGE
When a solid deforms at high temperatures, its structure changes. Holes or cracks may nucleate and grow inside the solid; its grain size may increase (because of grain growth) or decrease (because of dynamic recrystallisation); precipitates or dispersoids within it may coarsen or dissolve; a substructure introduced by prior working may be destroyed; or the microstructure may, in some other sense, be altered. All these changes we refer to as "creep damage". This damage usually accelerates the creep-rate, which in turn accelerates the rate of appearance of damage, until the specimen or component fails. The tertiary part of the creep curve shown in Fig. 1 is a direct result of this coupling of creep-rate and damage-rate. The figure defines the other quantities which interest us here: the time-to-fracture t~- (s), the strain-to-fracture %,, and the steady-state creep-rate ~, (s- 1). This paper examines one class of creep damage: holes and cracks which will be collectively termed "creep voids". We shall not discuss their nucleation (for which see Goods and Brown, 1979; Argon et al., 1980), but include it merely as an additive nucleation time (t,) or strain (E~).We focus attention on void growth and the way in which stress, temperature and stress-state affect it. We combine results from two earlier papers (Cocks and Ashby, 1980, 1981) with newly derived solutions to formulate differential equations for creep-rate and for the rate of growth of void-damage. The results are presented as "void growth rate maps", and are compared and related to earlier microscopic models of Hull and Rimmer (1959), Beere and Speight (1978), Edward and Ashby (1979), Chuang et al. (1979), and Dyson (1976, 1978), to the finite-element calculations of Needleman and Rice (1980) and to the continuum-damage mechanics of Kachanov (1958).
STRAIN TERTIARY STRAIN SECONDARY
= Pccstf ELASTIC* PRIMARY = TIME ~-
tf
=
FIG. 1. A schematic creep curve.
192
PROGRESS
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SCIENCE
These damage growth equations are then integrated for conditions of constant stress, constant load and simple load histories, to give the time and strain to fracture. A modelbased procedure for extrapolating creep data is developed. The effect of stress state is discussed and illustrated by constructing isochronous fracture surfaces. Where possible, the results are compared with experiment and their implications for the extrapolation of creep data are indicated. It must be remembered throughout that creep fracture is a very complicated phenomenon, only one aspect of which is analysed fully here. The creep life of creep-resistant alloys like the Nimonics or the low-alloy ferritic steels, is influenced strongly by the ease or difficulty of void nucleation and by microstructural changes other than void growth, both of which are considered at only the most elementary level. But in other ways the conclusions reached here are encouraging: we believe a general consensus is emerging on how holes grow and link; a single set of equations is presented which, at an approximate level, describe power-law creep growth, growth controlled by boundary or surface diffusion and growth involving the coupling of pairs of these. Further, a direct connection appears between the microscopic theories and the continuum-mechanics of creep damage; and the method leads to a new, model-based, extrapolation procedure which looks very promising.
2. MECHANISMSOF CREEP AND OF HOLE GROWTH Above 0.35 TM (or thereabouts) a crystalline solid creeps. At high stresses, the solid suffers power-law creep, described by the constitutive law d,~, = do
(2.1)
where ae is the equivalent (tensile) stress, defined by tl tze = L:r[(al - a2) 2 + (a2 - a3) 2 + (a3 - al)2]} 1/2
(2.2)
and d,., is the equivalent (tensile) strain-rate d,~. = ~g[(e12 " - d 2 ) 2 + (d2 - d 3 ) 2 + (da - d l ) 2 ] } 1/2.
(2.3)
The quantity do depends exponentially on temperature do ~c exp
Qc
RT
where Qc is the activation energy for creep. It is discussed further in Section 7. The quantities n and ao(MN/m -2) are creep constants and ala2a3 and dld2d3 are the principal stresses and strain-rates. At lower stresses and in small-grained solids, deformation is by diffusionalflow, and the strain-rate is controlled instead by diffusion across or around grains (Nabarro, 1948; Herring, 1950; Coble, 1963; Raj and Ashby, 1971):
12oo~Ge( 1 + rc~Da~ dD~ /
ds~, '" ~
(2.4)
where Dv and 6Ds describe the rates of latticeand of grain boundary diffusion,and d is the grain size. These regimes of deformation are shown for copper (which typifiesmany f.c.c. metals) in Fig. 2, which also shows the conditions under which dynamic rccrystallizationis
ON
CREEP
FRACTURE
BY VOID
TEM PERATURE I0"
-200
0
200
,
•
,
ml mBIIDEAL I m qml!'HEAR el i
400 .
('C) 600
800
I000
I
i
I
STRENGTH I m II m m ~ I m ~
mD
PURE
COPPER
d -
KS~ w:}')',
193
GROWTH
PLASTICITY
O.I mm
79
,o'
DYN~R~CsTALUSATIO N
b
I \\\~",~13 EAK
U~ Ld n,
I-- J6 ~
tO n,< UJ "r"
(L.T. CRE"EP)~ x"
~
~)1~
o
POWER ILAW
C
LLI
W
n."
m_ ._1 <
0 z
DIFFUSIONAL
KS:
16
F LOK-_.
' D,.'}.,o., O
O. 2
0.4
0.6
0.8
1.0
HOMOLOGOUS TEMPERATURE, T/'rM FIG. 2. A deformation-mechanism map for copper, showing the regimes of plastic deformation.
observed, and the transition region (labelled "Breakdown") between power-law creep and classical (rate-independent) plasticity. Creep fracture is more complicated. Under creep conditions, fracture most commonly occurs by the growth and coalescence of voids which lie on grain-boundaries. The growth can be controlled by boundary diffusion (Hull and Rimmer, 1959; Speight and Harris, 1967; Raj and Ashby, 1975); by surface diffusion (Chuang et al., 1979); by power-law creep (Edward and Ashby, 1979; Cocks and Ashby, 1980); or by a coupling of diffusion and power-law creep (Beere and Speight, 1978; Edward and Ashby, 1979; Needleman and Rice, 1980; Cocks and Ashby, 1981). At any instant in time, a particular growth mechanism dominates, so that a map, analagous to Fig. 2, can be constructed for void growth (examples are given later). But the time to fracture reflects the entire history of the growth of voids, from nucleation to final linkage. Voids usually grow by diffusion when they are small, but as they become larger, power-law creep takes over as the dominant growth mechanism: this coupling of mechanisms must be taken into account in calculating the time to fracture. And the stress state matters too: creep fracture depends not only on the equivalent stress (tr,) but also on the magnitude of the maximum principal stress (al) and the hydrostatic pressure p = -~(trl + tr2 + a3).
(2.5)
194
PROGRESS IN MATERIALS SCIENCE
The next section describes approximate ways of calculating damage rates b.y each of the void growth mechanisms, under multiaxial stresses. Section 4 shows how these can be combined and integrated to give times to fracture. 3. THE MODELLINGOF VOID GROWTH Voids may grow in a way controlled by boundary diffusion, or by surface diffusion, or by power-law creep or by combinations of these. When diffusion is involved, the voids grow on grain-boundaries, but when power-law creep is dominant they may grow within the grains as well. We now consider each growth mechanism in turn. For the purpose of analysis we isolate a cylindrical element of material of diameter 21 (the void spacing) and height d (the grain size) centred on a grain-boundary void of diameter 2rh, shown in Fig. 3. The surfaces of this unit are subjected to the stress field at, a2, a3 as shown. We calculate the void growth rates caused by this stress field, remembering that it may differ from the remote (applied) field cry, a~, a~ because of grain-boundary sliding. Finally, the two fields are related (Section 3.10) to give the rates of void growth caused by a given remote field. The calculations which follow are approximate. In solving diffusion problems, refinements which influence the results by less than a factor of two are dropped. In solving problems involving power-law creep, bound-methods are employed which also involve errors; but comparison of our results with the finite element calculations of Needleman and Rice (1980) shows agreement to within a factor of about two, giving confidence in the method. The value of the approach given here is that it is very simple, it deals with multiaxial stress states, and it gives analytical results which can easily be integrated to give times and strains to fracture for any given history of load and temperature. By far the largest source of error in using these results is the material data: power-law creep data can
ttftfil
---X
tt
" Y--Illlli
o',~
Ill
I
o-,
Fro. 3. Void growth on a grain-boundary, showing the unit of structure.
ON
CREEP
FRACTURE
BY VOID
GROWTH
195
,.Jim
~.= (~O(~o)nI ~-~-
RIGID GRAIN IOUNDARY
SURFACE DIFFUSION ,,.
(i
~n
%--..q arh
- ~ 2rh~-
- - - - - ~ 2,~ - - - - - - ~ RIGID GRAIN
RIGID GRAIN
..q-.
•
..i,-
)
)
(a)
)
t
)
;
;
t
.
o-n
)
t
....
¢=%1~o) )
(b)
;
(c)
FlG. 4. The three simple mechanisms which limit void growth.
generally be trusted to within a factor of about two, boundary diffusion coefficients, when known at all, are accurate only to a factor of about four; surface diffusion, to a factor of perhaps ten. 3.1. Void Growth Controlled by Boundary Diffusion Alone Hole growth controlled by boundary-diffusion alone is shown in Fig. 4a. Matter diffuses by boundary diffusion out of the growing void and plates onto the grain-boundary. The void remains spherical because surface diffusion rapidly redistributes matter within it. This diffusion problem can be solved with more than enough precision for our purposes (Hull and Rimmer, 1959; Speight and Harris, 1967; Raj and Ashby, 1975). There is general agreement that the volume of matter flowing out of the void per second is well approximated by
(
r23
dr _ 2~D,6Balfl \1 12"] dt kT ln(//rh)
(3.1)
where 2rh is the hole diameter and 21 their spacing. (The equation becomes inaccurate if the void size is near the "sintering limit" at which surface tension balances the driving force due to the applied stress. Then al/(1 - r2/l 2) must be replaced by [~1/(1 - r2/I 2) - 2?s/rs], where ?, is the surface energy of the solid.) The other components of local stress field (a2, a3) have no important influence on void growth by this mechanism. We now define the area fraction of holes, fh, by f~ _ r~
12
(3.2)
2De6Bft ao k Tl3 #-o
(3.3)
and we define a new material property, ~bo,by
C~o =
196
PROGRESS
where So and 4o were defined both because atoms flow out boundary over the area 7r(I2 of the void caused by diffusion
IN
MATERIALS
SCIENCE
in eq. (2.1). We further note that the void grows in volume of it by diffusion and because these atoms plate onto the r 2) jacking the two half-crystals apart. The total growth-rate plus jacking is
dt
dt
and the growth-rate of the damage, dfh/dt, is easily shown to be
df. 1 dV d--t = 2zrrh~ dt" Substituting these into eq. (3.1) gives the final equation for the damage-rate; and straightforward manipulation of eq. (3.1) gives also the strain-rate caused by void growth.
ldf
4o dt
4,o
-
f]/2
ln(1/~)
(a)
(3.4)
l de 4o & = In(l/A) \d,]\0"o,]"
(3.5)
[To this must be added the strain-rate due to diffusional flow, eq. (2.4) which, when boundary diffusion is dominant, becomes: 1 de = 6n4~o ae 40 dt -d3 ~o "
(3.6)
Power-law creep does not contribute to pure diffusional growth, of course; 0-0 and 40 appear only because they are convenient normalizing parameters.] The dependence of damage-rate on the local stress state can be illustrated by plotting surfaces of constant dfh/dt in principal stress space. Such a surface for plane stress and plane strain are shown in Fig. 5a. The axes are normalized stress: 0-1.
0"2.
X3
-~- 0"..3_3
0-0
where 0-0 is the simple tensile stress that produces the given damage growth rate. The surfaces are those of constant maximum principal stress. When grain-boundary sliding is included, the surfaces become more complicated: they are illustrated, and discussed, in Section 3.10. Under certain circumstances, voids growing by boundary diffusion or by a coupling of boundary and surface diffusion (Section 3.6) can become unstable. The perifery of the void develops perturbations which grow into finger-like extensions. These fingers tunnel ahead of the main void, increasing the damage rate (dfh/dt). The instability and subsequent growth can be analysed (Fields and Ashby, 1976) but seems rare in practice; we shall ignore it here. 3.2. Void Growth Controlled by Surface Diffusion When surface diffusion within the void is slow, it ceases to grow as a sphere. Matter flows out of it at the equator, causing it to become flatter and more crack-like (Fig. 4b) until the curvature difference between the poles and the equator is sufficient to drive a surface flux
ON
"
CREEP
FRACTURE
,,,~,,4
ANO STRAIN
O.S
~
~I~o.s
20
197
GROWTH
F~WER-LAW CREEP
PLANE STRESS
El
BY VOID
tr
-,.o
-O.S
PLANE STRESS
~ o \o.,
o
-O.SX
2.C
2.0
"~l~°'s !,,
2.0
,.;3.0
- 1.0" ~
//~
f..o
•
r~3.o
- 1.01" ~"
FIG. 5. Fracture surfaces for void-growth by diffusion, and by power-law creep, ignoring the stressconcentrating effects of grain-boundary sliding. These are introduced later and shown in Fig. 18.
which matches that leaving, by boundary diffusion, at the equator. This problem has been analysed in detail by Chuang and Rice (1973) and Chuang et al. (1979). This section provides a simple, approximate, analysis of the limit in which surface diffusion completely controls void growth. The coupling of this with boundary diffusion and power-law creep are considered later. The shape of a crack-like void is one with continuously changing curvature, as shown by the broken line in Fig. 6. We idealize this as a penny-shaped crack of constant width and semicircular tip of radius r0 a s s h o w n by the full line. The governing equation for the surface-diffusion current is:. I, =
O-k--T~\d-S]
(3.7)
where I, is the number of atoms flowing across unit length of surface per second driven by the surface potential gradient (d#/dS), and D,h, is the surface diffusion coefficient times
198
PROGRESS
IN MATERIALS SCIENCE
o-,
I
DIFFUSION
i_ ~-
r h ________~
I I I
I
I_ If
1
II
1
Fro. 6. Void growth limited by surface diffusion.
its diffusion thickness. The potential gradient is given by: d#_ d dS dS [-7sf~(xt + x2)].
(3.8)
Here S is a coordinate measured along the surface, and xl and x2 are the principal curvatures at S. We now approximate the differential by the difference in curvature between the crack tip and the crack flank (where, in our approximation, the curvature is zero) divided by the distance between them, which we take to be (Tr/2)ro. This gives:
dS
rr ro
where 1
Xl = - ro
and
x2 = -
1
rh
are the principal curvatures at the crack tip and 7, is the surface free energy. If the crack advances with velocity dr,/dt and rh >> to, then conservation of matter requires that:
drh 2ro -~- = 2I, f/.
(3,9)
But if surface diffusion totally controls the growth, the potential gradient in the grain boundary is negligible; and for the potential at the void tip to be continuous we require that:
1
ro
-
al
(3.10)
~,(1 - A )
Combining these equations gives:
dra at
2 -
D,6,f~
7r kTy~,(1 - A P
a~.
(3.11)
ON CREEP FRACTURE BY VOID GROWTH
199
This is identical with the equation derived by Chuang et al. except for the constant n/2; they find 2 x/~ instead. This difference is slight, but we can amend eq. (3.11) to include it. We define a second material property ~ko, analagous to ~bo [eq. (3.3)], ¢,o =
(3.121
and using
dfh dt
2rh drh 12 dt
and
& 8xrorh drh a t = d12(1 - J h ) d t we obtain
1 dfh d/of~/2 (trl~ 3 ~o d--t-= (1 - ~ \ ao---/ 1 dl~ do
(3.13)
41~of I/2 ~s (o'_1_1~2
(3.14)
=
To this must be added the strain-rate due to diffusional flow [eq. (3.6)], if appropriate. As before, it should be realized that power-law creep makes no contribution here; ao and do appear as normalizing parameters only. The surfaces of constant growth-rate in stress space, for this mechanism, are identical with those for boundary-diffusion limited growth (Fig. 5a).
3.3. Void Growth Controlled by Power-Law Creep Alone A void can grow by the power-law creep of the surrounding matrix as shown in Fig. 4(c) (Hellan, 1975; Edward and Ashby, 1979; Cocks and Ashby, 1980). The growth-rate can be computed by using finite element methods (Needleman and Rice, 1980); or it can be calculated approximately (Cocks and Ashby, 1980) by using a bound theorem due to Martin (1966). This second method is approximate; but it leads to analytical expressions for the void growth-rate, it can be adapted for multiaxial loading, and it can be checked against the results for the finite element calculations. In simple tension, the result is particularly simple: the shaded slab containing the void in Fig. 4(c)extends at a rate determined by the net section stress a d o -fh), while the rest of the cylinder (unshaded) extends at a rate determined by tre alone. The result is that the shaded slab dilates, causing the holes to grow. The application of the continuity equation (Cocks and Ashby, 1980):
27r-ff-rh dt
d
)
- fhi:ss
(3.15)
200
PROGRESS IN MATERIALS SCIENCE
leads immediately to expressions for the damage rate (dfddt) and the strain-rate (d~/dt): 1 dA
E
1
(1 - A )
(3.16)
~o Z = # (i _7~),
]a~
{
~o ~ =
1
+
2r._F 1 -z~L(1
t-l~(o2}"
---Ay'
(3.17)
jo,,oo/
where fl is a constant ( ~ 0.6) and ae is the von Mises equivalent stress (eq. 2.2), equal to the axial stress al in simple tension. The behaviour under combined axial and hydrostatic tension is less obvious, but can be treated by the bound method (Cocks and Ashby, 1980), and by finite element methods (Needleman and Rice, 1980). The results show that a hydrostatic tension -p/ae accelerates void growth. The damage-rate and the strain-rate are still given by eqs. (3.16) and (3.17) with fl--sinh-
2 ~ - ~
.
(3.18)
The result can be illustrated (as before) by plotting surfaces in stress space on which the hole growth-rate has a constant value. They are shown in Fig. 5b, c and d for plane stress, plane strain and axisymmetric loading. 102
~o =O.I n =5 t h - i O -2
I01 -
r_,,~
ool
COUPLED GROWTH
/
/
-
.,//
/ 7 Idt
//
/
~o-
/
I
I
/
/
//
/ OIFF'USIONAL
GROWTH
/
/"~-POWER- LAW CREEP GROWTH
I, %,
FIG. 7. The damage-rate as a function of stress, when voids grow by diffusion and power-law creep.
ON CREEP FRACTURE BY VOID GROWTH
201
3.4. Void Growth by Coupled Boundary Diffusion
and Power-Law Creep If the damage-rates given by eqs. (3.4) and (3.16) are plotted against stress, the behaviour shown in Fig. 7 is found. When the stress is low, the voids grow by diffusion; when high, the3; grow by power-law creep; in between, they grow by a combination or coupling of both (full line, the derivation of which is described below). If, instead, the damage-rates (dJ~/dt) are plotted against fh the behaviour is as shown in Fig. 8. When J~ is small, the voids grow by diffusion; when large, they grow by power-law creep; in between, they grow by a coupling of both (full line). The growth-rate has a minimum at the point at which the two curves intersect. This is important: the time-to-fracture is largely determined by the depth and position of this minimum, and therefore depends on the rates of both mechanisms. Hole growth by coupled boundary diffusion and power-law creep has been analysed in several recent studies (Beere and Speight, 1978; Edward and Ashby, 1979; Needleman and Rice, 1980; Cocks and Ashby, 1981). An analytical calculation of the rate is summarized in Appendix A. The equations given there have been solved and lead to the full lines shown on Figs 7 and 8. It is obvious from these two Figures that a reasonable approximation for the coupled rate of growth is simply given by adding the rates of the two simple mechanisms: boundary diffusion [eq. (3.4)] and power-law creep [eq. (3.16)]. This is such a convenient approximation that much of the following development will be based on it. The point at which the broken lines intersect on the two figures can be calculated by equating eqs (3.4) and (3.16). This is the point at which the dominant void
102
<~,., = 0.1 n
= 5
l dfh
/
COUPLED MECHANISM
DIFFUSIONAL GROWTH
\ \ ~
,o-'
.j/
/
//
/
/
/ ~ W E R - LAW CREEP / GROWTH /
/
/
JdK~e
I
Id s
I
Io-~
i
Id 3
/
i
Id 2
I
Io-'
fh FIG. 8. The damag¢-rate as a function of damage, wh¢n voids grow by diffusion and power-law creep.
202
PROGRESS
IN MATERIALS
SCIENCE
growth mechanism changes. The result is
~b0 = #f~/21n
(I)[ ~
I
(1 - J ~ ) "
(I- J~)l(e'y(e°'~ d\eo/\ell"
(3.19)
(The result for simple tension is obtained by setting fl = 0.6 and ee = at.) This is the equation of the line, in stress/temperature space, separating the field in which boundary diffusion is dominant from that in which power-law creep is dominant. It can be solved to give the critical area fraction, f~, at which the change of mechanism occurs (Cocks and Ashby, 1981). It is
I f ~ = [a(ln a - 1)] a/2
(3.20)
where
4(n + 1)(ae~"(e0~//. a =
(3.21)
3cb~ \go/ \ell
3.5 Void Growth by Coupled Surface Diffusion and Power-Law Creep When the damage-rates given by eqs (3.13) and (3.16) are plotted against stress (Fig. 9) and againstfh (Fig. 10), they are seen to intersect. As before, diffusion controls the early part ~ro = O . I n = 5 fh = IO-2
/
/
/
/
/
/
/
/
GROWTH
/ SURFACE DIFFUSION CONTROLLED
j6 ~
16'
/•//11 I /
I I I
1.0
%
IO
FIG. 9. The damage-rate as a function of stress, when void growth is limited by surface diffusion,and
power-law creep.
ON
CREEP
FRACTURE
BY VOID
GROWTH
203
I01
,o=O, . io 0
/!
/I
=s
~'= I %
/ ,/ t
I0 i I
df h
io dt
,d
SU~rAC~
Ol~~ ///' /Y/l/it,
1111 I
i0 "l'
/
I
iO-e io-S io-~
I
I
9 3 io-Z
I I 0 -~
i0 °
fh FIG. 10. The damage-rate as a function of damage, when void growth is limited by surface diffusion and power-law creep.
of void growth, but power-law creep takes over as the voids become larger and the net section stress rises. The coupling of these two mechanisms is treated in Appendix B. For most practical purposes it is adequate to assume that the coupled rate is simply the sum of the rates of the two individual mechanisms: sur]'ace diffusion [eq.(3.13)] and power-law creep [eq. (3.16)]. The point at which the broken lines intersect on Figs 9 and 10 is important in calculating the fracture time. It is the point at which the dominant mechanism changes, and is obtained by equating eqs (3.13) and (3.16). The result is
+o
=
ill2
'
(1 - - A ) n + '
,l:°'Y(=ol _J\~oo./ \~tt]' "
(3.22)
(Simple tension is described by setting ~ = 0.6 and o, = or.) This is the equation of the line, in stress/temperature space, separating the field in which surface diffusion is dominant from that in which power-law creep is dominant. It can be solved to give the critical area fraction, f~, at which the change of mechanism occurs. Provided f~ << 1 (as it usually is) the result is
[ +o (ooi?o,71'
I: ~- (,,+ I)/~,,~i ,,~o: j
(3.23)
204
PROGRESS
IN
MATERIALS
SCIENCE
3.6 Void Growth by Coupled Boundary and Surface D!ffusion When surface diffusion is very rapid, the growing void can maintain its equilibrium shape (two spherical caps that meet in the boundary plane). All the driving force is available to drive the boundary diffusion required to make the damage grow, which it does at a rate described by eq. (3.4). When, instead, surface diffusion is slow, the void becomes crack-like (Section 3.2). In the limit that all the driving force is used to drive surface diffusion, the damage-rate is given by eq. (3.13). Between these two limits, the void adopts a shape which reflects the relative rates of boundary and of surface diffusion, and the driving force is partitioned in such a way as to drive them at matching rates. This problem has been analysed in detail by Chuang et al. (1979). Appendix C contains a simple, approximate treatment which gives the void shape and damage-rates in this coupled regime. Obviously, both surface and boundary diffusion are necessary for void growth, so it is not simply the faster of the two which is dominant (as it was in the last two sections). We require some other criterion for the transition from surface to boundary diffusion controlled growth. A sensible criterion is that the void maintains its shape as it grows. This is examined in Appendix C, where we show that the transition occurs when a_l = 1.5 + 9A0 ln(1/fs) aCAV 2 X/2(1 -- A)
(3.24)
where acAP is the capilarity stress at the crack tip aCAP --
27,(1 - A)
(3.25)
rh
and Ao is a new dimensionless material property Ds~s A° = DatSs"
(3.26)
This is the equation of the line in stress/temperature space, separating the field of dominance of surface diffusion from that of boundary diffusion.
3.7 Interface-Reaction Control of Void Growth If grain boundaries are imperfect sources of vacancies (or if the void itself is an imperfect sink) then the boundary conditions of the diffusion problems are changed. Instead of equating the chemical potential of atoms (or the concentration of vacancies) to that in equilibrium with a grain-boundary on which a mean local stress at acts, the potential (or concentration) differs from the equilibrium value by the amount necessary to drive the "interface reaction"--in this case, the generation of vacancies. This explanation was proposed by Ashby (1969) to explain the observed inhibition of diffusional flow in dispersion-strengthened alloys. It applies equally to the inhibition of void growth. Since then other models (Harris, 1972; Burton, 1977) have been proposed which extend this sort of approach. All can lead to a threshold stress for void growth; and all can lead to a simple diminution of its rate. Interface-reaction control can be readily incorporated into the treatment of diffusional void growth (for the method, see Ashby, 1969) and is
ON
CREEP
FRACTURE
BY
VOID
205
GROWTH
0 C)
0
0
0
iI
C)
2rh0
0
0
FIG. 11. A h o m o g e n e o u s distribution of voids, growing by power-law creep.
likely to be important when voids are sharply facetted, or when alloys contain a fine dispersion of a stable second phase. It is not discussed further here.
3.8 Transgranular Creep Fracture Voids that lie within a grain grow only by power-law creep. It is true that lattice diffusion from the void to the free surfaces or to grain-boundaries will enlarge the void, but under all realistic circumstances, the rate of such diffusional growth is negligibly small. Figure 11 shows how a material containing a random array of voids within its grains can be divided into slabs, each containing a single void. The slabs are no longer separated by the grain size d but are stacked on top of each other. The growth equations of Section 3.3 can be adapted to describe the growth of this array. The damage-rate equation is unchanged; but the strain-rate equation is altered because the slab height 2rh replaces the distance d, giving the approximate results
1 dfs
1
(1 - fs)
1 d~ ~fl_ l ( a e ~ " Co ~ = L1 - fl + (1 -A)"_[\~oo] "
(3.27)
(3.28)
In simple tension fl -~ 0.6 and ~e = 31 as before. These results allow a direct comparison of the mechanistic approach of this paper with the continuous damage mechanics of Kachanov, discussed in detail in Section 6. J.PM.S.27 3-4 a
206
PROGRESS
IN MATERIALS SCIENCE
iO-'[
COPPER
~1 GROWTH BY LOWI0-'~- TEMPERATURE
0
Z
t
/
DATA
CONTROLLED GROWTH
--6[
~°o
POWER-LAW H
I
0.2
I
0.4
I
OF RAJ I
0.6 o.a TEMPERATURE (T/T M)
HOMOLOGOUS
~.o
FIG. 12. A void growth map for copper with 1 = 12#m andf~ = 10 -1.
3.9 Void Growth Maps The results developed in the previous sections are conveniently displayed as void growth maps (Figs 12-16)'I". The axes are the local tensile stress, normalized by Young's modulus, ~I/E, and temperature, T/Tu, (where Tu is the melting temperature). Each map is divided into fields in which a single mechanism is dominant. Superimposed on the fields are contours of constant damage-rate, (df~/dt). The damage-rate contours are calculated from the equations for simple power-law creep [eq. (3.16)] and simple boundary diffusion [eq. (3.4)], (taking the faster of the two) together with the full equation for crack-like growth by coupled surface and boundary diffusion (Appendix C)
I dfh
3~bo
( ~ "~3~'[1
2w/-~trliAof~/Zln(i/fh)]i/2 }a.
"o dt = 2 x/2fh(ln l/fh)3Aos \hTo// (k +
Y-~ ~-~)3"
..j
- 1
(3.29)
In the limit of surface-diffusion controlled growth, this reduces to eq. (3.13), which properly describes void growth provided
2 v/2f]/2 ln(1/J~) O'l/AO ~ 1. -
-
(3.30)
?A similar concept has been developed simultaneously by Svensson, L. E. and Dunlop, G. L. (1980) Proc. I U T A M Meeting, Leicester, England.
_
l1 Z-,~,~,h-,o' I POWER'LAW
Ul (Jr) LIJ rY
~,~
CREEP GROWTH
I
,o-~ Z
.
C
O
N
T
R
O
~
_u) <
Q:" O Z °
DATA OF RAJ
..t . . . . .
I
0.2
I
.............
0.4
1 ........
O,6
I
0.8
HOMOLOGOUS TEMPERATURE (T/T M)
1
1.0
Fro, 13. A void growth map for copper with I = 12#m and]~ = I0 -~.
~'
"U3 " "~
~
I CoPPER
............................
POWF.R..LAW TEMPERATURE PLASTICITY ~. .r...~ p GROWTH
~d~
m
"3 O"51 <1[ IZ I
DATA OF / NIEH & NIX
BOUNDARYDIFFUSION CONTROLLEDGROWTH
O Z
vo
0.2
0.4
o.~
o.s
HCMOLOGOUS TEMPERATURE (T/TM ) FIG. 14. A void&rowth map for cop_per with I = 1.6~um a n d f , = 0.2.
207
~.o
tot
I SILVER
-
I
?~ '~ J ~ A S T I C I T Y
Z
............
4
......!:~
,o
~to
,
BOUNDARY DIFFUSION
CONTROLLEDGROWTH
nf
O Z
i0 ~ O
i O. 2
1 O.4
I O.6
I 0.8
I.O
HOMOLOGOUS TEMPERATURE (T/TM ) FIG. 15. A void growth m a p for copper with 1 = 2 . 2 # m andfh = 0.2.
Io-'
S I LVE R .,Z =ll#mifh= O.8
,~, io-"
GROWTH BY LOWTEMPERATURE P L A S T I C I T Y
LLI -3 I~ I 0 ~
C R E E P GROWTH
""-
DATA OF
~
~
I
~ , \ "-,o-.~
,o i S U R F A C E DIFFUSION
iO'6 :;:;;~':/.:':!::::::!'~i:~'i/i~i~!~!i
O
0-2
O.4
0.6
O-8
HOMOLOGOUS TEMPERATURE (T/TM) FIG. 16. A void g r o w t h m a p for silver with I = 11 # m andfh = 0.8.
208
I.O
ON
CREEP
FRACTURE
BY
VOID
209
GROWTH
The field boundaries (heavy lines) separate the regions of dominance of each mechanism. The boundary between the two diffusion mechanisms is given by eq. (3.24). The boundaries between the diffusional mechanisms and power-law creep are given by equating the simple damage-rates [as described by eq. (3.19 and 3.22)], except when surface and boundary diffusion couple; then the damage-rate due to power-law creep (eq. 3.16) is best equated to eq. (3.29) giving an implicit equation of ~rl and T which can be solved by normal iterative methods. Equation (3.30) is shown on the maps as a broken line. To the left of this line. crack-like growth becomes dominated by surface diffusion so that eq (3.29) can be replaced by the simpler eq. (3.13). The shading on the maps shows the width of the transition between pairs of mechanisms. We have based the calculation of this width on the discussion given in the earlier sections. The transition from sphere-like to crack-like growth occurs over the range (Appendix C) 3 In(l/A) or1 6 In(l/A) 1 + (~ ~ - ~ Ao <~ aCAP <~ 2 + ~-f --~-j~ Ao
(3.31)
where aCAP is given by eq. (3.25). The transition between boundary diffusion and power-law creep has width (Appendix A): In(l/A) f qb0 "~o/,-1, a~ //~o ~,~/n-,, [1 + 2 In(l/A)] ~"/"- ~) \ f ~ / 2 ] <~ --ao <~ (1 - fh) \~37h3/~] .
(3.32)
That between surface diffusion and power-law creep has width (Appendix B): [(3)"(1 - A ) " - ' f ~ / 2 ~ o ] "/"-3~ ~< a--L ~ eo
f]/2( 1 - A)A
(1 -¼A).
(3.33)
The maps show three main fields: within the unshaded regions, simple power-law creep, simple boundary diffusion CHull-Rimmer growth"), and simple surface diffusion control the growth-rate. The area of boundary-diffusion control is relatively small. That of surfacediffusion is generally larger and dominant at low temperatures and slow growth-rates. Power-law creep tends to become dominant at high temperatures and, as one would expect, gives high growth-rates. These are separated by (shaded) regions in which the coupling of two mechanisms is important: they can be regarded as broad transitions between one "simple" mechanism and another. The bottom line on the figure shows the capilarity stress. When the local tensile stress is less than this, the voids sinter instead of growing. Interfacereaction controlled growth can be included on such a map, but it is not shown here because there is no evidence for it in pure copper or silver. The positions of the boundaries between the mechanisms depend on A (compare Figs 12 and 13). Because of this the boundaries sweep across the maps as the voids grow, and more than one mechanism has a term of dominance during the life of the component, even under conditions of constant stress and temperature. This is taken into account in Section 4 where times to failure are calculated. The growth maps of Figs 12-16 have been constructed for given values of void spacing, 21 and area fraction of voids, j~. In all of these experiments, voids were either present in the material at the start of the tests (Goods and Nix, 1978a, b; Nieh and Nix, 1980) or were nucleated early on in the tests (Raj, 1978). So in each situation the problem becomes one of void-growth controlled life. Figures 12 and 13 show the range of dominance of each mechanism for pure copper for I = 12 vm and two values of A. (The additional data used to construct these and subsequent
210
PROGRESS
IN
MATERIALS
SCIENCE
Table 1 Copper n A D o v ( m 3 s -1) Qv (KJ m o l e - 1) b (m) # o ( M N m -2) Tm (K) A#r(K -l) t2(m 3) D o n f s ( m a s -1) Qn (KJ m o l e - 1) y, (J m -2) Do,6,(mas -1) Q, (kJ m o l e - 1) ao (MN m -2)
4.8 2.1 x 10 -9 6.2 x 10 -5 207 2.56 x 10 -1° 4.21 x 104 1356 3.97 x 10 - a 1.18 x 10 -29 5.12 x l0 - i s 105 1,72 6 x 10 -1° 205 39.0
Silver 5.3 2.1 × 10 -9 4.4 x 10 - s 185 2.89 x 10 -1° 2.64 x 104 1234 4.36 x 10 - a 1.71 × 10 -29 6.94 x 10 -15 89.8 1.12 6 x 10-1°? 186? 61.0
?The values of Do,f, and Q, for silver were calculated by averaging over a n u m b e r of f.c.c, metals ( N e u m a n n and N e u m a n n , 1972).
maps is given in Table 1). The choice of I corresponds with that found by Raj (1978) in his experiments on copper bicrystals. The range of stress and temperature used in the experiments is illustrated on the maps. At low stresses, Raj (1978) showed that the times to fracture were consistent with the boundary diffusion model. At high stresses he observed that growth occurred by a plastic mechanism and failure times were found to be less than that predicted by the boundary diffusion model. These observations are borne out by the maps which show that the experimental range bridges the boundary diffusion and powerlaw creep mechanisms. The map for pure copper shown in Fig. 14 was constructed using l = 1.6/~m andA = 0.2, corresponding to the initial values found in the experiments of Nieh and Nix (1980). The map shows that experiments were conducted well within the regime where surface diffusion controls void growth. This is consistent with their finding that the times to failure agreed with the surface diffusion model. Figure 15 shows a void growth map for pure silver using I = 2.2/~m and f, = 0.2, together with the range of stress and temperature used in the experiments of Goods and Nix (1978a). Again, it is expected that surface diffusion should control void growth, as they observed. Goods and Nix (1978b) performed additional experiments on specimens containing large closely spaced voids. An appropriate map is shown in Fig. 16 on which the range of stress and temperature used in their experiments is drawn. Failure was found to be controlled b'y power-law creep. The experiments, however, seem to have been conducted in a regime where surface diffusion controls growth. This discrepancy is probably due to the large uncertainty in the data for surface diffusion coefficients. In constructing these maps the effects of stress concentration due to grain-boundary sliding was ignored; the stress shown here is the local stress. Taking grain-boundary sliding into account results in a translation in the experimental data upwards by an average factor of 1.3 on the figures, though local peaks in stress can be much larger. This shift would, in fact, move the experiments of Goods and Nix (1978b) towards the regime where power-law
ON CREEP FRACTURE BY VOID GROWTH
211
creep controls growth. The overall effects of grain-boundary sliding are the subject of the next section. 3.10
Importance of Grain-Boundary Sliding
The growth equations describe the growth of voids on a single, flat boundary like those shown in Fig. 4. Real engineering solids are made up of nesting polyhedral grains. At high temperatures the boundaries between these polyhedra slide, relaxing shear stresses there. Such sliding is essential to accommodate diffusional void growth; without it, the holes cannot grow. It has the effect of concentrating stress onto some boundaries and it also induces a locally triaxial stress field. The magnitudes of these effects can be calculated for an hexagonal array of grains (as in Fig. 17) subjected to a remote stress ~ in the 1-direction and a~ in the 3-direction (Appendix D). If these directions are coincident with the 1' and 3' directions then by considering equilibrium of the element ABCA'B'C' it can be shown that the traction normal to the boundary BC is
1,3o~ In practice the grain can be orientated at any angle to the applied stress system. By averaging this over all values of 0 the mean values of the maximum tractions, ol, is found to be =
s(~r2 + a~)
(3.34)
where a~, ~r~2 and tr~ are the applied principal stresses. Void growth by diffusion depends on the normal traction acting across the boundary on which the voids sit. Therefore, identical voids in specimens subjected to the same value of o~ will grow at the same rate. If the voids in a specimen subjected to a uniaxial stress ai grow
II~ I, O'L
~
3t
3,0" 3
FiG. 17. The unit used in the calculation of the stress concentrationdue to grain-boundarysliding.
212
PROGRESS
IN MATERIALS
SCIENCE
POWER-I.AW CREEP PLANE STRESS, n-5
DIFFUSION PLANE STRESS
z;
z °
~o
-21.O
z;"
-I.0
I
-,7..O
-IIO ~ ~" 11 -LO
-2.O to q
11 \
I.O POWER-LAW CRI~P PLANE STRAIN, n . 5 z.
DIFFUSION PLANE STRAIN
£**
I to -5.3
-2'.o -~o -I.O
-2(:
FIG. 18. Fracture surfaces for diffusionand power-lawcreep, incorporating the stress redistribution due to grain-boundarysliding.
dfh/dt,
at a rate then the voids in a specimen subjected to a stress state a ~1, aT, a~ will grow at the same rate, if ~I-Y-2 + Y~J:'I = 1 where E~, y.~ and Y.~ are the applied principal stresses normalized by ai. This result, plotted in stress space, describes a surface of constant growth-rate. It is shown in Fig. 18 for plane stress and plane strain loading conditions. The element ABCA'B'C' of Fig. 17 is constrained by the surrounding material to deform as it does. This constraint induces a local triaxial stress field within the grains. In the vicinity of the grain boundary an estimate of its magnitude can be found by equating the local and distant (applied) deviatoric components of stress (Appendix D). The result is
[Pl - ( 4La=~ + a ~ + a6a~ ')~
(3.35)
Void growth by power-law creep in a multiaxial stress field was discussed in Section 3.3. Growth is determined by the equivalent stress, a e, biased by the hydrostatic pressure, p. When grain-boundaries slide, the local value of is different from the applied value. By using eq. (3.35) to calculate the local value and substituting it into eq. (3.•8) for fl, the void growth-rate for this mechanism can be calculated. The result can again be plotted as a
p/ae ofp/ae
ON CREEP FRACTURE BY VOID GROWTH
213
surface of constant growth-rate in stress space. Surfaces for plane stress and plane strain loading conditions are shown in Fig. 18.
4. THE T I ~
AND STRAIN TO FRACTURE
We now integrate the differential equations of Section 3 to calculate times and strains to fracture. For simplicity we assume that the voids all nucleate at time t. (and strain c,); but if some other nucleation rate were known to apply, it could be combined with the growth equations (Raj and Ashby, 1975) and integrated to give times and strains to fracture for this case also. To evaluate the integrals we require a fracture criterion: a numerical value f¢ for fh at the instant of failure. Clearly, the sample has failed when J~ -- 1. But in reality it will fail sooner than this because, asfh approaches one, the true stress on the remaining ligaments increases rapidly and they will fail by ductile tearing, or by cleavage, or by some other fast-fracture mechanism. We have adopted the fracture criterion fc = 0.25, except (as explained in Section 5) when constrained growth is considered.
4.1. Boundary Diffusion Alone To find the time to fracture we integrate eq. (3.4) between the limits Jh=fi
at
t=t,
fh = f ~
at
t=
t:.
(4.1)
Here f~ is the initial area fraction of the holes, and t, the nucleation-time; fc is the value of fh at which the holes coalesce. The result is
ty = t, + ~
f3/2 In
+
-
•
In
+
--
(4.2)
when f~ is small ( < 10- z) as it almost always is, and f~ = 0.25, this simplifies to
(4.3)
tf ~ tn + ~
The shape of the creep curve, and the strain to failure, E.r, can be obtained by integrating both eqs (3.4) and (3.5) as a coupled set. But it is obvious that if diffusion acts alone, the strain is that due to the growth of the voids. If a hole of volume 4~ r h3 is inserted into a grain boundary of area (rtl2 - nr~), the relative displacement of the grains is 4 3
"srh
rtl2 and the strain at failure is u/d, giving 4-3/2 I
Ef ---- 3J c
(l"
(4.4)
This is the tertiary strain when voids remain roughly spherical. It depends on the area
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fraction of voids at fracture, f~. Taking this to be 0.25 gives 1
Es ~ 0.2 ~.
To this must be added the nucleation strain, ~., and a contribution from diffusional creep, if appropriate. When boundary-diffusion provides the only growth mechanism, strains to fracture are usually small. Then the fracture times at a constant load will be almost equal to those at constant stress. 4.2. Surface Diffusion Alone The time to fracture is given by integrating eq. (3.13) between the limits of eq. (4.1). At small values ofJ~, eq. (3.13) is well approximated by ,
_
~-o d t
\aol
Integrating at constant stress gives
ts
=
t,
(4.5)
7o o
+
If f~ is small and f, is set equal to 0.25 this simplifies to
1 - 2~(ao] 3 t:
=
t.
+
~o~
\~,i
(4.6)
"
Because the void is flat and crack-like, it contributes almost no strain as it grows, Since ~f is small, we expect fracture times at constant load to be almost equal to those at constant stress. ~o is given by eq. (3.12).
4.3. Power-Law Creep Alone If eq. (3.16) is integrated between the limits given by eq. (4.1) at constant stress, we obtain (Cocks and Ashby, 1980)
,
tf = t. + fl(n + 1)do In
[1-(,-
(1 --f~)~+ ' J \ e , / I "
(4.7)
When f~ is small and f~ is large (of the order 0.25) this simplifies to tf = t, + [3(n
+,
Voo .
1)doIn (n + 1 ) A ] \ e J "
(4.8)
Fracture times at constant load are obtained by noting that (in simple tension) if the initial stress is ~ri, then after time t it has increased to (7 I
a = (1 -
n¢ j-'-,,t "1/"
(4.9)
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215
giving (in this instance) t~, = [(n + l)J]]("/"+nt. + 1( - [(n t +rneo l)J~]("/"+') ~ "
(4.10)
which properly reduces to Hoff's result (Hoff, 1953) 1
t~ -- n~,,
(4.11)
when ~ goes to zero. The shape of the creep curve, and the strain to failure cf are given by integrating eqs (3.16) and (3,17) as a coupled set. But E~, can be obtained by inspection, It is the strain due to creep in the specimen as a whole (ts~,,) plus that due specifically to the voids [eq. (4.4)] ef - Q'Co where
+ 0.2
(4.12)
tf is given by eq. (4.8) and includes the nucleation time.
4.4. Coupled Boundary Diffusion and Power-Law Creep The time to fracture is obtained by integrating eq. (A4) (Appendix A) using the boundary conditions given by eq. (4.1). The results for both constant stress and load are given by Cocks and Ashby (1981), who compare them with the finite element computations of Needleman and Rice (1980): for the range of computed results available, agreement is good. The time to fracture obtained in this way depends on the quantity ~o and on the stress. In simple tension, for instance, Cocks and Ashby (1981) show that, at one limit (low 4)o, large ~r), it reduces to the value calculated for diffusion alone [eq. (4.3)]; at the other it reduces to the result for power-law creep alone [eq. (4.8)]. The entire behaviour is well approximated if each mechanism is integrated over the range for which it is dominant, and the two times are added to give Q.. Diffusion is dominant from f~ to f~ (Fig 8) and power-law creep fromf~ tof~, wheref~ is given by eq. (3.20). Then, from eqs (4.2) and (4.8) with f~ inserted in the obvious way, we obtain
tf = t. + ~ ~o~---~ In
+ 3J\~,]
+ B(. + I)~0 In (n + -l)f{ \ o r , ] "
(4.13)
This result has the proper limits at low and high stress and 4)o. Its value for simple tension, given by setting fl = 0.6 and (re = th is plotted in Fig. 19 (broken line) which shows that it closely approximates the more detailed calculations. The shape of the creep curve and the strain to fracture are obtained by integrating eqs (A4) and (AS) as a coupled set. But cs can be obtained, as before, by inspection. It is given by eq. (4.12) with tI given by eq. (4.13).
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NE~OLEMAN
RICE
tf
• t : L : ~ .~ ..~. -
ANALYTICAL RESULT/
J
~.7..""--,
0
,o
n-5 ,
tOM.rED
ft = 10"3
RESULT I~.SULT
"Jd -N% \
,o"
,,, ~
~o-3
I
io-2
I
io-~
I0-:
I0 0
I I01
10 2
FIG. 19. The creep life, plotted against the parameter ~, comparing the approximate analysis with the numerical computations of Needleman and Rice (1980).
4.5. Coupled Surface Diffusion and Power-Law Creep The problem is analysed in Appendix B. But as in the last section, the time to fracture is well aiaproximated by adding the time taken to grow from fi to f~ [eq. (3.23)] by surface diffusion to the time taken to grow from f~ to f~ by power-law creep. These two times are given by substitutingf~ in the obvious way into eqs (4.5) and (4.8). The result (assumingf~ is small andf~ is of order 1) is
tf = tn +
2[(f7),/2 _ f~,2] (O.o~3 -3tIn F _ i I .l( oy C:O----~O \¢~1.] [3(n + 1)~o L(n + 1)fTd\aJ"
(4.14)
The result for simple tension, given by setting fl = 0.6 and ae = trl, is shown in Fig. 20. Since crack-like voids contribute almost nothing to the strain, the failure strain is close to E'f = ~ss tf. 4.6. Coupled Surface and Boundary Diffusion Surface and boundary diffusion are not alternative mechanisms (in the way that diffusion and power-law creep are); both must operate if the void is to grow by diffusion. But regimes exist in which one or the other totally controls void growth (Sections 3.1 and 3.2). Between these limits, the behaviour must be treated as a coupled diffusion problem. The results are given as eqs (C7) and (C8) of Appendix C. These equations are readily integrated numerically to give the time and strain to fracture
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f t ,. I o -3
n=S
0.4 n=4
i0-1
~'II.~
"NI
lot,.,
I
to 2
I
I°3
"~ t
I
i°~
iilfo t "
-0.4
-0.6
-0.8
FIG. 20. The time-to-failure as a function of stress for various creep exponents.
by this coupled mechanism, for given values of Ao, f~ etc. Strains to fracture are calculated in Sections 4.1 and 4.2. 5. N O N - U N I F O R M DISTRIBUTIONS OF VOIDS: CONSTRAINED CAVITY G R O W T H
It is sometimes observed that voids nucleate on some grain facets but not on others. In previous sections it was assumed that voids were uniformly distributed on all boundaries which lay normal to a given direction (Fig. 21a). But it may be that some boundaries have many voids on them while others have fewer, or none (Fig. 21b). Then a pair of grains which have voids on a shared boundary may be surrounded by a shell of grains which are unvoided. This introduces a new aspect to the problem. Dyson (1976) calls it "constrained cavity growth" because, if a void is to grow, the surrounding shell of grains must deform also, and this requirement may under certain circumstances restrict void growth. The problem is best tackled by introducing a new variable; the spacing 2L of the clusters of cavities (Fig. 21b); if the grain size is d the quantity ( 2 L / d - 1) measures the thickness of the constraining shell. A number of regimes of behaviour are now possible; but to an adequate approximation, the rate of strain and of void growth in each can be obta!ne d b y th e use_of equations already given. We shall illustrate-this by considering clustered voids which grow by boundary diffusion, while the surrounding cage of grains deforms either by diffusional creep or by power-law creep (an analysis including surface diffusion follows the same method). The quantitiesfh and 21 are redefined so that they now describe the area fraction and spacing of voids on a cavitated boundary. We consider growth in uniaxial tension only, although the analysis can readily be extended (by the methods already given) to multiaxial stress states.
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(al
<
(b) FIo. 21. (a) Voids distributed uniformly on one set of boundaries. (b) Voids on some boundaries only.
5.1. Both Void Growth and Grain Deformation by Diffusion Alone Consider first void growth and grain deformation by diffusion alone. In the damaged region the strain-rate is due mainly to the plating out of material from the growing voids, so that grains A and B move apart at a rate which is given by eq. (3.5). Allowing for grain-boundary sliding (which increases the local stress by a factor 4/3) we have
1 d ~ _ _ 8 ~bo a t l ~o dt 3 In 1/fk ao d
(5.1)
where a¢ is the mean stress in this cavitated region. The surrounding shell of grains deform by diffusional flow I-eq. (3.6)] at a rate 13 tr. 1 de = 18~bo d3 ~o dt ao
(5.2)
where a. is the mean stress in the unvoided grains. To satisfy equilibrium (2L - d)a. + dab = 2La ~.
(5.3)
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219
Equating the two strain-rates, and using eq. (5.3) gives
(2L/d)a ® 4d(2L - d)" 1+ 27• 2 In 1/.~
(5.4)
Substituting this into eqs (3.4) and (5.1) gives
do dt
4 dpo (a*~r (2L/d) 3 fl/2 In(I/A) k~oo,] [1 + 4dX(2L/d - 1) 2712 In 1/fh
]
1dE 8 ~bo ( l ~ ( a ~ ' ~ [ (2L/d) "~ -= 4d2(2L/d - 1)|" C:o dt 3 ln(1/fh) \ }k, ~o 1 + 27--~n-l-~fh j
(5.5)
(5.6)
When all boundaries are cavitated (2L = d), this pair of equations reduces identically to those given earlier for diffusional growth [eqs (3.4) and (3.5)], as they should. But when L >> d, the strain-rate [eq. (5.5)] becomes that due to Coble creep [eq. (3.6)] and the void growth-rate becomes controlled by the rate of this creep in the surrounding shell of grains. This regime of behaviour exists only when the rate of Coble creep in the undamaged region [eq. (5.2)] exceeds the rate of power-law creep there. For this range of stress
_
-
o.~:,
so that the boundary of the regime is defined by
(5.7) When all boundaries are cavitated (2L = d) no constraint occurs. But when 2L >> d, constraint is possible. We assume (for the purpose of further calculation) that fracture occurs when voids on cavitated boundaries link. Then the time to fracture is calculated (as before) by integrating eq. (5.5) between the limits given by eq. (4.1). Iff~ is small, and fc = 0.25t and assuming a constant distribution of voids (so that ! and L do not change with time) we find the remarkable result
tf = t, + 2-Ll_\~o4oJa--iJ +
1 - 2Ljk2712dpo~ ° ~-; .
(5.8)
?In evaluatingthe integral leading to the second term in eqn (5.8), we take f, = 1. This is because separation can take place in the voided region only if the strain in the surrounding,unvoided,region is sufficientto give the full tertiary strain. When calculatingtimes to failure,the cut-offf~is not critical, and our choice of 0.25 is as good as any; but when calculating strains to failure, the choice of cut-offfc is important and must match the physical situation as closelyas possible.
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The time to fracture is the sum of two terms. The first is the time boundaries to fracture by unconstrained diffusion under a stress a ~ fraction of boundaries which are cavitated. The second is the time to strain" Ieq. (4.4)] in the uncavitated region under a stress er~, weighted uncavitated boundaries.
for the cavitated weighted by t h e give the "tertiary by the fraction of
5.2. Power-Law Creep Alone At the other extreme, the voids grow by power-law creep alone. We use the same method as that of Section 5.1. In the cavitated region (grains A and B, Fig. 21b) the strain-rate is given by eq. (3.17) with the stress set equal to ac. In the surrounding grains it is given by eq. (2.1) with the stress set equal to au. Equating the two, and taking J~ << 1, gives ~.-~ac
1 +--j- BA .
Substituting this into the equilibrium equation [eq. (5.3)] gives 2(L/d)a ~
ac =
(5.9)
1 + (-~ - 1)(1 + ~21 flf~/2)" In the limit that all boundaries are cavitated (2L = d), ac = a~; for all other values of 2L > d, the equation states that a, is very close to a ~. So in this case, the rate of void growth and of strain are given by
E {
1 dA = # 1 ~o dt (1 -J~)" ldE oZ=
1+
(1 - A )
2r, F 1 -1 dl_I1-A)"
1(;o) ]}(°7
(5.1o)
"
(5.11)
Note that the growth-rate of the voids is independent of the degree of clustering, and depends only on the local area fraction, f,, on the boundary. The strain-rate, of course, decreases a little as the cavitated boundaries become fewer. The condition for this pure power-law behaviour has already been considered in Section 3.5. If the void growth is to be predominantly by such creep, then eq. (3.19) must be satisfied, When ~ << 1, this simplifies to
- - >/ ao
k(n + 1)f~ :z In(i/A)_]
(5.12)
The time to fracture, in this case, is given by the result calculated earlier [eq. (4.8)]. Remember that fh has been redefined in this section as the area fraction on the cavitated boandaries. The result shows that the growth-rate and time to fracture, when power-law creep only is involved, depend on the local volume fraction and are not changed if neighbouring boundaries are uncavitated.
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GROWTH
5.3. Coupled Growth The two sets of equations developed above describe two extremes of behaviour: diffusion alone, and power-law creep alone. Between these extremes, voids grow by various couplings of two processes. Of these, the most obvious and interesting is that brought about when voids between grains A and B grow by diffusion alone, but the surrounding cage of grains deforms by power-law creep. Then, proceeding as before, we first impose compatibility by equating the diffusive strain-rate in the cavitated region [eq. (5.1)] to that in the power-law creeping region [eq. (2.1) with tre --- tru]; and then use the equilibrium equation [eq. (5.3)] to calculate the stress tr, in the uncavitated region. This results in the implicit equation for tr,
tro/
8°0
,)o.]
3 In UA J
The solution is wall approximated by
= { tro
1_
1
\1 - d/2L ] ao
ac
(2L/d - 1) ao
where ~
(1 - d/2L)" \ Cro}
ao-[(1
- d/2L)"l
1dk](a~]"-'ao} + 3d In (1/fh)81'°
(5.13)
The void growth-rate and strain-rate in this coupled region are then given by 1 dfh 4*0 {a~'~ do -~ -- 3f~/2 ln(1/fh) \~oo]
(5.I4)
d~ d-t = 3 ln(1/J~) d
(5.15)
with trc given by eq. (5.13). Note, first, that if every boundary is cavitated (2L = d) then cc = or" and the equations become identical with those derived earlier [eqs (3.4) and (3.5)]; there is no constraint. Second, at low stresses or when *o is large, the second term on the bottom of eq. (5.13) is dominant and eq. (5.14) reduces to do d~- = 2f~/2(I -
d/2L)"
"
(5.15)
The damage-rate is then completely controlled by the deformation-rate of the surrounding cage of uncavitated grains.But when tr__.~~_= ( 8•*0 ')(1/,-,) ao 3d In 1/f•]
(5.16)
ac = a ~ and growth is unconstrained. If a ~ is greater than that given by eq. (5.16) the J.p.M.S.27 3-4--c
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rate of deformation of the damaged region is primarily due to the power-law creep of grains A and B, and void growth is unconstrained. Equation (5.16) therefore represents the transition to constrained growth. It can be expressed in the form 81 ~bo a ~ ~--~ ~< 3--dIn 1/fh a-~"
E,s
(5.17)
This is exactly the expression given by Dyson (1979, p. 36, with his 22 = nl2) to define the transition to constrained growth. The time to fracture is once again given by integrating eq. (5.14) between the limits given by eq. (4.1). Again, taking f~ = 0.25 we find t; = t, + [ 1 -
(1
- ~ ) l + ( 1 - d/2L), [3~-~o 41 \( aao'~"~. ~JJ d/2L)"][ ~ " ( aao
-
(5.18)
As with pure diffusional growth, the time to fracture is the sum of two terms. The first is the time to fracture by diffusion alone, under the stress a ~, weighted (in a non-linear way) by the fraction of cavitated grains. The second is the time to accumulate the "tertiary strain" [eq. (4.4] in the uncavitated grains by power-law creep under the stress a ~, weighted (non-linearly) by their fraction.t 5.4. Creep Rupture Plots These results are plotted in Fig. 22, for 2L/d = 2 (every second boundary is cavitated). The curve has an unusual shape: over a wide range of stress it is concave upwards.
%
qbo = 0.I fi
IO PURE
10-3
=
I00
21.=
POWER-LAW
I(32
=
d n
I0"
I\
I
I
I0
I02
i
I
I0 ~
2
=
5
L
I04
IOs
a L
Lt,
I06 ~ •
~..~-ON CON STRAI N ED ~BOUNDARY DIFFUSION
CONTROLLED FAILURE I0' -
•
~
/DIFFUSIVE GROWTH ~
,/
CONSTRAINED BY REEP
\
-
DIFFUSION
\
FiG. 22. The time-to-failure as a function of stress when constrained growth is important. tVery recently, Rice, J. R. (to appear in Acta Met., 1981) has also examined this problem, and reached a similar conclusion,
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5.5. Other Distributions of Voids The discussion so far relates to a particularly simple, constant, non-uniform distribution. More complicated constant distributions can be dealt with in the manner outlined by Cocks (1980). But it appears to us to be unlikely that a void distribution remains constant with time. The nucleation of voids, during the test, on the previously void-free boundaries, would appear to be of importance. When the number of voids increases with time, it is still possible to calculate the damage rates and times to fracture. An example of the method is given by Raj and Ashby (1975), but does not, in general, lead to analytical solutions for the times to fracture. 6. COMPARISONWITH THE CONTINUUMTHEORYOF CREEP DAMAGE MECHANICS In this Section we compare the rate-equations for void growth described in Section 3 with the corresponding equations of continuum damage mechanics. We find that the two are very closely related, but that there are, nonetheless, discrepancies, so that their predictions differ. 6.1. The Continuum Theory of Kachanov (1958) and
Robotnov (1969) Creep damage-mechanics (Kachanov, 1958; Robotnov, 1969; Leckie and Hayhurst, 1977) has been developed as a continuum approach to the analysis of creep fracture. The damage (which can include all the processes listed in the Introduction: loss of section by cavity growth, degradation of the microstructure by strain, etc.) is measured by the scalar parameter co which varies from zero (no damage) to one (failure). Both the damage and the strain are assumed to grow with time in a way which depends on stress and on temperature, and on the current extent of the damage, so that in simple tension dco rb = - - = g(~l, T, o)
dt
d~
(6.1)
= d t = f ( a l , T, co). The creep life, the strain to failure and the shape of the creep curve are obtained by integrating the equations between appropriate limits. This procedure exactly parallels that developed in Sections 3 and 4; thus far, the mechanistic model and the continuum model are identical. To proceed further, functions are assumed for f a n d g. The functions chosen by Kachanov and others who have extended his work, are generally
•
¢-O= COO
~1
\no/\l
1
- co/
(6.2) (6.3)
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Here d~o is a temperature-dependent rate constant (like ~o) and v is a constant exponent (like n); the quantities tro,~o and n have already been defined [eq: (2.1)]. When the stress state is triaxial, the continuum eqs (6.2) and (6.3) are modified. Leckie and Hayhurst (1977), following Kachanov (1958), do this by writing
•
~,., = ,Ot~oo) t ~
\" )
1
•
cb = 09ot~---o-o)t ~ 1
(6.4)
"r
)
(6.5)
where tre and 7~ are defined by eqs (2.2) and (2.3) and tr + is identified either with the equivalent stress, ae, or the maximum principal stress, 61, depending on the material. To use the equations, of course, it is necessary to know whether the component is made of an "equivalent stress" material or a "maximum principal stress" material. 6.2. Comparison of the Continuum and the Mechanistic Models: Power-Law Creep When a random array of voids grow by power-law creep alone, the damage-rate and strain-rate are given by the mechanistic model as eqs (3.27) and (3.28). If we identify the area fraction of voids, fh, with the damage, 09 and restrict ourselves to stress states near simple tension (for which fl -- 1), then the strain-rate equations of the continuum and the mechanistic treatment are identical. At the same level of approximation, the mechanistic model (eq. 3.27) gives the damagerate _dt_ =
_Tfh)
This equation resembles the continuum damage-rate equation [eq. (6.2)] with r = n and &0 = 4o, and becomes identical to it when fh is large (since the - ( 1 --fh) term can then be neglected). But there is an important difference: when fh is small, the term [1/(1 - 09)]" is replaced by [1/(1 -.~)" - 1]. Physically the continuum model states that the damage-rate is finite even when there is no damage (09 = 0); the mechanistic model says the damage-rate is zero under these conditions: non-existent holes do not grow. Because of this, results calculated from the continuum and the mechanistic equations differ. If the time to fracture, for example, is derived by integrating the continuum eq. (6.2) with v = n and ~'Oo= ~o, the result is
(1- J~) '--'['°°~l".
(6.7)
t: = t, + (n + 1)~o \ t r y / By contrast, the mechanistic model leads to the result [cf. eq. (4.8)]
I
tf
---- t n d-
[
I
l(ao'~" '
fl(n + 1)~o In (n + 1)f~lktrt/
(6.8)
These two times are almost identical when J~ is large; but when f~ is small, the mechanistic model predicts a much longer life because the holes (when small) are growing more slowly. There are corresponding differences in the strain to fracture.
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225
For multiaxial loading, the mechanistic and continuum models differ in additional ways. First, the mechanistic model shows that the strain-rate as well as the damage-rate are accelerated by hydrostatic tension, and which enters through the term fl [eq. (3.18)]; the continuum model does not have this feature. When voids grow by power-law creep on boundaries [eqs (3.16) and (3.17)] the comparison is made more complicated because the grain size now enters the equations. But the same close resemblance and the same fundamental differences are still found. 6.3. Comparison of the Continuum with the Mechanistic Model: Diffusion Controlled Growth The mechanistic equations for hole growth by boundary diffusion alone were given by eqs (3.4) and (3.5). When fh approaches 1, we may write f~/2 ln(1/fh)~ (1 --fh), and the damage-rate equation becomes dfh ~--
. trl 1 ~oEo(~00) (1--~fh).
(6.9)
If, as before, we identify fh with damage, to, this equation is identical with the first of the continuum equations (eq. 6.2), with t.bo = ~boE0 and v = 1. But when fh is much less than 1, the two equations differ significantly; in particular, eq. (3.4) leads to a damage rate which decreases as damage grows (see Fig. 8), whereas the continuum equation predicts a steadily increasing damage rate. The strain-rate equation, too, becomes identical with the second of the continuum equations when fh is close to 1. In this limit In 1/fh ~ (1 -J~), and if holes are randomly distributed, l = d, giving ~-~ - 2 ~ o ~ o
(1 - fh)"
(6.10)
But, as before, the two equations differ significantly when fh is small--and this is the regime in which diffusional growth is important. Exactly the same conclusions hold when voids grow in a way which is limited by surface diffusion [eqs (3.13) and (3.14)]. When fh "- 1, the mechanistic and the continuum equations are identical, but when fh is small, the two sets of equations diverge. In the regime in which we expect surface diffusion to be important, this divergence leads to large differences between the two approaches. 6.4. Results of the Comparisons In summary, all the mechanistic equations have a close resemblance to those of continuum damage mechanics, and become identical to them when fh (or co) approaches the value 1. But whenfh is small (as it is for most of the life) there are important differences. The comparison highlights what we believe to be certain fundamental weaknesses of the continuum equations: first, the prediction that the damage-rate is finite even when there is no damage; second, the prediction that the damage-rate always accelerates with damage; and finally the incorporation of triaxiality in a way which is not consistent with (approximate) physical models. It must be emphasized that the mechanistic equations, too, are approximate. But they are
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PROGRESS IN MATERIALS SCIENCE
based on models and have limits, which are consistent with the physical processes which are thought to lead to fracture. They resemble the continuum equations, and can be thought of as a mechanistically-based extension of them to describe specific damage mechanisms. For these mechanisms we believe they are a better description than the simpler (though very flexible) equations of the continuum theory.
7. EXTRAPOLATIONOF CREEP DATA The engineer, when designing high-temperature structures, is often faced with the problem of extrapolating creep data. In particular, he may be given uniaxial creep data at constant load and for relatively short times, and find it necessary to extrapolate the data in three ways: (i) to longer times, (ii) to histories of non-steady load and temperature and (iii) to stress states other than simple tension. The models described in Section 3 of this paper give ways of doing this. Some of the results are limited (at present) to materials in which damage takes the form of voids and for which the void-growth time (rather than the time for voids to nucleate) controls the life of a component. But the procedure for extrapolating creep data to longer times is more general than this and appears to be valid for many sorts of creep damage.
7.1. Present Methods for Extrapolatin9 in Time Current extrapolation procedures for uniaxial creep data are empirical. In using the Graphical Method, data for log t.r are plotted against trl or log a x at a constant temperature and the resulting curve extended by eye. The method is formalized in the Reoression Analysis Method (e.g. Booker, 1976, 1977): the data are fitted to one of many possible empirical equations, of which a typical example is A2
log t s = At + T
A3
+ T log at
(7.1)
and the constants A1, A2, A3 are determined by a multivariable linear regression analysis (Draper and Smith, 1966; Daniel and Wood, 1971). The extrapolation by eye is now replaced by the use of the fitted equation, but no physical modelling is involved. The Time-Temperature Parameter Method (Larson and Miller, 1952; Conway, 1969; Penny and Marriott, 1971) has the merit that physical understanding of the fracture process can more readily be introduced. It breaks down the fitting procedure into two steps. A parametric combination of t.c and T is chosen and assumed to be a function of stress only
P(ts, T) = 9(al).
(7.2)
Some 20 or more such parameters have been suggested, among which the most widely used are the Larson-Miller Parameter (Larson and Miller, 1952)
P(tI, T) = T(C + log t r)
(7.3)
and the Dorn, or Fisher, parameter (Orr et al. 1954; Fisher, 1959)
P(ts, T) = log ts + C/T.
(7.4)
(This last equation is based on kinetic theory, but without any clear model for the fracture mechanism.) The data are plotted against the parameter P to give a master-plot, selecting
ON
CREEP
FRACTURE
BY V O I D
GROWTH
227
the constant C to minimize the spread. Then a polynomial (usually a quadratic in P) is fitted to the master plot, and this is extended to longer times and lower stresses, to give a stress corresponding to the design life. Finally a safety factor is applied to this stress. The parametric approach is made more general in the Minimum Commitment Method of Manson (1964 and 1968) although contact with physical models is thereby lost. The data are fitted to a basic equation of the form log ty + A P(T)log ts + P(T) = G(o.~)
(7.5)
where P(T) is to be determined from the data, and G(o.t), usually a polynomial in o.1, then describes the shape of the master curve. The equation obviously includes the two parameters described above as special cases. Numerical procedures have been devised to obtain P(T) and G(o.1), which then allow the equation to be used in design. All these methods must be regarded as completely empirical; where physical understanding is incorporated, it is at too vague a level to give the method itself any real physical basis. All assume that the physical mechanism which controls creep or fracture in the extrapolated region is fully represented in the available data, and that the presence in this data of information about other mechanisms exerts no influence on the extrapolation procedure. Further, none have the capacity to predict the creep life or creep strain if the load or temperature change during the life. To do this requires a differential method. The Kachanov-Rabotnov equations and the equations developed in this paper, are examples of differential equations for damage and strain. All have the general form (for simple tension) of eq. (6.1). If such equations are known for each damage-mechanism and for each mechanism of creep, then they can be integrated to give the strain at any time, and the time to fracture, for any history of loading. The great strength o f this method is that it allows the progress of damage-accumulation to" be followed throughout the life. Its weakness is a practical one: the data needed to determine the functions f and g of eq. (6.1) are particularly difficult to obtain. But if the models are good enough to give the forms of f and g, this difficulty is removed and the method leads to powerful extrapolation procedures. The remainder of this paper illustrates this approach. 7.2. Model-Based Procedure for Extrapolating in Time In Sections 4 we showed that the time to fracture in tension by any single mechanism or by the coupling of any pair of mechanisms of void growth gave a normalized fracture time to tI which depends on normalized stress (th/o.0), on a material property (q~o, ~ko or Ao), on f/ and (weakly) on n. Boundary diffusion coupled to power-law creep, for example, gives fracture times described by ~otj- =
o , - - , n,
.
(7.6)
O'o
For any given material, n a n d ~ are constant. Then to ts depends only on ~bo and o.1/tro. Fracture by this coupling of mgchanisms, or of either acting alone, at all stresses and temperatures, is described by a family of curves, each characterized by a value of ~bo [defined by eq. (3.3)]. At this point, it is necessary to introduce an explicit temperature-dependence for power-
228
PROGRESS IN MATERIALS SCIENCE
law creep. We shall use an expression? which is standard in the materials science literature (e.g. Sherby and Burke, 1967)
where Dc is the creep diffusivity: Dc = Do~exp(-Q
i° - AD
(7.8)
and 2flDs~ ( ~ f f -I
dP° = AD
"
(7.9)
We may choose go and ~o in any way which is consistent with the constitutive law (eq. 2.1). We may therefore redefine a0 to give a~, and Eo to give ~,, such that ~bo is independent of temperature. Then if creep rupture data at different temperatures is normalized by plotting it on axes of log al/a* and log ~* t:, all points must fall on a single master curve. Eq. (7.9) can be rewritten
where Qc and QB are the activation energies for creep and grain-boundary diffusion respectively and B is a dimensionless material constant. We then define a , in such a way that t~0 is constant and equal to unity
I- Qc - Q~ -I :~F, B.,.-,, expLtn- r)~r~J
(7.11)
=
and, from eq. (7.8)
i~ - COo<.b kT
exp -
[ ~-
O.c]
(7.12)
where C is a dimensionless material constant and Doc is the pre-exponential of the diffusion coefficient. A single master curve should, therefore, be obtained when data is plotted on axes of P1 = loglo tri/a~j and P 2 --- loglo t/i*. From eqs (7.11) and (7.12)
,, =
-
.(1
. -
logio-
/~o
"" loglo ai
-
H
(7.13)
and P2 = logio tf
--
J
.
1
.
logto. To
. logio .
~o
"" loglo tf
J
To
.
(7.14)
i'The detailed form of this expression is unimportant. Exactly the same result is obtained if. for example, we choose instead i = A' exp - QJR T (tr/a,)" where A' is a constant and a, a referencestress.
ON CREEP FRACTURE BY VOID GROWTH
229
Here To is a reference temperature (that of the centre of the range of data, for instance) and #0 is the value of the modulus at that temperature. The quantities H and J are explicitly given by the model H =
Qc - Qs
2.3(n - 1)R j
_
"QB -
Qc
2 . 3 ( n - 1)R
(7.15)
though in practice they should be chosen to minimize the spread of data. The procedure for doing this is illustrated by Fig. 23; it is adapted from the method of Manson and Ensign (1978). The creep rupture data, plotted on axes of logto aa and logao t I are traced onto a transparent sheet containing no grid work. Smooth lines are drawn by eye through the data at each temperature. The sheet is placed over a fine grid and rotated
SUBSHEET (3ONTAINING GRIDWORK
_I
I / /
I,~. /i~
/
~
/ I
\\ "%
d
i
.
• m
m
m
. m
. m
~' m
m
m
m
m
m
m
m
ami==-__Immmmmmm mmmmmmmmmi==im ISOTHERMALS
BUT NO
GRIDWORK
"To
o TI
To>T, >T~>Ts
°T3 FIG. 23. Illustration of the method used to determinethe scaling numbers H and J.
230
PROGRESS
IN
MATERIALS
SCIENCE
clockwise about the point A until the horizontal distances L o between two isothermals T = T~and T = Tj is as nearly constant as possible along their length, and the ratios of the lengths is consistent with 1
1
1
1
When these conditions are met, the angle of rotation 0, and the lengths Lo, are noted, and used to form the quantity
L =
Lij
The two scaling numbers, H and J, are then given by H = L sin0 J = Lcos0
(7.17)
The data are replotted on axes of PI and P2 (eqs 7.13 and 7.14) to give a master curve for the material. Finally, a curve of constant ~bo, is fitted through, and extended beyond, the data, giving a modelbased extrapolation. As an example of the use of the method, we apply it to Williams' (1968) data for the creep fracture of 304 stainless steel at a number of stresses and temperatures. The raw data is shown in Fig. 24a. The same data, replotted using the parameters P1 and P2 with H = 419 K and J = 15,000 K, (chosen by the fitting procedure just described) are shown in Fig. 24b. The solid line is that predicted from the theory for n = 9. Data for the creep-rupture of a steel, S-590 (Grant and Bucklin, 1950) are shown in Fig. 25a. The same data, replotted according to the method described above with H = 860 K and J = 14,000 K, are shown in Fig. 25b, together with the computed curve for failure resulting from precipitate coarsening (Appendix E). As a third example, data for O F H C copper (Goldhoff, 1974) are shown in Fig. 26a, The same data are replotted with H = 811 K and J = 3900 K in Fig. 26b. In all three examples the procedure works well. Further, the quantities H and J obtained in this way can be related back to fundamental quantities (n, Qc, Qn) through eq. (7.15). The merit of this procedure is its basis in a physical model: it is not an empirical procedure (like almost all other extrapolating methods), but one which fully incorporates a mechanistic understanding. It is readily seen that the same procedure is valid for any one, or any coupled pair of the three growth mechanisms discussed in this paper; and Appendix E shows that it is valid also for creep fracture caused by certain sorts of microstructural instability. It is also valid for simple instances of nucleation dominated creep fracture. For instance, if void nucleation occurs catastrophically after a strain En, where En is a material property, the failure time can still be written in terms of ~bo, a/tro and material properties and the extrapolation procedure holds. 7.3. Extrapolation to Non-Steady Histories of Stress
and Temperature When the stress or the temperature vary during a creep test, it is normal engineering practice to calculate the life by using the linear cumulative damage rule (or Robinson's rule;
ON
CREEP
FRACTURE
&A
0
A
BY
O
VOID
D o
_
O
F
o
A~
a A
i~
231
GROWTH
~O. o 5380(:: O 593~C; A 649oC 0 7040(: v 816oc 87toc o 927°(:; ":-
v
0
eC K~4
6
TIME-TO-FAILURE (hrs) 2.4
!~ q:~
4
STAINLESS
2.~
! -I~-
t-!
o To - $93 °C -866K g
Jr
t.~
O" ,, Ilvl~l tf ,, (hrs)
I
I 3
I
I
1 6
I 7
FIG. 24. The extrapolation procedure, applied to data for 304 Stainless Steel of Williams (1968); raw data above, normalised data below.
Robinson, 1952). If the time spent at stress tr~ is t~, and the failure time at that stress is tyi, the rule states that failure occurs in a time ty where
~
ti
C
tfi
or
f[~
dt
tf
C
(7.18)
with C = 1. It is easily shown (Appendix F) that each of the equations describing void growth by a single mechanism [eqs (3.4), (3.13) and (3.16)] satisfies the life-fraction rule with C = 1. But if the voids grow by a coupled mechanism, or if they grow by a single mechanism at a low stress and by another at a higher stress, then changing the stress changes the mechanism,
232
PROGRESS
IN
MATERIALS
SCIENCE
and C is no longer equal to unity. Consider a sample in which voids grow by diffusion alone at a low stress and that a change of stress (or temperature) occurs such that for the remaining part of their life they grow by power-law creep alone (Fig. 27). If the voids grow by boundary diffusion until their area fraction isf~, the elapsed time t") is given by setting f~ = f~ in eq. (4.2). At this point the stress (or temperature) changes so that they now grow by power-law creep. A further time elapses as the holes grow from f~ to f~ (when failure occurs) by power-law creep. This time, t(2), is given by settingf~ = f~ in eq. (4.7). We have formed the sum t(z)/t~ I) + t(2)/t~2), and plotted the result in Fig. 27a. The sum is less than 1 unless one mechanism controls the entire life; for the conditions of the plot it can be as low as 0.3, so that Robinson's Rule is unsafe. If, instead, we cause the holes to grow first by power-law creep and then by diffusion, the result shov, n in Fig. 27b is obtained. The life-sum is greater than 1, so that Robinson's Rule is conservative.
03
- ~S-590 ALLOY o
o
°° 0
ee
a.
o
o
v
•
0
u 0
•
~
0
3Eeo 2
•
,
o
o o V
eO
U~ uO UJ rY
•
I0
0
0
v
•
o 650"C o 73TC
A 816"C
0 871eC v 927"C • I038* C
I lo-~
I
~
IO ~
TIME-TO-FAILURE
s-sN
'
IO4
(hrs)
AL-C-C-
g g, e To-e7ve - 1144K
--
O'(MPo) t~(hrs)
0 -d
-2
0
t,-J oo (+
2
4
FIG. 25. The extrapolation procedure applied to data for an alloy steel, S590, of Grant and Bucklin
(1950); raw data above, normalised data below.
ON
CREEP
FRACTURE
BY VOID
233
GROWTH
[ OFHC COPPER • ;IlI'C • 316"C o 371"C :,4g~
_ ,d
. St)T C A Mg'C eTO~C • 760"C
+ 48TC ~ 871"C o S38"C x
UJ rY I-u~
0
o
O
I0
+
V
•
X
e
0
6
•
•
+
000
•
I)
x
Aa
13Q
O
O O
+
TIME-TO-FAILURE (hrs) 3
"[OFHC GOPPER To - S,I~ C
I
- 8IlK 0'lldPa)
v
o
-2
-I
I
o
2
,o%,,- 3.900
3
(+-+o)
4
$
FIG. 26. The extrapolation procedure applied to data for OFHC Copper of Ooldhoff (1974); raw data above, normalised data below.
POWER-LAW CREEP GRow'rI~
POWF-q-I.~W
(7 lorT~
CnEEP G~OwT.
DIFFUSIONAL GROWTH
l
|
I'
OtFFUSIONAL GROWTH
1'
fs
fh
fh
fs n-9
sI
0.5~ I.O
.
I I °o-"
.
.
I Io -3
.
.
n-'9 I Io -2
.
I lo-'
I<)
0.5
fs
-
0_4
1 IO-]
I IO'2
I iO ~
FIG. 27. An illustration of deviations from the linear cumulative damage rule.
fs
234
PROGRESS IN MATERIALS SCIENCE 7.4. Extrapolation in Stress Space
When a pressure vessel or pipe supports an internal pressure, or when a component with a change of section is loaded in tension, the stress state is a multiaxial one. Current design practice assumes either that the equivalent stress, ae, controls fracture ('Won Mises materials") or that it is controlled by the maximum principal stress at ("Maximum Principal Stress materials"). Explicit equations were given earlier for the growth-rate of voids, and the time and strain to fracture, under local multiaxial stress fields. When grain boundaries slide, the local field is related to the remote (applied) field by the equations given in Section 3.10. Combining the two sets of equations gives expressions which show how the time and strain to fracture vary with the triaxiality of the remote field. This was the approach used in constructing the second set of constant growth-rate surfaces (Fig. 18). If the nucleation time, tn, is short, then these figures become identical with the isochronous fracture surfaces for the material (surface in stress space on which the fracture time has a constant value) provided one mechanism is dominant throughout life. But if the nucleation process dominates the fracture time, it is necessary to know also how tn depends on stress state. This problem is outside the scope of the present paper.
8. SUMMARY
Void growth by boundary diffusion, surface diffusion and power-law creep, acting either singly or together, is analysed. Equations for rate of growth of void damage and for the strain-rate are developed in Section 3, for multiaxial stress states. The results arc illustrated by growth-mechanism maps by plotting surfaces of constant growth-rate in stress space. The time and the strain to fracture arc obtained by integrating the rate-equations. The results of doing so are given in Section 4. When voids are non-uniformly distributed, their growth may be constrained by the need to deform adjacent grains with fewer or no voids on them. Some simple results for fracture under these conditions are given in Section 5. The results of the mechanistic approach of this paper bear a close relationship to the equations of continuum-damage mechanics. This relationship is discussed in Section 6, where it is shown that, in spite of their similarities, the results of the two methods differ. Simple modifications to the continuum method are possible which would make it consistent with the mechanistic approach. An important aspect of an analysis like that given here is the help it gives in extrapolating creep fracture data. A new procedure for extrapolating failure times to longer times, and for producing creep-rupture master curves, is given in Section 7, which also includes a discussion of the validity of Robinson's Rule and of creep fracture under multiaxial stress states.
ACKNOWLEDGEMENTS We wish to acknowledge financial support from the S.R.C. (in the form of a studentship) and from the U.K.A.E.A., Harwell. We are particularly grateful to Dr. B. F. Dyson for his critical comments and suggestions.
ON
CREEP
FRACTURE
BY V O I D
GROWTH
235
REFERENCES ARGON,A. S., CHEN, L-W. and LAU, C. W. (1980) Creep and Fatigue Fracture, (ed. Pelloux, R. M. N. and Stoloff, N.) A.I.M.E., N.Y. ASHBY, M. F. (1969) Scripta Met. 3 837. BEERE,W. and SPEIGHT, M. V. (1978) Metal Sci. 4, 172. BOOKER, M. K. (1976) ORNL Report ORNL/TM-5329; (1977) ORNL Report ORNL/TM-5831. BROWN, A. M. and Astray, M. F. (1980) Acta Met. 28, 1085. BURTON, B. (1977) Diffusional Creep in Polycrystalline Materials, Trans. Tech. Publications No. 5, p. 62. CHUANG,T-J. and RICE, J. R. (1973) Acta Met. 21, 1625. CHUANG, T-J., KAGAWA,K. I., RICE, J. R. and SILLS, L. B. (1979) Acta Met. 27, 265. COBLE, R. L. (1963) J. Appl. Phys. 34, 1679. COCKS, A. C. F. (1980) Ph.D. Thesis, Cambridge University. COCKS, A. C. F. and ASHBY,M. F, (1980) Metal Sci. 14, 395. COCKS, A. C. F. and ASHBY,M. F. (1981), CUED Report No. CUED/C/MATS/TR 27; to be published. CONWAV, J. B. (1969) Stress-Rupture Parameters: Origin, Calculation and Use, Gordon and Breach, N.Y. DANIEL, C. and WOOD, F. S. (1971) Fitting Equations to Data, John Wiley, N.Y. DRAPER, N. R. and SMITH, H. (1966) Applied Regression Analysis, Wiley, N.Y. DvsoN, B. F. (1976) Metal Sci. 10, 349. DVSON, B. F. (1978) Canad. Met. Quarterly 18, 31. EDWARD, G. H. and ASHBY, M. F. (1979) Acta Met. 27, 1505. FIELDS, R. J. and AstraY, M. F. (1976) Phil. Mag. 33, 33. FISHER, W. A. P. (1959) RAE TN Structures, 270. GRANT, N. J. and BUCKLIN,A. G. (1950) Trans. Am. Soc. Metals 42, 720. GOLDHOFF, R. M. (1974) J. Testing Evaluation 2, 387. GOODS, S. H. and BROWN, L. M. (1979) Acta Met. 27, 1. GOODS S. H. and NIX, W. D. (1978a) Acta Met. 26, 739. GOODS, S. H. and Nix, W. D. (1978b) Acta Met, 26, 753. HARRIS, J. E. (1972) Met. Sci. J. 7, 1. HELLAN, K. (1975) Int. J. Mech. Sci. 17, 369. HERRING, C. (1950) J. Appl. Phys. 21,437. HOFF, N. J. (1953) J. Appl. Mech. 20, 105. HULL, D. and RIMMER,D. E. (1959) Phil. Mag. 4, 673. KACHANOV,L. M. (1958) lzv. Akad. Nauk. SSSR No. 8, p. 26. LARSON, F. R. and MILLER, J. (1952) Trans. Am. Soc. Mech. Engrs 74, 765. LECKIE, F. A. and HAYHURST,D. R. (1977) Acta Met. 25, 1059. MANSON, S. S. (1964) Proc. Inst. Mech. Eng. 178, Part 3A. MANSON, S. S. (1968) Am. Soc. Metals Publication D8-100, p. 1. MANSON, S. S. and ENSIGN,C. R. (1978) Am. Soc. Mech. Engrs Publication MPC-7. NARARRO,F. R. N. (1948) Strength of Solids, Phys. Soc. London, p. 75. NEEDLEMAN,A. and RICE, J. R. (1980) Acta Met. 28, 1315. NEUMANNG. and NEUMANN,G. M. (1972) Surface Self-Diffusion of Metals, Diffusion Inf. Centre. NIEH, T. G. and NIX, W. D. (1980) Acta Met. 28, 557. OOQWST, F. K. G. and HULT, J. (1961) Ark. Fys. 19, 379. ORR, R. L., SHERBY,O. D. and DORN, J. E. (1954) Trans. Am. Soc. Metals 46, 113. PENNY, R. K. and MARRIOTT,D. L. (1971) Design for Creep, McGraw-Hill, pp. 192-238. RAaOTNOV YU. N. (1969) Proc. X I I I U T A M Congress, Stanford (ed. M. H6tenyi and W. G. Vincenti); Springer, Berlin, p. 342. RAJ, R. (1978) Acta Met. 26, 341. RAJ, R. and ASHBY,M. F. (1971) Met. Trans. 2, 1113. RAJ, R. and AsI-mY, M. F. (1975) Acta Met. 23, 653. ROBINSON,E. L. (1952) Trans. Am. Inst. Min. Engrs 74, 777. SHERBY,O. D. and BURKE, P. M. (1967) Prog. Mat. Sci. 13, 325. SI-mWMON,P. G. (1969) Transformations in Metals, McGraw-Hill, pp. 299-302. SPEiGtrr, M. V. and HARRIS,J. E. (1967) Metal Sci. l, 83. SWlNKELS` F. B. and ASHBY,M. F. (1981) to appear in Acta Met. WILLIAMS,W. L. (1968) Am. Soc. Metals Publication D8-100, 351.
APPENDIXA : Void Growth by Coupled Boundary Diffusion and Power-Law Creep The method of dealing with coupled diffusion and power-law creep is illustrated by Fig. A1. All current treatments (Beere and Speight, 1978; Edward and Ashby, 1979; Cocks and Ashby, 1981) assume that voids grow, by boundary diffusion, on a grain-boundary across which a mean normal traction a~ is applied. The material
236
PROGRESS
i
#
IN
#
MATERIALS
SCIENCE
i
#
t
EFFECTIVE, x DIFFUSION HOLE SIZE) / t ZONE ~'x-it
'I
..\ ~/
-
-~
-- -'~/"_ . ~
-~
t
•
-
--
t
A~OWER - LAW CREEP ZON~
/
~'.~~~:.~'~:..~:~::..'.'~
~.:::!::..~:~:.:.~.:.;..::.:....::.?.:.;....~..:...:~.:..:.::::.....yv.~:::.::::::~ ~.'-"-:':'.."!:y!:.~:..~'..!..~-..? :'!:!':..:'..':'!'....:-: '.:..'._-.~:.~".'-":':
.~
~.:~.'"':':"': ": :').:!~'~.'#~:'":';':.:"""~;':':~':" r:-:~"~'.'~.:~...'::~-!~-'~:.: ~':~.~
I
.----- r e ---.~
rd
Ji
i
¢
i
i
i i
i I
t
t
i
i
i
i'
FIG. A1. A void growing by coupled boundary diffusion and power-law creep.
leaving the void plates out onto the boundary over a ring of outer radius r~ (occupying an area fraction fj). This diffusion zone is an important feature of the model; if its radius r4 extends to the hole half-spacing I, hole growth is purely by diffusion. But, if it shrinks until r~ = rk the surrounding power-law creep zone (I > r > rj) totally controls hole growth. When rj lies between these extremes, growth is by a coupling of two mechanisms. Within the diffusion zone, the stress is partly relaxed by the material which deposits on the boundary plane. This sheds load onto the surrounding power-law creep zone, which behaves as if the slab contained a hole of effective size r e (with corresponding area fraction re). The method of Cocks and Ashby (1981) is to show that the axial creep-rate in the power-law creep zone is bounded by ~, = ~o(a'~"
1
(A1)
\ a o / (1 - fe)"" This must match the axial strain-rate of the central part of the slab, caused by diffusive plating. A straightforward application of eq. (3.1) for the annular ring of radius rd shows this to be
~z =
2flDB3al 1 kTl 3 f~/zfa In(fdfh)
(A2)
where f~ = r~/I 2. Equating the two gives
¢o
(~t)'-'f~lZf,
=
ln(fdf,)
(A3)
(1 - f.r
where ~o has already been defined by eq. (3.3). If this is now optimized with respect to fa (to give the value of fd which maximizes the strain-rate), an expression for fd is obtained
,o(,Oy-', .
=
f,f~,'El + 21n(f;f~)]"
{. - sJ1 + (in L
t, JlJ
,n
(A4)
l,,J
For known values of $0, *t/ao and .~, the quantity f~ can be calculated. The quantity fe is then found using eq. (A3). In the diffusion limit, the diffusion zone extends halfway to the next void. Settingfd = 1 in eq. (A4) gives
ON
CREEP
FRACTURE
BY VOID
237
GROWTH
In the power-law creep limit the diffusion zone shrinks onto the void. Setting fj = fh gives _
\ai}
(1 - fhP- l"
These two equations are used to give the range of the coupling (shaded band) on the growth maps. By reasoning which exactly parallels that leading to eq. (3.16) and (3.17) the damage-rate and strain-rate are found to be
i~o dt -
(1
i~dt =
1+
f~)"
(l
(AS)
d L(1 -f~)~
dJ\ao/
This approach is the most refined of the recent analytical treatments of this problem, all of them approximate. It gives results which agree well with the finite element calculations of Needleman and Rice (1980), over the range their calculations cover. All these calculations have in common the feature that the hole growth-rate depends only on al/ao and on the quantity t~o, which appears in the work of Needleman and Rice (1980) as =
\ol/
(A7)
and in Edward and Ashby (1979) as the parameter P
1 V2 P = ~L
{ao~n-I~] 2In ~°~-lJ
~
.
(AS)
This has the important consequence, used in Section 7, that a parametric plotting method can be prescribed which leads to a single "creep-rupture master curve" when fracture is caused by the mechanism analysed here. The quantity q~o,then, is the important material property which controls void growth. APPENDIX B: Void Growth by Coupled Surface Diffusion and Power-Law Creep Fig. B1 shows a void growing by coupled diffusion and power-law creep. Surface diffusion limits void growth,
causing the void to become flat and penny-shaped. Matter is transported along the void surfaces to the tip, where it enters the boundary. Here it diffuses up to a distance rd from the centre of the void, and deposits as a layer of uniform thickness. The radius of the crack tip is r0 and the stress in the diffusion zone is aj. They are related by 7s ro = --. (B1) ad
t
t
t
#
I
~
t
l
DIFFUSION
L
ez
/ZON~
! #
--~.,~
- -
--
- r -i- - - - T
--P
. . . . .
~ "~-.
I
,
re
~_
' i
.
~
rd
,-
POWER-LAW
t /CREEP ZONE
_I
£z
-,
Z
i
l
i
iT|
l
i
i
i
FIG. B1. A void growing by coupled surface diffusion and power-law creep.
J.P•.s. 27 3-4--D
238
PROGRESS
IN
MATERIALS
SCIENCE
Matter flows out of the penny-like void at the rate dv dt
--
dr h 8nrorh - - . dt
(B2)
The velocity drh/dt is given by eq. (3.10) and the form of eq. (3.11) given by Chuang et al. (1979). Combining these with the last equation gives dv 2 v ~ n DdS,?,~rh dt = r~k T
(B3)
This is deposited uniformly over a disc of area n(r~ - r~) giving 1 dv 4, = 4nro(r~ -- r 2) ~ "
(B4)
Defining r~
A-
12
(B5)
and using the definition of ~'o [eq. (3.12)-I we find
~-z fl/2~O ( O'a'~3 ~o
(B6)
8(:,-A)~,}Too "
The power-law creep zone sees a void of effective radius re, and the strain-rate within the shaded zone is therefore ~o
(1 - re)""
(B7)
Equilibrium requires that °'a = (1
--
f~) \ f d - f h /
If we now equate G in the two zones, and substitute for a,, we find 8(fa--fh) 4
~bo = f~/2(1 _ f,,),,_3(fd _ f , ) 3
( O I ) n-3 ~o "
(B9)
This is a relationship between f, and f~. The optimum value off~ is that at which df,,/df4 = O. Differentiating the last equation w.r.t, fj, setting the result equal to zero and solving for f~ gives f, = ¼A + ~ .
(B10)
Inserting this into the equation for ~o gives ~o
.
(0"1~(n-3)
(fd -- ,~)
. . . 27 f~/2[1 - (¼f~ + ~fh)]"-3 \ a o /
(BI 1)
In the diffusion limit, the diffusion zone extends halfway to the next void, a distance I. Setting fa = 1 gives
~o =
f~/'(l - A t - " \ ~ o /
If ¢'o is greater than this, power-law creep is unnecessary, and the voids grow by diffusion alone, at a rate controlled by surface diffusion. In the power-law creep limit, the diffusion zone shrinks down onto the void. If fj is set equal to fh in the equation for ~b0, we merely obtain ~o = 0 meaning that a small diffusion zone exists. If, instead, we substitute fj = 2J~ we obtain the condition for substantial power-law creep control
(o,?-, ~,o ~< 2-~ (1 - ~ f , r - 3 \ a o }
(BI3)
Equations (BI2) and (B13) give the range of the coupling, shown as the shaded band on the void-growth maps. As was the case for the coupled mechanism described in Appendix A the void growth-rate, and the fracture time, by this coupled mechanism, are completely determined by stress and the value of ~o. This has the important consequence, used in Section 7, that a parametric plotting method can be prescribed which leads to a single "creep-rupture master curve" for fracture by this coupled mechanism also. In this case, the quantity ~#o is the
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important material property controlling void growth and fracture, but otherwise the method of Section 7 applies unchanged. APPENDIXC: Void Growth by Coupled Surface and Boundary DiJJi*sion This problem is treated in some detail by Chuang et al. (1979). Here we present a simple, approximate analysis which gives the necessary results. Consider a periodic array of voids of spacing 21 and diameter, in the boundary plane, of 2rh (Fig. 6). Material flows by surface diffusion to the tip, where it enters and diffuses along the boundary, depositing uniformly. The driving force is distributed so as to drive these two processes at matching rates. We allow for this by introducing a tip radius, r* (to be determined) which adjusts itself so that the surface flux equals the boundary flux at the tip. Then the normal traction or* in the boundary where it meets the tip and the chemical potential ~* at the tip are given by ~*
=
- #*fl =
- 7~fl
+
""
r*
provided r s >> r*. The volume of matter arriving at the tip per second is dv - - = n~ r . rj, drs -~- = 4nr*l 2 dfh dt dt
(C2)
2x/-2nD,6,7,f~rh k Yr .2
(C3)
-
using eqs (3.10) and the Chuang et al. (1979) form of eq. (3.11). This must be matched by the boundary flux. The volume of matter flowing into the boundary is given by eq. (3.1), but since the tip radius is near the sintering limit, we must replace O'l/(l - J~) by [al/(l - fh) - ys/r*], giving
Equating the two equations and solving for r* gives r.
( 1 - f~)z
I~
r*
,,/-2 In ( ~ ) Ao
+ ~.(1
-
A)3
-I
(C5)
where Ao is a further material property
Ao
=
D,6, DI~e"
(C6)
The equations for coupled growth arc obtained from eqs (C2) and (C3).
q,3 ~'1/2
1 dA Co r.jh do d~- = ~o3,*~
(c7)
I d~ 4~00f.t/27~ do dt ffi (1 - J~) d ~ r .2
(C8)
where r* is given by eq. (C5), and Co by eq. (3.12). Equation (C7) is the same as that given by Chuang et aL for the crack velocity. When 2,J/2~.r. In ( ~ ) A o ~< 1 ~,(I - A ) s
(C9)
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eq. (C5) reduces to 1
(71
r*
y,(1 -- A)
Then the two rate-equations (C7) and (C8) reduce identically to those given as eqs (3.13) and (3.14), as they should. .Equation (C-x)) is that of the broken line on the growth maps. When Ao is large, the void becomes more nearly spherical, and the calculation breaks down because r* is no longer small compared to r,. In the limit of very large Ao, the boundary diffusion limits the rate of diffusion, the damage-rate and strain-rate are given by eqs (3.4) and (3.5). We require a definition for the boundary separating crack-like growth from equilibrium void-growth. We adopt the criterion that r* is a fraction of the void radius r* rh
1 3
giving [from eq. (C5)] 9:ln(~) o_~_t = 1.5 + a~p
4
A° (CIO)
(1 - j~)2
where a~p -- 2y,(! - f,)/r, is the capilarity stress. This is the equation of the boundary between the regime of crack-like growth and boundary diffusion controlled growth. The transition from crack-like to equilibrium growth is, in fact, a gradual one. We assume that the void grows in a crack-like manner if r* 1 ~< (Cll) rh 4 and it maintains an equilibrium shape if r* ~< 1 rh 2"
(C12)
Substituting these limits into eq. (C5) gives stress ranges over which the transition occurs [eq. (3.31)].
APPE~qDIX D: Stress State Caused by Grain-Boundary Slidino Consider the hexagonal array of grains shown in Fig. 17, subjected to remote stresses a~: and a~: in the 1 and 3-directions, which are rotated through the angle 0 with respect to the 1' and 3' axes. Resolving forces in the 1' and 3' directions gives d~ =
a~
2
-
cos 20
+ ~
2
2
cos 20
~ 3 = o~ - o~ sin 20. 2
(D1)
Isolating the microscopic element of material ABCA'B'C' gives the average stresses shown in Fig. (DI). The boundaries AB, BB' and A'B' slide freely and cannot therefore support any shear stresses. Equilibrium requires that a~c = ½ ( 3 ~
-
~)
(D2)
O'bb = O'3 + N. 3r~ 3.
Substituting for o'~, a'3 and r~3 using eq. (D1) gives abo = ½ ( a ~ + O D + ( a ~ -- a ~ ) C O S
20
abb = ½(or + a]:) - (a~ - a]~)cos(20 + 60°).
(D3)
abe and Obb are the mean tractions on the grain-boundaries. As the stress system is rotated, the maximum traction shifts from the BB' boundary to the AB (and A'B') boundary. The magnitude of the maximum traction is repeated every 30 ° of rotation. In calculating the mean value of the maximum tractions, ax, only 30 ° of rotation
ON
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Crab
A-- ¢ab
241
GROWTH
C
l
B t
i
C
,rac ~
---%c b TOC
C*
'rob- ~
%c
O'ob
FIG. D1. The analysis of grain-boundary sliding.
need be considered. Therefore,
•[16
~b~ dO
~x
--
(D4)
nt6
do 00
which gives or, = 4~al~t - . a-3- . 3
(D5)
It is equally probable that the grain could lie in the 12 plane. For this orientation the mean of the maximum tractions, a r is 3 "
(D6)
Averaging over both orientations gives the overall mean of the maximum tractions, at, as a l = ~(r~: - ~(cr~ + a;:).
(D7)
The element ABCA'B'C' of Fig. DI is constrained by the surrounding material to deform as it does. This constraint can be satisfied if the deviatoric components of stress remain constant and equal to the applied components throughout the element. A measure of the level of triaxiality in the vicinity of the grain-boundary can be found by equating the principal deviatoric component there with the principal applied component, If S~: is the applied principal deviatoric component, then S; ~ = { a ; ~ - ~ ( ~
+ a~)
(D8)
equating this with the local principal component, $1, where ~ 1 = ~C~a~ 2 4 ~: -
~(e~ + ~ ) ]
- ~(~
+
~)
(D9)
a~ and a~ are the other two local principal stresses, gives a local level of triaxiality of tP/ L
where a [ is the applied effective stress.
= _ (4(;~ -i-(r~ q- (r~) 6e~:
(DIO)
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APPENDIX E: Creep Fracture Resulting from Precipitate Coarsening If do is taken as a reference strain-rate in the creep law [eq. (2.1)] then Oo can be interpreted as the stress that results in a strain-rate do. If the material ages during the course of a test then the stress needed to maintain a strain-rate do decreases, i.e. Oo decreases. At low temperatures the strength of a material is found to be inversely proportional to the precipitate spacing Ip. It therefore seems reasonable to assume that ~ro is inversely proportional to lp. Then ~o -- %0
(El)
where at the start of the test ¢0 -- aoo and lp = Io. The mean radius of a spherical precipitate after a time t is given by (Shewmon, 1969) r~
ro3 + 8DpypC°I'22
9RT
t
(E2)
where ~,p is the surface free energy of precipitate material, Dp is the diffusivity of precipitate material through the matrix, and Co is the equilibrium solubility of large precipitates, and since the volume fraction of precipitates must remain constant, i.e. fv = r~/l~ = r~/l~, we obtain 1~ = 1 + 0odot
(E3)
Oo - 8D#pC°D2 9R Tffl~d o "
(E4)
where
From eqs (El) and ( E 4 ) (E5)
Ooo
ao
=
(1 + Oodot)1/3"
Substituting this into the creep law [eq. (2.11] gives d =
(1 + OodotY'/3.
do
(E6)
Now, consider a uniaxial creep test conducted at constant load. During the course of the test, the volume of the specimen must remain constant.
Aoho = Ah = const
(E7)
where A is the cross.sectional area and h the height of the specimen. Differentiating eq. (E7) w.r.t, t gives 1 dA A dt
ldh hdt
. . . . . . .
d
(E8)
and if o~ -- P/Ao is the initial uniaxial stress then eq. (E6) can be written as 1 dA / o ' " / A \" A dt =do~-~ooo)~A)(1 + 0odotY '3.
(E9)
The time failure is found by integrating eq. (E9) between the limits A/Ao = 1 at t = 0; and A/Ao = 0 at t = ty, giving
dot:=
I + ~
\ - -o~ / 3
-
1 .
(EIO)
Equation (El01 can be plotted as a creep rupture curve using axes of al/aoo and dots. Curves for 0o = 1.0 and a range of values of n are shown in Fig. El. Equation (El0) can be written as o'l Oo is a material parameter. It can be defined in the same way as 4~o in Section 7.2 to give a model-based procedure for extrapolating creep data (Cocks, 19801. The result is again that data should be plotted using axes PI = loglo ol - H
-
ON
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243
GROWTH
% % n=5
~
~o=1.0
n = 5
JO- 6
2
~.0
o.5
1I0l -l.
IC) I~
~
i
~
I0 0
102
-o._c
-I.O
-I.5 FIG. El. Creep rupture plots when fracture is a result of precipitate coarsening.
and
(, ,)
P2 = loglotf - d ~ -
Too
where H and J are now defined by H
-Q,+Q.+Qo 2.3nRT
j _ Qp + Qo 2.3RT
and Qc, Q~ and Qo are the activation energies of the exponentially varying quantities D, D r, Co. APPENDIX F Consider any mechanism which causes damage to accumulate in a way which can be written dfh dt
1 h(a)k(T)#(fh)
(F1)
that is, stress, temperature and damage are (mathematically) separable. The eqs (3.4), (3.13) and (3.16) are all of this type. Then the time-to-fracture for any history of loading is given (ignoring nucleation) by h[o(t)]k[T(t)]
-
o(fh) dfh.
(F2)
At constant stress and temperature, the time-to-fracture is t r = h(a)k(T)
fo*
g(fh) dfh.
(F31
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We now form the life-fraction [eq. (7.18)] giving
ft's ~ - . f l h(a)k(T)g(fk) dfh = 1. do •/o ty h(tr)k(T) g(fh)dfh
(F4)
Io'
A proof of this sort was first given by Odqvist and Hult (1961) who, however, limited themselves to a simple power-law for h(a) and g(A).