BACKSCATTER FROM A SPHERICAL INCLUSION WITH COMPLIANT INTERPHASE CHARACTERISTICS. M. Kitahara Tokai University Shimizu, Shizuoka 424, Japan

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1 BACKSCATTER FROM A SPHERICAL INCLUSION WITH COMPLIANT INTERPHASE CHARACTERISTICS M. Kitahara Tokai Univerity Shimizu, Shizuoka 424, Japan K. Nakagawa Total Sytem Intitute Shinjuku, Tokyo 162, Japan J. D. Achenbach Center for Quality Engineering and Failure Prevention Northwetern Univerity Evanton, IL INTRODUCTION In tudie of cattering by an incluion it i generally aumed that the incluion i perfectly bonded to the urrounding matrix material. The actual bond between two material i, however, generally effected by a thin layer, which may be called an interphae, rather than an interface. It i well known that the mechanical behavior of uch an interphae may ignificantly influence the overall mechanical behavior of a olid containing incluion. In thi paper it i invetigated to what extent an interphae affect the cattered field generated by an incident ultraonic wave. Both a completely intact but compliant interphae, and an interphae which doe not tranmit traction over part of the area between the incluion and the matrix, have been conidered. The interphae i generally very thin. In thi paper, it i aumed that the radial and the tangential traction are continuou acro the interphae, but the diplacement may be dicontinuou from incluion to matrix. The traction are aumed to be proportional to the correponding diplacement dicontinuitie. The proportionality contant characterize the tiffne and trength of the interphae. On the bai of thi interphae model, which correpond to a ditribution of pring between the incluion and the matrix, a rigorou analyi ha been carried out of the backcattered field generated by an incident longitudinal wave. Within the context of the preent model the cae of a partially defective interphae, i.e., the cae of a crack over part of the urface between the incluion and the matrix, i eaily included by etting the pring contant identically zero over the cracked urface. The analyi ha been carried out by deriving a et of ingular integral equation for the traction and diplacement acro the 47

2 interphae. Thee equation have been olved by the boundary element method, and the cattered field ha ubequently been obtained by the ue of the elatodynamic integral repreentation. An analyi for a imilar configuration, but by the ue of the null field approach, and for a completely intact interphae, ha been preented by Datta, Olon and Botrom [1]. FORMULATION Let A and~ be Lame' elatic contant, and p the ma denity of the incluion, and let A,~.p be the correponding quantitie of he urro~nding matrix material, a hown in Fig. 1. In thi figure, u and u are the incident and the cat ered 5 wave, repectively. Thetotal wave field i defined a u - u + u. In the following, the time-harmonic factor exp(-iwt) ha been uppreed, and the upper barnotation i ued for quantitie related to the incluion. Boundary Integral Equation for the Matrix and the Incluion form The boundary integral equation for the matrix material take the Cij(~)uj(~) = { Uij(~ l)tj(l)dsy- { Tij(~ l)uj(l)dsy + ui(~), x S, where Si the interphae boundary at the matrix ide and U.. (x,v) i the fundamental olution for 3D time-harmonic elatodynamic: u (x ) - _L [ei~r 6 + L _g Q_ rei~r- eiklr l] (1) 1J - "- ij _,y 4~~ r ij kf axi axj t r r J ' (2) while T.j(~ l) i the correponding traction. The boundary integral equatioft for the incluion i of the form cijc~>ujc~> = u ijc~.y>tjcy>dy f Ti.(x,v)u.(v)dS, x, (3) - J-"- J"- y Fig. 1 Scattering by an incluion with a compliant interphae. 48

3 where S i the interphae boundary at the incluion ide. In matrix form Eq.(l) become after the dicretization: H u - G t + ui (4) ::::::::- :::::::- - Next we operate with AG-l on both ide of Eq.(4), to obtain :::::::::::::::: where K and F are defined a "" K - A G-l H, F - A t ::::::: ::::::: ::::::: ::::::: - ::::::: - (6a,b) Here it ha been aumed that in taking Q, the frequency doe not coincide with an eigenfrequency of the incluion. The matrix ~ ha the following diagonal form -1 A- "" Here the ubcript M denote the total number and the a-th ub-matrix A ha the form A _ [~a za 0 za (7a) of boundary element, (7b) where A i the area of the a-th element. a In the ame way, Eq.(3) for the incluion may be reduced to where K u- F :::::: ::::::: :::::::::- ls - ~ ~ ~ F - A t Spring Equation to repreent the Interphae (8) (9a,b) The interphae between the incluion and the urrounding matrix i now modelled by a continuou ditribution of pring, ee, Fig. 1. The pring are aumed to be linear and they have three component. The ytem of equation defining the pring connection can be written down in the following form [ g _ g l { ~SP } { ~SP } -S S usp FSP ::::::: ::::::: - - where g i the pring contant matrix, which i defined a "" [ ~1 ~~ R ~ l ~ =a.. ~ -SP -SP SP SP (10) (lla) In Eq.(lO), u and F (u and F ) are the d~placement and force on the pring at the incluion (matrix) ide. The upercript SP deignate quantitie in the pring. In Eq.(lla), the a-th ub- 49

4 matrix ~ i the pring contant matrix of the a-th element; it ha the form "':.. l (llb) where Rla' R2a' and R3a are pring contant per unit area in the x1,x 2,_and x3 direction, repectively. The under-bar for the ubcr1pt a mean that no ummation i implied. In Eq.(lO) and (11) the pring layer i aumed to be of zero thickne, and hence A i equal to A. Thi mean that R - R and S a a "'0 "'0 "' - ~- The zero thickne aumption alo implie that the effect of inertia of the interphae i neglected. Interaction Condition and the Sytem of Integral Equation For the eight vector quantitie, ~and E in Eq.(S), ~and fin SP SP -SP -SP Eq.(8), and u, F, u and F in Eq.(lO), the following interaction condition-hold - - SP u -u FSP + F- 0 SP u u FSP + F - 0 (12a,b) (12c,d) From Eq. (S), (8), (10), and (12), we obtain the following ytem of integral equation (13) For the given incident field, u 1, thi ytem can be olved for u at the incluion ide and u at the matrix ide. We can ubequently obtain F and F from Eq.(8) and (5). The cattered field can then be calculated from the original integral repreentation. PARTIAL DEBONDING OF THE INTERPHASE It hould be noted that the pring contant per unit area in pring Fig. 2 Incluion with a crack over part of the interphae. 50

5 Eq.(llb), Ria (i- 1,2,3; a- 1,2, M), may change for each direction i( - 1,2,3) and they may alo change in each element a(-1,2, M). A a pecial cae, we can put Ria= 0 (i- 1,2,3; a- k+l, k+n), where N i le than M. Thi i a model for partial debonding of the interphae. Part of the boundary (N element) i a crack-type urface, while in the remaining part (M-N element) contact between incluion and matrix i maintained by the pring connection, a hown in Fig. 2. Numerical Example We retrict our attention to cattering by a phere of radiu d. The incident wave i choen to ~e a plane longitudinal wave which travel in the x3-direction: ~ (~) - exp(iklx 3 )~ 3. In the following calculation, the non-dimenional wave number kld wa fixed a kld 1.0, and the material propertie were taken a cl/cl - lj}2, II - II 1/4, PIP - 1. To diplay general trend for the preent pring model, the abolute value of the total diplacement on the both ide of the interphae are hown in Fig. 3, for the cae of pring contact over the whole boundary. The pring contant per unit area are R /R - R 8 /R - R~/R for all element, where R - pcl2/d r o o "~' o o 2. 5 lu I, 1-;; I r r 2.5 Fig. 3 Total diplacement at the incluion and matrix ide of the interphae, Rr/R 0 - R8/R 0 - R,p/R

6 (for implicity, we writer intead of R ). The left hand ide how polar plot of the raaial diplacem~gt lurl on the matrix ide (half black circle) and lui on the incluion ide (open circle). The r right hand ide how the tangential component lu 9 1 and lu 9 1. For thi cae of complete pring contact, we can obtain the exact olution by a light extenion of Pao and Mow' expreion [2]. The olid line of Fig. 3 how thee exact olution. Figure 4 how the component of the cattered field for the boundary diplacement, where lul and lul are plotted on the left- hand ide, and lu8 1 and lu 9 S S r r S I 1 on the right-hand ide. The cattered field i defined a u u - u. For thi calculation, the pring contant Rr/R 0, R 8 /R 0 and R /R 0 were et equal to zero in the lower half (x 3 ~ 0) of the interphae. Thi correpond to the cae that the lower half of the interphae i crack urface. Over the upper half of the pherical interphae, Rr/R 0 - R8/R 0 - R /R for all element I u I, lu I r r Fig. 4 Scattered diplacement at the incluion and matrix ide of the interphae, at the inonified ide of t he incluion; pring contant over the hadow ide: Rr/R 0 = R9/R 0 = R /R

7 0.8 u I lu 1 1 () Real Imaginary o x/d Fig. 5 Real and imaginary part of the backcattered diplacement field for the cae of a crack at the inonified ide of the incluion; pring contant over hadow ide: Rr/R 0 - R8/R 0 - R~/R Figure 5 and 6 how the real and imaginary part of the back cattered field, u~ - u3 - u~, along the x3-axi. Figure 5 i for the cae of a crack over the lower half of the interphae (x 3 ~ 0), and Fig. 6 i for the cae of a crack over the upper half (x ~ 0). Over the half of the pherical interphae which remain intac~, the pring contant are Rr/R 0 - R 8 /R 0 - R~/R for both figure. Becaue of the ymmetry with repec to the x1 - axi, he b~ckcattered field on the x3-axi ha only a u 3 component, ince u 2 - u 1 = 0 along the x 3 -axi. The backcattered diplacement u~ ha been plotted at a ditance of 30d from the front face of the incluion. It i noted that a comparion of the real and imaginary part of the cattered wave form of Fig. 5 and Fig. 6, how a phae hift in the amount of about one half wave length. 53

8 u 3 lu r () Real 0 Imaginary JO x 3 /d Fig. 6 Real and imaginary part of the backcattered diplacement field for the cae of a crack at the hadow ide; pring contant at the inonified ide: Rr/Ro - R8/R 0 - R /R ACKNOWLEDGMENT The work of two of the author (M. Kitahara and J. D. Achenbach) wa carried out in the coure of reearch under.contract DEFG02-86-ER13484 with the Department of Energy, Office of Baic Energy Science, Engineering Reearch Program. REFERENCES 1. S.K. Datta, P. Olon and A. Botrom, in Wave Propagation in Structural Compoite, edited by A.K. Mal and T.C.T. Ting, AMD-Vol. 90, (The American Society of Mechanical Engineer, 1988), pp Y.H. Pao and C.C. Mow, Diffraction of Elatic Wave and Dynamic Stre Concentration, (Crane, Ruak & Co., New York, 1971). 54