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A mathematical model for calculating stresses in a four-layer carbon-silicon-carbide-coated fuel particle
JOURNAL
OF NUCLEAR
32 (1969)
MATERIALS
A MATHEMATICAL
322-329.
0 NORTH-HOLLAND
MODEL FOR CALCULATING
PUBLISHING
CO., AMSTERDAM
STRESSES IN A FOUR-LAYER
CARBON-SILICON-CARBIDE-COATED
FUEL PARTICLE
J. L. KAAE chclf Benera+?Atomic
Incorporated, San Diego, California 92112, USA Received
A
previous
stresses
mathematical
in the
model
coatings
silicon-carbide-coated
on
carbon, The
are : low-density
silicon carbide, assumptions
mathematics of
the
coating.
used
in
model.
the
model
has been considered,
En g&&al,
the
a possibility
and in that
initiale
couche
of
interface
de carbone
interne
possibilit6
be considered
de contraintes
Spannungen
densification of the carbon
auf Vierfachschichten bestand
it raises the possibility
of high
particule
de combustible
permettant
lea revetements
de
aus:
mathematisches
betrachtet,
B 3 couches de carbone et
a &A Btendu au cas d’un
rev6tement
mit niedriger Dichte,
& 4 couches.
Lea 4 couches consid&+es
Verfiigung
dense, le carbure de silicium et le carbone isotropique
gerufen
dense.
des
traitement utilis6s
mathematique
dans
sont
le
modele
semblables
le modAle pr6c6dent.
et
le
Es wurde jedoch
stehonde Volumen
Behandlung wie bei den
die MGglichkeit
durch
Graphits,
nicht unabhiingig
scheinen
Spannungen,
bestrahlungsinduzierte
zur von
in
der
inneren
hervor-
Verdichtung
Graphitschicht
eine
zu spielen ala in der Busseren. Diese
Spannungen kijnnen durch Abtrennung
une
von
von der inneren Schicht.
allgemeinen
grdssere Rolle
8. ceux
Cependant,
gemacht,
dass sich die innere Graphitschicht
der Ent,fernung Im
dans
und
dichtem
Sic last. In diesem Falle ist das den Spaltgasen
sont : le carbone & faible densit& le carbone isotropique
utilis6es
Model1 urn
aus Graphit
ausgedehnt. Die Vierfachschicht
Graphit
Dreifachschichten.
sur une
de silicium
hypotheses
dens la couche de
Im Model1 und seiner mathematischen
de carbure
Lea
mais se pose alors la
&e&es
wurden iihnliohe Nliherungen pr&6dent,
de
interne
isotropen Graphit, Sic und dichtem isotropen Graphit.
stresses in the inner carbon at a later time.
dans
lea
SIC auf beschichteten Teilchen zu untersuchen, wurde
at the inner interface can relieve
lea tensions
par
dans le cas de la
que dans la couche
in Dreifachschichten
are more severe in the inner carbon than in the outer
TJn modele mathematique
dues I la
induite
of the inner
carbon. Separation
calculer
du
carbone interne dans un dernier stade. Ein bereits vorhandenes
but
carbone
peut relhcher ces contraintes,
In general, it appears that the stresses due to the
these stresses,
que lea tensions
du
carbone externe. Une separation 8, l’interface
coating. initial fast-neutron-induced
du dbplacement
neutrons rapides sont plus s&&es
case the volume
of the displacement
il semble
den&cation
are similar to those
However,
to the flssion gases cannot
to be independent
and
et dans ce cas
disponible pour lea gaz de fission ne pout
paa &re consid& indhpendant rev&ement interne.
dense isotropic
separation at the inner carbon-silicon-carbide available
le volume
The four layers
carbon,
196Q
carbure de silicium a Bt6 consid&&,
carbon-
and dense isotropio carbon.
of the calculations
previous
calculating
a three-layer
fuel particle haa been extended
to the case of a four-layer considered
for
6 January
der Schiohten
abgebaut werden, aber es ergibt sich die MGglichkeit,
possibilit6 de s6paration B l’interface interne carbone-
dass sich Spannungen
1.
layer, and a dense isotropic pyrolytic-carbon layer. That model has now been altered to consider the different coating design shown in fig. 1. As can be seen, it includes a second dense isotropic carbon layer inside the silicon carbide. This paper is intended to describe the mathematical
Introduction
carbon
In a previous paper, a mathematical model for calculating the stresses developed in composite silicon-carbide-carbon coating on a nuclear fuel particle during service was described 1). The model considered a three-layer coating consisting of a low-density pyrolytic322
layer,
zu spiiteren Zeiten aufbauen.
a pyrolytic
silicon-carbide
A
Fig.
1.
MATHEMATICAL
MODEL
FOR
a. Croes section of a four-layer
alterations in the model, to show some typical results obtained using the model, and to discuss some of the implications of these results.
2.
Model
The assumptions involved in the model have been described in detail in the previous paper 1). In summary, it is considered that: carbon does not possess 1. The low-density mechanical strength. Thus, it does not restrain the outer coatings and does not transfer loads from the swelling fuel to the outer coatings. 2. An internal pressure is generated due to released gaseous fission products. The volume available to the gases is that in the porosity in the low-density carbon. Solid and condensed fission products decrease this volume during the life of the fuel particle. 3. Stresses are generated in the coatings due to differential thermal expansion between the dense carbon and silicon carbide, due to the internal pressure, and due to dimensional changes in the dense carbon induced by fast-neutron irradiation. 4. The silicon carbide is dimensionally stable under fast-neutron irradiation and undergoes no creep. irradiation 5. Creep induced by fast-neutron
CALCULATINQ
coated
fuel particle;
323
STRESSES
b. Model
fuel particle.
of the dense carbon tends to relieve the stresses generated in the carbon coatings. The internal pressure was calculated by considering gaseous, condensed, and solid fission products as described in the previous paper. The method used to calculate the stresses in the coatings is to consider each layer independently, calculating displacements caused by the internal pressure, thermal expansion, creep, and fast-neutron-induced dimensional changes, and then to calculate interface pressures from the mismatch at the interfaces. The stresses and displacements induced in the carbon layers by the fast-neutron-induced dimensional changes and creep have been calculated using the equations developed by Prados and Scott 2). The dimensional changes as a function of fastneutron fluence (E> 0.18 MeV)* used in these calculations, have been taken from Bokros, Dunlap and Schwartz 3) and from Bokros, Guthrie, Dunlap and Schwartz 4). These authors have shown that under fast-neutron irradiation, carbons initially densify isotropically and subsequently anisotropically expand in the radial direction and contract in the tangential direction. Values used for the irradiationinduced creep constant have been calculated * made
In the subsequent to
fast-neutron
will be understood.
discussion where reference is exposure
this
cutoff
energy
324
J.
L.
from the data of the same authors using the suggested by Kennedy, among expression
KAAE
The interface pressures have been calculated from the equations
others, for polycrystalline graphite 5). As in the previous model, the equations have been set up to solve for stresses since it has been assumed that a maximum principal stress
and
failure criterion
where
applies for the materials
in the coatings. Using the method
described
above,
tangential and radial stresses generated coatings are given below. 1. Inner dense carbon : oTl(r)
= PO{r,5/(2+)}((r;
+ 2+)/(r:
used the
p; = (&/r3)/
in the
- $)}
-~l{~~/(2~3)}{(2~++~)/(~93--~)}+~Tl(~)ADC, cRl(T) = -PO(@3){(ri:
- @)/(r;
- r;)}
In these equations, 11 and 12 are moduli which represent the constraint across each of the interfaces :
-Pl(r~/~)((~-r~)/(r~-r~)}+(TRl(T)ADC.
2.
Silicon
carbide:
I1=
[(q
+ ri)( 1 -p2)/{2Ez(r:
- Q}]
+ (p2/E2)
+ [(2~~+~~)(l-~1)/(~~1(~~-~~)31-(~1/~1)~ 12 = [(2$
+ r;,( 1-
p3)/{2E3@
- r:))]
+ (p3/E3)
+ [(2~:+~~)(l-~2)/{2~2(+~~)}1-
3.
where
Outer dense carbon:
E is Young’s
modulus,
(/@2), and ,B is Poisson’s
ratio. oT3(r) =P2{r;/(21.3)}{($+
2r3)/(r; - r:)> + CT3(T)ADC,
0R3(9) = -P2(+3){(r;
-r3)/(r3,
- T;)} + CRS(r)ADC>
where GT(T)ADC= tangential stress induced in the carbon by anisotropic dimensional changes and creep caloulated from the equations of Prados and Scott 2), CTR(Y)ADC =radial stress induced in the carbon by anisotropic dimensional changes and creep calculated from the equations of Prados and Scott 2), PO = fission-gas pressure, PI =pressure generated at the inner carbon-silicon-carbide interface, P2 =pressure generated at the outer carbon-silicon-carbide interface.
The
mismatch
be considered
at the inner
to be made
up
interface,
61, can
of a sum of three
terms, 8; + Sy + S;‘, where S; is the expansion of the inner carbon due to the fission-gas pressure,
13; is the mismatch
due
to thermal
layer
expansion
differences,
TC2 is the ture,
and
silicon-carbide
deposition
the LX’Sare the coefficients
temperaof thermal
expansion
; ~3;’ is the mismatch and
creep
in the
&‘=T3[@Tl(r3)
due to dimensional inner
carbon
layer,
+ ((1 -~l)/&}‘JTl(~S)ADC],
changes
A
MATHEMATICAL
MODEL
BOR
and G*(r) is the accumulated radiation-induced and creep strain in the tangential direction. The mismatch at the outer interface, 82, can be considered to be made up of sum of two terms, &’ + Si’, where Sl is the mismatch due to thermal expansion differences,
CALCULATING
STRESSES
the other carbon layer through
326 the elastic dis-
placements produced in the silicon carbide. The interaction was solved by calculating the creep strain distribution in the inner layer by the usual procedure holding the outer layer constant, calculating a new creep-strain distribution in the outer layer using the inner layer values, recalculating the creep-strain distribution in the inner layer, and continuing until
and
Tc~
is
the
outer
carbon
deposition
temperature ; 8;’ is the mismatch due to dimensional changes and creep in the outer carbon layer, s;‘=
-r4[G~&4)+
((1 -j&%}~T3(~4)ADC].
In uniaxial tension, creep in the carbon was assumed to follow the steady-state expression for polycrystalline graphite 5) : PC= Kuj, where K is a creep constant, c is the stress, and p is the fast-neutron fluence. This creep law was extended to the triaxial stress state in the coating by assuming an equivalence of the octahedral sheer stress and shear strain so that
and .&a = K(CJR- CT)?. The method employed in calculating the creep strain distribution in each carbon layer was similar to that employed in the previous model, following
the type of iteration procedure
described by Mendelson, Hirshberg, and Manson 6). A transient creep constant is not employed in these calculations because the experimental technique used to determine the creep constant for pyrolytic carbon does not involve a state of constant stress; thus, an average value under conditions of increasing stress was measured. Also, consideration of transient creep at low fast-neutron fluences alone does not greatly affect the calculated stresses at higher doses. In the present case, it was necessary to consider the interaction between the two carbon layers in the creep calculation since creep in one carbon layer could affect the stresses, and thus creep, in
consecutively calculated values for the outer layer converged to within a preselected limit. Actually, the interaction was small because of the relatively high elastic modulus of silicon carbide. With an inner dense pyrolytic carbon layer, the fast-neutron-induced dimensional changes cause tensile stresses across the carbon-siliconcarbide interface and thus give rise to the possibility of interfacial separation if a weak bond is formed between the two materials. Such a possibility does not occur with the outer pyrolytic carbon because, except for small tensile stresses caused initially by differential thermal expansion, only compressive stresses are developed across this interface. The possibility of loss of bonding along the inner interface has been considered in the model by allowing separation at the interface if the stress across the interface, -PI, exceeds an arbitrary critical value. This is accomplished, once PI< -ctict, by setting PI= 0 and PZ = &/(r4Iz) as long as PI-c 0. If at any subsequent time 6i> 0, i.e., if the separation closes, the previous method of calculating PI and PZ is employed. Also, separation of the interface would allow considerable change in the volume available to fission gases by deformation of the single carbon layer; this is taken into account in a fashion similar to that described by Prados and Scott for a two-layer carbon-coated particle 2). The method involves recalculating the internal volume and the fission-gas pressure after each completed calculation of the Lreep strain distribution, then recalculating the creep strain distribution using the new pressure and continuing until consecutively calculated values of the pressure converge.
326
J.
L.
EAAE
.3. Results
from
In the four-layer coating, there are five positions at which the tangential stresses might
conditions the stresses in the inner carbon layer rise to high values after a rather low fast-neutron
lead to fracture of one of the layers. These positions are the inner and outer surfaces of
fluence and burnup, and then subsequently decrease. The rapid increase in stress is due
the inner dense carbon, the inner surface of the silicon carbide, and the inner and outer
to the initial densification of the carbon under fast-neutron irradiation, and the decrease is
surfaces of the outer carbon. Values of tangential stresses at these five positions as a function of
due to the increasing anisotropy of the dimensional changes and to creep. The stresses
fast-neutron fluence are shown in fig. 2 for the fuel particle and coatings with the assumed dimensions and properties shown in table 1. The assumed operating conditions are also shown in table 1. In addition, a strong (ati*,t > 8 000 psi) carbon-silicon-carbide interface .was assumed, as was lOOo/o fission gas release
in the outer carbon do not rise to as high values initially and do not subsequently decrease as rapidly. In both layers the stress on the outer surface eventually increases over that on the inner surface because of the anisotropy of the dimensional changes. The silicon carbide is placed in compression by the dimensional
the core.
As can
be seen, under
these
40
OUTER
SURFACE
OF
INNER
PyC
25 OUTER
INNER
SURFACE
OF
OUTER
PyC
INNER
SURFACE
OF
INNER
PyC
INNER
SURFACE
SURFACE
OF
OF
OUTER
PyC
SIC
-60
NEUTRON
Fig.
2.
Tangential
DOSE
etreasea in the ooating aa a fhnation ailioon-carbide
(NVT
X
lo*‘)
of fast-neutron interface.
fluence asmming
a strong carbon-
A MATHEMATIUAL
MODEL
BOB
OALCULATINB
STRESSES
327
TABU 1 Assumed fuel-particle dimensiona, prop&&a,
and operating aonditions.
A.
: Diameter ................. Th:u ...................
200 pm 3:l
Low-density oarbon Thickness ................. Density ..................
60 pm 1.2 g/cm*
Inner dense carbon Thiokness ................. Density .................. Modulus of &&i&y ............ Poisson’s ratio ............... Coefficient of thermal expansion ....... Deposition temperature ...........
26 pm 1.76 g/oma 4x10s psi 0.33 6.4 x 10-e (“C)-1 2000 W
Silicon carbide Thickness ................. Modulus of elasticity ............ Poisson’s ratio ............... Coefficient of thermal expansion ....... Deposition temperature ...........
20 ,um 60 x 108 psi 0.33 6.6 x 10-S (“Q-1 1700 ‘C
Outer dense carbon Thickness ................. Density .................. Modulus of elaatioity ............ Poisson’s ratio ............... Coefficient of thermal expansion ....... Deposition temperature ........... B.
26 pm 1.70 g/am* 4 x 10s psi 0.33 6.4 x 10-6 (“C!)-1 2000 “C
Operating conditions Temperature ................ Burnup ..................
1260 “C Linear with fast-neutron fluence, 4 fkione per 100 initial metal 8tmm for mh 1x10= nvt
Pyrolytio carbon creep constant at this temperature
............
changes of the carbon, but because the silicon carbide carries almost all the load imposed by the internal pressure, and because creep subsequently tends to relax the stresses imposed by the carbon, the compressive stresses decrease after an initial increase. The above conclusions hold only for the coating properties and conditions shown in table 1. For example, for a different density isotropic pyrolytic carbon at the same tempera-
3.0 X 10-s’ (pai*nvt)-1
ture, different dimensional changes apply, and for the same carbon at a different temperature, different dimensional changes and creep constant apply 4). In both of these cases quite different stresses would be calculated. Tests have indicated that the strength of the bond of pyrolytic silicon carbide to an asdeposited pyrolytic carbon surface is greater than 8 000 psi. For most coating designs, this bond strength is sufficient to withstand the
328
J. L. KAAE
tensile radial stresses developed at the interface up to fracture of the carbon due to the tangential stresses. Inclusion of a thin low-density, low-
entire load due to the internal pressure, at least as long as creep and dimensional changes do not return it to contact with the silicon
strength carbon layer between the inner carbon and the silicon carbide, however, should allow separation at the interface during irradiation.
carbide. It can be shown that under some combinations of burnup, fast-neutron fluence,
The effect of separation
and temperature, interfacial separation would be beneficial to the performance of the coatings,
at the carbon-silicon-
carbide interface on the stresses developed in the coatings on the fuel particle assumed above
and under other combinations
is shown in fig. 3. The bond strength of the interface in this case was assumed to be 3000 psi. Separation at the interface allows a decrease in the stresses induced by the initial densification of the carbon under fast-neutron irradiation. However, it also raises the possibility of high stresses in the inner dense carbon layer at a subsequent time since after interfacial separation the inner carbon must support the
tend to be beneficial are high temperatures, low burnups, and low fast-neutron exposures. Coated particles designed to allow interfacial separation are currently being irradiation tested. In the case of the previous model, the results of a number of experimental irradiation tests of fuel particles were available to compare with results obtained from the model, and the theoretical results correlated well with the
conditions
where interfacial
detrimental. separation
The
would
40
35
-
OUTER
INNER
SURFACE
OF
SURFACE
INNER
OF
INNER
PyC
PVC
25 t
OUTER
SURFACE
OF OUTER
INNER
0
I
SURFACE
2 NEUTRON
J?ig.
3.
PyC
OF OUTER
4
3 DOSE
PyC
(NVT
5
6
X 102’)
Tangential stresses in the coating as a function of fast-neutron fluence assuming a weak carbonsilicon-carbide
interface.
A
MATHEMATICAL
MODEL
FOR
experimental results even through the data describing the behavior of pyrolytic carbon under fast-neutron irradiation were incompletei). Subsequent calculations using more complete data where variations of the creep constant and dimensional changes with temperature have been included have not greatly altered the correlation 7). At the present time insufhcient experimental tests have been carried out to either confirm or deny the four-layer model. 4.
Conclusions
In summary, a previous mathematical model for calculating stresses in the coatings of a three-layer carbon-silicon-aarbide-coated fuel particle has been extended to the case of a four-layer carbon-silicon-carbide coating. The effects of increasing internal fission gas pressure, fast-neutron-induced anisotropic dimensional changes in the carbon after an initial isotropic densification, and fast-neutron-induced
CALCULATING
STRESSES
329
creep in the carbon on the stresses in the coatings during the life of the particle have been calculated for a typical particle. Fracture at the inner carbon-silicon-carbide interface has been considered, and the implications in terms of stresses and the overall integrity of the coatings have been discussed. References 1) J. L. Kaae, J. Nucl. Mat. 29 (1969) 249 2) J. W. Pradoa and J. L. Scott, Nucl. Appl. 3 (1967) 488 8) J. C. Bokros, R. W. Dunlap and A. S. Schwartz, Effect of high neutron exposures on the dimensions of pyrolytic carbon, to be publ. in Carbon 4) J. C. Bokros, G. L. Guthrie, R. W. Dunlap and A. S. Schwartz, J. Nuol. Mat. 31 (1969) 26 6) C. R. Kennedy, 26 Conf. Industrial Carbon and Graphite, London, 1965 (Society of Chemical Industry, 1966) 6) A. Mendelson, M. H. Hirshberg and S. S. Manson, J. Basic Eng. 810 (1959) 685 7) J. L. Kaae, D. W. Stevens and C. S. Luby, Gulf General Atomic Incorporated, unpublished data
OF NUCLEAR
32 (1969)
MATERIALS
A MATHEMATICAL
322-329.
0 NORTH-HOLLAND
MODEL FOR CALCULATING
PUBLISHING
CO., AMSTERDAM
STRESSES IN A FOUR-LAYER
CARBON-SILICON-CARBIDE-COATED
FUEL PARTICLE
J. L. KAAE chclf Benera+?Atomic
Incorporated, San Diego, California 92112, USA Received
A
previous
stresses
mathematical
in the
model
coatings
silicon-carbide-coated
on
carbon, The
are : low-density
silicon carbide, assumptions
mathematics of
the
coating.
used
in
model.
the
model
has been considered,
En g&&al,
the
a possibility
and in that
initiale
couche
of
interface
de carbone
interne
possibilit6
be considered
de contraintes
Spannungen
densification of the carbon
auf Vierfachschichten bestand
it raises the possibility
of high
particule
de combustible
permettant
lea revetements
de
aus:
mathematisches
betrachtet,
B 3 couches de carbone et
a &A Btendu au cas d’un
rev6tement
mit niedriger Dichte,
& 4 couches.
Lea 4 couches consid&+es
Verfiigung
dense, le carbure de silicium et le carbone isotropique
gerufen
dense.
des
traitement utilis6s
mathematique
dans
sont
le
modele
semblables
le modAle pr6c6dent.
et
le
Es wurde jedoch
stehonde Volumen
Behandlung wie bei den
die MGglichkeit
durch
Graphits,
nicht unabhiingig
scheinen
Spannungen,
bestrahlungsinduzierte
zur von
in
der
inneren
hervor-
Verdichtung
Graphitschicht
eine
zu spielen ala in der Busseren. Diese
Spannungen kijnnen durch Abtrennung
une
von
von der inneren Schicht.
allgemeinen
grdssere Rolle
8. ceux
Cependant,
gemacht,
dass sich die innere Graphitschicht
der Ent,fernung Im
dans
und
dichtem
Sic last. In diesem Falle ist das den Spaltgasen
sont : le carbone & faible densit& le carbone isotropique
utilis6es
Model1 urn
aus Graphit
ausgedehnt. Die Vierfachschicht
Graphit
Dreifachschichten.
sur une
de silicium
hypotheses
dens la couche de
Im Model1 und seiner mathematischen
de carbure
Lea
mais se pose alors la
&e&es
wurden iihnliohe Nliherungen pr&6dent,
de
interne
isotropen Graphit, Sic und dichtem isotropen Graphit.
stresses in the inner carbon at a later time.
dans
lea
SIC auf beschichteten Teilchen zu untersuchen, wurde
at the inner interface can relieve
lea tensions
par
dans le cas de la
que dans la couche
in Dreifachschichten
are more severe in the inner carbon than in the outer
TJn modele mathematique
dues I la
induite
of the inner
carbon. Separation
calculer
du
carbone interne dans un dernier stade. Ein bereits vorhandenes
but
carbone
peut relhcher ces contraintes,
In general, it appears that the stresses due to the
these stresses,
que lea tensions
du
carbone externe. Une separation 8, l’interface
coating. initial fast-neutron-induced
du dbplacement
neutrons rapides sont plus s&&es
case the volume
of the displacement
il semble
den&cation
are similar to those
However,
to the flssion gases cannot
to be independent
and
et dans ce cas
disponible pour lea gaz de fission ne pout
paa &re consid& indhpendant rev&ement interne.
dense isotropic
separation at the inner carbon-silicon-carbide available
le volume
The four layers
carbon,
196Q
carbure de silicium a Bt6 consid&&,
carbon-
and dense isotropio carbon.
of the calculations
previous
calculating
a three-layer
fuel particle haa been extended
to the case of a four-layer considered
for
6 January
der Schiohten
abgebaut werden, aber es ergibt sich die MGglichkeit,
possibilit6 de s6paration B l’interface interne carbone-
dass sich Spannungen
1.
layer, and a dense isotropic pyrolytic-carbon layer. That model has now been altered to consider the different coating design shown in fig. 1. As can be seen, it includes a second dense isotropic carbon layer inside the silicon carbide. This paper is intended to describe the mathematical
Introduction
carbon
In a previous paper, a mathematical model for calculating the stresses developed in composite silicon-carbide-carbon coating on a nuclear fuel particle during service was described 1). The model considered a three-layer coating consisting of a low-density pyrolytic322
layer,
zu spiiteren Zeiten aufbauen.
a pyrolytic
silicon-carbide
A
Fig.
1.
MATHEMATICAL
MODEL
FOR
a. Croes section of a four-layer
alterations in the model, to show some typical results obtained using the model, and to discuss some of the implications of these results.
2.
Model
The assumptions involved in the model have been described in detail in the previous paper 1). In summary, it is considered that: carbon does not possess 1. The low-density mechanical strength. Thus, it does not restrain the outer coatings and does not transfer loads from the swelling fuel to the outer coatings. 2. An internal pressure is generated due to released gaseous fission products. The volume available to the gases is that in the porosity in the low-density carbon. Solid and condensed fission products decrease this volume during the life of the fuel particle. 3. Stresses are generated in the coatings due to differential thermal expansion between the dense carbon and silicon carbide, due to the internal pressure, and due to dimensional changes in the dense carbon induced by fast-neutron irradiation. 4. The silicon carbide is dimensionally stable under fast-neutron irradiation and undergoes no creep. irradiation 5. Creep induced by fast-neutron
CALCULATINQ
coated
fuel particle;
323
STRESSES
b. Model
fuel particle.
of the dense carbon tends to relieve the stresses generated in the carbon coatings. The internal pressure was calculated by considering gaseous, condensed, and solid fission products as described in the previous paper. The method used to calculate the stresses in the coatings is to consider each layer independently, calculating displacements caused by the internal pressure, thermal expansion, creep, and fast-neutron-induced dimensional changes, and then to calculate interface pressures from the mismatch at the interfaces. The stresses and displacements induced in the carbon layers by the fast-neutron-induced dimensional changes and creep have been calculated using the equations developed by Prados and Scott 2). The dimensional changes as a function of fastneutron fluence (E> 0.18 MeV)* used in these calculations, have been taken from Bokros, Dunlap and Schwartz 3) and from Bokros, Guthrie, Dunlap and Schwartz 4). These authors have shown that under fast-neutron irradiation, carbons initially densify isotropically and subsequently anisotropically expand in the radial direction and contract in the tangential direction. Values used for the irradiationinduced creep constant have been calculated * made
In the subsequent to
fast-neutron
will be understood.
discussion where reference is exposure
this
cutoff
energy
324
J.
L.
from the data of the same authors using the suggested by Kennedy, among expression
KAAE
The interface pressures have been calculated from the equations
others, for polycrystalline graphite 5). As in the previous model, the equations have been set up to solve for stresses since it has been assumed that a maximum principal stress
and
failure criterion
where
applies for the materials
in the coatings. Using the method
described
above,
tangential and radial stresses generated coatings are given below. 1. Inner dense carbon : oTl(r)
= PO{r,5/(2+)}((r;
+ 2+)/(r:
used the
p; = (&/r3)/
in the
- $)}
-~l{~~/(2~3)}{(2~++~)/(~93--~)}+~Tl(~)ADC, cRl(T) = -PO(@3){(ri:
- @)/(r;
- r;)}
In these equations, 11 and 12 are moduli which represent the constraint across each of the interfaces :
-Pl(r~/~)((~-r~)/(r~-r~)}+(TRl(T)ADC.
2.
Silicon
carbide:
I1=
[(q
+ ri)( 1 -p2)/{2Ez(r:
- Q}]
+ (p2/E2)
+ [(2~~+~~)(l-~1)/(~~1(~~-~~)31-(~1/~1)~ 12 = [(2$
+ r;,( 1-
p3)/{2E3@
- r:))]
+ (p3/E3)
+ [(2~:+~~)(l-~2)/{2~2(+~~)}1-
3.
where
Outer dense carbon:
E is Young’s
modulus,
(/@2), and ,B is Poisson’s
ratio. oT3(r) =P2{r;/(21.3)}{($+
2r3)/(r; - r:)> + CT3(T)ADC,
0R3(9) = -P2(+3){(r;
-r3)/(r3,
- T;)} + CRS(r)ADC>
where GT(T)ADC= tangential stress induced in the carbon by anisotropic dimensional changes and creep caloulated from the equations of Prados and Scott 2), CTR(Y)ADC =radial stress induced in the carbon by anisotropic dimensional changes and creep calculated from the equations of Prados and Scott 2), PO = fission-gas pressure, PI =pressure generated at the inner carbon-silicon-carbide interface, P2 =pressure generated at the outer carbon-silicon-carbide interface.
The
mismatch
be considered
at the inner
to be made
up
interface,
61, can
of a sum of three
terms, 8; + Sy + S;‘, where S; is the expansion of the inner carbon due to the fission-gas pressure,
13; is the mismatch
due
to thermal
layer
expansion
differences,
TC2 is the ture,
and
silicon-carbide
deposition
the LX’Sare the coefficients
temperaof thermal
expansion
; ~3;’ is the mismatch and
creep
in the
&‘=T3[@Tl(r3)
due to dimensional inner
carbon
layer,
+ ((1 -~l)/&}‘JTl(~S)ADC],
changes
A
MATHEMATICAL
MODEL
BOR
and G*(r) is the accumulated radiation-induced and creep strain in the tangential direction. The mismatch at the outer interface, 82, can be considered to be made up of sum of two terms, &’ + Si’, where Sl is the mismatch due to thermal expansion differences,
CALCULATING
STRESSES
the other carbon layer through
326 the elastic dis-
placements produced in the silicon carbide. The interaction was solved by calculating the creep strain distribution in the inner layer by the usual procedure holding the outer layer constant, calculating a new creep-strain distribution in the outer layer using the inner layer values, recalculating the creep-strain distribution in the inner layer, and continuing until
and
Tc~
is
the
outer
carbon
deposition
temperature ; 8;’ is the mismatch due to dimensional changes and creep in the outer carbon layer, s;‘=
-r4[G~&4)+
((1 -j&%}~T3(~4)ADC].
In uniaxial tension, creep in the carbon was assumed to follow the steady-state expression for polycrystalline graphite 5) : PC= Kuj, where K is a creep constant, c is the stress, and p is the fast-neutron fluence. This creep law was extended to the triaxial stress state in the coating by assuming an equivalence of the octahedral sheer stress and shear strain so that
and .&a = K(CJR- CT)?. The method employed in calculating the creep strain distribution in each carbon layer was similar to that employed in the previous model, following
the type of iteration procedure
described by Mendelson, Hirshberg, and Manson 6). A transient creep constant is not employed in these calculations because the experimental technique used to determine the creep constant for pyrolytic carbon does not involve a state of constant stress; thus, an average value under conditions of increasing stress was measured. Also, consideration of transient creep at low fast-neutron fluences alone does not greatly affect the calculated stresses at higher doses. In the present case, it was necessary to consider the interaction between the two carbon layers in the creep calculation since creep in one carbon layer could affect the stresses, and thus creep, in
consecutively calculated values for the outer layer converged to within a preselected limit. Actually, the interaction was small because of the relatively high elastic modulus of silicon carbide. With an inner dense pyrolytic carbon layer, the fast-neutron-induced dimensional changes cause tensile stresses across the carbon-siliconcarbide interface and thus give rise to the possibility of interfacial separation if a weak bond is formed between the two materials. Such a possibility does not occur with the outer pyrolytic carbon because, except for small tensile stresses caused initially by differential thermal expansion, only compressive stresses are developed across this interface. The possibility of loss of bonding along the inner interface has been considered in the model by allowing separation at the interface if the stress across the interface, -PI, exceeds an arbitrary critical value. This is accomplished, once PI< -ctict, by setting PI= 0 and PZ = &/(r4Iz) as long as PI-c 0. If at any subsequent time 6i> 0, i.e., if the separation closes, the previous method of calculating PI and PZ is employed. Also, separation of the interface would allow considerable change in the volume available to fission gases by deformation of the single carbon layer; this is taken into account in a fashion similar to that described by Prados and Scott for a two-layer carbon-coated particle 2). The method involves recalculating the internal volume and the fission-gas pressure after each completed calculation of the Lreep strain distribution, then recalculating the creep strain distribution using the new pressure and continuing until consecutively calculated values of the pressure converge.
326
J.
L.
EAAE
.3. Results
from
In the four-layer coating, there are five positions at which the tangential stresses might
conditions the stresses in the inner carbon layer rise to high values after a rather low fast-neutron
lead to fracture of one of the layers. These positions are the inner and outer surfaces of
fluence and burnup, and then subsequently decrease. The rapid increase in stress is due
the inner dense carbon, the inner surface of the silicon carbide, and the inner and outer
to the initial densification of the carbon under fast-neutron irradiation, and the decrease is
surfaces of the outer carbon. Values of tangential stresses at these five positions as a function of
due to the increasing anisotropy of the dimensional changes and to creep. The stresses
fast-neutron fluence are shown in fig. 2 for the fuel particle and coatings with the assumed dimensions and properties shown in table 1. The assumed operating conditions are also shown in table 1. In addition, a strong (ati*,t > 8 000 psi) carbon-silicon-carbide interface .was assumed, as was lOOo/o fission gas release
in the outer carbon do not rise to as high values initially and do not subsequently decrease as rapidly. In both layers the stress on the outer surface eventually increases over that on the inner surface because of the anisotropy of the dimensional changes. The silicon carbide is placed in compression by the dimensional
the core.
As can
be seen, under
these
40
OUTER
SURFACE
OF
INNER
PyC
25 OUTER
INNER
SURFACE
OF
OUTER
PyC
INNER
SURFACE
OF
INNER
PyC
INNER
SURFACE
SURFACE
OF
OF
OUTER
PyC
SIC
-60
NEUTRON
Fig.
2.
Tangential
DOSE
etreasea in the ooating aa a fhnation ailioon-carbide
(NVT
X
lo*‘)
of fast-neutron interface.
fluence asmming
a strong carbon-
A MATHEMATIUAL
MODEL
BOB
OALCULATINB
STRESSES
327
TABU 1 Assumed fuel-particle dimensiona, prop&&a,
and operating aonditions.
A.
: Diameter ................. Th:u ...................
200 pm 3:l
Low-density oarbon Thickness ................. Density ..................
60 pm 1.2 g/cm*
Inner dense carbon Thiokness ................. Density .................. Modulus of &&i&y ............ Poisson’s ratio ............... Coefficient of thermal expansion ....... Deposition temperature ...........
26 pm 1.76 g/oma 4x10s psi 0.33 6.4 x 10-e (“C)-1 2000 W
Silicon carbide Thickness ................. Modulus of elasticity ............ Poisson’s ratio ............... Coefficient of thermal expansion ....... Deposition temperature ...........
20 ,um 60 x 108 psi 0.33 6.6 x 10-S (“Q-1 1700 ‘C
Outer dense carbon Thickness ................. Density .................. Modulus of elaatioity ............ Poisson’s ratio ............... Coefficient of thermal expansion ....... Deposition temperature ........... B.
26 pm 1.70 g/am* 4 x 10s psi 0.33 6.4 x 10-6 (“C!)-1 2000 “C
Operating conditions Temperature ................ Burnup ..................
1260 “C Linear with fast-neutron fluence, 4 fkione per 100 initial metal 8tmm for mh 1x10= nvt
Pyrolytio carbon creep constant at this temperature
............
changes of the carbon, but because the silicon carbide carries almost all the load imposed by the internal pressure, and because creep subsequently tends to relax the stresses imposed by the carbon, the compressive stresses decrease after an initial increase. The above conclusions hold only for the coating properties and conditions shown in table 1. For example, for a different density isotropic pyrolytic carbon at the same tempera-
3.0 X 10-s’ (pai*nvt)-1
ture, different dimensional changes apply, and for the same carbon at a different temperature, different dimensional changes and creep constant apply 4). In both of these cases quite different stresses would be calculated. Tests have indicated that the strength of the bond of pyrolytic silicon carbide to an asdeposited pyrolytic carbon surface is greater than 8 000 psi. For most coating designs, this bond strength is sufficient to withstand the
328
J. L. KAAE
tensile radial stresses developed at the interface up to fracture of the carbon due to the tangential stresses. Inclusion of a thin low-density, low-
entire load due to the internal pressure, at least as long as creep and dimensional changes do not return it to contact with the silicon
strength carbon layer between the inner carbon and the silicon carbide, however, should allow separation at the interface during irradiation.
carbide. It can be shown that under some combinations of burnup, fast-neutron fluence,
The effect of separation
and temperature, interfacial separation would be beneficial to the performance of the coatings,
at the carbon-silicon-
carbide interface on the stresses developed in the coatings on the fuel particle assumed above
and under other combinations
is shown in fig. 3. The bond strength of the interface in this case was assumed to be 3000 psi. Separation at the interface allows a decrease in the stresses induced by the initial densification of the carbon under fast-neutron irradiation. However, it also raises the possibility of high stresses in the inner dense carbon layer at a subsequent time since after interfacial separation the inner carbon must support the
tend to be beneficial are high temperatures, low burnups, and low fast-neutron exposures. Coated particles designed to allow interfacial separation are currently being irradiation tested. In the case of the previous model, the results of a number of experimental irradiation tests of fuel particles were available to compare with results obtained from the model, and the theoretical results correlated well with the
conditions
where interfacial
detrimental. separation
The
would
40
35
-
OUTER
INNER
SURFACE
OF
SURFACE
INNER
OF
INNER
PyC
PVC
25 t
OUTER
SURFACE
OF OUTER
INNER
0
I
SURFACE
2 NEUTRON
J?ig.
3.
PyC
OF OUTER
4
3 DOSE
PyC
(NVT
5
6
X 102’)
Tangential stresses in the coating as a function of fast-neutron fluence assuming a weak carbonsilicon-carbide
interface.
A
MATHEMATICAL
MODEL
FOR
experimental results even through the data describing the behavior of pyrolytic carbon under fast-neutron irradiation were incompletei). Subsequent calculations using more complete data where variations of the creep constant and dimensional changes with temperature have been included have not greatly altered the correlation 7). At the present time insufhcient experimental tests have been carried out to either confirm or deny the four-layer model. 4.
Conclusions
In summary, a previous mathematical model for calculating stresses in the coatings of a three-layer carbon-silicon-aarbide-coated fuel particle has been extended to the case of a four-layer carbon-silicon-carbide coating. The effects of increasing internal fission gas pressure, fast-neutron-induced anisotropic dimensional changes in the carbon after an initial isotropic densification, and fast-neutron-induced
CALCULATING
STRESSES
329
creep in the carbon on the stresses in the coatings during the life of the particle have been calculated for a typical particle. Fracture at the inner carbon-silicon-carbide interface has been considered, and the implications in terms of stresses and the overall integrity of the coatings have been discussed. References 1) J. L. Kaae, J. Nucl. Mat. 29 (1969) 249 2) J. W. Pradoa and J. L. Scott, Nucl. Appl. 3 (1967) 488 8) J. C. Bokros, R. W. Dunlap and A. S. Schwartz, Effect of high neutron exposures on the dimensions of pyrolytic carbon, to be publ. in Carbon 4) J. C. Bokros, G. L. Guthrie, R. W. Dunlap and A. S. Schwartz, J. Nuol. Mat. 31 (1969) 26 6) C. R. Kennedy, 26 Conf. Industrial Carbon and Graphite, London, 1965 (Society of Chemical Industry, 1966) 6) A. Mendelson, M. H. Hirshberg and S. S. Manson, J. Basic Eng. 810 (1959) 685 7) J. L. Kaae, D. W. Stevens and C. S. Luby, Gulf General Atomic Incorporated, unpublished data