A rigid triangular inclusion partially bonded in an elastic matrix

Theoretical and Applied Fracture Mechanics 8 (1987) 173-185 North-Holland

173

A RIGID TRIANGULAR I N C L U S I O N PARTIALLY B O N D E D IN AN ELASTIC MATRIX

E.E. GDOUTOS, D.A. EFTAXIOPOULOS and M.A. KATTIS School of Engineering, Democritus University of Thrace, GR-671 O0 Xanthi, Greece

The two-dimensional problem of a rigid rounded-off angle triangular inclusion partially bonded in an infinite elastic plate is studied. The unbonded part of the inclusion boundary forms an interfacial crack. Based on the complex variable method for curvilinear boundaries, the problem is reduced to a non-homogeneous Hilbert problem and the stress and displacement fields in the plate are obtained in closed form. Special attention is paid in the investigation of the stress field in the vicinity of the crack tip. It is found that the stresses present an oscillatory singularity and the general equations for the local stresses a r e derived. The singular stress field is coupled with the m a x i m u m circumferential stress and the m i n i m u m strain energy density criteria to study the fracture characteristics of the composite plate. Results are given for the complex stress intensity factors, the local stresses, the crack extension angles and the critical applied loads for unstable crack growth from its more vulnerable tip or two types of interfacial cracks along the inclusion boundary.

1. Introduction

Extensive experimental studies on the fracture of composite materials have revealed that a common failure mechanism is the debonding of the different phases due to manufacturing a n d / o r loading conditions. This type of failure mode is usually modelled by the problem of an inclusion partially bonded in an elastic matrix. In analyzing such problems, certain idealizations concerning the geometry and the mechanical properties of the constituent materials are usually made. Thus, for composites with a great value of the elastic modulus of the reinforcing materials relatively to the matrix, it appears reasonable to consider the reinforcing materials as rigid. Furthermore, they usually assume a simple geometrical shape in order to be tractable in mathematical analysis. Within this framework, a number of elasticity solutions have appeared in the literature by assuming the inclusion to be rigid and of a circular or elliptical shape [1-7]. Curvilinear inclusions have also been considered and analyzed using linear elasticity. Thus, Sendeckyj [8,9] studied the problem of two symmetrical interface cracks at rigid curvilinear inclusions which otherwise are perfectly bonded to an elastic matrix under longitudinal shear. The case of a rigid inclusion with cuspidal points on its boundary and perfectly bonded to a matrix was analyzed by Panasyuk et al. [10]. For this situation, the problem of fracture

initiation from the cuspidal points of the inclusion was studied by Gdoutos [11,12] using the strain energy density failure criterion. Numerical results were given for the cases of rectilinear, astroidal and hypocycloidal inclusions. In the present communication, the problem of failure initiation from an interfacial crack placed along the boundary of a rigid triangular curvilinear inclusion embedded in an infinite plate is studied. The plate is subjected to an uniform biaxial stress system at infinity. A closed form elasticity solution is first obtained by using the conformal mapping technique in conjunction with the method of analytic continuation of the complex potentials [13]. The stress solution is then coupled with the maximum circumferential stress and minimum strain energy density criterion to obtain the fracture characteristic quantities of the composite plate.

2. Theoretical background

Consider an infinite elastic plate which occupies the z-plane and contains a rigid curvilinear inclusion which is perfectly bonded to the plate along the part A o of its boundary (Figure l(a)). The remaining part A s of the inclusion boundary forms an interracial crack. The plate is subjected to an uniform biaxial stress system N and T and a rotation E~ at infinity. Let the inclusion boundary

0167-8442/87/$3.50 ~ 1987, Elsevier Science Publishers B.V. (North-Holland)

174

E.E. Gdoutos et aL / A rigid triangular inclusion

A be described in a clockwise sense so that the matrix is on the left when moving on A in the positive direction. Consider now the conformal mapping of the z-plane on another plane ~ by the equation ~'+ ~- + F , '

z=m(~)=R

(1)

where the quantities R, b I and b z are real constants, which transforms the inclusion boundary A on the unit circle F( 1~"] = 1) of the ~'-plane (Figure l(b)). The end points t I = rle i°', t z = rxe i°2 of the interracial crack are mapped on the points o I = e ie' and o2 = e i02 of the circle F. The bonded part of the inclusion A n and the interracial crack A s are mapped on the parts FD and F s of F. It is easily seen from eq. (1) that the inclusion and the plate in the z-plane are mapped on the regions R and L inside and outside, respectively of the circumference of the circle F in the ~'-plane. The mapping function (1) defines a system of orthogonal curvilinear coordinates (~, n) in the z-plane for g"= e ~+i". The stress components ot~, oon and o~ in the system (~, ~) and the displacement can be expressed in terms of the two complex potentials W(~') and zcU(~) by the following equations [13]

so that F ( z ) = F(~).

(6)

The definition of the function W(~') from the L region outside the unit circle F in the f-plane is now extended to the region R inside the circle I" by the equation i

l

S*

Equation (7) ensures the analytical continuity of W(~') from the left and right of /" through the unloaded parts of the boundary. From eq. (7) by writing { for 1/~ and t / ~ for .~ and taking the complex conjugate,

dI('

m'(f)~q/'(f) = - ~

N ~)W(~).

i

is obtained. Introducing this value of ¢U(~) into eq. (2), 2~'m'(~')( oee + ioe. )

2(o~ + ion.) = W(~) + W(~)

~m (~') ~ , ( ~ ) ~m---~

fm'(f)

- ~'m'(~') Y'~(~)'

lm.

(2)

i

l

m ,

fm(~)

2(%. - io~n) = W{~') + W(~) + ~m--~) W ' ( ~ )

+ ~.m,(~----~~ ( f ) ,

(3)

aD 4/.t--~- = [~¢W(~')- W(~)]i~'m'(~') +[m(~)W'(f)

+~'(f)~(f)]if,

(4)

where D = u + iu is the complex displaccrncnt in the Cartesian coordinate system (x, y) a n d ~ = 3 - 4 v or x = ( 3 - v ) / ( 1 + v) for plane strain or generalized plane stress conditions respectively, with v representing the Poisson's ratio of the material of the plate. In eq. (2), the following notation has been introduced F(z) = F(~'),

which expresses the complex stress o~ + ion. in terms of W([) defined in the region L and its analytic continuation into R.

(5)

3. Hiibert formulation Consider now that the point ~" remaining in L tends to the point o of the circle F. Then, 1/~ tends to the same point o from R and eq. (9) takes the form 2 o m ' ( o )( o~,, + iO~no)

= o r n ' ( o ) [ W L ( o ) -- W n ( o ) ] .

(10)

E.E. Gdoutos et al. / A rigid triangular inclusion

175

tN r ...........................

'

matrix

\ " " ~ i ~ r/ac k

I

L

ol

T

T inclus~

x

Z= m(~)

z - plane ...........................

J

- plane

1" (a)

(b)

Fig. 1. A rigid triangular inclusion partially b o n d e d to an elastic matrix and its conformal m a p p i n g on to the unit circle.

where wL(o') and WR(o') are the limiting values of W(~) as ~ tends to o" from L and R respectively and o'~o and o'~no are the boundary stresses on A. In a similar manner, it can be shown for the displacement that

4. Solution of the problem

4 / ~OD - = ~ [m'(o.)W(o)] t. io"

+ [m,(o')W(o) l n io'.

(11)

For the case of our problem, we have the following conditions for the stresses and displacements along the inclusion boundary

o'~o + io'~no= O,

o" ~ Is,,

(12)

D(o')=iem(o'),

o ' E F D.

(13)

The first equation denotes that the crack faces are stress free, while the second expresses the condition that along the bonded part of the inclusion boundary, the displacements of the points of the matrix and the inclusion are the same, where represents the rigid body rotation of the inclusion. Introducing eqs. (12) and (13) into eqs. (10) and (11) it follows that W ( [ ( o ) - WoR(o)=0,

a ~ F s,

Equation (14) shows that the unknown function W0(~) is continuous along the arc I s, while eq. (15) suggests that its values from the left and from the right along the arc F D satisfy the nonhomogeneous Hilbert problem.

The unknown function W0(~") is determined from the solution of eq. (15) in conjunction with its behavior at infinity. It is first observed that Wo(~ ) has at ~"= 0, a pole of order not greater than three since the mapping function m(~') has at that point a pole of order two. Furthermore, the function W0(~ ) should have at infinity a pole of the first order. From eq. (15) it is obtained for wo (~')[131 Wo(~") = 4ip" x(~')I(~') + R(~')X(~), where

1 [ i( 1 =

o'~Fo,

_ f)

+ Ao) + @

do,

(18)

A_ 2

+--

(14)

+...+-- A 3 ~3

K

(15)

-

-

where it has been put

Wo( ~ ) = m'(~ )W(~ ).

m'(~) xL(o)(o

R(~)=(A~

W(((o) + 1 WoR(o") _ 4/~ i e m ' ( o ) , K

(17)

K

o2)" o,)"

(16)

r=O.5+iX,

(19)

'

'

(20) ,

X= logK

2'~ "

(21)

176

E.E. Gdoutos et a L / A rigid triangular inclusion

The function X(f), known as the Plemelj function, is holomorphic in the whole E-plane cut along the arc F o on which X R ( o ) = --KXL(o). Only the branch of the function X ( f ) for which lime ~ ~ [ f X ( f ) ] = 1 is considered. For the calculation of the integral I(E), a contour A surrounding the arc F D is considered and taking into account that on I ' , X~(o -~¢XL(o), it is obtained

where d. =- e x(:~ o,~ -io, dj = d0(cos ~o~- 2X sin '~c0) c

(29)

From eqs. (17) and (24). it is obtained for the function W0 (£) 4/~i~ [ W{,(~)= 1 + ~ m ' ( ~ ) - - [ , g l ( ! ; )

.~ ~ ~4:(S )]X(.~')j

+ R(£ ) X(~ ). I(~') - 2,~i(T + ~) .

do,

which contains the unknown coefficients .4~. 4 ~ A ~. A a and A 3 ° f e q . (19) aim the unknown rotation of the inclusion {

where it was put

F(E) =

~3oi,

(22)

m'(E) X(E) "

(23) 5. Determination of the quantities A i and {

Using the well-known properties of the Cauchy integrals, it is obtained for I ( f ) I(E)= 1--~ [F(E)-g,([)-g2(E)],

(24)

where gl(E) and g2(~) are the principal parts of F(E) at E = oo and E = 0 given by

For the determination of the unknown coefficientsA/(j=l,0, -1, -2, 3) and the rotation of the inclusion {, use is made of the condition that the complex potentials W ( f ) and ~/'(~) are holomorphic in the E-plane and have a particular behavior at infinity. For ~ = oc. where z = m(~} =

RE I131 g,(E) = s d +

S~ g2(E) =

So, S_ 2 S t + U + ---~-E '

(25t

w(E) .

.2 .

+

#"(.~)- ( N- T)

with: S1 =R, S O= - R ( c o s ½{o + 2X sin ½w) e '°'',

(26}

and

S 3

2Rb 2 do ,

S_ 2

Rb~ 2Rdlb 2 d~ + d~ R ( dlb I + 2d2b2) do2

.

~o[

1

iM + N ' (_, "1 ¢rR2~2 + }! R_~f:~ }.

i a- l )

(32)

In these equations, e~ is the rotation at infinity, M is the moment of the stresses applied on A about the origin and N ' is a real constant. Consider first the series expansion of the function W0(E)= m'(E)W(E) given by eq. (30) at ~"==

Di

D2

(27) where

2Rd2b2 do

D 1--1

and

D 2 = ( c o s } ~ + 2 X sin ~ 0 ) e '°'. (34)

with ~0 = 02 - 0 a and 00 = ½(01 + 02). In eq. (27), d o and d~ are the first two coefficients of the series expansion of X(E) at E = 0. We have

Thus, from eq. (30), it ts obtained for tile expansion of Wo(E ) at ~"= oc

X(~)=do+dlE+

w°(E)=a°+T-

"",

(28)

+ """

(35)

177

E.E. Gdoutos et al. / A rigid triangular inclusion

where

where

a 1 = A o + A1D2.

a 0 = A 1 and

(36)

From eqs. (31) and (1),

(

b12b2

Wo(f)=R 1

~2

~,

do = eX(2.~ ,~) i0o d~ = d o [cos 1.0 _ 2)~ sin lw] e ioo,

)

d 2 = ¼do[(1 + 4~.2) + (3 - 4 ) k 2) COS 60

( 1 )1

__~___+ T + T4i#eoo × IN ;-;;+o

___=~_~

8)t sin co] e -2i°o,

-

, (37)

d 3 = ~d0[(15 - 36X2) cos 3,0 -

)t(46 - 8)~2) sin 3*0

or

(N+T 4i/~G~)(1)

Wo(~) = n - T -

+ ~

+ o

~;

.

Comparing eqs. (35) and (38) and using eqs. (36), it is deduced that (N+T A o = -RD 2 ~ -

4i#%o) + ~

'

A1 --

+ (1 + 4X2)(9 cos 1.0

(38)

-62t sin ½ *0)] e 3i00. (44) Then, eq. (30) in conjunction with eqs. (19), (25) and (43), gives

A0

, B~ B2 B3 m (~)W(~') = flo + T + - - + + ...

D2.

(45)

(39) For the determination of the remaining unknowns Aj ( j = - 1, - 2, - 3) and c the behavior of the other complex potential Yf'(~') in the entire plane, including the point at infinity, is studied. Following eq. (8), the expansion of the quantities m(1/~)W(~') and ~'(1/~)W(1/~) at infinity are needed. It can easily be established that in the neighborhood of infinity

4t~ie ( R

80 = ~

-- S o d O - S

+Aod o + A

ldl - S 2 d 2 - S 3d3)

l d l q- A _ 2 d 2 + A

81 = A - l d o + d _ 2 d l

3d3,

+A_3d2,

82 = .'4 _2do + A _ ldl,

r3 = A _ 3do.

~'(1/~') -- b2~ 2 + bl~ + bib 2 m'(~"

+

where

(46)

Equation (45) renders

1 + b 2 + 2b 2 ~2 + ...,

(40)

and using eqs. (16), (35) and (40), it is obtained d~ N

W(~') = 2K2~" + K 1 +

~.--5-'

(41)

Substituting eqs. (41) and (47) into eq. (8), we obtain for the expansion of the function rn'(~')zCK(~') at infinity, the following equation

(42)

m'(~')Y#'(~') = (fi3 -- 2K2)~ + ( ~ 2 - K , )

where K2 = b2ao,

K1 = b2a 1 + b l a 0,

(47)

K_,=b2a3+b,a2+blb2al + ( l +b2 + 2bZ)ao .

+/.:t, +=

/~o+K , + ...

(48)

For the__ determination of the expansion of N ' ( 1 / f ) W ( 1 / f ) at infinity, the expansion of m'(()W(f) at ~'= 0 is first considered. For the Plemelj function XG'), we obtain at ~"= 0

The function m'G')zcK(~) should be hoiomorphic, and eq. (48) gives

X(~) = d o + d , ~ + d2~"2 + d3~ 3 + . - . ,

~3 - 2K2 = 0.

(43)

(49)

178

E.E. Gdoutos et al. / A rigid triangular inclm'ion

Furthermore, from eqs. (1) and (48)

6. Local stress distribution

m'(~')~F(~')

( =

R

bl

2b2 )

~-2

1

~3

I ~R2~ × [(N-T)iM+N ' + "2

O

¸

1

(50)

Comparing eqs. (48) and (50) f12- 2Kl = R ( N -

T),

(51)

fli = O,

fl° + K-1 -

iM+N' ~rR

The analysis of the stress distribution in the neighborhood of the crack tip is of particular importance and allows study of the growth characteristics of the interfacial crack. Consider the tip t 2 of the crack and the tangent t~x of the inclusion at the point t 2 (Fig. 2). Using the asymptotic expansion of the function Wc~(~) and its analytic continuation for I~'1 < 1 around the point t2 in conjunction with eqs. (3). (4) and (8), and after lengthy algebra, the following equations of the curvilinear stress components o~> o,,( and o~,~ are obtained e xa

Rb,(N- T).

[cos(½a + X In :Trt, ) + e 2a{~ ~'cos(~a

From eqs. (49) and (51), it is obtained for the unknown coefficients A_~, A 2, A_3 and the rotation of the inclusion

÷ ~/]+4X 2 sin a × sin(~o~ + X In

eX'~ A-3 = 2zb2,

(52)

A_2=~b,-

<+~.

d1 A_,=A_2-~o-A

d-, ,~,

a ln2~rp)

+ 2dmO

(53)

[sin(

2"~O-~)]h',

+ X in :.,, t

--

e 2a(~ "' sin( ~ -- a In 2~rO)

-

(1-+ 4~e sin ,~

× cos(~. + X In

(54)

2~O-,~)]JV:. (57)

M(I+K) ( l + x ) Im L = + 4R~r/x Re T 4ix Re T

(55)

where yi

8 = R ( N - T),

matrix

L = AI(1 + biz + 2b~)+ Ao(D2b I +

D3b2) x

+ A _ l ( D i b 1 + D2b2) + A _ 2 D l b 2 + Rbl( N -

ip T ) + Lo,

Qv

i

3

L o = Y'~ A _id,, i=0

i

T = ( Rb 1 + S _ I D 1 + SoD 2 + SID3)bl +(2Rb 2+S_2D 1+S + S1D4)b2 + To, 3 To = R - Y" S_,d,. i=0

t2

11)2 + SoD 3

rack it (56)

inclusion . . . . . . . . . . . .

~'~ ]l

Fig. 2. Local polar coordinate system.

L

.....

x

179

E.E. Gdoutos et al. / A rigid triangular inclusion

07171 --

_ _eX~

A= (Re[m'(oz)]Re[P(o2) ]

[3 cos(½a + ~ In 2"frO)

2 2V/2V/~

+ Im[ m'(o2)] Im [ e(o2)] } - e 2x~È-') cos(½a - ~ In 2,rrp) -V/1+4~

X { Ira'(02)] 3/2 }

2 sina

x { [m'(o2)] 3/2 } -1,

[3 sin(½a + X In 2~0) + e 2x(~-~) sin(½a - ?t In 2~0)

2,

(58) o~-

e x~ [sin(la + ~, in 2~rp) 2 2V/2V/~ -

e 2x~'~-'~) sin(½a - X In 2,~p)

+ V/1 + 4X2 sin × cos(~a + X In 2'n'0- q~)] K,

[cost½, +

V~- + 4X2 sin

x sint ,, +

(66)

The functions R(~') and gl(~') and g2(~) are given by eqs. (19) and (25). Numerical results were obtained for the case of an equilateral triangular inclusion with an interfacial crack. The triangle is symmetrically located with respect to the x-axis and the plate is subjected to uniform stresses T and N at infinity along the directions of the x- and y-axis, respectively (Fig. 3). The mapping function, z = m(~), in this case, has the form

In 2 0)

+ e 2xe~-') cos(½a - X In 2~p) -

(65)

with 4/~ci P(~)=R(~)-~-~x[gl(f)+g2(~)].

+l/l+4X 2 sina × cos(3a +X In 2 v 0 - ~ ) ] K

(64)

B= {Re[m'(o2)]Im[P(o2)] -Im[m'(o2)]Re[P(02)]}

× sin(3a + X In 2~rp- cp)] K, + 2 2eXa ~

-1,

In 2 p-

(59) where cp = tan-a2X.

(60)

The stress intensity factors K~ and K 2 are given by K~ = e-~2x'~/ ~-----~--sin ½c0 (A cos 0 + B sin 0),

(61)

K 2 = e- '2x,o~ / ~ s i n½~0 (A sin 0 - B cos 0 ),

(62)

Two locations of the interfacial crack along the inclusion boundary were considered. In the first case, the crack is symmetrically located with respect to the bisector of the angle subtending between the rays starting from the triangular height and terminating on one of its sides (Fig. 3(a)), while in the second case, the crack starts from the point of the triangle at x-axis and extends along one triangle side (Fig. 3(b)). Figures 4 and 5 present the variations of the non-dimensional quantities K1 =K1/(T~-R) at crack tip B versus half crack angle ~60 1 for the following values of the ratio s = N/T of the applied stresses N and T: s = 0, 0.25, 0.5, 0.75 and 1.0. The value of the Poisson's ratio u of the matrix material was taken equal to 0.30 and plane strain conditions were assumed to dominate in the plate. Analogous results for the case of Fig. 3(b) are presented in Figs. 6 and 7 for the crack tip A.

where 7. C r a c k g r o w t h

O=0o + +_._..___~ 2~r +~, ln[4,~ sin ½,.,Im(o2)l] 4

(63)

Failure of the composite plate would take place by propagation of the interfacial crack. Due to

E.E. Gdoutos et al. / A rigid triangular inclusion

180

tN=sT r . . . . . . . . . . . . . . . . . . . . . . .

ty

t N=sT

•

matrix

y

matrix

K,U

T

!

T

2

4

. . . . . . . . . . . . . . . . . . . . . . . . .

(a)

(b)

Fig. 3. Geometrical configuration of two interfacial crack locations.

2.0

1.5

t 1.0

0.5-

0.0

I 3°

'

150

1

t

3 0 ~'

45 °

60"

0~12 ~

=

-0.5-

-1.0 /

Fig. 4. Variation of the dimensionless stress intensity factor tk;1 versus half crack angle ~ 0.75 and 1.0.

for the case of F~g. 3!a) s = 0. ~).25, {kS,

2.0

,i"l / i

1.5

t ,2

/

ilii i

1.0-

s=o.5

d~

20 °

1

400

60 °

¢o/2 ---~

Fig. 5. Variation of the dimensionless stress intensity factor K2 versus half crack angle 12o~ for the case of Fig. 3(a). s = 1), 0.25, 0.5, 0.75 and ] .0.

E.E. Gdoutos et al. / A rigid triangular inclusion

181

2.5

20"

t 1.5"

iff 1.0-

0.5-

0.0 0°

I

I

I

300

60 °

90 °

1200

Fig. 6. Variation of the d i m e n s i o n l e s s stress intensity factor J~l versus crack angle ~0 for the case of Fig. 3(b). s = 0, 0.25, 0.5, 0.75 a n d 1.0.

mixed-mode loading conditions which dominate in the vicinity of the crack tips the crack, generally speaking, will extend from either of its tips into the matrix. For the prediction of crack growth, the maximum circumferential stress [14] and the minimum strain energy density [15] criteria are used. According to the maximum circumferential stress hypothesis, the crack grows from one of its tips along the direction in which the circumferential stress o,~ becomes maximum. The direction of

crack growth is determined from the following relations o`'`"

- -

~a 2

~ 2o,,,,

= 0,

- -

< 0,

~a

o,~,~ > 0.

(68)

The stress as.. is calculated from the curvilinear stress components o~, %~ and o~n by

ooo ~(o~ + o~n) + =

½(o. -

%)

- o ~ sin 2a.

cos 2,~

(69)

2.0-

S~I:035 t 1.5-

S~0.5

1.0-

$=0.25 0.5-

s:o 0.0 30 °

60 °

90 o

120 °

Fig. 7. Variation of the d i m e n s i o n l e s s stress intensity factor K2 versus crack angle w for the case of Fig. 3(b). s = 0, 0.25, 0.50, 0.75 and 1.0.

E.E. Gdoutos et al. / A rigid triangular inclusion

182

V=0.1

~

~

t~=40 ° =

.

1.0

t

0.5

I

OmO O0

I

300

60 °

I

90 °

I

120 °

I

150 °

'

'

180 '~

Ot--~ Fig. 8. V a r i a t i o n of the n o r m a l i z e d c i r c u m f e r e n t i a l stress o*~ versus the p o l a r angle a for the tip B of the case of Fig. 3(a). ~ = 40 °. s = 0.5 a n d ~ = 0.1, 0.2, 0.3.0.4 a n d 0.5~

A small circular core region surrounding the crack tip is usually introduced for the determination of the stress a~. This arises from the inability of the analytical solution of the stress and displacement fields based on theory of continuum mechanics to describe the state of affairs in the close vicinity of the crack tip. The variation of the quantity o~,~= 2~/~0 o ~ ( p 0, ot)/T versus the polar angle c~ for ~0 = 40° and s = 0.5 for the tip B of

s:l

the crack of Fig. 3(a) is shown in Fig. 8. The radius P0 of the core region was taken equal to P0 = 0.005 R. The curves of Fig. 8 correspond to the values of the Poisson's ratio ~ = 0.1, 0.2, 0.3, 0.4 and 0.5. Analogous results are presented in Fig. 9 for ~ = 0.3 and s = 0, 0.25, 0.5, 0.75 and 1.0. The minimum strain energy density criterion is based on the assumption that crack growth takes place along the direction of the minimum strain

o:40 o

1.5- ~

v=03 s:O.?5

$ 0! 5 -

1.0-

f

t

s • 0.25

0.5

0.0 0o

I 300

60 °

90 °

I 120 °

f 150 °

180 °

Q--.Fig. 9. V a r i a t i o n of the n o r m a l i z e d c i r c u m f e r e n t i a l stress a*,~ versus p o l a r angle a for the tip B of the case of Fig. 3(a). ,3 = 40 ° . v = 0.3 a n d s = 0, 0.25, 0.5, 0.75 a n d 1.0.

183

E.E. Gdoutos et al. / A rigid triangular inclusion

2.0-

2.0

s=l

1.5

1.5

S=l

t

,J

s

1.0

1.0

s=0.5 • 0.25

0.5

0.0

_•

s=o

o,o 0°

150

300

45 ~

60 °

0°

300

60 °

90 °

120 o

~12 " - -

Fig. 10. Variation of the dimensionless m i n i m u m strain energy density factor Srm, versus half crack angle ½ w for the case of Fig. 3(a). The crack always starts from its tip A. s = 0, 0.25, 0.5, 0.75 and 1.0.

Fig. 11. Variation of the non-dimensional m i n i m u m strain energy density factor Smi,~ versus half crack angle ~2o~ for the case of Fig. 3(b). The crack always starts from its tip A. s = 0, 0.25, 0.5, 0.75 and 1.0.

maximum

stress

minimum strain

........

/

energy /

60 o

//

t

$=1

I

it

II

o ,:~

30

o-

.."

0 °

/

0o

"'

'

I

150

'

,

~

30 °

,

,

$1

,,"*4

'~

450

I

/

t

I

t

,~ / /

"

I

t'~"

60 o

'"12 Fig. 12. Variation of the crack growth angle act versus half crack angle ~o~ ] for the case of Fig. 3(a) according to the m a x i m u m circumferential stress and the m i n i m u m strain energy density criteria.

184

E.E. Gdoutos et aL / A rigid triangular incht~ion

60 o /

ii

0"

/

I s

/ /

," "

40°-'~

2 o

//

$ 0.25

"-

$ ('

o,t.

-

--...a....~_.._.

maximum o

. . . . . . . . . . I

30 °

O

stress

minimum

strain

F

I

60

90"

L9

energy '

"

120

'--~

Fig. 13. Variation of the crack growth angle ac, versus half crack angle L2~ for the case of Fig. 3(b) acc¢~rding to the maximum circumferential stress and the minimum strain energy density criteria.

energy density factor S given by dW S = r dV '

(70)

The strain energy density d W / d V elastic problem is expressed by dW 1 [1 dV = ~ ~(K+l)(o~+o,,)

for the plane

2

half crack angle ½0~ is shown in Fig. 10. Analogous results for the case of Fig. 3(b) where the crack again first starts from its tip A are shown in Fig. 11. Finally, Figs. 12 and 13 represent the critical crack growth angles ac, for the cases of Figs. 3(a) and 3(b) respectively, according to the maximum circumferential stress and minimum strain energy density criteria,

(7 1 where # is the modulus of rigidity of the matrix material. The direction of crack growth is determined by OS Oa - 0 ,

--

0aS > 0. Oa 2

--

(72)

Unstable crack growth occurs when the minim u m value S ~ of S becomes equal to a critical value S~, which is a material constant. Thus, the critical load for unstable crack growth is determined by Smi, ~- Scr.

(73)

It is evident that the crack would grow from its tip with the m a x i m u m strain energy density factor. Following these considerations, it was found for the case of Fig. 3(a) that the crack always grows from its tip A. The variation of the non-dimensional quantity Smin = 4.171"tSmin/(T2R) versus

8. C o n c l u d i n g

remarks

In an attempt to model the mechanical behavior of certain particulate composites, the problem of a crack lying along the interface of a rigid triangular inclusion embedded in an infinite elastic plate was considered. The boundary problem was solved in closed form based on the complex variable method of the two-dimensional theory of elasticity. The inclusion was mapped into the unit circle and the problem was reduced to a non-homogeneous Hilbert problem. The local stress field in the vicinity of the crack tip was analyzed and used m conjunction with two failure criteria to predict the unstable crack propagation. The angle of crack growth into the elastic plate and the critical applied failure loads were determined for various geometrical configurations of the interracial crack and load combinations. The results of the paper shed light into the

E.E. Gdoutos et al. / A rigid triangular inclusion

complicated problem of modelling the microstructure of particulate composites whose reinforcing constituents are of irregular shape, like the various inorganic fillers, the metal or boron filaments, the aggregate or sand particles in concrete. In such cases, the debonding of the different phases constitutes a fundamental failure mechanism of the composite. In this work, the strength of the bond was assumed to be high enough so that the crack grows into the matrix. The problem of growth of the debonded part of the interface will be addressed in a separate communication.

References [1 ] A.H. England, "An arc crack around a circular inclusion", J. Appl. Mech. 33, 637-640 (1966). [2] A.B. Perlman and G.C. Sih, "Elastostatic problems of curvilinear cracks in bonded dissimilar materials", Internat. J. Engrg. Sci. 5, 845-867 (1967). [3] M. Toya, "A crack along the interface of a circular inclusion embedded in an infinite solid", J. Mech. Phys. Solids 22, 325-248 (1974). [4] B. SundstrOm, " A n energy condition for initiation of interfacial microcracks at inclusions", Engrg. Fracture Mech. 6, 483-492 (1974). [5] M. Toya, "Debonding along the interface of an elliptic rigid inclusion", lnternat. .L Fracture 11,989-1002 (1975).

185

[6] E. Viola and A. Piva, "Two arc cracks around a circular rigid inclusion", Meccanica 15, 166-176 (1980). [7] E. Viola and A. Piva, "Fracture behaviour by two cracks around an elliptic rigid inclusion", Engrg. Fracture Mech. 15, 303-325 (1981). [8] G.P. Sendeckyj, "Debonding of rigid curvilinear inclusions in longitudinal shear deformation", Engrg. Fracture Mech. 6, 33-45 (1974). [9] G.P. Sendeckyj, "Elastic inclusion problems in plane elastostatiscs", lnt ernat. J. Solids" Structures 6, 1535-1543 (1970). [10] V.V. panasyuk, LT. Berezhnitskii and 1.1. Trush, "'Stress distribution about defects such as rigid sharp-angled inclusions", Problemy Prochnosti 7, 3-9 (1972). [11] E.E. Gdoutos, "Fracture of composites with rigid inclusions having cuspidal points", in: Analytwal and Expertmental Fracture Mechanics, ed. by G.C. Sih and M. Mirabile, Sijthoff and Noordhoff, Alphen aan den Rijn, 943-958 (1981). [12] E.E. Gdoutos, "Fracture phenomena of composites with rigid inclusions", in: Mixed Mode Crack Propagation, ed. by G.C. Sih and P.S. Theocaris, Sijthoff and Noordhoff, Alphen aan den Rijn, 109-122 (1981). [13] N.I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, translated by J.R.M. Radok, Fourth edition, Noordhoff, Leyden (1975). [14] F. Erdogan and G.C. Sih, " O n the crack extension in plates under plane loading and transverse shear", J. Basic Engrg. 85D, 519-527 (1963). [15] G.C. Sih, "'Strain energy density factor applied to mixed mode crack problems", Internat. J. Fracture 10, 305-321 (1979).