Crack Nucleation Due to Elastic Anisotropy in Polycrystalline Ice

8/11/2019 Crack Nucleation Due to Elastic Anisotropy in Polycrystalline Ice

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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--

Crack Nucleation Due to Elastic Anisotropy in Polycrystalline Ice

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