crack nucleation due to elastic anisotropy in polycrystalline ice
TRANSCRIPT
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3 0 S S H Y A M S U N D E R A N D M S W U
are characteristic of purely brittle behavior.
The dislocation pile-up mechanism has subse-
quently been adopted by Schulson (Schulson, 1979;
Schulson et al., 1984) and Cole (1986) to explain
crack nucleation under tension and compression,
respectively. The study of dislocat ion etch pits in
deformed ice by Sinha (1978) which shows that
dislocations do pile-up at grain boundaries appears
to be the only microstructural evidence which sup-
ports the operation of this nucleation mechanism.
Schulson et al. (1984) explicitly associate nuclea-
tion-controlled tensile failure with the brittle be-
havior of ice, while Schulson (1987) goes on to state
that nucleation occurs under the same uniaxial stress
in compression as in tension.
During compressive creep tests on columnar-
grained polycrystalline ice, Sinha (1984) found no
evidence of dislocation pile-ups around micro-
cracks in hundreds of grains. He concluded that
stress concentrations resulting from dislocation pile-
ups may not be the mechanism responsible for crack
nucleation. In turn, Sinha (1982, 1984) postulated
that grain boundary sliding causes stress concentra-
tions at triple points between grains and at jogs in
the boundaries and that these stress concent rations
are responsible for crack nucleation. Since grain
boundary sliding is responsible for the anelastic or
delayed elastic component of strain in the creep
model of Sinha ( 1979 ), he goes on to sta te that this
mechanism of crack nucleation does not involve
purely elastic deformat ion and hence does not cor-
respond to truly brittle behavior. There is as yet no
consensus in the ice literature as to which of the two
mechanisms (dislocation pile-up versus grain
boundary sliding) is responsible for the nucleation
of strain-dependent microcracks.
The internal stress and strain fields in a polycrys-
talline aggregate are very inhomogeneous due to
crystal anisotropy and dislocation movement. Both
these sources of irregularity can lead to stress con-
centrations particularly at grain boundaries and
cause crack nucleation. Cole (1986) proposes that
the elastic anisotropy of ice crystals is specifically
responsible for the st rain-independent crack distri-
bution observed by Gold (1972) at high rates of
loading.
Cole ( 1988 ) has subsequent ly compared the dis-
location pile-up mechanism with the elastic aniso-
tropy mechanism. His analysis shows that while the
basal plane contains a sufficient density of glissile
dislocations to form the pile-up, the time required
to move the dislocation into the appropriate config-
uration (approximately 125 s) is much greater than
the experimentally observed times for the nuclea-
tion of first cracks (typically less than 20 s). He
concludes that the first cracks to nucleate are not
a result of the pile-up mechanism, but rather a re-
sult of the elastic anisotropy mechan ism . His sub-
sequent analysis shows that the two mechanisms
cannot be readily distinguished on the basis of either
the nucleation stress or the grain size dependency.
The analysis of the elastic anisotropy mechanism
by Cole ( 1988 ), however, is based on a highly sim-
plified model which considers just two isolated ice
crystals subjected to a uniaxial stress.The elastic
strain energy of each crystal is computed by assum-
ing that its response is uncoupled from the other
crystal. The difference in strain energy of the two
grains drives crack nucleat ion which, in effect, cor-
responds to a nucleation event driven by shear on
the grain boundary. The analysis also implicitly
predicts that crack nucleation occurs at the same
uniaxial stress in compression and tension.
This paper presents a theoretical analysis of crack
nucleation due to elastic anisotropy in polycrystal-
line ice. The microstructural stresses due to the
elastic anisotropy of individual crystals in an oth-
erwise isotropic polycrystal are analyzed using a
first-order approximation of the approach devel-
oped by Eshelby (1957). A similar first-order ap-
proximation has been used by Evans (1984) to
analyze the grain-boundary residual stresses in iso-
tropic ceramic polycrystals stemming from the
thermal expansion anisotropy of individual crys-
tals. The singularity of the stress concentration near
a grain-boundary facet junction due to elastic an-
isotropy provides the mechanism for inducing mi-
crocrack precursors, if similar nuclei do not already
exist as proposed by Gold (1972). The precursors
can nucleate into microcracks through the local in-
tensification of the applied and microstructural/in-
ternal stress fields.
The analysis of the stress required to nucleate mi-
crocracks and the incipient growth direction is based
on a solution to the general problem of an extending
precursor in a combined stress field. This solution
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CRACK NUCLEATION 31
u s e s t h e m a x i m u m p r i n c i p a l t e n s il e s t re s s M P T S )
c r i t e rion fo r mixed -mo de c rack ex t ens ion wh ich has
b e e n p r o p o s e d b y E r d o g a n a n d S i h 1 9 6 3 ) . T h e lo -
ca l ma te r i a l re s i s t ance m ay be cha rac t e r i zed in t e rm s
o f th e s u r f a c e en e r g y o f e i t h e r t h e g r ai n b o u n d a r y o r
t h e s o l i d - v a p o r i n t e r fa c e , b o t h o f w h i c h a r e s i m i la r
i n m a g n i t u d e f o r i ce K e t c h a m a n d H o b b s , 1 9 6 9 ) .
The e f fec t o f Co u lom bic f r i c t iona l re s i s t ance i s i n-
c luded in de f in ing the e f fec t i ve s t re ss d r iv ing the
p r e c u r s o r i n t h e s h e a r in g m o d e o f d e f o rm a t i o n .
Fur the r , i t i s pos tu l a t ed t ha t t he f i r s t c rack to nu -
c l ea t e i s a ssoc i a t ed wi th t he mos t favo rab le o r i en -
t a t io n s o f b o t h t h e p r e c u r s o r a n d t h e a d j o i n in g
grains.
T h e a n a l y s i s p r o c e d u r e i s a p p l i e d t o s t u d y t h e
p rob lem o f c rack nuc l ea t ion in po lyc rys t a l l ine ice
und er un i ax i a l a nd b i ax i a l s t a t e s o f l oad ing invo lv -
ing bo th t ens i l e and compress ive s t re sses . The e f -
fec t s o f g ra in s i ze on the nuc l ea t ion s t re ss a re a l so
exp lo red . The p red i c t i ons o f t h i s m ic ros t ruc tu ra l
m o d e l a r e th e n c o m p a r e d w i t h th e p h e n o m e n o l o g -
i ca l mode l based on a l im i t i ng t ens i l e s t ra in c r i t e -
r i o n t h a t h a s p r e v i o u s l y b e e n p r o p o s e d b y S h y a m
Sund er and Ting 1985 ) .
N L Y S I S O F M I C R O S T R U C T U R L
S T R E S S F IE L D
The ana lys is o f m ic ros t ruc tu ra l s t re sses due to t he
e l as t ic an i so t ro py o f i nd iv idua l c rys t a ls i n an o th -
e rwi se i so t rop i c po lyc rys t a l is based on a f i r s t -o rde r
a p p r o x i m a t i o n o f t h e E s h e l b y 1 9 5 7 ) p r o c e -
du re .T he f i r s t s t ep , in t he com ple t e t h ree -s t ep Esh -
e lby p roce dure see F ig . 1 ) , is t o sepa ra t e f rom the
so l id t hose m ic ros t ruc tu ra l fea tu res wh ich c rea t e t he
ma jo r pa r t o f t he m ic ros t ruc tu ra l s t re sses o f in t e r -
e s t. To ca l cu l a t e t he s t re sses i n t he v i c in i t y o f g ra in
boundar i e s due to e l a s t i c an i so t ropy in a two-d i -
men s iona l sys t em , th i s s t ep co r res pond s to sepa ra t -
i ng the g ra ins su r round ing the p recu rs o r o f i n te res t
s ince t hey wil l dom ina t e t he s t re ss fi e ld . The b ehav-
io r o f each o f t hese sepa ra t ed g ra ins i s de f ined in
t e rms o f t he e l a s ti c s t re ss -s t ra in re l a t ions fo r s ingl e
ice cry stals, i .e. :
~,=s,o I)
where a i s t he ap p l i ed s t re ss vec to r , c , is t he vec to r
ISOTROPIC M TRIX
X
n
0
Fig. 1. Schematic of three-step Eshelbyprocedu re for analyz-
ing m icrostructural stress field.
o f s tra in s i ndu ced in ea ch o f t he i nd iv idua l g ra ins
expressed in eng inee r ing no ta ti on as c = [ ~ Em ~=,
~ y z , 7 z x , 7 x y ] r
wi th T deno t ing the t ranspose ope ra -
t i on , and Sg is t he com pl i ance ma t r ix o f a s ing l e
c rys t a l i n t he chosen f ram e o f re fe rence . I f t he cho-
s e n f r a m e o f re f e r e n c e d o e s n o t c o i n c i d e w i t h t h e
s t andard f rame o f re fe rence fo r the hexagona l i ce
crysta l , then Sg i s g iven by:
S~ = Rs TS, ,Rs 2)
where Rs i s t he t h ree -d imens iona l ro t a t i on ma t r ix
and S , , i s t he co mpl i ance ma t r ix o f t he s ingl e ice
c r y st a l i n t h e s t a n d a r d f r a m e o f r e f e r e n ce b o t h o f
w h i c h a r e d e f i n e d i n A p p e n d i x A . T h i s c o m p l i a n c e
t enso r i s an i so t rop ic s ince i ce possesses a hexagona l
c rys t a l s t ruc tu re w i th f i ve i ndep ende n t e l a s ti c con -
s t an ts , no t j u s t tw o as fo r iso t rop i c m a te r ia l s . T he
dyna mic e l a s ti c cons t an t s o f s ing l e i ce c ryst a l s have
been de t e rm ined by seve ra l i nves t i ga to rs see , e .g .
G am m on e t al ., 1983 ) .
The rem ain ing m a t r ix i s a ssum ed to ac t a s an i so -
t rop i c med ium, hav ing the same e l a s t i c p roper t i e s
as the polycr ysta l l ine bod y, i .e .:
f fm ~ -- S m O ( 3 )
where Cm i s t he v ec to r o f st ra in s i ndu ced in t he ho -
mo geneo us ma t r ix a nd Sm is the e l a s t ic comp l i ance
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32 S . SHYAM SUNDER AND M .S. WU
tensor of the isotropic polycrystal given in Appen-
dix A and has the same components in all orthogo-
nal coordinate frames. The polycrystal elastic con-
stants can be determined through independent
testing or from the monocrystal data as outlined by
Gammon et al. 1983).
In the second step, the separated grains are de-
formed to fit into the cavity in the mat rix by apply-
ing surface tractions on each grain. Microstructural
elastic stresses are induced in the grain due to the
misfit strains and are given by:
O 0 ~,~-C g ( E m - - E g ) 4)
where Cg is the elastic stiffness matrix for single ice
crystals in the chosen frame o f reference which can
be dete rmined from the crystal stiffness matrix Cgs
in the standard frame of reference and the associ-
ated ro tation matrix Rc given in Appendix A. Using
Eqs. 1 and 3, Eq. 4 can be rewritten in the following
more convenient form:
~ r o = C g S m - - I ) o 5 )
The surface tractions on each grain can then be ob-
tained from the equilibrium conditions on the
surface.
In the last step, the separated grains are welded
back to the matrix and the surface tractions on their
boundaries are relaxed. This step creates an addi-
tional stress field trc in the whole body. The result-
ant microstructural stress field due only to the ef-
fects of elastic anisotropy can then be expressed as
the sum of,70 and tr. The total stress field crt is the
sum of the applied stresses and the microst ructural
stresses.
A complete solution obtained with the Eshelby
procedure by Evans 1984) for the problem of ther-
mal expansion anisotropy in ceramic potycrystals
shows that the microstructural stress distribution in
the vicinity of a grain-boundary facet junc tion may
be expressed as:
a x ) = [ I + F l n l / x ) ]ao,avg 6)
where I is the identity matrix, F is a diagonal matrix
with the elements representing shape functions that
take into account the crystallographic orientations
of the adjacent grains, angle between facets, etc., and
~ro,avg is the average of the microstructural stresses
given by Eq. 4 for the two grains adjoining the pre-
cursor under consideration. The variable l is the
length of the grain boundary facet which Cole
1988) has estimated to be about 0.56 times the
grain size d, and x is the distance from the grain
boundary facet junction.
Due to the logarithmic term in Eq. 6, the stresses
become singular near the grain junc tion as x tends
to zero. Unlike thermal stress concentra tions, stress
singularities caused by a geometrical discontinuity
along a bimaterial interface in elasticity are often
much more powerful than
l n l / x ) .
These singular-
ities are generally expected to be of the form
l / r ) ~ ,
where 0 < 2 < 1 and r is the radial distance from the
grain boundary facet junc tion see, e.g. Bogy and
Wang, 1971 ). The in tact material cannot sustain a
singular stress field since it exceeds the molecular
forces of cohesion. Consequently, the stress singu-
larity at the grain boundary junction provides the
mechanism for inducing microcrack precursors. In
the case of highly irregular grain boundaries as is
the case for an actual material specimen, stress sin-
gularities may occur at geometric discontinuities
such as steps and ledges on the grain boundary in
addition to those occurring at junctions.
Although the average microstructural stress does
not depend on the grain facet size, the scale effect
contained in the singularity allows the stress to be
sustained over a larger area o f grain facet as the facet
length increases. Consequently, the microcrack pre-
cursor will tend to be larger for larger grains. Al-
though the details of this process are not well-known
even for ceramic materials, it is reasonable to con-
sider ratios of precursor size 2a to facet length or
grain size that vary in the approximate range given
by 0.1
< 2 a / l < 0 . 2
or 0.05
< 2 a / d < O . l O .
This is the
typical length over which the influence of the sin-
gularity is a maximum.
The first-order approximation of the Eshelby
procedure is based on the average microstructural
stress field given by tro,avg, .e. the contribution o f ~rc
to the microstructural residual stress field which is
responsible for the singularity term of Eq. 6 is ne-
glected. Once a precursor has formed, the asymp-
totic stress field ahead of the precursor is taken to
follow the well-known square-root singularity of
linear elastic fracture mechanics and not the singu-
larity due to geometric stress concentrations which
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C R C K N U C L E T I O N
is derived from the analysis of an intact or defect-
free material.
The precursors can nucleate into microcracks
through the local intensification of the total stress
field which is the sum o f the applied stress field and
the first-order approximation of the microstruc-
tural/i nter nal stress field given by Eq. 5. I f the mag-
nitude of stress intensity is small, then either the
formation of the precursor may be prevented or if
the precursor does form it may not be able to nu-
cleate into a microcrack a crack typically of the size
of a grain). Equation 5 predicts that the precursor
is in general subject to both normal and shear
stresses. For example, when polycrystall ine ice is
loaded in uniaxial compression the equation pre-
dicts a normal t nsi l stress on a precursor that is
parallel with the loading axis and when the basal
plane orientation lies in the range given by
0 < < 25 or 65 < < 90 see Fig. A.1 in the
Appendix).
C R C K N U C L E T IO N C R I T E R IO N
The microcrack precursor or nucleus is in general
subject to a combined stress field, not only tension
or pure shear. In addition, its extension may take
place at an angle with respect to the orientation of
the precursor. Thus, the analysis of the stress re-
quired to nucleate microcracks and the incipient
growth direction must be based on a solu tion to the
general problem of an extending precursor in a
combined stress field.
s y m p t o t i c s t re s s f ie l d
As previously stated, the asymptotic stress field
ahead of the precursor is taken to follow the well-
known square-root singularity of linear elastic frac-
ture mechanics, not the more general geometric
stress singularity which is derived f rom the analysis
of an intact or defect-free material. Further, the pri-
mary effect of crystal anisotropy is to determine the
total remote stresses acting on the precursor; its in-
fluence on the local asymptotic stress field is con-
sidered to be negligible.
In the case of thermal anisotropy, Evans 1978 )
recognizes that the superposition of the logarithmic
C
O
It.
> -
I---
Z
li t
I
z
CO
iii
t r
o~
a
ii i
d
0
and nega t ive w hen 12 < 0 . The e f fec t ive shea r s t re ss
g iven in Eq . 9 is se t t o ze ro whe neve r the m agn i tude
of the s um of the f i rs t tw o terms , i .e ., 12 i s smal ler
than o r equa l t o t he f r i c t iona l s t re ss te rm .
The asympto t i c s t re ss f i e ld i n t he v i c in i t y o f t he
p r e c u r s o r c a n b e e x p r e s s e d a s th e s u m o f t h e s t r es s
f i e l d s d u e t o t h e d e f o r m a t i o n s i n m o d e s I a n d I I ,
respect ively :
a i j = K x / 2 z c r l / 2 f i j O +K II / (27rr)l/2filij(O ) ( 1 0 )
wh ere K~ and KII are th e s t ress - in tensi ty fa ctors in
mo des I a nd I I , re spec t ive ly , wh i l e f~o and f i j r e
t r igonom etric fun ct ions o f the angle 0 defin ed in F ig .
3 . The s t re ss i n t ens i t y fac to rs fo r s t and ard c rack ge -
om etrie s are avai lable in the l i tera ture (see , e.g. S ih ,
1973 ) . Fo r a t h rou gh th i ckness p re cu rso r o f l eng th
2 a i n an i n f in i te p l a t e , t hey a re l i s ted be low:
K I = 7 p p 7 ~ a ) 1 /2 1 1 )
K n = O'pr Tt'a)I/2 (1 2 )
I f avp i s com press ive , i t i s a ssum ed tha t t he c rack i s
c losed and conse quen t ly KI = 0 .
T h e a s y m p t o t i c d i s tr i b u t io n o f th e t a n g e n ti a l a n d
shea r s t re sses g iven by Eq . l 0 can be expressed as
fol lows:
C r o o = 2 7 ~ l r ~ 7 ~ c o s O ) [ g , c o s 2 O ) - 3 K i i s i n O ]
( 1 3 )
o )
t rr0 2 (2 r t r ) t / 2 cos [KI s in 0+Kxx(3 cos 0 - 1 ) ]
( 1 4 )
whe re r is the rad i a l d i s t ance f rom the t i p o f the p re -
cu rso r and 0 i s t he ang le measu red an t i -c lockwise
f rom a l i ne ex t end ing a long the p recu rso r i n t he so l id
as show n in F ig . 3 .
Th e s t ress aoo wi l l be the princ ipal s t ress i f t rio= 0 .
Th i s i s t he case fo r 0=0m where 0m i s found by
equ at ing Eq. 14 to zero (see , e .g . Brock , 19 86):
2 1/2
t n
Om K t~ + _ [ K ~ .,_~q
2
K n
L ~ K I I , ] A
( 1 5 )
The p r inc ipa l s t re sses co r resp ond ing to t he tw o va l-
ues o f 0m are then g iven as:
1 2
O 1 O 2 - ( 27 ~ r)T ~ CO S ( ~ ) I g I c s ( ~ ) -
3 KI, s in ( ~ ) 1
( 1 6 )
Accord ing to t he max imu m pr inc ipa l tens i le s t ress
c r i t e r ion , t he p recu rso r beg ins t o g row when the
ma x im um va lue o f t he p r inc ipa l t ens i le s t re ss g iven
by Eq . 16 has t he sam e va lue a s tha t fo r g rowth in
an equ iva l en t mode I p rob lem, i . e . equa l t o K l c /
/ t r ) 1 /2 . T h e g r o w t h c o n d i t i o n i s o b t a i n e d b y e q u a t -
ing Eq. 16 wi th th is q uan t i ty , i .e .:
K i c = K , c o s S ~ ) - 3 K i t c o s 2 ~ - ~ ) s i n ~ )
( 1 7 )
For g rowth under pu re mode I l oad ing , t he c r i t i ca l
va lue o f the s t re ss i n t ens i t y i s g iven by :
KI = p lane s t ra in (1 8)
and :
Kic = (2E y) w: p l ane s t re ss (19 )
where y i s t he g ra in bo un dar y su r face ene rgy Y~b, f
ex t ens ion occu rs a long a boundary , face t o r t he s o l id -
vap or su rface energy 7sv , i f the p recurs or ex tends in to
the crysta l .
Va lues fo r t hese two su r face ene rg i es have been
d e t e r m i n e d b y K e t c h a m a n d H o b b s ( 1 9 6 9 ) , i.e .
7gb=0 .065 J m -2 an d 7sv=0 .109 J m -2 . Fo r ex t en -
s ion a long the g ra in bou nda ry Ki t is equa l t o 36 .83
K P a m I /2
und er p l ane s t ra in and 34 .83 KPa m t /2
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36 S. SHYAM S U N D E R A N D M . S . W U
under plane stress with E= 9.33 GPa and v = 0.325.
The corresponding numbers for extension into the
crystals are 47.69 and 45. l0 KPa m ~/2. Since both
surface energies are similar in magnitude (and
stresses derived from the two estimates dif fer only
by about 29.5 ), precursors on the grain boundary
can easily extend in to the crystals if there are obsta-
cles to their growth on the grain boundary. Typi-
cally precursors may grow either parallel or perpen-
dicular to the basal plane under these conditions.
Although the surface energies are in general ex-
pected to be temperature sensitive, experimental
data on this va riation is unavailable at the present
time.
Once the crack nucleation criterion is satisfied,
the precursors start growing until either (a) the rate
of energy supply is inadequate, or (b) they are ar-
rested at obstacles such as neighboring junc tions or
crystals with unfavorable orientations and form a
stable microcrack. The microcracks nucleated in this
manner generally have a length which is propor-
tional to the grain size and which coincides with the
wavelength of the microstructural stress field. Cole
(1986) has found that the average crack length is
slightly smaller than (viz. approximately 0.6 times)
the mean grain size.
The instability condition for the propagation of
microcracks (as opposed to their nucleation) must
be based on a driving force supplied by the applied
stress field that can overcome the material resis-
tance offered by obstacles. This resistance may be
characterized in terms of conventional polycryst l
fracture toughness measures and is known to be
about 2-3 times the surface energy based resistance
for ice dete rmined in this paper. In general, there is
a transition from the smaller to the greater resis-
tance as the length of the nucleated crack increases.
Even if the applied stress field is biaxial, it is in
general not necessary for the tota l stress f ield to be
biaxial due to the contribution of the microstruc-
rural stress field. However, in the specific case where
the c-axes of the grains adjoining the precursor lie
in the reference two-dimensional plane the total
stress field is biaxial. Analysis of the microstruc-
rural stress field under an applied biaxial load shows
that out-of-plane s h e a r s a r c n o n e x i s t e n t s e e Figs.
18 and 19 in the Appendix), while the out-of-plane
normal stress has no effect on a two-dimensional in-
plane precursor. The present analysis relates di-
rectly to this situation, but is a good approximation
for more complex conditions (where the c-axis may
be oriented arbitrari ly) since the applied rather than
the microstructural stresses dominate the total stress
field.
D I C ~ O N O F ~ L P R E D I C T I O N S
C r a c k n u c l e a t i o n u n d e r u n i a x ia l t e n s i o n a n d
c o m p r e s s i o n
Cole (1988) has compared the stress levels at
which cracks nucleate due to the d islocation pile-up
mechanism and the elastic anisotropy mechanism.
Figure 4, reproduced from this paper, shows the
variation o f crack nucleation stress with grain size
for the two mechanisms. From this figure he con-
0 . .
O 3
O 3
L U
I -
O 3
z
0
t--