Crack bifurcation in brittle solids

J. Mech. Phys Solids, 1968,Vol. 16,pp. 329to 336. PergamonPress. Printedin Great Britain.

CRACK

BIFURCATION

IN BRITTLE

SOLIDS

Ry E. SMITH Central Electricity Research Laboratories, Leatherhead, Surrey (Received 10th May 1968) SUMMARY CLASSIC thermodynamic considerations are employed to derive the condition for bifurcation of a crack situated within a brittle solid deforming in an anti-plane strain mode. Criteria are determined for bifurcation along circular segments at right angles to the initial crack, and also along planar segments inclined at an arbitrary angle with respect to the original crack, the results being used in a brief discussion of physical situations where bifurcation occurs.

1. SUBSEQUEKT

to the classic

been given to determining when

various

work

of

GRIFFITH

the criterion

(1%X),

considerable

for a crack to propagate

attention

in a brittle

has solid

stress or geometrical

was based on thermodynamic been employed in discussions little

INTRODUCTION

or no attention

conditions are imposed. Griffith’s approach and although this has frequently considerations, of the behaviour of continuously planar cracks,

has been given

to the determination

of the criterion

for a

Crack bifurcation is clearly an important practical problem, crack to bifurcate. since it is intimately related to the fracture behaviour of inhomogeneous solids, for example materials containing second phase particles or fibre composite structures. In view of this importance

the present paper, by extending

the Griffith approach,

determines the condition for bifurcation of a planar crack situated within a body Section 2 derives the criterion for deforming under anti-plane strain conditions. bifurcation along circular segments at right angles to the initial crack, and also considers bifurcation along planar segments inclined at an arbitrary angle with respect to the original crack, while Section 3 discusses the results in relation to physical

situations

where bifurcation 2.

2.1.

Mathematical In anti-plane

occurs. THEORY

preliminaries

strain deformation

the displacement

at all points of a solid and satisfies Laplace’s

there existing

a complex

function

ze, is normal

to the xy plane

equation

F (x), with z = z + iy*, such that w =kRe[F(z)]

*Although z is used in this context, there should be no confusion with tbe z arising as a suffix in p,, etc. 329

(2)

15. SxrTw

330 where TVis the shear modulus

of the material.

‘I’he appropriate

stress components

are given h-y

or, expressed

in polar co-ordinates

(r, 8).

In employing potential theory techniques to investigate anti-plane strain deformation problems, the standard procedure is to transform the x plane into some other t (I- E +- in) plane where the determinatjon of conformal

transformations dF

dF dt =z’&

N being the outward 2.2.

Bijwcation

of F is facilitated,

using the properties

: dr

normal to the ET plant.

along circlclar segment3

The model is illustrated in Fig. I. A crack EA’ .BCL)A”, whose surface is stress-free, is situated within an infinite elastic body (shear modulus CL)subject to the externally applied shear stress

Fm. 1. A crack EA’ BCDA”, whose surface is stress-free, situated within an infinite bodl subject to the externally applied shear stress f)v2 = 0. The complete crack is a planar segment BA’ A” of length 2c, together with circular segments extending to the points TI and D. which subtend an angle 2a at the centre of a rircle of radius (6.

puz = @. The complete crack consists of a planar segment ISA rl” of length 2e, together with circular segments extending to the points R and II), which subtend an angle 2a at the centre of a circle of radius n. Thr ronformal tr~nsforIl~ation mapping the mmplet~e crack in t.he z plane into

Crack bifurcation

in brittle solids

331

FIG. 2. The t (= E + in) plane obtained from the z (= ;L’ + iy) plane of Fig. 1 by the conformal transformation described by relations (6) and (7) ; corresponding points are as indicated.

a planar crack in the t plane (Fig. 2) is given by the pair of equations

I_%

I 1 zs”

(7)

X

corresponding points transforming as shown (Figs. 1 and 2); for large z and t (6) and (7) reduce to z = 21 sin (a/Z). The boundary conditions in the t plane are P&J = 0, pa, = 20 sin (a/2) at infinity, iV being the outward normal to Fig. 2, while the surface of the planar crack along the e axis is stress-free. The complex function giving the displacement and stresses at all points of the original solid is therefore E’ = -

2i0 sin (a/2) [(t -

cr)s -

(a + cr)a]:,

(6)

t and z being related by (6) and (7). Thus, consider a point z = ael(a-*a) on the crack surface near the tip B (Fig. I), i.e. 0< &~
20 sin a

a [u sin (42)

6r 8

fz tan* (a/2)

0a

[a” Cns2(42)

+ cr]

+ 2acr (1 - sin (a/2))]4-

’

(9)

cr being expressed in terms of c by the pair of equations (I -t 2c n + 2c and (IL + 2c,)

0 <

a + 2ce -

free2a ae-ia -[ a

= -

2

iue~~/a 2

(10)

a + 2cs + iae-la/a 1

(a + 2co)

I -

n

U -

-5 a f

2cs) I

.

(11)

LUternatively, consider a point z = ae *(a+&) directly ahead of the crack tip B (Fig. l), i.e. 6or < a, the corresponding point near Br in the t plane being t = -

r~sin (42)

+ ia 00s (a/2) [&/tan

(a/2)]+.

E.

332

SMITH

The general relations in Section 2.1 and expression distance 6r directly ahead of the crack tip B as

(8) give the shear stress at a point a small

rrsina t6 + [a sin (u/2) i_ cl] ~_ _ prz (r) = 2 tan-8 GL0 6r [a2 C0S2(cc/B) + 26zcr(1 - sin (~/Z))]fi With a planar crack, the displacement 1957; BARENBLATT 1961) the relations

discontinuity

.

(12)

and local shear stress satisfy

(IRWIN

and K

prz (F) = --

(27&)*

K being referred to as the ‘ stress intensification these relations, the appropriate value of K being K=

’

factor.’

Expressions

(9) and (12) also satisfy

2-3 s (an)3 sin ot [a sin (912) + cr] tan* (E/2) [a2 eoss (a/2)

-+ 2acr (1 -

sin (a/2)}]*

.

W)

Adopting the procedure originally employed by Griffith, the crack extension condition is determined by equating to zero the change in the total energy of Lhc system for an infinitesimal extension of the crack: this condition may be expressed (IRWIN 1957) directly in terms of the stress intensification factor IC by the relation hT’! = ‘4/&Y where Y is the free surface energy. extend at the tips B and D when 02 =

Accordingly

8t~Y tan (a/2)

(13)

with the present model the complete

* [as co@ (@)

ira sine a

$2ac1(1

[u sin (a/2)

For the special case where the complete and relation (17) reduces to

- sin -_. (CL/~))] + ~11s

crack will

(17)

crack has no planar segment, c and cr are both zero

1 $ = 4PY . __-I___. Im sin G(sins (a/2) However, wben the lengths of the two circular segments are small compared with that of the planar crack, i.e. abc Q c, relations (10) and (11) give c = cr cf and relation (17) simplifies to o2

=

8PY , --I 7TC

which is therefore the condition crack of length 2~. 2.3.

Bifurcation

for bifurcation

along circular segments at right angles to a planar

along planar segments

The model is illustrated in Fig. 3. A crack QMBPDN, whose surface is stress-free, is situated within an infinite elastic body (shear modulus p) subject to the externally applied shear stress p,, = S; the complete crack consists of a planar segment QMN of length 2c, together with two other planar segments MBP and NDP each of length L and inclined at an angle np to QMN. The conformal transformation mapping the complete crack in the z plane into the unit circle in the t plane (Pig. 4) is (DARWIN 1950) 1 (t3

I)38 (t -

&)1-B

(1 -

e-ta)l-8

(20)

--7 t

Crack bifurcation

in brittle solids

333

@‘m. 3. A crack QMBPDN, whose surface is stress-free, situated within an infinite body subject to the externally applied shear stress p, = (I. The complete crack is a planar segment QMN of length 2c, together with two other planar segments MBP and NDP each of length L and inclined at an angle V@ with QMN.

corresponding

points transforming

as shown (Figs. 3 and 4), with sins (4/2) = /I sin2 (a/2), c = 26 E (1 + cos a)l-8,

(21)

I, = UfiB(1 - /3)1-8 sins(cq2). For large a: and t relation (20) simplies to z = It. The boundary conditions in the t phne are = 0, PIN = 20 at infinity, N being the outward normal to Fig. 4, while the surface of the unit

Pt.v

i

FIG. 4. The t (= E f formal transformation

iv) plane obtained from the z (= z f 8) plane of Pig. 9 by the condescribed

by relation

(20);

corresponding

points

are as indicated.

E.

334 cylinder is stress-free. The complex the original solid is therefore

SMTH

function giving the displacement

and stresses at all points of

(22) 1 and z being related by (20) and (21). Thus, consider a point z = (L - 6r) e’rfl on the crack surface near the tip B (Fig. 3), i.e. 0 < Sr Q L; the corresponding point near Br in the t plane is t = et(++e+) with IS+1c< I#+relations (20) and (21) giving (23)

L”‘i;)‘r]*tan(%),

S+=*

the two values corresponding to points situated on opposite faces of the original crack. The general relations in Section 2.1 and expression (22) show that the displacement discontinuity across the crack surface a small distance 6r from the tip B is

Similarly, by considering a point z = (L -t 6r) e”vp directly ahead of the crack tip B (Pig. 3), i.e. 0 < 6r Q L, it may be shown that the shear stress at this point is

cot t+/w L 1. lo cos fj ___

pra (r) =

2 (1 ____

/?) *

L/3GT

(25)

Expressions (24) and (25) satisfy the general relations (13) and (14) for the displacement discontinuity and local shear stress, the appropriate value of the stress intensification factor K being K = 21a cos 4

x (1 _____.

cot (+/a 1 Relations when

#E) t

L,B

1

(16) and (26) therefore show that the complete

P-Y LB

a2= 7712(1

crack will extend at the tips B and D

cot2 (4/Z) -

(26)

8) COG 1#1’

(27)

4 and 1 being given by (21). When the lengths of the two inclined segments are small compared with that of the segment QMN (i.e. L < c), (27) simplifies to 8PY

*2,_

7rc

~

P

s (28)

L 1-P

1

which is therefore the condition for bifurcation along two planar segments each inclined at an angle T,!Ito a planar crack of length 2c. If /3 = 4, and the new planar segments are at right angles to the original crack, (28) reduces, as expected from physical considerations, to relation (IS), the condition for bifurcation along circular segments at right angles to the initial crack. Condition (28) may be compared with the corresponding one [obtained from SIH’S analysis (1965)] for a planar crack to fork along a single segment at an angle n/3 with the original crack : o2

=

4PY --

xc

1 + g 6 -

[ 1-p

1

As expected, it is clearly more difficult (with p Q $) for bifurcation to branch along a single segment.

to occur than it is for a crack

Crack bifurcation in brittle solids 3.

885

Drscussro~

Crack bifurcation criteria have been derived in the previous Section by applying classic thermodynamic arguments. The procedure was to allow short crack seg ments to form at a tip of the original crack, determine the condition for extension of such segments and then, by letting their length tend to zero, derive the appropriate bifurcation criterion, This approach was adopted in discussing bifurcation along circular segments at right angles to the initial crack, and also along planar segments inclined at an arbitrary angle with respect to the original crack; as anticipated, the results for the two models are in exact agreement when the new planar segments are perpendicular to the original crack. As indicated in the Introduction, crack bifurcation is an important practical problem, since it is intimately related to the fracture behaviour of inhon~ogeneo~~s solids, especially fibre composite materials. For example, it has been suggested (COOK and GORDON 1964; EMBUILY, PETCH, WRAITH and WRIGHT 1967) that an attractive means of increasing a solid’s fracture resistance is to introduce a plane of weakness parallel to the applied tensile stress and normal to the anticipated dirertion of crack propagation, thereby causing the crack to bifurcate and preventing continued propagation in the transverse direction. In such a situation it is necessary to compare the bifurcation stress with that for transverse propagation. If the antiplane strain models of the present paper are used to discuss the problem, for the special case where the plane of weakness is perpendicular to the original crack, the bifurcation stress is given by

where YB is the effective fracture energy of the plane of weakness. This stress must be compared with that given by the classic Griffith expression : ,2-4pYT XC

(91)

where YT is the fracture energy for the transverse cleavage plane. Bifurcation is clearly easier if the fracture energy of the plane of weakness is low compared with that of the transverse cleavage plane, whereas it is made more difficult by virtue of geometrical co~iderations associated with branching along planes inclined to the original cleavage crack and also because two additional segments form as a consequence of bifurcation. The present analysis shows that the geometrical and multiplicity effects are of similar importance, and that bifurcation is appreciably easier than transverse propagation only when the fracture energy of the plane of weakness is significantly (i.e. an order of ma.gnitude) less than that of the transverse plane. It must be emphasized that the present paper has derived bifurcation criteria by assuming that an anti-plane strain model of an elastically homogeneous solid adequately represents real situations, and since the Griffith approach has been adopted dynamic effects have automatically been neglected. The first assumption is reasonable, although it is worth justifying by comparing the results with those obtained from plane strain analyses; these are obviously more difficult than that

386

F

of the present equations. The With

paper because

neglect

of physical

presumably be propagating and a fully comprehensive

SMITH .

of the complexity

of dynamic

the type

1.

effects

is likely

problem

of the governing to be a more

envisaged

partial

hazardous

in the present

paper,

differential procedure. a crack

will

with a high velocity as it strikes the plane of weakness, analysis should take the associated kinetic effects into

account.Although BERRY

attention (YOYFE 1951; BARENBLATT and CHEREPANOV 1960; JOHNSOX and HOLLOWAY 1966; CONGLETON and PETCII 1967) has

1960;

been given to dynamic crack problems, incorporated into a rigorous bifurcation

it is not clear how kinetic effects can be analysis. The present paper’s approach

has been to persevere with the well-established thermodynamic procedure developed by Griffith, while recognizing that some modifications will be forthcoming when it becomes clear approach.

how dynamic

effects

can be encompassed

by the thermodynamic

h!KSOWLEDGMENT The author thanks the Central Electricity Generating Board for permission to publish this

paper. HEYEKENCES

BAHENBLATT, BARENBLATT,

G. I. G.

I.

1961

Problems of Continuum

1 WI)

Philadelphia), p. 21. Prik. Math. Mech. 24, 993.

and CHEREPANOV, G. 1'. RERRY,J.P. 1960 CONGLETON,.J. 1967 and PETCK,S.J. 1964 COOK, J.and and GORDOS, J. 13. I)ARWIX, C. 1950 F,MBuRY,J.D., PETCII,N. J.1967 WRAITH, A. E. and WRIGHT, E. S. 1921 GRIFFITH,A..4. 1957 IRWIN,G.R. 1966 JOHNSON, J. Iv:. and HOLLOWAY, 1).G. 1965 SIH, G. C. 1951 YOFFE, IL

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239, 114.

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