Elastic-wave scattering and Stoneley wave localization by anisotropic imperfect interfaces between solids

Transcription

1 Geopfiys. 1. Int. (1994) 118, Elastic-wave scattering and Stoneley wave localization by anisotropic imperfect interfaces between solids u'. Huang and S. I. Rokhlin Nondestructioe Evaluation Program, The Ohio State University, 19 esr 19th Avenue, Columbus, Ohio 432 1, USA Accepted 1944 February 15. Received 1994 February 2 1 INTRODUCTION Elastic-wavz interaction with layered solids and the effect of interfaces between two solids on their behaviours have important applications to geophysics, acoustics and non-destructive evaluation. The pioneering work of Stoneley in this area has stimulated a great deal of research up to the present. A relatively new aspect of wave interaction in elastic media is the effect of imperfect interfaces and the description of boundary conditions (BC) on such interfaces. Typical examples for (a) a fractured interface with preferred crack orientation, and (b) an interphase with cylindrical-like pore inclusions are shown in Fig. 1. To model such imperfect interfaces non-classical BC are generally used. There are two approaches to obtaining the BC that describe the interface imperfection. The first is to use rnicr+mechanical analysis to define the BC directly, such as the quasi-static (spring) model for cracked interfaces (Baik & Thompson 1984; Margetan, Thompson & Gray 1958). These studies and more recent theoretical and experimental work (Margetan et al. 1992; Nagy 1992) SUMMARY In this paper, elastic scattering and localization of guided waves on a thin anisotropic imperfect interfacial layer between two solids are studied. e have proposed a second-order asymptotic boundary condition approach to model such an interfacial layer. Here, using previous results, we derive simple stiffness-matrix representations of stress-displacement relations on the interface for the decomposed symmetric and anti-symmetric elastic motions. The stiffness matrices are given for an off-axis orthotropic layer or, equivalently, for a monoclinic interfacial layer. For the problem of scattering on such a thin anisotropic la,yer between identical isotropic semi-spaces the scattering matrices are obtained in explicit forms. Analytical dispersion equations for Stoneley-type interfacial waves localized in such a system are also given. Additional results are included for irnperfect interfaces, such as fractured interfaces, modelled by spring boundary conditions. The applicability of the stiffness-matrix approach to the layer model is analysed by numerical comparison between the approximate and exact solutions. The numerical examples, which include reflection transmission on the interphase and dispersion curves of the interfacial waves, show that the stiffness-matrix method is a simple and accurate approach to describe wave interaction with a thin anisotropic interfacial layer between two solids. Key words: guided waves, scattering, second-order asymptotic boundary conditions, stiffness matrices, thin interfacial layer. suggest that fractured interfaces, and in particular fractured interfaces between crusts (Pyrak-Nolte, Myer & Cook 199; Pyrak-Nolte, Xu & Haley 19921, can also be modelled by such spring BC. The other approach is to use ii thin multiphase interfacial layer to model a solid-solid interface with complex properties, and use an asymptotic expansion to substitute for the interphase by equivalent interface BC (Rokhlin & ang 1991a). Besides this method of asymptotically expanding the boundary-problem solution, other asymptotic methods have recently been proposed by Bostrom, Bovik & Blsson (1992), using series expansions of the governing differential equations, and by ickham (1992), using approximations with boundary integral methods. A mathematical analysis of these BC, with regard mainly to uniqueness, has been given by Martin (1992). In this paper we describe and compare both interfacial layer and spring models and derive analytical solutions for imperfect anisotropic interfaces between identical isotropic semi-spaces. In some cases. the interfacial layer model is straightforward, when for example, an adhesive joint or an embedded thin crust is present. In other cases, it is used as 285

2 286. Huang and S. I. Rokhlin (b) Figure I. Typical examples of imperfect interfaces. (a) A fractured interface with a distrihution of cracks. (b) An interphase with inclusions or pores. q is the rotation angle of the incident (1.3) plane with respect to thc crack or pore orientation. an approximation to an actual imperfect interface, such as an interphase with inclusions or porosities (Rokhlin & ang 1991a). As shown in Fig. 1, the interface imperfections can have preferred orientations that induce anisotropy on the interface. hen the incident plane (the (1,3) plane in Fig. 1) is rotated with respect to the imperfection preferred orientation, the in-incident plane and out-of-plane elastic motions are coupled due to the interface anisotropy, even for isotropic semi-spaces. To study wave interaction with a layered interface between solids, one can use transfer-matrix approaches: see, for example, monographs by Brekhovskikh (198) and Aki & Richards (198) for the isotropic case, and papers by Ingebrigsten & Tonning (1969), Takeuchi & Saito (1972), Fahmy and Adler (1973), Mandal & Toksoz (199) and Nayfeh (1991) for the anisotropic case. However, with these methods, the role of the different interfacial parameters can be understood only by intensive computation. Rokhlin et al. have proposed using equivalent asymptotic BC to model a tbin (compared to the wavelength) interfacial layer (Rokhlin & ang 1991b, 1992; Rokhlin & Huang 1992; Huang & Rokhlin 1992). Here, on the basis of a second-order asymptotic BC approach (Rokhlin & Haung 1993), we demonstrate a stiffness-matrix representation of the asymptotic BC for decomposed symmetric and anti- symmetric fields, and derive second-order analytical solutions for scattering and dispersion equations of interfacial waves on a thin anisotropic layer between identical isotropic semi-spaces. Additional results are obtained for fractured interfaces described by spring BC. The organization of the paper is as follows. In Section 2 we briefly review the second-order asymptotic BC, as described by Rokhlin & Huang (1993), which yield second-order approximate wave solutions in the nondimensional thickness parameter kh, where k is the wave number and h the thickness of the layer. In Section 3 we derive the stiffness-matrix representation of the secondorder BC for decomposed symmetric and anti-symmetric fields. The symmetric and anti-symmetric stiffness matrices are given for an off-axis orthotropic, or equivalently a monoclinic, interfacial layer. Physical meanings of matrix elements and conditions for further simplification are discussed. In Sections 4 and 5 explicit equations for the scattering matrices and the dispersion of interfacial waves for a thin ansiotropic layer between identical isotropic semi-spaces are obtained. Similar results for fractured interfaces modelled by spring BC are given in Section 6. Assessments of the accuracy of the stiff ness-matrix approach, and other simplified approximations. are given in Section 7 by numerical comparison with the exact solutions. The calculated results include reflection transmission and dispersion curves of interfacial waves. 2 THE SECOND-ORDER ASYMPTOTIC BC Let us consider a thin anisotropic layer positioned on an interface between two elastic solids of general anisotropy and assume the z-axis to be the interface normal as shown in Fig. 2. To describe the elastic motion of the layer one UPPER SEMISPACE INCIDENT AVE r SLO SHEAR ux uy uz FAST SHEAR O zx ozy Ozz LONGITUDINAL >X ANISOTROPIC INTERFACIAL LAYER h u: u; u: LONGITUDINAL a:, a:y a:, FAST SHEAR LOER SEMISPACE 1 SLO SHEAR Figure 2. Elastic system with an anisotropic interfacial layer between two generally anisotropic semi-spaces.

3 ave scattering and localization 287 relates the particle displacements and stresses, U and U' respectively, on the top and bottom surfaces of the layer by a transfer matrix B: given by Rokhlin & Huang (1993) as Eq. (1) is the starting point for deriving equivalent BC to replace the interfacial layer. For a system with density and elasticity varying only in the z direction. one can represent the governing equations for the elastic field vector U in the form of a system of differential equations: - C 3.3 where k = to/v = (w/v,) sin 8, is the projection of the wave vector on the interface, V, is the speed of the incident wave,, the incident angle and V the tracing speed (or wave speed of the guided mode for interfacial wave problems). Eq. (2) is a six-dimensional form of the equations of motion of the medium. This type of representation was used by Alterman, Jarosch & Pekeris (1959) for the isotropic case, and by Stroh (1962) and Ingebrigsten & Tonning (1969) for the generally anisotropic case. The differential equation (2) has the well-known matrix exponential solution U(z,, + Az) = exp (-ikaza)u(q,). (3) Therefore, for an interfacial layer of thickness h, eq. (3) becomes U = exp (-rkha)u'. (4) The exact transfer matrix B in eq. (I) is B = exp (-iitha) = I - ikha - i(kha)' + + O[(khA)']. (5) A simple form of the 6 x 6 matrix A for an off-axis orthotropic interfacial layer (as shown in Fig. 3) has been z m INCIDENT PLANE / ikq Ih w"p,, ik where po is the density and C, the elasticity tensor of the interfacial-layer material under the rotated coordinate system (x, y, z). In eq. (6) the compliances Sd4, S,, and S, can be written in terms of the elastic constants as I and Q, are the stiffness defined under the condition of plane stress (uz, = a:; = ): u: u: u: I 1 1 (Jzx a& (Jzz OFF-AXIS ORTHOTROPIC INTERFACIAL LAYER Figure 3. Rotated (x. y,z ) and material (x", vo, z) coordinates for an off-axis orrhotropic interfacial layer. C,,C,, where Q,] = C,, - ~. c33 Eq. (8) is Hooke's law for a very thin anisotropic free plate. The off-axis orthotropic layer can be used to model a very general anisotropic imperfect interphase since the only axis of symmetry for the elastic system is the interface normal. For example, it has been used to model an interphase with cylindrical pores (see Fig. lb) through a two-phase model by Christensen (1991) (see also Rokhlin & Huang 1993). In Fig, 3. (x", y", z) is the material coordinate system of the

4 288. Huang and S. 1. Rokhlin orthotropic layer whose elasticity tensor C" has the form and (x, y, z) is the rotated coordinate system formed by the incident plane, (x, z), and the interface plane, (x,y). One sees that the interface plane is a plane of symmetry, but the incident plane deviates from the plane of symmetry, the (xo, 2 ) plane, by a rotation angle QI. The elasticity tensor C under the rotated cjordinate system has the form and its relation to C'" is given in Appendix A as a function of the rotation angle QI. Note that C, given by eq. (lo), is equivalent in form to the elasticity tensor for a monoclinic material. To use the solution (4) directly, one must compute exp (-ikha). This can be done by standard matrixdiagonalization procedures, solving for the eigenvectors, which equivalently requires finding wave solutions of the interfacial layer (e.g. Fahmy & Adler 1973; Mandal & ToksGz 199; Nayieh 1991). The purpose of introducing equivalent BC is to reduce a multilayered problem to a standard boundary problem. To do this, we follow Rokhlin & Huang (1993) in taking a finite-difference approximation to the exact differential equation (2): Rewriting eq. (11) in the transfer matrix form (l), we have ( ik;a)-'( ikta) U' = B,,U', u= I+- I -- where the approximate transfer matrix B,, is B,, = (I +?) ikha -I (I- ikf) = 2(1+ 1)ikhA -'- I = I - ikha - $(kiza)' + +i(kha)'+ U[(khA)4]. (13) Comparing the expression (13) with the series expansion (5) of the exact transfer matrix B in the small parameter kh (thin interfacial layer), we note that the transfer matrix B,, is identical to second order to the exact transfer matrix B. Thus, BC (1 I ) or (12) give a second-order approximation to the exact solution for the elastic field. Another important feature of this new representation IS that the second-order BC are symmetric with respect to the elastic field vectors on the top and bottom surfaces of the layer. This will be crucial for its decomposition into symmetric/anti-symmetric parts, as will be discussed in the next section. (9) As shown by Rokhlin & Huang (1993), the scattering solutions based on the second-order BC are unique and satisfy energy balance, as well as being highly accurate. They also give zero scattering for a homogeneous medium when the interfacial layers is made of the same material as the semi-spaces. 3 STIFFNESS-MATRIX REPRESENTATION OF THE SECOND-ORDER BC Let us take both semi-spaces to be of the same material and the interphase plane to be a plane of symmetry of the elastic system. Then the wavefield can be decomposed into symmetric and anti-symmetric parts as illustrated in Fig The anti-symmetric part For the anti-symmetric part, the elastic fields on the opposite surfaces of the interfacial layer must satisfy the conditions u, = -u;, u, = -u;, u, = u: Uzx = ULX, or" = u;,, UZZ = -Uzr. ' 1 (14) Substituting eqs (14) into the second-order asymptotic BC (ll), we have ikh (u~, u,,,,, uzz).1 = - 7 A(,, u,, u,,, a,,, )"'. (15) I Using the matrix A given by eq. (6) for an off-axis orthotropic layer, one obtains [ = KA[ I One sees from (16) that each element in K, has the units of stiffness (elastic constant per unit length), so we call K, the anti-symmetric stiffness matrix. It is clear that the second-order asymptotic solution for the anti-symmetric case depends on only three of the thirteen elastic constants for a monoclinic interfacial layer (C',,, C,, and C,,). The first two diagonal terms in K, are in-plane (h 'C,,) and out-of-plane (h-'c,4) shear stiffnesses. (Note that 'plane' here means the wave propagation plane, which is perpendicular to the interface. hk'c4, is a shear stiffness coupling between the in-plane and out-of-plane shear stress and displacement.) The element ikc',, couples the normal stress or displacement to the in-plane shear displacement or stress; ikc,, couples the normal stress or displacement to the out-of-plane shear displacement or stress, so these two are called the coupling terms. The (3,3) diagonal entry in K, ik'h(c,, -pov') = -$wzpf,h{ 1 - (V&,l,/V2)}, includes an inertia part, -~o'poh. which is not zero at normal incidence (k = ). Here V,,, = (C5s/p,J"2 is the speed of the in-plane shear wave propagating along the x direction in the interfacial-layer material.

5 ave scattering and localization 289 INCIDENT AVE (2A) / I INCIDENT AVE (A) INCIDENT AVE (A) lnt RFAClAL LAYER ANTISYMMmIC CASE + SYMMETRIC CASE INCIDENT AVE (- A) INCIDENT AVE (A) Figure 4. Decomposition into antisymmetric and symmetric fields. The semi-spaces are identical and the interface planc is ii symmetry plane of As each row in K, determines a stress component, to examine the effect of different elements in that row, we write eq. ( I 6)2 in the form Let us first consider the case when C4, << CSs. Then K, be approximated as c,, C,, - ikh 2 (f:). (17) can Eq. (18) can be used when either the anisotropy of the interfacial layer or the rotation of the incident (x, z) plane from the plane of symmetry is small (i.e. ~1 is small in Fig. 3). Eq. (18) is exact when the incident (x, z) plane coincides with the plane of symmetry of the orthotropic layer (C,5 = at QJ =O). In this case the in-plane and out-of-plane (SH) elastic motions are decoupled: 2c:, or" = u,. h r l ikh 1 (2) One sees that the second-order asymptotic BC for the in-plane components (19) of the anti-symmetric motion depend only on C&, while the condition (2) for the out-of-plane components depends only on CY4. For the isotropic case, C:, and C& in eqs (19) and (2) are simply the shear modulus, plo, of the interfacial layer. The term +kh in eq. (17) can be represented as --_ kh - w nh. h sin, = -sin 6, = E, sin O,, 2 2y A, where A, is the wavelength of the incident wave and E, = nh/a, can be considered as a non-dimensional thickness. One sees that tkh is small when either E, or sin, is small, i.e. at low frequency or small incidence angle. Now we can rewrite the matrix K, in eq. (17) as K, = "/ h 2 c,, C55 ic, sin 6; where cs,,,) = nh/a,y,,,, = wh/(2vyv,,) is the non-dimensional layer thickness parameter normalized by the wavelength A,y,, of the SV wave in the layer material. Equation (22) contains terms of different orders in E; sin,: zeroth, first and second. The second-order term cf sin2 ; (the incident-wave-dependent part) in the (3,3) element can be neglected if E; sin 6, << E, ~ ~ ~ i.e.,, sin f3,(asvo/aj) << 1 or sin 6,( VyV(,/&) << 1. This implies that the shear-wave speed Vs,,(, on the interphase is much less than the incident wave speed v, or else that the incidence is nearly normal (6,-). For interfacial guided waves, V = V,/sin 6; is simply the phase speed of the guided mode. If one neglects, in addition in K, all the first-order terms in E, sin 6, (or tkh), one has I 1 - cs5 r - 7 c45

6 29. Huang and S. I. Rokhlin and the BC (16) simplify to To examine the effect of different elements, we rewrite K, as L The BC (24) are valid when either E, or the incident angle Oi is small. They are equivalent to the second-order BC in the case of normal incidence (, = in eq. 22 or k = in eq. 16). Such a BC model is a generalized sti#ness-rnass model for the anti-symmetric case of an anisotropic interface as only the effective inass and stiffnesses of the interphase are considered. Neglecting further the inertial term on the right-hand side of eq. (24) results in the BC for the anti-symmetric motion that account for only the shear stiffness of the interphase: 2c,, 2C,5 uzr =--u, +--u,, CT,, =- u, f-u,, h h h h uzz =. The normal stress urz on the interface now vanishes. The BC (25) constitute a generalized stiffness model for the thin interfacial layer. They can only be applied for a very thin interfacial layer with very low density or for a cracked interface under anti-symmetric loading. 3.2 The symmetric part The symmetric part of the elastic fields on the two surfaces of the interfacial layer must satisfy u, = u;, u, = u;, u, = -u, uzr 'I (26) = -CTi2, u:,. = -u;", uzz = u;=. Thus the second-order BC (11) are ikh (,, u,, u ~ uzv, ~ ())*'=--A(, 2 u,, uy,, O,O, uzz)7'. (27) For a thin off-axis orthotropic layer represented by eq. (6), this reduces to I - L L J J I 'L 33-1 h (28) where Ks is the sjrnmetric stiffness matrix, which depends on a different and larger set of elastic constants of the interfacial layer, (C,,, C3,, C,,, C,,, C,, and C,,), than K,. Here the (3,3) entry is the normal stiffness h-'c,,, the off-diagonal coupling terms are all incident wave dependent and both the (1, 1) and (2,2) entries are incident-wave and inertia dependent. X For small anisotropy of the interfacial or small rotation angle q, C,,/C,, << 1, C3,/C3, << 1 and Ks becomes 1 -- " O ikh C,, 2 c33 Eq. (29) reduces to (3) when the incident (x, z) plane coincides with the plane of symmetry of the orthotropic layer, the (xo, z) plane. In this case the in-plane and out-of-plane (SH) elastic motions decouple; and k2h uzy = -(cg, - pv2)u,. 2 One sees that the second-order asymptotic BC for the in-plane components (31) of the symmetric motion depend on Cy,, C:, and C:,, while the out-of-plane components (32) depend only on C:,. For the isotropic case, C?,, Cy,, C:, and C&, in eqs (31) and (32) are replaced by combinations of the Lam6 constants, A,, and po, of the interphase. Using (21) one can rewrite eq. (29) as LL33 Ks=h X 1

7 where E,~) = nh/l = wh/(2vn,), VnO = (C,,/P~J ~; is the non-dimensional thickness parameter normalized by the longitudinal wave propagating normal to the interface in the interfacial layer material. One sees from (33) that, like the anti-symmetric stiffness matrix K, given by eq. (22), the symmetric stiffness matrix K, also includes terms of different orders in E, sin ;. The incident-wave dependent part with second-order terms E; sin ; in the (1,l) and (2,2) elements can be neglected if E; sin2 8,/& << C,,/CI1 and C,3/C,. These conditions correspond to c11 Vz = V:/siri2 ; >> - = V;O and Po (34) where V,,, and V,,,, are approximately the speeds of longitudinal and SH waves propagating along the x direction in the interphase (exact when rp = ). If (34) are satisfied, w2ph the (1,l) and (2,2) entries in (28) become -~ 2 independently of the incident wave. By neglecting in K, all the second-order terms in E: sin2 Oi we have c,, V >> - = ViHOr In the BC (35) the coupling between the x and y components is omitted. If one further neglects all the first terms in E; :;in, (or ;kh), one has implying that PO. (35) (37) Eqs (35) and (37) are valid when either E, or, is small. The BC (37), with only the diagonal terms remaining in K,, are called the stiffness-mass BC for the symmetric case. They are exactly the second-order BC for normal incidence (, = in eq. 33 or k =, V- in eq. 28). The x, y and z components all decouple. This contrasts to the stiffness-mass BC (24) for the anti-symmetric case, where the coupling between the x and y shear motions remain, due to the terms in CdS. If one further neglects the mass (po=o) in (37), one obtains the stiffness BC for the symmetric field that involve only the normal stiffness: a,,=a,,=o, c3.3 uzz=2-u,. h (38) The shear stresses o,, and uzv on the interface become zero. ave scattering and localization SCATTERING COEFFICIENTS ON AN ANISOTROPIC LAYER BETEEN IDENTICAL ISOTROPIC SEMI-SPACES Using the stiffness matrix representations (16) and (28) for the anti-symmetric and symmetric parts of the second-order asymptotic BC, we will derive analytical solutions for scattering coefficients for a thin anisotropic layer between identical isotropic semi-spaces. Let us assume that the incident field is a combination of longitudinal and transverse (SV and SH) waves with displacement vectors u,; = 2A,P, exp (ik,n - r), uvi = 2AvPv, exp (ik,n * uhi = 2A,PHj exp (ik,n r), (39) where 2A,, 2A,, 2A, are the amplitudes and P,,, P,,,, P,, the polarization vectors of the incident waves. I stands for longitudinal, V and H for SV and SH waves respectively; k, and k, are the wave numbers of the longitudinal and transverse waves and n and r are the wave normal and position vector. Owing to the isotropy of the semi-space, an incident transverse wave of arbitrary polarization can always be decomposed into two transverse waves, the in-plane part, SV, and out-of-plane part, SH. The time factor exp (-iwr) is omitted throughout. e assume that the three different incident waves satisfy Snell s law, k = k, sin 8, = k, sin,, where 8, and, are the incident angles of the longitudinal and transverse waves. Let us first decompose the elastic field into anti-symmetric and symmetric parts, as shown in Fig. 4, and solve each part of the problem separately using the corresponding stiffness matrix. There are in general three reflected and three transmitted waves in each decomposed field for each of the three types of incident waves, due to the interface anisotropy. Thus a combination of scattered waves for all three incident waves gives the anti-symmetric or symmetric part of the total scattering. To describe this, we introduce scattering matrices, R and R for the reflected field, T and T for the transmitted field, as Tjg. p$) T(rr.5) vv where R, k and R;k are the anti-symmetric and symmetric parts of the reflection coefficient, and T,,k are the transmission coefficients. The first subscript denotes the type of incident wave and the second subscript the reflected wave. From reciprocity, one obtains (from Fig. 4) T = -R and T = R. Note that to use the decomposition procedure shown in Fig. 4, the polarization vectors for waves of the same type in the upper and lower semi-spaces should be chosen to be symmetric about the interface plane. The convention used in this paper is that for a longitudinal wave P, = n, for an SH wave P, = (, 1, and for an SV wave P,=nXP, in the upper semi-space and P, =P,,Xn in the lower semi-space. The scattering matrix R(.sJ can be found by satisfying the corresponding anti-symmetric and symmetric BC, (16) and

8 292. Huang and S. I. Rokhlin (28). A detailed derivation of the system of boundary equations is given in Appendix B. The scattering coefficient equations imply that Here c,, 2 c,, P(I + -- (a, q P P + q2 y )]h which satisfies reciprocity of the energy conversion coefficients between different modes (Rokhlin, Bolland & Adler 1986). In (41)!he non-dimensional parameters are cr = sin 8, = k/k,, = V,/V, (V, and V, being, respectively, the longitudinal and transverse wave speeds in the semi-space), p = ('s 8, = (1- and y = E cos 8, = (E2- cr2)li2. There are only six independent elements in R(.S)., the corresponding elements of R" are given below and in Appendix C: AO(-p, -6) R;; = a"(p, h) ' (43) where p is the density and p is the shear modulus of the semi-space, and A" = is the second-order characteristic equation for anti-symmetric interfacial modes localized in the interphase, discussed in detail in the next section. In the above equations, h = k,h, = 2a2-1, a, = a' + By, a2 = - - 2py. Ar = 2 + 4cv'py is the characteristic function for Rayleigh waves in the semi-space. The analytical solutions (42) and (43) are accurate only to second order in h, the third-order terms in h in A" arising from coupling of the second-order in-plane- and out-of-plane components. They are not the complete third-order terms, but they are required for the energy balance (see Rokhlin & Huang 193). e obtain for R'. (45) P CI I c33 - c:3 where the mass factors are ql = 1 - C33PV2 c33c66 - c:, P and q2=1-. As = is the second-order disper- GPO V2 sion equation for symmetric interfacial modes localized in this thin interphase and also discussed in detail in the next section. Solutions for other elements of are also given in Appendix C. The total scattering matrices for the reflected and transmitted waves are the sums of the symmetric and anti-symmetric scattering matrices: R = +Ra + i, (48) T=;T~+;~=-;R~+;w. (49) In general, the rank of the system of equations is halved when the second-order asymptotic BC are applied if the elastic system is symmetric about the mid-plane. The stiffness matrix representation of the second-order asymptotic BC greatly simplifies the derivation and study of the wave solutions. As discussed in the following section, this representation also has advantages in studying interfacial wave phenomena, since the symmetric and anti-symmetric modes are interfacial waves of different speeds and are naturally decoupled. 5 LOCALIZATION OF STONELEY-TYPE GUIDED MODE BY A THIN ANISOTROPIC LAYER BETEEN THE SOLIDS Since Stoneley (1924) found theoretically that an interface between two isotropic elastic solids can support a localized interfacial wave, there have been many studies of conditions for localization of generalized Stoneley waves on interfaces with welded and non-welded BC (e.g. Scholte 1947; Ginzbarg & Strick 1958; Lim & Musgrave 197; Chadwick & Currie 1974; Barnett et al. 1985; Murty 1975; Xu & Datta 199; Pyrak-Nolte et al. 1992) or thin interfacial layers between solids (Schoenberg 198; Rokhlin, Hefets & Rosen 198; Ma1 1988; Huang & Rokhlin 1992). It was found that

9 ~ for isotropic solids a thin interfacial layer with a shear modulus less than those of the semi-spaces will tend to localize thc elastic energy and, in particular, that such a layer between identical solids can always support interfacial waves (Rokhlin et al. 198; Rokhlin, Hefets & Rosen 1981). Here we will discuss the conditions for localization of interfacial guided modes in a thin anisotropic layer between identical isotropic semi-spaces. Suppose that an interfacial wave with displacement uj = Af;(z) exp [ik(x- Vt)] propagates in the x direction along an anisotropic interface layer, where V is the interfacial wave speed. The boundary equations (1) must be satisfied by the interfacial wave alone, which is composed of non-homog.eneous bulk waves. The waves in the semi-spaces are all evanescent, three in each semi-space due to the coupling effect of the interface anisotropy, while the waves inside the interface layer can be either propagating or evanescent. Let the wavelength of the interfacial wave be much greater than the thickness of the interfacial layer, so that the interfacial-wave problem can be solved using the second-order BC approach. 'The characteristic equation for the interfacial wave can be found either by setting the determinant of the boundary condition matrix to zero (Huang & Rokhlin 1992) or, equivalently, by setting the denominator in the scattering coefficients to zero. The method of calculating the exact solution for the interfacial wave speed and nunierical comparison of the exact and asymptotic solutions for the anisotropic case are given by Huang & Rokhlin (1992), where dissimilar anisotropic semi-spaces have been considered. Here we focus on the second-order characteristic equations obtained in the previous section for a thin off-axis orthotropic layer between two identical isotropic semispaces. The dispersion equations for the anti-symmetric and symmetric interfacial modes are obtained by setting the characteristic functions (44) and (47) to zero: The elastic motions of the anti-symmetric and symmetric modes on the interfacial layer are illustrated in Fig. 5. If the wave-propagation direction is not an axis of symmetry, i.e. if the x direction does not coincide with either the xo or yo direction, the interfacial-wave characteristic equation (5) cannot be decomposed into dispersion equations for modes with in-plane and out-of-plane SH-type polarizations. The 'plane' here is defined by the interfacial wave normal and interface plane normal, i.e. the (x, z) plane. In this case, the interfacial modes are not pure modes but quasi-modes. The particle motion for the interfacial wave is ellipsoidal and frequency dependent with strong coupling between the in-plane and out-of-plane displacement fields: see Huang & Rokhlin (192). However, when the interfacial wave propagates along a symmetry axis of the orthotropic interfacial layer, the in-plane and out-of-plane vibrations are decoupled. Thus the charactcristic equation for guided waves can be factored into two equations, one for the in-plane polarization and one for the SH polarization. Let us consider first the anti-symmetric case. The in-plane part of the second-order dispersion equation for an anti-symmetric mode propagating ave scattering and localization 293 (a) Anti-symmetric mode (b) Symmetric mode Figure 5. Guided waves along a thin inierfacial layer. (a) Anti-symmetric mode, (b) symmetric mode. in the plane of symmetry along the x axis is The dispersion equation for the SH-type anti-symmetric mode is c:'t A&, = Ph + 2i- =, P P (52) which yields no propagating wave solutions since a propagating mode requires that (Y > 1 and p = i(a' - 1)"' be positive pure imaginary. It is interesting to note that the characteristic function A", given by eq. (44), for an interfacial mode propagating along an off-symmetry axis direction (the x axis) can be represented as 1.2

10 294. Huang and S. I. Rokhlin where C, are the clastic constants in the rotated coordinate system (x, y, z) and the functions A: and A&, are given by eqs (51) and (52). The term in C,, is due physically to coupling between in-plane and out-of-plane vibrations. It is plain from c:q. (51) that in the plane of symmetry the interfacial wave speed V = V,/m for the anti-symmetric mode depends on only one elastic constant C:, (the in-plane shear modulus) of the interphase, on the interfacial layer density p(l and on the non-dimensional layer thickness k,h. Physically this is due to the particle motions of an anti-symmetric wave on the two sides of the layer being in phase in the z direction (see Fig. 5), so that the wave speed is insensitive to the normal stiffness h-'c;, (Rokhlin et al. 198). Eq. (51) is of second order with respect to h, compared to the first-order equations obtained by Rokhlin et af. (198, 1981) for the isotropic case, which have been applied to measurements of the shear modulus on thin films between solids. As follows from (51), the in-plane shear modulus C!, of the interfacial layer can be found from the experimentally measured speed of the anti-symmetric interfacial wave for an anisotropic layer: (54) Here recall that p is the shear modulus of the semi-spaces. For an isotropic interphase, the in-plane shear modiilus C& is simply the shear modulus po of the interphase. It is also interesling to discuss the results when simplified BC models are used. hen the stiffness-mass model is used for this interfacial layer, as given by BC (24), the dispersion equation for the anti-symmetric mode becomes A"(C,) = A:(C.5,)A:H(C4,) where &(C5?) has the same form as the characteristic function for the ca!,e of symmetry: Po Arh-i-yh -, 1 P (56) and A&,(Cj4) has!he same form as in (52). If we further neglect the inertia terms, the dispersion equation for the simple stiffness model (25) is (47) can be represented as (59) The additional terms on the right-hand side of (59) are due to coupling between the in-plane and out-of-plane vibrations. The in-plane part of the characteristic function for a symmetric mode propagating along an axis of symmetry (the xo axis) is and the characteristic function for the SH-type symmetric mode in this case is The characteristic equation A:," = yields a real solution for a propagating wave, since 6 is a positive pure imaginary number and the interfacial wave speed V> Vy:.,, where V(:.., = (C&/po)1'2 is the bulk SH-wave speed of the interphase. e can also consider cases when simpler BC models are used. If the stiffness-mass model with BC (37) is used, the dispersion equation becomes where and where and the form of A&,(C,J stays the same. One sees that the second-order term in h vanishes in (58). Similarly, for a symmetric interfacial mode propagating along an off-symmetry axis the characteristic function A' in One sees from eqs (62) to (64) that As depends only on C,,, which is independent of the interfacial-wave propagation direction. The in-plane and out-of-plane parts decouple due to neglect of the coupling terms in (37). The in-plane part of eq. (63) depends only on C;, = C,,, and the out-of-plane part involves no elastic modulus. In contrast, for the anti-symmetric case, due to the presence of a coupling

11 ave scattering and localization 295 spring tern], h-'c,,, which is wave number independent and thus of order zero, the decoupling of in-plane and SH-type vibrations is not possible as long as such stiffnesses are retained in the BC (24). If we further neglect the mass, the dispersion equation for the simple stiffness model (38) is the plane of symmetry. the propagating SH-type anti-symmetric mode does not exist. Next consider the symmetric mode. By substituting (Y = 1, p = into the right-hand side of (47) and equating to zero, one obtains Ph, - QhZ = where where cl' 3 A:, = i4y -> + 2A,h =, P - (66) and &)H=2/3, which gives the bulk shear wave in the semi-space. Now we consider conditions for the localization of propagating interfacial modes by a thin anisotropic interfacial hyer. If the speed V of a guided mode is less than the shear-wave speed V, in the semi-space, we have a 'propagating' mode. If the interfacial wave speed is higher than that of at least one bulk wave in the semi-space, the guided mode leaks energy into the substrate and is called a 'leaky' mode. ith increasing frequency, the phase speed of a given mode decreases, approaching the slowest bulk wave speed allowed in the interfacial layer, V,,,. Thus, for an interfacial layer to support propagating modes in a certain frequency range, the interfacial wave speed must be slower than the shear-wave speed of the semi-space and greater than that of the interphase: V,,, < V < V,. This implies that the interphase should be a 'slower' material than the semi-space. A mode that is leaky at low frequencies may become a propagating mode as the frequency increases. In this case, the frequency at which the propagating guided mode appears is called the cut-off frequency. To find the cut-off frequency, one can set V = V, (a = 1, p = ) in the dispersion equations and solve for hc. The dispcrsion equations derived from the exact layered approach are transcendental functions in h that give, in addition to the fundamental modes, an infinite number of higher ordcr modes. The analytical dispersion equations shown above are second order in h and therefore should accurately describe the dispersion of the fundamental modes, espccially for predictions of the cut-off frequencies h<, which are the lowest frequencies allowed for the propagating modes. Let us first consider the localization of anti-symmetric interfacial waves. Making the substitutions (Y = I, p =, y = i(l - E2)I/' = i{2(1- v)}~"', a, = 1, a2 = -1 and Ar = 1 into the right-hand side of (44) and equating to zero, on<: obtains where v is Poisson's ratio,,u the shear modulus and p the density of the semi-space; C, and po are the elastic constants aiid density of the interphase. Eq. (67) has two solutions: cine is h, =, showing that the fundamental anti-symmetric mode does not have a cut-off frequency; the other is always negative and thus non-physical. As one can see from thc characteristic equation (52) for the SH mode in +c,,c,-c:, c c:,c33-2c,,c,,c3, + G C M IL2 ($-?I- P3 (68) Since h,. = is a root, the fundamental mode of the symmetric wave also has no cut-off frequency. As one sees from the characteristic equation (61) for the SH mode in the plane of symmetry. the solution with 6,. = corresponds to the propagating SH-type symmetric mode. The second positive solution. h,. = P /Q, depends on the modulus and density ratios of the semi-space and interphase. For the plane of symmetry, the solution hc = P /Q simplifies to h, = P P For the isotropic case, Lame constants A,, and pi, should be used to replace C',',, CY3 and Cy3 in eq. (69). For h < h,, the lowest symmetric mode is a leaky wave with complex wave number. ith increase of frequency the speed decreases so that, beyond a cut-off frequency, it becomes a propagating mode. e can also write the expressions for the cut-off frequencies for the stiffness-mass and stiffness models, given by eqs (63) and (66) respectively, as h,.= 1-v 6 SCATTERING AND STONELEY-AVE LOCALIZATION BY ANISOTROPIC FRACTURED INTERFACES. SPRING BC MODELS In the previous two sections a thin anisotropic interfacial layer between identical isotropic semi-spaces was considered. The layer approach should be used when the imperfect interface can be modelled by a non-homogeneous multiphase interfacial layer. The properties of the interfacial layer can be obtained using micromechanical models for multiphase materials (Rokhlin & ang 1991a). However, a different approach can be used if interface imperfections can be described by the spring model (Baik &L Thompson 1984).

12 26. Huang and S. I. Rokhlin This approach is more natural for the description of fractured interface:,. such as that shown in Fig. l(a). Here we consider analytical solutions for two types of spring models, stiffness niass and stiffness. Let us start with the stiffness-mass model. In general the BC can be expressed as (Rokhlin Sr Huang 1993) -IIJ?M -u2m -w'm s, s,. sc sh s, symmetric part is a,,r = 2K,u, + 2Kcuv, o'm uzr = -- 2 u,, a:,, = 2K,.ur + 2K,~t,,, where K, = sf{/(svsh - Sf), KH = S,/(S,S,, - Sf) and K,. = -S,./(S,S, - Sf). One sees from eqs (74) that the normal stiffness K,, is not involved in the anti-symmetric pait of the BC. Eqs (74) have the same form as the stiffness-mass model (24) for an interfacial layer if one defines M = poh and the stiffnesses by the corresponding interfacial layer parameters in (24). The symmetric part of the stiffness-mass BC is where S, and S,, are the in-plane and out-of-plane shear compliances, S<. i:; a coupling compliance between the in-plane and out-of-plane shear stresses and displacements, S,, is the normal compliance, M is the mass surface density of the interface and u is the angular frequency. One sees that both the stresses and displacements are discontinuous across the interface. Eq. (72) generalizes the Baik- Thompson quasi-static model (Baik & Thompson 1984; Margetan et uf. 1988) by adding a coupling shear spring compliance S,. in the BC. This anisotropic coupling spring S,. appears naturally when the fracture (crack) orientation deviates from the incident plane, as demonstrated in the asymptotic expansion of the BC for the layer model. hen the incident plane coincides with the crack orientation (q = " in Fig. la), the compliances in BC (72) are simply the inverses of the corresponding stiffnesses: S,=St-l/K';, S,5=SL=l/K&, S,=S:=I/K:andS,=, where Sc, SL, S: are the compliances along the xo,y" and z directions respectively. In the Baik-Thompson quasi-static model ibaik & Thompson 1984; Margetan et al. 1988), the stiffnessc:~ Kt, Kt, K: are obtained for a cracked interface using fracture mechanics. If q #, the anisotropic compliances S,, S,, S, and S, are given by S, = cos2 qst + I sin2 q~:, S, = sin' qs, + cos' q$,, S,.= sin q cos ip(s':, - St), S,, = S: = l/k:. (73) By applying tc. eq. (72) the same decomposition procedure as in Section 3, we obtain the anti-symmetric and symmetric parts of the stiffness-mass BC. The anti- where K, = l/s,, = K::, which is independent of the crack orientation. Here the shear stiffnesses are not involved. The x, y and z components of the elastic motions are decoupled from each other. Compare with (37) for a thin interfacial layer. Next consider the stiffness model where the mass M is assumed to be zero in eq. (72). In this case the stresses are continuous across the interface. In general the BC can be written as The anti-symmetric part of the stiffness BC is a,, = 2K,u, + 2K,u,. uzv = 2K,u, + 2K,uV, a,, =. (77) One sees that the normal stress is zero for the anti-symmetric field on the interface as in the stiffness model (25) for an interfacial layer. The symmetric part of the stiffness BC is Here the shear stresses are zero, and the normal stress is proportional to the normal displacement on the interface. e have obtained characteristic equations for Stoneley waves and scattering coefficients described by the stiffnessmass and stiffness models. Let us first consider the solutions for interfaces with stiffness-mass BC. The dispersion equation for an anti-symmetric mode localized on such an interface between identical semi-spaces is where A;(K,,, M) and A;,(K,) have the same forms as the characteristic functions for the anti-symmetric mode when waves propagate along or perpendicular to the crack orientation. In the last case, K, = Kt, K, = KL, K,. = and A: = 2K$(2i/?pkf + a,w2m) + 2A,p2k: - iyw2mpk, =, 1 A& = pk,/? + 2iKL =. J (8)

13 The second equation in (8) does not yield a real solution since /3 = i(a' - 1)"2 is a positive imaginary number for a propagating mode (a > 1). The scattering coefficients are obtained in forms similar to those for the interfacial layer: A:A;lH(-P) + 4Kf(2Ppkr - iu,w2m) R&ff = - - A;lA&(/3) + 4K:(2Ppkr - ialw2m) ' A,;(-P)A;lH - 4K?(2Ppk, + ia,w2m) R",= -- At(p)A& + 4K5(2/3pkr - ia,w2m) ' R:,l = - 4a8_5pkl(4ipktfl + w2m)k,, A" 4iP R$H =- pkr(2pk, + iyw2m)k,, A" + 2p!k:(2K, + ipk,p)). For the symmetric mode the dispersion equation is A'(K,,, M) = AXKtZ 3 M)A:,,,(M) =, (82) where A;) = 2K,,(2;ypkl + a,w2m) f- 2A,p2k: - i/3w2mpk,, and ( c,, c,, G 3 c:4 c:5 C$ Po (83) Ai)H = 2pkl/3 - iw'm. (84) Eq. (84) depends only on the mass M and yields a real solution. The decoupling of the in-plane and SH modes shown in eq. (82) is due to the decoupling of in-plane and SH components shown in (75). The scattering coefficients for the symmetric part are Table 1. Elastic constants C:: (GP?) and density (g cc-') of a porous interphase (porosity 3 per cent) in its material coordinate system (x", y". z) I I I I I I 1 ave scattering and localization 291 For interfaces with BC (76), where the effect of mass is neglected, the dispersion equation for an anti-symmetric mode is A"(Kv, KH, K,) = AG(KV)A&(KH) + 4K:P =, where A: = 2iPK'b + pk,a,, (86) (87) and A:, is given by eq. (8). For the symmetric mode on an interface with stiffness BC (78), the dispersion equation is As = 'LiyK,, + pkrar. (88) This depends only on the normal stiffness K,. Since the shear stresses of the symmetric mode vanish, there is no localization of the SH-type mode. The scattering coefficients for the anti-symmetric and symmetric parts are obtained by substituting (86), (88) and M = into eqs (81) and (85). 7 NUMERICAL RESULTS To investigate the accuracy of the second-order BC, we calculate the scattering coefficients and dispersion curves of interfacial modes on a thin orthotropic interfacial layer between identical isotropic semi-spaces. The material properties of the semi-space are V, = 6.2 km s-', V, = 3.24 km s-' and p = 2.7 g cc-'. The imperfect interphase is modelled by a parallel row of cylindrical pores, as shown in Fig. l(b). The effective elastic moduli of the interphase are calculated from Christensen's two-phase model (Christensen 1991); the Cl: for an interphase with 3 per cent porosity are listed in Table 1. One sees that the elastic modulus of the interphase along the pore orientation (the x" direction in Fig. 3) is greater than in the other two directions. Further, we assume that the incident (x, z ) plane deviates from the pore orientation by a rotation angle q. Calculations have been carried out using both secondorder BC and simplified models. The terms stiffness-mass and stiffness models refer to the BC (24), (37) and (25), (38) where the stiffnesses and mass are defined through the interfacial layer parameters. 7.1 The scattering problem First, we consider mode conversion of an incident SH wave into reflected and transmitted SV waves when q = 45". The elastic constants C,, for the interphase with 3 per cent porosity in its rotated coordinate system (x, y, z) are listed in Table 2. The excitation of the SV wave is due purely to the anisotropy of the interface about the incident plane. Recall that the anti-symmetric part of the reflection Table 2. Elastic constants C,, (GPa) for the same material as in Table 1 but in the rotated (9' = 45") coordinate system (x, y, I). c11 c22 c33 cl2 c13 c23 c44 c55 c c45 c16 c26 c

14 298. Huang and S. I. Rokhlin coefficient RL,, is, from (C3) and (41), (Y = sin 8,, /3 = cos 8,, y = cos o,, w = -cos 28,. ) (89) One sees from eq. (89) that the mode conversion coefficient R(;lv is proportional to the modulus C45 of the interfacial layer. Note that the SV and SH waves have the same speed due to the isotropy of the semi-space, and physically constitute a single transverse wave with its polarization vector rotated from that of the incident SH wave. However, to demonstrate the effect of coupling between the in-plane and out-of-plane parts of the elastic field due to interface anisotropy, we treat them separately here as two different transverse waves. The energy conversion coefficients rhv = R& and fhv = TL,, and the phase shifts of the reflected and transmitted SV waves obtained using different BC models are plotted in Fig. 6 as a function of h/ao at incident angle 8, = 5". A()= v,/f is the wavelength of the slow bulk transverse wave in the layer propagating normal to the interface, where f is the frequency and v, = (C~4/p,,)"2 is the speed. In Fig. 6 the exact solutions are represented by solid lines, calcul;ited using an algorithm obtained by Rokhlin & Huang I 1992), the solutions calculated using the second-order asymptotic BC (16) and (28) by open circles, the solutions using the stiffness-mass model (2) and (32) by solid circles and the solutions using the stiffness model (25) and (38) by crosses. One sees from Fig. 6 that the second-order BC give close approximations to the exact solutions for both the energy coefficients and the phase shifts up to h/a =.2 in both the reflected and transmitted fields, while other models give solutions deviating significantly from the exact solutions for h/ao >.5 for most cases. This demonstrates the necessity of retaining the coupling and inertial terms in the approximate BC models, and the advantage of using the second-order BC approach. One sees also from Fig. 6 that the transformation between the in-plane and out-of-plane transverse waves is significant, even when the interface anisotropy is not marked, as in the present example where C,, = -2.2 compared to CS5 = Note that there is no transformation to the longitudinal modes in this case due to high incidence angle. To illustrate the effect of interface anisotropy on the mode conversion between SV and SH transverse waves, we calculate the energy conversion coefficients rf,,, and t,,, of the reflected and transmitted SV waves as functions of the pore orientation angle Q, for the same incident SH wave. The non-dimensional layer thickness parameter h /Ao is fixed at.1. The results obtained using different BC models are plotted in Fig. 7 with all the symbols the same as in Fig. 6. Fig. 7 shows that only the second-order BC give a satisfactory approximation to the exact solution for both the reflected and transmitted SV waves. One sees that in the plane of symmetry (Q, =" or 9") there is no mode conversion because there is no coupling between the in-plane and out-of-plane elastic fields (C4s = C,, = C,, = O), while at Q, = 411'. where the interfacial coupling between normal and out-of-plane shear components is the strongest, both the mode-conversion coefficient and the deviation of the approximation from the exact solution reach their maximum values. Next, let us consider the reflection of a longitudinal incident wave on the same porous interphase. The energy coefficient r,/ = R:, and the phase shift of the reflected wave of the same mode are shown in Fig. 8 as functions of h/ao at the same incident angle, 8, = 5". One sees from Fig. 8 that all the BC models give reasonable approximations to the exact solutions for small h/ao (4.1). but for larger h/hl, the second-order solution gives the best approximations, up to h /Al, =.4. To judge better the performance of the second-order BC and other simplified models, we calculate the same reflection as a function of incident angle 8, at the relatively large value.2 of h/al,. The results are shown in Fig. 9 for the same porous interphase. One sees that the second-order BC give good and homogeneous approximations to both the energy coefficient and the phase shift for all incident angles. hen this angle is less than 2", there is little difference between the second-order solutions and those obtained using the stiffness-mass model, since the coupling terms and the angle-dependent part of the inertia terms are negligible. The stiffness model does not give a good approximation at this value of h/a,. Note that at normal incidence r, --.2 due to the existence of the porous interphase, compared to zero reflection if h/ho =. 7.2 The interfacial wave problem e calculate the dispersion curves of the interfacial guided modes in the same interphase to assess the applicability of the second-order BC. First, we consider propagation in the plane of symmetry, assuming that the interfacial wave travels along the pore direction (i.e. the x axis). The wave speed V/V,, normalized by the shear-wave speed V, in the semi-space, of an anti-symmetric mode is given in Fig. lo(a) as a function of h/ho, ho= V,,/f being the wavelength of the slow shear wave propagating normal to the interface. The exact solutions (solid lines) have been calculated as described. by Huang & Rokhlin (1992), the solution calculated using the second-order dispersion eq. (51) is represented by open circles, the first-order solution (neglecting h' terms in eq. 51) by a dashed line, the solution using the stiffness-mass model (56) by solid circles, and the solution using the stiffness model (58) by crosses. One can see that the second-order approximation works much better than the first-order approximations, while the solutions using the simplified models have large deviations, even for small values of h/a,), since, at the incident angle 8, = 9, the coupling effect between the normal and shear components cannot be neglected. This once again demonstrates the necessity of retaining the coupling and mass terms in the approximate BC models and the advantage of using second-order approximate BC. In Fig. 1(b) V/V, is given as a function of the pore orientation angle cp at fixed h/a,,=.2. This time the interfacial wave normal deviates from the x direction by Q, (the pore orientation angle). The second-order solution, shown by open circles, comes from the analytical dispersion equations (44) or (53). The first-order solution also comes from eq. (53), except that in the right-hand side the h2 terms

15 ave scattering and localization EXACT SOLUTION k OOOOOSECOND ORDER E.a. STIFFNESS-MASS x x x x x STIFFNESS Y u v) p: g?.1 u sr p: z w ti 3 v E E.5 w.3 1 u 2 Y.2 : w 3 u s4.1 1 EXACT SOLUTION OOOOOSECOND ORDER xx x x x STIFFNESS b p: z w ho h 1.o (4-3 - EXACT SOLUTION z ooooosecond ORDER d /' 5 Figure 6. Mode conversion of an incident SH wave into reflected and transmitted SV waves on a porous interphase. h/a,, is the non-dimensional layer thickness. The pore orientation angle p = 45" and the incident angle, = 5". (a) Energy conversion coefficient r,,, for the reflected SV wave. (b) Phase shift in radians for the reflected SV wave. (c) Energy conversion coefficient f, for the transmitted SV wave. (d) Phase shift in radians for the transmitted SV wave. are dropped, including those in A:(Cs5) (see eq. 51). One wave propagation direction as ~i increases, i.e. as the sees from Fig. lo(b) that the second-order approximation interface normal deviates more from the pore direction. closely matches the exact solution while the first-order For given material parameters it is found that for solution gives some discrepancy. The interfacial wave speed h/ao <.6 there are no higher order symmetric or drops due to a decrease of interfacial stiffness along the anti-symmetric modes for both the exact and second-order

16 ~ EXACT 3. Huang and S. I. Rokhlin k E U E.3 R.2 u 53 2 U g.1 u * a SOLUTION OOOOOSECOND ORDER * * STIFFNESS-MASS x x x x x STIFFNESS p: k 4!2 z u Y PORE ORIENTATION ANGLE /k\ ("I U U z p:. - EXACT SOLUTION OOOOOSECOND ORDER STIFFNESS-MASS x x x x x STIFFNESS w Oi = 5" ****** ** I 1 I PORE ORIENTATION ANGLE 'p Figure 7. Energy conversion coefficients of SH to SV waves as functions of pore orientation angles Q, for different BC models. The incident SH wave angle 8, is 5" and h/k,,=.1. (a) r,,". (b) fhv. solutions. The cut-off frequency for the symmetric mode is given by eq. (69) as h,. = 4.85 (h/l(, = 1.23). For such a large value of h,. the approximate equation cannot be used to predict the cut-off frequency. The SH-type interfacial mode does exist, but its speed is extremely close to the shear-wave speed of the semi-space in the frequency range of the calculation, and, as shown by Huang & Rokhlin (1992), the I a h h E -&!z.12 2 E u si.8 t2 i E z.4 ooooosecond ORDER ma om STIFFNESS-MASS CG w E g h -.5 a w rn EXACT SOLUTION ooooosecond ORDER ma moo STIFFNESS-MASS ho Figure 8. Reflection of a longitudinal wave on a porous interphase. The pore orientation angle 47 = 45" and the incident angle, = 5". (a) Energy coefficient r,, for the reflected longitudinal wave. (b) Phase shift in radians for the reflected longitudinal wave. asymptotic solution does not predict the dispersion behaviour of the SH modes well. It has been found that a much softer and slower interphase tends to localize symmetric interfacial modes at much lower frequencies (Rokhlin et af. 198, 1981). Thus we take the interphase to be extremely light, soft and isotropic, with V:) = 1.2 km s-', V:) =.5 km s-.' and p(, =.3 gcc-', and calculate the wave speed of the fundamental symmetric mode for the same semi-spaces as previously. The

17 ~ ave scattering and localization 31 Y EXACT SOLUTION ooooosecond ORDER STIFFNESS-MASS x x x x x STIFFNESS u 's P; z.d a - v.4 -{.2. h/ao =.2 cp = 45" INCIDENT ANGLE ei (b)., - EXACT SOLUTION SECOND ORDER STIFFNESS-MASS x x x x x STIFFNESS -.4 E E E INCIDENT ANGLE ei Figure 9. Reflection of a longitudinal wave on the same porous interphase. The pore orientation angle q = 45" and h/a,, =.2. (a) Energy coefficient r,, for the reflected longitudinal wave. (b) Phase shift in radians for the reflected longitudinal wave. normalized symmetric wave speed is given in Fig. 11 as a function of h /Al,. The second-order solution (open circles) is calculated using eq. (6), and the first-order solution (dashed line) in the same way except that the second-order terms in h are neglected. One sees from Fig. 11 that both the second- and first-order solutions closely approximate the exact solution for h/a,) <.2, with the second-order solution slightly better. The cut-off frequencies for both the secondand first-order approximations are the same and can be calculated directly from eq. (69). The value, h, =.389 or h/a,, =.41, is extremely close to the exact value h/a,, =.4, found numerically. 8 CONCLUSIONS Second-order asymptotic BC accurately describe wave interaction with a thin anisotropic interfacial layer between ma i a n w 5.92 ; : r" Kl I z.9 w n 1. m w.98 p:! n msecond ORDER Ommm -- FIRST ORDER xxxxx STIFFNESS mmmom STIFFNESS-MASS I I 1 1 I 1 ' I ~ I " " " " ' I " " " 7-TT EXACT SOLUTION ooooosecond ORDER FIRST ORDER PORE ORIENTATION ANGLE 'p Figure 1. Normalized wave speed of an anti-symmetric interfacial model guided by a porous interphase (porosity 3 per cent). (a) Dispersion curves obtained using different BC models as functions of h/al, (q = "). (b) ave speed as a function of pore orientation angle q (h /A,, =.2). two anisotropic solids. The solutions generated using these BC satisfy energy balance and uniqueness, and predict no scattering from an interfacial layer having properties equal to those of the semi-spaces. It has been shown that retention of coupling and mass terms in the asymptotic models greatly improves tbe accuracy of the approximation. which is especially critical for interfacial waves. The advantage of the asymptotic BC compared to the exact solution is their relative simplicity, since there is no need to describe the wave behaviour inside the interfacial layer. 3

18 32. Huang and S. I. Rokhlin n w 1.oo.98 - FI El.92 3 h/ho =.4 SECOND ORDER FIRST ORDER Figure 11. Normalized symmetric interfacial wave speed as a function of the non-tiimensional thickness parameter h/&, for an extremely light soft interphase between the same semi-spaces as previously. For a thin anisotropic layer between two identical semi-spaces, the second-order BC can be decomposed, for symmetric and anti-symmetric cases, into stiffness-type relations described by 3 X 3 matrices. Analytical solutions for wave scattering and dispersion equations for interfacial modes on an off-axis orthotropic interfacial layer between isotropic semi-spaces are given. They can be applied to predict the effect of interface anisotropy on wave scattering and interfacial mode dispersion. The characteristic equations for interfacial modes relate explicitly to the interphase elastic moduli and can be utilized for moduli determination from interfacial wave-speed measurements. They are also related to the interface stiffness parameters in the case of fractured interfaces. ACKNOLEDGMENTS The authors would like to thank Professor P. Chadwick for his helpful comments on the manuscript. REFERENCES Alterman, Z., Jarosch. H. & Pekerls, C.L., Oscillations of the earth, Proc. R. SOC. Lond., A, 252, Aki, K. & Richards, P.G., 198. Quantitative seismology: theory and methods, H. Freeman and Company, San Francisco, CA. Baik, J.M. & Thompson, R.B., Ultrasonic scattering from imperfect interfaces: a quasi-static model, J. Nondestruct. Eval., 4, Barnett, D.M., Lothe, J., Gavazza, S.D. & Musgrave, M.J.P., Consideration of the existence of interfacial (Stoneley) waves in bonded anisotropic elastic half-spaces, Proc. R. SOC. Lond., A, 42, lti \ Bond,., The mathematics of the physical properties of crystals, Bell Syst. Tech. J., 22, Bostrom, A., Bovik, P. & Olsson, P., Exact first order vs. spring boundary conditions for scattering from thin layers, J. Nondestruct. Eval., 11, Brekhovskikh, L.M., 198. aves in layered media, 2nd edn, Academic Press, New York. Chadwick, P. & Currie, P.K., Stoneley waves at an interface between elastic crystals, Q. J. Mech. appl. Math., 27, Christensen, R.M., Mechanics of composite materials, 2nd edn, Chapter 3, Kreiger, Malabar, FL. Fahmy, A.H. & Adler, E.L., Propagation of acoustic waves in multilayers: a matrix description, Appl. Phys. Lett., 2, Ginzbarg, A.S. & Strick, E., Stoneley wave velocities for a solid-solid interface, Bull. seism. SOC. Am., 48, Huang,. & Rokhlin, S.I., Interface waves along an anisotropic imperfect interface between anisotropic solids, J. Nondestruct. Eval., 11, Ingebrigsten, K.A. & Tonning, A., Elastic surface waves in crystals, Phys. Rev., 184, Lim, T.C. & Musgrave, M.J.P., 197. Stoneley waves in anisotropic media, Nature, 225, 372. Mal, A.K., Guided waves in layered solids with interface zones, Int. J. Eng. Sci., 26, Mandal, B. & Toksoz, M.N., 199. Computation of complete waveforms in general anisotropic media-results from an explosion source in an anisotropic medium, Geophys. J. Int., 13, Margetan, F.J., Thompson, R.B. & Gray, T.A., Interfacial spring models for ultrasonic interaction with imperfect interfaces: theory of oblique incidence and application to diffusion-bonded butt joints, J. Nondestruct. Eval., 7, Margetan, F.J., Thompson, R.B., Rose, J.H. & Gray, T.A., The interaction of ultrasound with imperfect interfaces: experimental studies of model structures, J. Nondestruct. Ed., 11, Martin, P.A., Boundary integral equations for the scattering of elastic inclusions with thin interfacial layers, J. Nondestruct. E~al., 11, Murty, G.S., A theoretical model for the attenuation and dispersion of Stoneley waves at the loosely bonded interface of elastic halfspaces, Phys. Earth planet. Inter., 11, Nagy, P.B., Ultrasonic classification of imperfect interfaces, J. Nondestruct. Eval., 11, Nayfeh, A.H., The general problem of elastic wave propagation in multilayered anisotropic layer, J. acoust. SOC. Am., 89, Pyrak-Nolte, L.J., Myer, L.R. & Cook, N.G.., 199. Transmission of seismic waves across single natural fractures, J. geophys. Res., 95, Pyrak-Nolte, L.J., Xu, J. & Haley, G.M., Elastic interface waves propagating in a fracture, Phys. Rev. Lett., 68, Rokhlin, S.I. & Huang,., Ultrasonic wave interaction with a thin anisotropic interfacial layer between two anisotropic solids: exact and asymptotic-boundary-condition methods, J. acoust. SOC. Am., 92, Rokhlin, S.I. & Huang,., Ultrasonic wave interaction with a thin anisotropic interfacial layer between two anisotropic solids. 11: second order asymptotic boundary conditions, J. acout. SOC. Am., 94, Rokhlin, S.I. & ang, Y.J., 1991a. Analysis of ultrasonic wave interaction with imperfect interface between solids. in Review of Progress in Quantitative Nondestructive Evaluation, 1A, pp , eds Thompson, D.O. & Chimenti, D.E., Plenum Press, New York. Rokhlin, S.I. & ang, Y.J., 1991h. Analysis of boundary