Interaction of a crack with a field of microcracks

hr. A&. Engng Sci. Vol. 21, No. 8, pp. Printed inGreat Bntain.

1009-1018.

002~7225/83/081009~10$03 Oil/O Pergamon Press Ltd.

1983

INTERACTION OF A CRACK WITH A FIELD OF MICROCRACKS

Alexander Chudnovsky*, Mark Kachanov** * Department of Civil Engineering, Case Western Reserve University, Cleveland Ohio,44106;*bepartment of Mechanical Engineering, Tufts University,Medford Massachusetts, 02155, U.S.A.

(Communicated by F. Erdogan) (received December 10, 1982) ABSTRACT Crack propagation in brittle materials is oftenaccompanied by intensive microcracking; being a major energy sink,this phenomenon can strongly affect the fracture process. A twodimensional problem of elastic interaction of a macrocrack with a field of microcracks is considered in the article. Consideration is based on the self-consistent method, generalized with the account of strong non-uniformity of the stress field in the vicinity of the macrocrack. The technique of double layer potentials is used. A closed form solution for the effective stress field is constructed.

Introduction Microscopic that of

“damage”

the crack fied

observations

in many cases

(generated,

tip.

of

presumably,

intensive

of microstructure

on fracture zones

In some cases

as a region

defects

fracture

(see

propagation

are very by high [2,31,

microcracking

can be reasonably

in brittle

different tensile

for

from ideal stresses)

example)

or crazing, modelled

phenomenon has been modelled as a "crack-layer" the crack-layer kinematics have been identified.

this

materials cuts. develops

[l-3]

indicate

Typically,

a zone

in the vicinity

zone can be clearly

in many other

as a field

cases

of

identi-

the developing

of microcracks.

This

[8,9] and driving forces conjugate to Further progress requires knowledge

of the stress field around the damage zone, i.e. understanding of the micromechanics of the crack-layer. Microcracks, distribution

interacting with the main crack and with each other, change the stress

in the vicinity of the macrocrack tip.

crack area is a major energy sink in the process

Also, since creation of the micro-

of fracture propagation, the presence of

a microcrack field can strongly affect the energy release rate and make characterization of the fracture toughness of a brittle material by a single parameter K Ic questionable. The scheme for the analysis of interaction between a macrocrack and a microcrack array is proposed here.

It is based on a self-consistent method modified with the account

of high stress gradients in the vicinity of a macrocrack.

1009

A. CHUDNOVSKY

1010 Formulation

of the Problem:

A two-dimensional,linear

and M. KACHANOV

Small Scale Microcracking Model

elastic solid containing a macrocrack (-!L.II)and an adja-

cent microcrack array is considered.

The statistics of the microcrack distribution will

be assumed given; its details are not discossed here.

The stress field in the solid with

cracks can be represented as a superposition

is the stress field due

where am I

to remotely applied loads in absence of cracks and 'Jo

is the stress field generated by the a-th crack with the faces loaded by traction $($B)a

where 2' is the unit normal to the a-th crack and b'_"B is the stress field that

would have been generated by the rest of the crackintheabsenceofofthea-th

crack; (

) a

denotes the value taken on the line Sa of the a-th crack. The basic idea of the self-consistent method (in the modification used here) is that each microcrack is assumed to be imbedded into a certain "effective" (not necessarily unieff form) stress field a taken as a superposition of grnand a sum eE of the field generated eff by all the microcracks imbedded into the 0 field ("self-similarity"). Thus, the supereff position representation of e(K) is substituted by the following representation of (!

:

eff(5)

(!

= g-(x)

+

&)

Note that the self-consistent method is widely used in physics of many particles systems; the "effective" field is usually assumed to be uniform. themethod-

Two modifications of

the'leffective field" model, with cracks being embedded into a (uniform)

effective stress field, and the "effective matrix" model, with cracks contained in the "effective" elastic matrix-havebeen

applied in finding the overall elastic properties

of cracked solids (see for example [4,5]).

In our case the "effective" stress field,

being affected by the macrocrack tip singularity , can no longer be assumed uniform. Therefore, a more general form of the self-consistent method is introduced here toaccount for the mentioned non-uniformity. First, we "ill consider the simpler case when the zone of microcracking Occupies Only a small vicinity of the macrocrack tip.

For simplicity of calculation we assume, in

addition, the Mode I (tensile)loadingatthemacrocrack; of thamicrocrackfieldiifsymmetrical

thisisnotaffectedbythepresence

with respect to the macrocrack axis.

In this case

the macrocrack tip-dominated stress field 2' plays the role of Q-so that eff 0

where g

z

(5) = $(l$) + &)

(1)

is the stress field induced by the entire microcrack array and ec can be repre-

sented in the usual form for a KI - dominated field, with the substitution of KI by a eff accounting for the influence of the microcrack array on the main crack: certain KI eff KI UC(x) = E 9(S) s KTff a'(x)

(2)

An assumption used here (and requiring, in principle, a further evaluation) is that the presence of microcracks does not alter the l/v'; character of the stress singularity.

Interaction of a crack with a field of microcracks

1011

, with crack opening displacementsbeing the A double layer potentialtechnique Potentialdensities,will be used for the representationof the stress field e'(x). This technique,being generallyconvenientfor the formulationof the crack problemsin &asticity (6,7], is particularlywell-suitedfor the incorporationofastatisticaldescriptio* of a random microcrackarray. Thus. the stress field inducedby the a-th microcrackis representedin the form,

(3) where b_'(t)is the a-th microcrack'sopening displacement,Q(x, x) is Green's function, and T is the stress operator transformingthe displacementfield ~(5) = IS

into a stress field.

b(S),$!(x. S)dS

The followingsimplifyingassumptionis introduced. Although the effectivestress field (1) is non-uniform,we assume that it does not change much within the lengthof each of the microcracksand can be taken approximatelyconstantalong each of them. (This assumptionis introducedmerely for simplification of calculationsand is not essential for the method employed.) Then the opening displacementof the a-th microcrackis ha(x) = f pa.oeff(no)ao

aa

(4)

where eafs) is the upper half of the ellipse of a unit maximal openingwith the extremities at the tips of the crack and E is the Young's modulus;go stands for the center of the a-th crack and x denotes a coordinateaxis along the crack line with the origin at a :o; p and 2aa denote a unit normal to the a-th crack and its length. (Note that nonuniformitiesof the effectivestress field along the microcrackscan be accountedfor by introducingthe shapes of the openingdisplacementthat are more complex then elliptical. These

refinementsare beyond the scope of the presentarticle.) The overall stress field inducedby the microcrackarray is a superpositionof the

fields (3) inducedby individualmicrocracks:

where &(s,P) is the tensor of Influenceof the a-th microcrack,definedby the relation

Thus, $

is expressedin terms of the values of the effectivestress field at the

microcrackcenters and the equationof self-consistency takes the form:

eff

0

(II1

=

,Fff oo(z)t

J1 ~(5,

$3: ~~~~~~~~

(7)

The second equationof self-consistency is formulatedfor the effectivestress ineff at the tip of the main crack (-II,E). The microcrackarray induces tensity factor KI tractionspo.~L(FJ~"oat the points 5 CC-%, &),

1012

and M. KACHANOV

A. CHUDNOVSKY (no is

a unit

normal

to the macrocrack)

so that,

(8) where KT is

the stress

intensity

factor

due to remote

loading

in the absence

of micro-

cracks. Tensorial geff

equation eff and K I *

(5)

(7)

Solution The equation S = 1,

(7)

and scalar

for

equation

the Small Scale

can be reduced

(8)

constitute

a system

Microcracking

to a system

of

equations

for

Model

of algebraic

equations

by taking

L[ = z*

. . . N:

eff($) =KTff

L?

co($)

+ ail

& ($.

$):cJ~~~($

(7’)

a#t3 SO that

the system

equations

is

reduced

to a system

6N + 1 unknowns - values u::’ eff and K I * This system, being introduced into (7), gives

(z’)

centers

solution,

Taking

into

the microcrack

consideration,

array

however,

(positions

ra,

becharacterizedonlyin

approach

to the system

as given

by (7)

tation

(8)

for

microcrack

usually

(7’).

back

of 0 eff

the effective

the effective

stress

N is

large

the integrand

in

field

and that

a a and orientations

Successive

(8).

stress

can be solved

that

sizes

linear

0

algebriac

field exactly; eff

the configuration

na of microcracks)

are obtained

They result

(7).

its

.

wesuggestanalternative

iterations

at the

of can

iterative

by introducing

in the following

0

eff

represen-

(5): N

eff(5)

- KFff $(x)

!!

+K

eff I

N

form,

this

+ KFff

Z &(x, a=1

$+:~“(z+

+

2ja2):go(za2)

+

N

z al=1 a&l 9

In compact

of

6N + 1 scalar

in principle,

probabilisticterms,

(7),

into

of

L(lc, $1):

LJzal,

. . .

(9)

+ a2

series

can be written

as. (10)

where i(m)

_ i(m)

(r

_._ xol s...s_ Pa)

=

z al=1

...

a2#aL, is

a “transfer”

function

characterizing

interaction

z L(x, am-1 - I

a3b2. of

$1):

: Lcxam-l _ I ,x am)

...

. . . amzO,_l

a microcrack

with

(11) the macrocrack

Thus, the o-th term in (10) successive coupled interactions of microcracks. eff The I-st order term gives stresses generated by the - dominated field. gives the KI eff - dominated miciocrack array, with each of the microcracks being embedded into the KI through

n-l

field

(more precisely, being loaded by a uniform traction eff 0 p (5) at the microcrack center). This by the field KI preted

as accounting

for

the interactions

between

each of

equal

to the traction

term can,

therefore,

the microcracks

induced be inter-

and the

Interaction of a crack with a field of microcracks macrocrack.

1013

The IX-nd terms accounts for couple interactionsbetween microcracks: it

gives stresses generatedby the microcrackarray, with each of the microcracksbeing subjectedto the stresses induced by other microcracks,the latter being embedded into eff the K - dominated field. Similarly,the III-rd term takes into considerationthe I "triple" interactionsof microcracks,etc. This is illustratedby the diagram in Fig. 1.

0-th

order term

I-st order term

term

term

FIG. 1

Introducing(10) into the second equationof self-consistency(8) and solving for eff we obtain: KI a1

eff _ KI

(1

l-

N g+!+f a_S (CoCo): c ;(c, $):[cO(fQ) + Ha=1 -R

Expre$sions(10) and (12) constitutea solutionof the system (7). (8) provided the geometryof the microcrackarray is known. Note that with the obvious modificationfor the macrocracktip field, a mixed type loading can be consideredwithin the same framework. We considerthe microcrackarray to be random (only probabilisticcharacteristics of it are known). Then (10) representsthe effectivestress field correspondingto a certain particularrealizationof the possiblemicrocrackgeometries. Then, introducing a configurationalspace Ra

of the u-th microcrack,with the elementswa = {za, aa, ?a]

characterizingits position, size and orientation,and an averaged "transfer"function

1014

and M. KACHANOV

A. CHUDNOVSKY


$(m+x.nal PK-1,

(x*x )> =

I...,

xa) f (Ual ,....lPm-1)

n%n=1

no1

dual (a #a #...#a 1 2 we can write

the following

stress

generated

tip

field

) Ill-1

. . . dw”m-1 (13)

expression

for

the mathematical

by a random microcrack

array

expectation

in the vicinity

of of

the effective

the macrocrack

: 0

=,tff[O”W+ KilaL,j eff(g)

<~(K)(15,xa),:~ocn*,fcwa,dwa

(14)

sza

and the corresponding intensity

factor

The infinite where d is This

form stress investigation9

is

(14)

between

improvement

of

highly of

to estimate

beyond

estimate

ones,the

the latter

will

Consider fit

into

and a’

< E < I

is the microcrack

by a crack

length.

embedded into

distance

as l/r.

this

article;

the possibility

of

the stress

field

a uni-

Further of

for

by “successive

It creates

can be expected.

However,

common, more “chaotic”

difficulconf igu-

improved. Case

in the form

The stress

field

(1).

with

the difference

that

0’

can no longer

(2).

a’(x)

as before

(15)

= (Em+ (Ec + QC

have the same meaning as in

be represented

(5)

c&>

case when the microcrack array is large enough and does eff dominated field and the macrocrack is subjected to the KI eff in the the stress field, we represent 0 Separating, as above, eff rz

pression

if

of

form

C

at least

now the general the region

a mix mode loading.

where a

stress

depend on the geometryofthemicrocrackfield.

by the microcracks

can be substantially

the effective

with

of

amplification

precisely.

and 2a

generated

the scope

of

(8)).

centers

field

General

not

expectation

into

to a and decreases

a convergence

estimate

the

the stress

is

“ordered”

(14)

can be shown to be convergent

proportional

the above given

transformation”

the mathematical

the microcrack

that

ofconvergences

For certain

rations,

in

on the fact

field

for

by introducing

series

the distance

is based

ties

expression

(obtained

generated

by the microcrack

and can be written

in the short

array

is

given

by the same ex-

form

(16)

with

the definition

of

(5).

Operator

rated

by a microcrack

M transforms

The stress the double

the operator

layer

field

the effective

array a’

M directly

embedded into

induced

following

stress eff

a

.

by the main crack

potential: II 0%

field

=

I -p.

b(S)+

from a comparison Of (16) with eff into the stress field genea

(z.S)dS

can also

be represented

in terms of

Interaction of a crack with a field of microcracks The crack

opening

correspond array,

displacement

to the remote

loading

correspondingly.

elastic

solid

with

k(c)

z as a sum Q = bCO+ kc where b” and b,

can be written

and to the stress field g’, induced by the microcrack I: b_ and Qc can be assumed knownfromthesrrlutionfor an

Relation

a crack

1015

loaded

by arbitrary

traction

and can be written

in the form

a. = 1 5%

$(Sf

E;.$ :&Sl)dc;l

(13)

R where R (x, field cracks)

This

y) is a certain

generated

qxpression

where definition Operator

of

embedded into embedded into

Thus,

with

(x)

=

loading

_ I bQ1(5)*(t)(F,5)d# (stress inkthe absence of micro-

gc:

form

MC and McC follows

from comparison

of

(18)

with

(19).

0” Into the stress field generated by the macroeff M :o is the stress field generated by the macro-cz array-generated field CC, with microcracks being

gW; similarly,

the account

for

&

loading

the microcrack eff to the stress field 2

subjected

of

.

( 16) and (19 1, the equation

of

self-consistency

(15)

the form,

Note that if

remote

Denotrng to remote

representation

in a short

the operators

MC transforms

crack

subjected

the following

can be wrFtten

crack

takes

known function.

by the macrocrack

WE!can write

R

the previously

the first

term is

taken

influence of

cansidered

term gW is omitted

case

to be the KI -dominated the microcrack

(small

from (201,

array

field

scale

microcracking

the second

term is left

KI~*(~X and the fourth

zone)

term,

on the main crack,is interpreted

is recovered

unchanged,

the third

reflecting

as substitution

the of

KI by KFff. The action is

shown although

of the operators the action

in

(20)

of MC and M 4

is

illustrated involves

all

in Fig,

2 (only

microcracks).

one microcrack

fOI6

A. C~ffD~~V~~Y

and

M.

KACHANOV

The same Iterative approach as before (alternative to two systems of

bN linear

algebraic equations obtained by substituting 5 = z B into (21)) can be applied to equation (20).

Successive iterations yield the following expression for the effective

stress field:

where the m-th order operator (McC + MC)m is additional

Note that the

(as compared with (7)) terms in equation (20) result in the appearance of

new terms inthesolution. involved if only

Diagrams inFig. 3 illustratethephysicalmeaningof

the first two members of the sum are retained.

into two columns. crack

defined similarly to (13).

the terms

These terms are divided

The first column contains the field g: = ?Jc,:c generated by the main

(subjected to the remotely

applied field @)

and the "secondar$' stress fields

successive transformations of 0; by the microcracke and by

the main crack. Thus, the

terms of the first column represent the main crack-dominated fields.

eff

i!

FIG. 3 The terw

af the eecond column show the ~tfeee field eE:p" generated by the

mfctocrack array (embedded into the ca- field) and the "secondaryW stress fields succasalve transformations of the field I$:om

; thus, these terms represent the micro-

crack array - dominated fields. 'Pwo physically different special cases can therefore be distinguished.

If the terms

of the first calumn are dominant , the effective stress field is determined, basically, by the main crack; micorctacks play only a "correctional" role.

(With the additional

assumption that Qc QD - I$** a'(x) the previously , canotdered case of emall scale microcracking, is then recovered.)

If, on the other hand, the terms of the second column are

dominant, then the presence of the main crack is so "shadowed" by the microcrack array

Interaction of a crack with a fieldofmicrocracks

1017

That the latter essentially determines the structure of the effective stress field.

Note

that such extensive zones of "diffuse" damage have been found to accompany fracture propagation in various materials [1,2,3,...]. To take into account a random character of the microcrack array, statistical averaging (similar to (13)) is to be introduced into the operators MT and McC involved in the transformations of a stress field by the microcrack array; then the solution (21) will represent a mathematical expectation of the effective stress field, modification,

With this obvious

(21) provides a general expression for the effective stress field in the

vicinity of a macrocrack propagating in a random field of microcracks. On the Energy Release Rates for the Crack-Microcrack Array System The above mentioned concept of crack-layer (a phenomenological model of a crackmicrocrack array system) implies additional "degrees of freedom" in the process of fracture propagation, thus giving the rise to additional terms in the overall energy release rate.

They are associated with translation, isotropic expansion, shape change and change

of the direction of propagation of a crack-layer and can be expressed in terms of path eff derived above make it possible to express the integrals [8,9]. The formulae for 8 energy release rates corresponding to the mentioned "degrees of freedom" in terms of the geometry of the crack-microcrack array configuration. Consider, for example, the J-integral, representing the part of the overall energy release rate associated with the translational component of the crack-layer propagation. Assuming x1 to be the direction of translation, we have

J =

[-g.(~~~~*(pO + e’),,

+ $(geff:

$: geff)nlldr

i r where

so and

sL are the displacement fields generated by the main crack and by the micro-

cracksrray,correspondingly,$$is

the elastic compliance tensor and c is the unit normal to

an (arbitrary) contour l'encircling the microcrack aone. In the case of a small microeff eff P ,eff * KY + AKI one obtains crack zone, substituting CI go + a' and assuming KI I the representation of J aa a sum:

J - J; + Jc + JCL

where

is the energy release rate associated with the macrocrack in the absence of microcracks

JI =

+ +_1: I

5: &lldr

is the part of J associated with the microcrack array and eff -Q KI e" zl

+

KTff $':$:~']nl}dr

corresponds to the crack-microcrack array interaction. Substitution of the above obeff tained formulae for oE and KI into Jz and JCL will result in further detalizations of J (in terms of double,

triple and higher order crack-microcracks interactions).

Note that a similar consideration of 3 is possible in the general case of non-small microcracking zone if geff in (22) is taken from (20).

1018

A. C~UD~OVSKY

and M. KACHANOV

Acknowledgement The authors wish to acknowledge partial financial support from the NASA Lewis Research Center Grant #NAG 3-223.

Gratitude is also extended to Professor Fazil

Erdogan for his useful discussions and comments.

References I..

Hoagland, R.G., Hahn, E.T. and Rosenfield, A.R., “Influence of Microstructure on Fracture Propagation In Rocks", Rock Mechanics, 2, 77 (1973).

2.

Takemori, M.T., Bankert, R.J., Moet, A., Chudnovsky, A., "Fatigue Crack Layer Propagation in Polypropylene", to be published.

3.

Botsis, .I.,Meet. A., Chudnovsky, A.. "Fatigue Crack-Layer Morphology in PolyStyrine", to be published.

4.

Kanuan, S.K., "Random Field of Cracks in Elastic Continuum", in "Issledovania po Uprugosti i Plastichnosti" (Papers on Elasticity and Plasticity), ed. I.. Kachanov, lo, 66 (1974) fin Russian).

5.

Budiansky, B. and O'Connell, R.J., "Elastic Moduli of a Cracked Solid", Int. J. of Solids and Structures, 2, 81 (1975).

6.

Kupradze, V.D., "Potential Method in the Theory of Elasticity", Israel Program for Scientific Translation, Jerusalem (1965). Lur'e, A.I. "Theory of Elasticity", Nauka. Moscow (1970) (in Russian). Khandogln, A., and Chudnovsky, A., "Thermodynamics of the Quasistatic Crack Growth", in "Dynamics and Strength of Aircraft Constructions", ed. L. Kurshin, ft. (in Russian) Novosibirsk. pp. 148-175, (1978). "ATheory of Long-Term Strength in Fatigue and Creed: Transactions of Damage Workshop (D. Stautterd ed.) Kentucky, pp. 79-87 (1980).