~
Acta metall, mater. Vol. 43, No. 5, pp. 1909 1916, 1995 Copyright ~ 1995 ElsevierSciencektd 09567151(94)003874 Printed in Great Britain. All rights reserved 09567151/95 $9.50 + 0.00
Pergamon
A M O D E L FOR D Y N A M I C E M B R I T T L E M E N T D. BIKAt and C. J. MeMAHON Jr~ Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, PA 19104, U.S.A. (Received 20 December 1993; in revisedJorm 1 September 1994)
AbstraetAmodel is presented to calculate the rate of crack advance in the stepwise decohesion process which can result from the stressinduced penetration of a surfaceadsorbed element into a solid, usually along grain boundaries. We call this process dynamic embrittlement. The model employs a diffusion equation containing both the usual randommixing term and a term reflecting the work done by a tensile stress when surface atoms diffuse inward. Given the diffusion constant of the surface species and the stress profile at the crack tip, the concentration buildup ahead of the crack as a function of time can be calculated. This can be combined with an empirical relationship between the interfacial concentration of the surface species and the stress to cause decohesion to give the crackgrowth rate. This model is applied to the case of sulfurinduced cracking of an alloy steel in the process known as stressrelief cracking.
1. INTRODUCTION A kinetic model is offered here to describe the intergranular cracking of a metallic alloy as the result of the stress induced grain boundary penetration of a mobile, surfaceadsorbed element. It is based on the understanding that, under certain conditions, any surface atom would have a tendency to enter a grain boundary under the influence of a mechanical force (dynamis in Greek) acting normal to the boundary, especially in the presence of a stress concentrator. The driving force is the same as that which operates in diffusion creep and diffusive growth of creep cavities. The ingress of such mobile surface atoms at the grain boundary could reduce the cohesive strength of the boundary and cause what we call dynamic embrittlement. The process is illustrated schematically in Fig. 1. In essence it requires the simultaneous effects of an embrittling species near or above its melting temperature and a high tensile stress. The result of this analysis, coupled with an empirically based fracture criterion, will be applied to the case of sulfurinduced cracking in a M n M o C r N i steel [1], and the applicability of the model to other systems will be discussed. In previous work, McClintock and Bassani [2] used a steadystate solution of the diffusion equation to describe diffusioncontrolled cracking due to an environmental effect in which the cracking process is envisioned as an atombyatom separation caused by the action of an embrittling species at the crack tip. They also foresaw that intermittent crack advance tNow at Procter and Gamble European Technical Center, Brussels, Belgium, ++To whom all correspondence should be addressed.
could occur as a result of the penetration of an embrittling species over a length in which the cohesive strength was reduced to the level of the cracktip stress. This approach was applied to oxygeninduced cracking of a nickelbase superalloy [3]. Bassani [4] later incorporated a stress term in the diffusion equation and treated the case of steadystate propagation of a crack under the influence of an asymptotic stress profile at the tip, which leads to a maximum concentration of the embrittling element at the tip. In the present model, a rising stress profile, characteristic of a deforming solid, is employed, and a steady state is not assumed. This places the maximum concentration ahead of the crack tip and makes the crack growth inherently discontinuous.
2. THE MODEL 2.1. Stressinduced d(fJusion along grain boundaries In the presence of a tensile stress, a, the chemical potential of any atom of volume ~ is reduced by Ap =  f ~ r when that atom is transported from a free surface into the interior of the solid. Such transport can occur relatively rapidly along grain boundaries, owing to their larger diffusion coefficients compared to lattice diffusion. The reduction of the chemical potential would be nonuniform along the boundary in the stress field of a notch or cracklike defect, and this would give rise to a diffusive flux toward the region of maximum stress, driven by the gradient in chemical potential v~ = ~V~r
(1)
where Vcr is the relevant stress gradient. It may be noted that, if surfaceadsorbed atoms were in chemical equilibrium with both the boundary and
1909
1910
BIKAandMcMAHON:
A MODEL FOR DYNAMIC EMBRITTLEMENT
ooO(o00 o° O o ~O O ) 0 0 O O0 00 000 0 0000 O000 000 000 0 n °( 0 O0 0 0 0 0 O~ ~ ) ~ 0 ~ 0
o Oo °o °o
Oo o oZgo
The diffusive flux in the boundary, J, due to the stressinduced chemical potential gradient would be
[51
Df~C
J = ~
Va
(2)
where D is the diffusivity and C the concentration of the species diffusing in the solid, k the Boltzmann constant, and T the absolute temperature. This stressinduced flux would establish a concentration gradient of the diffusing species in the solid; thus, the flux equation becomes QC
J=D(VC~Va).
(3)
The continuity equation with constant D leads to
0
° 0
0o° ©
0
000 O0 0 0 0 0 0 0 0 00 C 0 0 0 0 o ( () 0 o 0 o 0 0 0 0 0 00 o ) 00~00~0
oo
00() 0~00000 0 0 ( )C 0 0 0 0 0 0 OOOo 0 0 0 0 0 o 00 0 O000 0 ~ 0 0 O O0 00 O0 ° 0 o ° 0 0 ° 0 o 0 o0 0 o° 000 0 o 00 00 00 Q 00 00 00
QoO00 O0 0 0 0
00
000,, O 0 n O
_.
•
OYO O~
O00Q u 0 0 u 0 O ~ 0 00 O00 O00 O0 00 O00 O00 O00 0 o 0 o 0 o O00 00 00 0 00 O00 O00 O0 Fig. 1. Schematic representation of the process of dynamic embrittlement, in which a lowmeltingpoint surface species (black atoms) are induced to diffuse into a solid along a grain boundary by the application of a stress normal to the boundary; decohesion occurs when a critical combination of stress and concentration is reached.
the bulk prior to the application of the stress, any stressinduced diffusion of these atoms into a grain boundary would tend to be reversed upon the removal of the stress.
OC DV. VC at
kr
Va
(4)
This equation must be solved for the concentration profile of the species in the solid as a function of time with the appropriate initial and boundary conditions. In the case that the grain boundary diffusion is much faster than bulk diffusion, which is an appropriate assumption for a temperature of 0.40.5 of the absolute melting temperature (CoNecreep conditions), equation (4) can be written in one dimension, i.e. along the boundary, as follows 8C ~2C ot  D ~x 2
Df~ ( ~3 ~3a) k~ ~xC~x
(5)
where D is the grain boundary diffusivity. (An alternative derivation of this equation is given in Appendix A.) A grain boundary intersecting a free surface is considered. Surfaceadsorbed atoms (which can be an environmental species or solute atoms and which may include embrittling atoms), once entering the boundary driven by an applied tensile stress, would tend to assume a concentration profile that satisfies equation (5) at any time. If an initial distribution (uniform or nonuniform) of segregated atoms exists in a grain boundary before the application of the stress, then equation (5) can be applied to examine the redistribution of these atoms along the boundary. In this case, hostmetal atoms would also tend to diffuse in response to the applied stress, as in diffusion creep, and to "mix" with the solute species; this would cause the grain boundary composition to become uniform again at long times. This could explain the transient redistribution of intergranular phosphorus in steel reported by Shinoda and Nakamura [6]. 2.2.
The governing equations
The solution of equation (5) requires the knowledge of the stress field along the boundary as a function of time and distance. The stress profile is a function of time, because at elevated temperatures the stress relaxes by both dislocation motion and the
B1KA and McMAHON:
A MODEL FOR DYNAMIC EMBRITTLEMENT
wedging effect of the surface atoms which are being platedout in the grain boundary. It has been found experimentally in the case of an alloy steel [1] that the rate of cracking is highly nonuniform, as it should be in view of the expected strong dependence of the grain boundary diffusion coefficient on grain boundary structure. Thus, the front of the main crack is characterized by regions of rapid propagation along intergranular paths of especially fast diffusivity, and uncracked ligaments are left behind this front in regions where the intergranular diffusivity is especially slow. The constraint provided by these uncracked regions permits the crack tips to be quite sharp; fractographic observations on the steel indicate the crackopening displacement to be less than 1 #m. Therefore, it is considered appropriate to model the cracktip stress field in terms of a cracklike cavity embedded in a solid, rather than in the usual terms of a macroscopic crack, the tip of which is fully relaxed by unconstrained plastic flow. The stress field of an equilibrium cavity is parabolic, with the maximum stress at the end of some diffusion length, 2, where the diffusive flux is zero [7]. The cavity takes on the equilibrium shape because of rapid surface diffusion, and it grows by the "platingout" of hostmetal atoms over the diffusion length. Under creep conditions, the volume of material platedout would be accommodated by creep; i.e. the displacement rate due to this plating can be equated to the creep strain rate i [8] ( D b 6b ~)MO'b k T2 3 
BaG
(6)
1911
of these atoms would not affect the stress distribution ahead of the cavity in any other way. With the above considerations, the stress field ahead of an intergranular cavity can be written as cr'(x') = a~ + x'(2  x')(0~  or6)
(7)
where all stresses have been normalized with respect to the yield stress. That is O"
X
a ' =   , and x ' =  . 6y )t For global equilibrium, the average of this stress over the length )~ should be equal to the normal stress acting on the boundary, Orb, which can be assumed equal to the maximum stress applied to the boundary in the vicinity of a macroscopic notch; e.g. 68 = 3~ry or a(~ = 3. This condition a~, =
;01
o'(x')dx'
leads to an estimate for the constant ~. This stress distribution is illustrated schematically in Fig. 2. Substituting equation (7) i~to equation (5), the concentration of species in the boundary should satisfy at any time the following equation OC"
Ot'
O2C '
= / ~
 [2(1  x ' ) ,
~?C'
x (~  a 0 ) 1 ~ +
2(c~  a 0 ) C '
(8)
where C Ct~__
where ab is the stress acting normal to the boundary (assumed equal to the farfield stress), (Dbfbf~) M the grain boundary diffusivity and atomic volume of the host metal (M), n the stress exponent, and B = i / a " is the Norton coefficient. Equation (6) can be used to calculate 2 as a function of the applied stress and the hostmetal diffusivity. The maximum stress at the end of the diffusion length would be a multiple of the local yield strength, o* =0~oy, and would depend on the a m o u n t of strain hardening ahead of the tip. The stress at the cavity tip, or0, would be given by a0 = )'~(xt + K2), where ~ct and ~c2 are the principal curvatures of the surface of the cavity, and 7s is the surface energy. For an equilibrium (cylindrical) cavity ~q=sin~P/R, where
Cb
Gb
= 2 cos 1 (7b/2~/s) GB
G ob
r~
is the dihedral angle, and 2R the cavity diameter, and K2 = 0. This expression follows from the requirement that the chemical potential, /~, of an atom moving from the cavity surface into the boundary must be continuous [9]. The presence of surfaceactive atoms on the surface of the cavity would reduce the surface energy, 7~, and thus would decrease the stress at the cavity tip. However, it is assumed that the presence
O0
x Distance, x Fig. 2. Normal stress acting on the grain boundary ahead of a cavity.
1912
BIKA and McMAHON:
A MODEL FOR DYNAMIC EMBRITTLEMENT Table 1. Material parameters of ~ Fe related to creep fracture (Refs [22 24]) Parameter Value Unit f~ 1.18 x 10 29 m3 b 2.48 x 10 "~ m G (300 K) 64 GPa G (800 K) 48 GPa A* 7 x 10~3 n (stress exp.) 6.5~5.9 7b [27] 0.85 J/m 2 ~,,~ 2.1 J/m 2
==¢ 09
N IN
0 0 _J
i
[ c~min
p
I
i
i
C o n c e n t r a t i o n of the Embrittler, C '
Fig. 3. General dependence of the stress for decohesion on the planar concentration of an embrittling element. a n d Cb is the equilibrium grain b o u n d a r y concentration
2.3. The fracture criterion A critical c o n c e n t r a t i o n needs to be defined at which decohesion can occur at a certain stress level. This p r o b l e m has been previously addressed experimentally by the work of K a m e d a a n d M c M a h o n [10], Shin a n d Meshii [11], a n d K u m a r a n d Eyre [12]. In a c c o r d a n c e with these results, the present model employs an empirical relationship in which the cohesive strength varies inversely with the solute concentration, as follows
t tI=_ T
and .f
its freesurface equilibrium concentration, C~, for the t e m p e r a t u r e of the test at all times. T h a t is, the crack or cavity surfaces were considered as infinite sources o f embrittling atoms. The second b o u n d a r y c o n d i t i o n [equation (11)] follows from the r e q u i r e m e n t t h a t the flux of all a t o m s should be zero at the end o f the diffusion length 2.
kT2 2 D ~2~y'
and
[~= k T
O'y ~, 0"? / ~ , E q u a t i o n (8) was solved numerically for the timed e p e n d e n t c o n c e n t r a t i o n of the embrittling species using the initial condition C'=
1 for 0 < x ' <
1 at t ' = 0
(9)
a n d the b o u n d a r y conditions C'=C~forx'=0atanytimet'/>0
(C~ = C j C b )
O'fm i n 
l)t ~
) "}1
(12)
where art. . ., .O'frin , a n d c'min have the meanings indicated in Fig. 3. This equation, coupled with e q u a t i o n (8) represents, in effect, an a d v a n c i n g front of progressively weaker intergranular cohesion. A n estimate of the p a r a m e t e r s involved was based on the b e h a v i o r f o u n d for sulfur in iron [1 1] a n d for o t h e r impurities in temperembrittled steels [10].
(10) 3. APPLICATION TO SULFUR IN STEEL
and ~C' ~x'
= 0 for x ' =
1 at any time t ' > 0.
(11)
The initial c o n d i t i o n [equation (9)] m e a n s that the initial distribution of embrittling species in the b o u n d a r y is equal to the equilibrium grain b o u n d a r y concentration. The first b o u n d a r y condition [equation (10)] m e a n s t h a t the c o n c e n t r a t i o n of the embrittling element at the crack tip remains equal to
Species Fe Fe Fe S S
The numerical values o f the p a r a m e t e r s involved in the above equations are given in the Tables 13. These values were used for the c o m p u t a t i o n of the sulfur c o n c e n t r a t i o n a h e a d of a crack at 550°C as a function o f time. In view of the constraint provided by the uncracked regions, as described earlier, a n intergranular crack with a tip configuration similar to t h a t of a creep cavity is used to model the case of sulfur in
Table 2. Diffusion coefficients in ~Fe Path D O (10 4me/'s) Q(kJ/mo ) Lattice 2.0 240.9 Grain Boundary 2.5 167.4 0.02 105 Surface 5 × I04 232 3 143.8 Lattice 1.6 202.8 Grain Boundary 6 × 10 4 119.4
Reference [25] [26] [27] [22] [28] [29] [30]
BIKA and McMAHON:
A MODEL FOR DYNAMIC EMBRITTLEMENT
Table 3. Model parameters related to sulfur in steel at 823 K Parameter
Value
T kT D~ Dbv~ D~ D' try ~0
823 1.136 2.35 5.9 1.7 2.71 400 2
O"b G
K J/atom m2/s m2/s m2/s mS/atom MPa MPa
1200 47.3
/~
=
kT Oyf2s 
sulfur would become distributed ahead of the cavity tip in the manner shown in Fig. 4 [solution to equation (8), with equation (12), for the conditions of equations (9), (10) and (11)]. It can be seen that the maximum in sulfur concentration shifts with time toward the position of the maximum of the local applied stress at the end of the diffusion length, 5.. At 550:C, a sulfur concentration of 5000 Cb was reached at 0.5z, and saturation occurred at longer times ( ~ 1.5r) at a level of 7200 C~. The diffusion length 2 was calculated [using equation (6) and the data in Tables 2 and 3] to be in the range 1100nm, depending on the diffusivity of iron (4.3 x 10 135.9 × 1 0 ts m 2 / s ) and the stress exponent for creep (6.56.9). This corresponds to a time constant = 1.0482 2/D in the range 0.6 ms6 s. The local stress to fracture was calculated using equation (12) for the sulfur concentration as a function of distance ahead of the tip and the time, given in Fig, 4. The results are plotted, superimposed on the local stress distribution, in Fig. 5. Decohesion would start at some critical distance, x,~, as shown in Fig. 4, and the rate of crack growth would be
Units
x 10 20 x 10 20 x 10 ]5 x 10 Is x 10 ~s
MPa GPa
1.05

Cb C,
0.010 10 12 1.7 1.53
C rain
p a~,"~ /a~ i"
wt% wt% wt%
steel. A factor contributing to the sharpness of the crack tips in the steel is the high density of dislocations and fine carbides in the bainitic microstructure; these severely limit the tendency for the stress to relax by dislocation emission from the crack tip on a short time scale. The crack faces would be contaminated with segregated sulfur at the test temperature as a result of surface segregation from the bulk and sulfur spreading from reprecipitated fine sulfides on the intergranular facets. Under the influence of the stress field, this I
I
1913
i~ = %~i,_ 0.92 _ 2.25 (Db f~)~crY t~it 0.4z k T)~
0"~'I~,'3.(13)
= 2.25 (D~f~)~aY (Db~bf~)M kT kTB
1
'
I
I
I
5000
•
•
•
•
t=O.5
T = 550°C 0 0 0 0 0
•
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• •
4000 O
•
0
•
0
• i
0
o
o
•
o
0
;
2000 O
*
Z
•
.
o
++
o
~ oO ,
o
+° + + +
°
I
0.0
,
I
o
¢. *.
.
03 •
o
o
o
oo
0.2
*
+ + + + + + + + + + +
o
+++++ +++
++++ ++
+
. . . . . . . . . . . . . . . . . . . . .
1000
o
o o
O
o
o o
tO 0
•
O
o e
O
O
2
o
o
0
•
3000
o
0.4
0
• •
O
0
O0
0
•
t~
0
0 @
,
I
,
°°°° I
. . . . . . . ,
,I
0.2 0.4 0.6 0.8 Normalized Distance from Crack Tip, ~ X
°°oo ,
0.1 I
1.0
Fig. 4. Calculated concentration profiles of sul~r ahead of a cracklike cavity ~ r various normalized times at 550°C.
1914
A MODEL FOR DYNAMIC EMBR1TTLEMENT
BIKA and McMAHON: I
. 0
o
0
I
+ '
'
I
I
"r++++++++ 0.2
o 0
o o 0
o 0
Sh'ess Io fracture
*
o
°o oo °
el
0
rO
°o,
°°°°°°°°
oO
O
0.3
cD
.
g
terit = 0.4
o
Z
2
0
I
,
,
,
,
0.0
I
,
,
,
0.5
,
)
1.0
,
,
,
,
I
1.5
,
,
,
,
I
2.0
Normalized Distance from Crack Tip, x/£ Fig. 5. Superposition of the stress profile ahead of a cavity and the decohesion stress corresponding to several sulfur concentratrions (cf. Fig. 4). Decohesion may occur at x~r~ = 0.95 and t ~r~0.4. Using the model parameters given in Table 3, the rate of cracking was found to be in the range 10 7 1 0 5 m / s . This is consistent with the cracking rates observed in the steel [I]. 4. DISCUSSION Clearly, the crackgrowth rate depends on both the parameters related to creep deformation and to the mobility of the embrittling element. The strong dependence of the rate on the strength of the material is indicated by the combined stress exponent, [(n  1 ) / 3 ] + 1 . Physically, the material must be strong enough to withstand the relatively high stress needed to drive the diffusion of the embriuler into the boundary and to cause decohesion. This model comprises a straightforward diffusion equation and three materialsrelated factors: the decohesion criterion, the cracktip stress profile, and the mobility of the embrittling species. The first factor is not critical, since the general form of the cohesive strength vs impurity concentration is known, and any uncertainty in the magnitude of the embrittling effect does nothing more than introduce a scaling factor in the crack velocity. The other two factors are critical to the precision of the calculation, however, and the uncertainties associated with them are limiting factors. In particular, the stress profile at the crack lip can, at present, only be approximated. As pointed out
by Chert [l 3], several size scales at the crack tip have to be considered; these are shown schematically in Fig. 6. On a scale of tens of micrometers, the crack tip has a radius defined by the amount of plasticity ahead of the crack. This is a function of the yield strength of the material and of its shorttime creep behavior. On a scale of micrometers, the crack tip is much sharper and is modeled here as a creep cavity attached to the crack tip; this is justified in terms of
101 p00. ~
GB
Fig. 6. Schematic representation of a cracklike cavity extending from the tip of a macroscopic crack and the difl'usion zone ahead of the cavity, showing the different size scales involved in the cracking process.
BIKA and McMAHON: A MODEL FOR DYNAMIC EMBRITTLEMENT the constraints imposed by the surrounding uncracked ligaments. The diffusion zone extends from this cavity on a scale of tens of nanometers, and this defines the process zone for decohesion. The evidence for this in the steel is the striation spacing of about 100 nm. In order for the stress field at the tip of the cavity to be calculated adequately, the shorttime stress relaxation must be dealt with in a convincing way. This is beyond the scope of the present work, but it is argued that the choice of a diffusionrelaxed cavity provides a conservative lower bound to the characterization of the stress profile. The mobility of the embrittling element depends mainly on the structure of the grain boundary and on whether or not other solutes have segregated to the boundary from the bulk. The latter factor would, of course, inhibit the penetration of the embrittling element by decreasing the amount of free volume associated with the boundary. The structure of grain boundaries in polycrystals normally varies from one location to another, because the boundaries are generally curved. In addition, the diffusion coefficient normally varies with direction along the interface, since the grain boundary structure is often characterized by the presence of "pipes" in which free volume is concentrated. Thus, diffusion along the pipes would be faster than transverse to them. Finally, in the case of a steel which has transformed from austenite to bainite [1], the grain boundaries in question are actually the loci of the prior austenite boundaries. The microstructure during cracking comprises patches of bainite lathes, which are not continuous across the prior austenite boundaries, because they bear a crystallographic relationship with their parent grains. This adds further variation to the grain boundary structure in the steel. Because of the variations in mobility of the embrittling element and in the local microstructure of a transformed steel, it is meaningless to try to calculate a single crack velocity. One must deal with a range of velocities and expect that the cracking process would be highly nonuniform. The formation of patches of cracked boundaries along the tip of the advancing crack and the presence of uncracked ligaments behind this tip, as seen in the steel, are manifestations of this fact. As indicated by equations (6) and (13), the diffusion length 2 decreases and the crackgrowth rate increases as the material becomes more creepresistant, i.e. as the stressrelaxation rate decreases. This fact, coupled with the high dependence on stress [equation (13)], means that the susceptibility of a material to this kind of cracking increases strongly with its strength. It must be emphasized that the crackgrowth rate given in equation (13) refers to the rate of crack advance on the submicron scale. The rate of crack propagation through the material on a coarser scale depends on microstructural factors, such as the grain
1915
size. Owing to the variability of diffusivity along grain boundaries, some boundaries crack much faster than others. In a coarsegrained material, this has a large effect on the overall cracking rate because of the large local increase in stress intensity on the boundaries which surround a cracked boundary. In a finegrained material this factor is less important, so there should be a substantial effect of grain size, which is not considered in equation (13). The model is consistent with the crackgrowth rate found in the steel, and it has been shown also to agree with the observed cracking kinetics in a Cu8%Sn alloy, which has been found to reproduce the stressreliefcracking phenomenon in steels [14]. In this case, the segregation of tin from solid solution in the bulk provided the embrittling element on the surface of the crack, and the cracking occurred at temperatures several hundred degrees lower than in the steel. There is nothing in the model which would restrict it to cases in which the surfaceabsorbed element originates in the bulk. It could just as well have come from a gaseous or liquid environment or a solid surface coating. Moreover, there is no reason why this kind of cracking should not occur in any highstrength deformable solid. (The deformability gives the rising stress profile ahead of the crack tip, which drives the diffusion.) The various references to oxygeninduced cracking of nickelbase alloys, cited in {1], are cases in point. In general, any lowmeltingpoint element, which necessarily has a high mobility and a low binding energy, should present the danger of dynamic embrittlement to any highstrength structural alloy. From a practical point of view, it may be seen that socalled "hot shortness", or the tendency of a material to crack during hightemperature deformation processing, is a form of dynamic embrittlement. For example, unscavenged sulfur is known to be responsible for hot shortness in steels, and CuSn bronzes are notoriously hot short. A remedy for this condition is either to scavenge the responsible internal element, as done by manganese in steels, or to add a solute which segregates faster or more strongly than the embrittling element. Thus, it is known that the presence of phosphorus inhibits sulfurinduced cracking in steels [15], and similar additions have been shown to inhibit bismuthinduced cracking of copperbased alloys [16]. Finally it should be pointed out that the description of hydrogeninduced cracking of steels which is widely employed [17 21] is essentially similar to what we are calling dynamic embrittlement. The crucial difference with hydrogen is its extremely high mobility, which means that diffusive transport occurs mainly through the lattice, instead of along grain boundaries. It is believed [20, 21] that the intergranular nature of most hydrogeninduced fractures in steels comes from the prior segregation of other elements at concentrations which are not sufficient to produce brittle fracture in the absence of hydrogen.
1916
BIKA and McMAHON:
A MODEL FOR DYNAMIC EMBRITTLEMENT
5. SUMMARY It can be argued t h a t d y n a m i c e m b r i t t l e m e n t is a pervasive p h e n o m e n o n , but t h a t the u n d e r s t a n d i n g of it has been so fragmented that its generic n a t u r e has been obscured. It is modeled here with a diffusion e q u a t i o n containing the Fickian r a n d o m  m i x i n g term a n d a term accounting for the work d o n e by the applied stress when diffusion from a free surface occurs, as in creepcavity growth. This is c o m b i n e d with an expression for the dependence of cohesive strength o n impurity concentration, the stress profile at a crack tip, and the diffusion coefficient o f the impurity, to give an expression for the crack growth rate. The application to stressrelief cracking o f an alloy steel is consistent with recurrent decohesion on a scale o f a b o u t 100 n m at the tip o f the p r o p a g a t i n g crack, as evidenced by striations o n the grain b o u n d ary facets exposed by this cracking. The theory is also consistent with the m e a s u r e d cracking rates in a model C u  S n alloy.
AcknowledgementsThis work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, through grant no. DEFG0287ER45290 and by the American Welding Society through an AWS Fellowship to DB. Discussions with Drs J. L. Bassani, N. Aravas, M. Kantha and E. A. Delikouras are also gratefully acknowledged. REFERENCES
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APPENDIX
A
StressAssisted Diffusion Along a Grain Boundary When a normal stress, or, acts on a boundary, the chemical potential of the boundary species (solute and metal atoms) would be It = Ito + k T l n a crf2
where a is the activity of the species in the boundary and a = ~C, where C is the concentration and c~ the activity coefficient, k T the thermal energy, af~ the mechanical energy, and It0 the reference chemical potential at the boundary, corresponding to a = 1 (pure substance). The flux along the boundary due to this chemical potential would be
Db C 63//
J = kT
Ox 
Db k T C a x [it° + kT21n(ctC)al)]"
For the special case in which the activity coefficient and the atomic volume do not depend on stress or concentration, i.e. f~ or c~ ¢ f ( C , a) :Af ( x , t)
ac
Db~
06
J=Db~+~rc ~
which can be put into the continuity equation, i.e. aC
aJ
at
Ox
to result in the governing equation for C = (C (x, t ) and o = a(x, t) and Db ~ f ( x )
6qC at
a 2C Ob Ox2
Db~ ~ [ aly ] kT oxLC
J
I