Shear wave scattering from a partially debonded elastic elliptic inclusion

Medum~~

C.ommmmimtio~,Vot.22, No. I, pp. 7946, 1995 ~ © 1995l~mv~seiea~Ltd Primedia tl~ USA. Allfiglmhum'red 0093-6413195 $9.50 + .00

SHEAR WAVE SCATTERING FROM A PARTIALLY DEBONDED EI,ASTIC E L L I P T I C I N C L U S I O N

Y.S.Wang and D.Wang Department of Astronautics and Mechanics, P.O.Box 344 Harbin Institute o f Technology, Harbin 150001, P.R.China

(Received 14 July 1994; acceptedfor print 22 September 1994) Introduction

In a previous paper [I] we solved the problem of an elliptic arc crack subjected to anti-plane shear waves by using the wave function expansion method and singular inte. gral equation technique. In this paper we extend the analysis to the scattering of shear waves by a partially debonded elastic elliptic inclusion. The debonding region is modeled as an interface crack with non-contacting faces. A singular integral equation is derived in terms o f the dillocation density function and solved numerically to calculate the dynamic stress intensity factor (DSIF) and the scattering cross section (SCS). Singular integral equation Consider an elliptic inclusion with long radius a and short radius b. It is constituted by an elastic material of shear modulus ~l and density Pl, and is partially debonded from an infinite elastic matrix o f characteristics ~o and Po (cf. figure 1). An elliptic coordinate system (~,~) is used and the angular coordinates of the crack tips are ~1 and ~2. An incident anti-plane harmonic shear wave propagates in the direction 00 with the form

where K ~o = co / C To,C zo -- ~1P o / P o ,r o ----~ a 2 - b 2 and A is the amplitude. Since all motion is time harmonic of frequency co, the term e pose the total o u t - o f - p l a n e displacement as 0}

(o)

(])

Wo + w 0 + w 0 w(~,,D--~o) (1~ wl + w l

I

where

~0---- t a n h

-I

/~l

.

is omited for simplicity. Decom-

~>~o (2)

~<~0

( b / a ) . w~ ) (.]= 0,1) are the scattered fields for the case of a perfectly 79

80

Y.S. WANG and D.WANG

(i) bonded inclusion and can be found in [2]. w j are the additional fields due to debonding which, by the use of the wave function expansion method [2], can be expressed as

I

= E A ~ M c " (qo,OCe (qo,tl)+ ~Bo, ns~l(qo,OSe (qo,ri) m-o

m.I

(3)

At Mc~l(qt,~)ce(q,,ff)+

[ w (1'1(~,~) =

BimMs - (qo,~)se,(qo,tl)

m~O

with

l

~2

ql = 4 A f r o

2

m.I

and A ~

and B ~

unknown.

Mct,),Mst,),cem

and

se

in the above

equation are Mathieu functions defined in ref [2].

ZOo Figure 1: Partially debonded elastic elliptic inclusion If we denote Aw(tl) as the discontinuity of the displacement across the interface which is equal to the crack opening displacement (COD) when ~E(~1,~2) and to zero when r/ ¢(~1 ,~2), the mixed boundary conditions may be written as

f/s ~0°~ (~o,,1)_ = ~.~l (~o,~) !

where ,(o )J - r#1° aw( Jo~ ries in

ce

(r/) and

and J = x/(cosh2~- cos217)/2. We express Aw(t7) as an infinite se-

sem (tl): oo

Aw(rl) =

oo

~ AWc=Cem(qo,e)+ ~ A~s, se (qo,~l) m--O

(5)

m-I

where

A~cm -__l_f~2Aw(tl)cem(qo,tl)drI A~s, -_ l ~a~Aw(rl)sem(qo,tl)dr! -a t

~

(6)

~ -z I

Substituting eqs (3) and (5) into the first two equations of (4) and making use of the orthogonality

ofce=

and

se

[2], we have

SHEAR WAVE SCATrERINGFROM DEBONDEDINCLUSIONS

81

AIpCmMc fly , (ql,~o)~ M e .oy(qo,~o) --0 iA o ~ - /Lp~o " (i) o) Ao~-- ~AIpC Mcp (ql,~o)~Me ( q o , ~ o ) - - A w c . / M c ~ ) ( q o ' ~ o 5

(7a)

-

pmO

I Bo., - ~

Bt S Ms or p (ql,~o)/Ms,.or (qo,~o)--0 "

Born --

(7b)

(2)

~.

(3)

--

.

(3).

BIpS~Msp (q z,~o) ~ M s - (qo,~o)-- AW s., /.~as [qo,~o)

po|

where

'

"

t~=#l/ ~o , C~ = ' ~ _ ce.(ql,q)" ce.(qo,q)dq

and

S., =

ser(ql,q) •

se. (q o,ff)dff. Obviously, for q i -- q o,C ,- = S ~ = 6 ~,, eqs (7a,b) become Ao,. -- IzA i, Mc Bo,,' _#BI

(ql,~0)/Mc

Ms~Y(qI,~o)/Ms

where

Dc,. --/zMc

(ff(ql,~o)Mc

.. (I)'.

(q0,~o)= OY (qo,~o)_#Ds

(3)

.

(qo,~o)-Mc

(3)

~' M s OY -- s., ,. (q I '~o )Aw (2).

m [ql,~o)Mc (I) O

Ds., = IzMs [ql,~o)Ms (qo,~o)-Ms,. (~l,~o)MS

(3)'

(85

(qo'Go)

(3)'t't~

x,fo,~o)

(9)

For q~ # q0' eqs (7a,b) can not be solved exactly. However, if we consider the series of

ce,. and s e

in q0 or q i (ef.ref [2]), we may find that C and S . approach 6 as m ~ oo. Thus, for a given accuracy, we may choose a big enough number M. When m > M .4 ~. and B ~

can be expressed approximately by eq (8), while for m <~ M, the

approximate solutions of A ~ and B~. can be obtained by solving truncated versions of ¢qs (7a, b) and the results may be expressed as M

M

p~O

p~O

The choice of M depends on q o,ql and Iql - Oo I. When they are smaller, a n o t - t o o - b i g Mean ensure enough accuracy. The last equation of (4) gives rise to roJ--~1

(o) (~o 't/) ' ~(~1'°~2) [ ~oNc . m(q)Aw-cm+ . - , N s ' 0 1 ) A w s " -- - x~,l

(11)

where M

N c ,. (n) =

t p-~oMcp(1)"(q 1'4o)A~-p1. ce p(q i P]) , - i

(i)'

[Dc m Mc m (ql,~o)MC and

(3)"

m ~
(qo,~o)Ce (ql,q) , m> M

N s ( ~ ) have the similar expression (replacing Mc,ce,Dc and A~., in eq (125 by

82

Y.S. W A N G and D. W A N G

Ms,se,Ds and B-~,, respectively). J0 (rt) in eq (12) is ~/(cosh2¢ - cos2q) / 2. Introduce the dislocation density function

*(,1)

r oJo (~) oq

(aw)

(13)

Then equation (6) becomes

A~-Cm

ro f~2 Jo(~)~(~)ce,,(qo¢l)dq

m = 0,1,2.-(14)

r° f~ ~ Jo(q)~(q)sdm(qo,q)dq

A~-s.

/1~ -!

m = 1,2,3.--

where c=e'. (qo,q) = c e (qo,r/) and ~'m (qo,q) = cem(qo,q)" They can be found in ref [1]. Substitution of eq (14) in eq (11 ) leads to an integral equation iJ'% 7J o. .( 0 ~ ( 0 [ ~ ® 1 d~ = - x01(¢o,~/), (o) _ M l ( m ) ~ , . ( q o , O + =~.M2(m)se-m(qo,O n - " Jo(•) L.-o

rte(~l, %)

(15)

with M l ( m ) = i # i N c , , ( q ) and M 2 ( m ) = i#~Ns,,(rl). Considering the properties of the Mathieu functions as m ---} + oo[l], we have

M j ( m ) ~ (q o,0 ~ iBsinm(cosmq M 2( m ) ~ (qo,O ~ - iffcosm~sinmr/

m

+ oO

(1 6)

with p = - # l / (1 +#). We set i Jo(O e(~,~l)

-

7z

J o(~)

¢o

~=~.0[M I (m )~m (q o,O - i[3sinm~cosm~l ] -

+ ~ [g2(m)~,,(qo, 0 + i/~cosm~sinmr/]}

(17)

m-I

Equaton (15) reduces to a Hilbert singular integral equation

-

)J -6

(¢0,.),

(18)

which is the same form as eq (29) of ref [1] and may be further converted to a Cauchy singular integral equation of the first kind. The numerical solution of eq (18) was performed in detail in ref [1]. We will not give it here. Dynamic stress intensity factors The DSIFs at crack tips a t and n 2 are defined as

SHEAR WAVE SCATrERINGFROM DEBONDEDINCLUSIONS

I Ks, = lira

83

[4 2roJo("X~, -- 7)'c:, (~o,r/)]

''"

L K~2 = lim

(19) [4 2roJo("X" - ~2) z~, (Go,~)]

where z¢,(~o,~/) is the shear stress along the bonding region. Following the analysis similar to that presented in ref [1], we have

K m = -/34roJo(~,)c F( -

1) , K , 2 = - ~4

where F(z) = 4~(c~ + d)(1 - 2 ) 1 / 2 with c = (% - u l ) / 2

roJo(%) c F(I)

(20)

and d = ("2 + "1) / 2.

Scattering cross section

When

r----4x2+y 2 -,oo, ~--,oo.

Ms,.o}(qo,O

Considering the properies of

(3)

Mc (qo,~)

and

[2], we may derive the following asymptotic expression for the scattered

far-field displacement r e~"'-4)[Fto)(Oo,tl)+F°)(Oo,~l)], x w e(o}(~,T/)+w 0) o (~,t/),,, 8~K~_~_

r,oo

(21)

where F ¢0)(0 o,~/) is the scattered far-field pattern for a perfectly bonded inclusion and can be obtained from ref [2] and

F°)(Oo,tl)

is that due to debonding which follows, from eq

(3), as

Fen(Oo,~l) = 1 {~.o '- . A°~ce'(q°'tl)+ ~® i "Bo, se (qo,~l)}

(22)

m-I

The SCS is defined as [2] ~(oJ) =

(23)



where < e o > is the time average of the incident flux which may be written as [2] (D 2 < e 0 > - ~ ~oKrolAI

(24)

and < P" > is the total energy flux for the scattered fields defined in ref [2] If we use the far-field expression of the scattered displacement given by eq (21) and consider the f o l lowing expressions for very large value of ~[3]

roJ.,r=[x2

+

,

l ~ ~-; 0

ro-J ~

(25)

< p " > may be expressed as < P ' > - 4nm/%~" IF(O)(0o,r/)__ +

Ftl)(Oo,q)[2dq

(26)

84

Y.S. WANG and D.WANG

Numerical results The numerical results for the DSIF and SCS have been computed for a special combination of inclusion and matrix: aluminium, #1 =26.5 G P a , Pl = 2 . 7 g / c m 3 , and steel #0 =77.0 G P a , Po = 7 . 8 g / c m ~ . In this case, we choose M = 3 to ensure the accuracy of 10-2 for K ~ a < 2. The results are shown in figures 2 - - 4 in which the D S I F is normalized by ,o~r~-a(,0 = # o A K z ~ ) and SCS by a, and ~ represents the half angular width of the debond. The effects of ~ on the normalized D S I F and SCS for the debond symmetric about the major axis and minor axis are displayed in figures 2 and 3. A resonance appears at low frequency. As the debond is increased the resonance becomes more pronounced and occurs at lower frequency. This phenomenon is called low frequency resonance and was also demonstrated in refs [1] and [4] for an arc crack and in refs [5], [6] and [7] for a partially debonded circular inclusion. One may note that in 1986 Coussy [8] solved the similar problem and gave a far-field solution for the long-wavelength limit which did not show such a phenomenon. The effects o r b / a trated in figure 4 for ~ = 135 ° . As b / a

on the normalized D S I F and SCS are illus-

becomes larger the position of the resonance

peak appears at lower value of Kz0 a, that is, the normalized resonant frequency [K~oa ] R becomes larger. In ref [5] a simple spring-mass model was presented to explain the low frequency resonance behavior for a partially debonded circular inclusion. This simple model indicates that the resonant frequency is coR = ( K / m ) 1/2 where K is the equivalent elastic stiffness and ra the equivalent mass. The low frequency resonance was also discussed in far more detail in ref [6] and an explicit asymptotic expression of the resonant frequency was derived. The results show that the resonant frequency can be simply modeled as the frequency of a s p r i n g - m a s s system with twice the mass of a unit length of the inclusion. And it was also demonstrated that K depends on #1 / #0 and the ratio of the bonding area to debonding area. This spring-mass model can certainly be applied to a partially debonded elliptic inclusion for which m = 2 p l n a b

-'I K ]1/2 and thus [Kroa]R = C ro 2pl~r-/~/a . The

stiffness K will only change slightly with b / a

for a given unchanged a. Therefore

[Kroa]R will decrease as b / a is increased. In other words, the increase in b / a means

the decrease in the inertia of the inclusion, and thus resulting in the decrease in the normalized resonant frequency [Kroa]R. This betlavior is exactly what we have displayed in figure 4.

SHEAR WAVE SCATTERING FROM DEBONDI~D INCLUSIONS

85

Finally we indicate that as b / a - , I, i.e. the elliptic inclusion tends to a circular inclusion, the analysis and the results presented in this paper coincide exactly with those o f ref [7]. Furthermore when the inclusion and matrix are made o f the same material, the problem solved in this paper reduces to the problem we have dealt with in ref Ill.

, =.=o" " ~t ---..=,=' t~l

-~~.A:Iv.=o s

"?1

6

°'N'- A

=.¢

,

1~'/~ ~'/'X

,::~ ~\ Ji t,k,/;"-x ,,,=o~

//

o.o

o.~

o.~

~,,. o'.~ -;:~ -'~---~.5

D.O

0.3

~ -

0.6 ~..~s 0.9

1.2

1.5

<2

1.-'~

Figure 2: The effects of = on the normalized DSIF for b / a = 0.5

8.o[

,~,,l,~

4.5

~'~

--"~ ---o.-,~

0

0.3

/I \ \

/ I~

.
1.2

1.5

o; o

o.3

0.6 K.,.,,,o19

Fil~re 3: The effects of = on the normalized SCS for b / a = 0.5

86

Y.S. WANGand D. WANG 3 O.

~

&b~INl~

"~ I 00"80''

!

~

/ /i/~l\~ /

~ ~1.0

/ /~Y\o8

/ /'t,a

/ / ~\A\A

0.6

o.,

.-~

iii/I////V~~~ //// / // °,.o

I

0.3

o16 K.. o!9

.1 1.2

1.5

0

0.3

0.6 K.t~a 0.9

~ . 1.2

1.5

Figure 4: The effects o f b / a on the normalized D S I F and S C S for ~ = 135 °

References 1. Y.S.Wang and D.Wang, Eng. Fracture Mech. 48,289 (1994). 2. Y.H.Pao and C.C.Mow, Diffraction of Elastic Waves and Dynamic Stress Concentrations, Crane and Russak, New York 0973). 3. K.Harumi, J.Appl.Phys. 32, 1488 (1961). 4. Y.S.Wang and D.Wang, Int. J.Fracture 59,R33 0993). 5. Y.Yang and A.N.Norris, J.Mech.Phys. Solids 39,273 (1991). 6. A.Norris and Y.Yang, J.Appl. Mech. 58,404 (1992). 7. Y.S.Wang and D.Wang, Proceedings of the 2nd International Conference on Nonlinear Mechanics (ed.W.Z.Chien), p202, Peking University Press, Beijing (1993), 8. C.Coussy, Mech. Res. Commu. 13, 39 (1986).