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Theoretical model of guided waves in a bone-mimicking plate coupled with soft-tissue layers

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  1. Vietnam Journal of Mechanics, VAST, Vol.43, No. 1 (2021), pp. 91 – 104 DOI: THEORETICAL MODEL OF GUIDED WAVES IN A BONE-MIMICKING PLATE COUPLED WITH SOFT-TISSUE LAYERS Hoai Nguyen1, Ductho Le2, Emmanuel Plan3,4, Son Tung Dang5, Haidang Phan3,6 1Institute of Physics, Vietnam Academy of Science and Technology, Hanoi, Vietnam 2Faculty of Mathematics, Mechanics and Informatics, VNU University of Science, Vietnam 3Institute of Theoretical and Applied Research, Duy Tan University, Hanoi, Vietnam 4Faculty of Natural Science, Duy Tan University, Da Nang, Vietnam 5Sintef industry, S. P. Andersens veg 15B, 7031 Trondheim, Norway 6Faculty of Civil Engineering, Duy Tan University, Da Nang, Vietnam ∗E-mail: phanhaidang2@duytan.edu.vn Received: 21 December 2020 / Published online: 21 February 2021 Abstract. Quantitative ultrasound has shown a significant promise in the assessment of bone characteristics in the recent reports. However, our understanding of wave interac- tion with bone tissues is still far from complete since the propagation of ultrasonic waves in bones is a very challenging topic due to their multilayer nature. The aim of the cur- rent study is to develop a theoretical model for guided waves in a bone-mimicking plate coupled with two soft-tissue layers. Here, the bone plate is modeled as an isotropic solid layer while the soft tissues are modeled as fluid layers. Based on the boundary condi- tions set for the three-layered structure, a characteristic equation is obtained which results in dispersion curves of the phase and group velocities. New expressions for free guided waves propagating in the trilayered plate are introduced. The amplitudes of wave modes generated by time-harmonic loads applied in the plate are theoretically computed by reci- procity consideration. As an example of calculation, the normalized amplitudes of the lowest wave modes are presented. The obtained results and equations discussed in this study could be, in general, useful for further applications in the area of bone quantitative ultrasound. Keywords: guided waves; bone plate; trilayered structures; reciprocity; quantitative ultra- sound. 1. INTRODUCTION Guided wave propagation in layered structures plays an important role in the study of quantitative ultrasound (QUS), a method of great potential in the assessment of bone characteristics. Bone QUS takes advantage of mechanical waves that are more sensitive © 2021 Vietnam Academy of Science and Technology
  2. 92 Hoai Nguyen, Ductho Le, Emmanuel Plan, Son Tung Dang, Haidang Phan than conventional X-ray method to the determinants of bone strength [1]. Unlike X- rays, QUS is safe for newborn babies and pregnant women because it is a non-ionizing method. Moreover, QUS approach can provide information about the elastic properties and defects of bones [2]. Numerous studies have been considered to understand how ultrasound interacts with the bone structure, see, for example, [3,4]. Lowet and Van der Perre [5] studied simulation of ultrasound wave propagation and the method to mea- sure velocity in long bones. Numerical simulations of wave propagation and experiment measurement were used to gain insights into the expected behavior of guided waves in bones [6]. The velocity dispersion and attenuation in a tri-layered system, which con- sists of a transversely-isotropic cortical bone plate sandwiched between the soft-tissue and marrow layers, were computed using a semi-analytical finite element [7]. However, wave propagation and scattering in bones is a very challenging topic due to the bones’ multi-layer, anisotropic, and viscoelastic nature. The understanding of wave interaction with bones is, therefore, still quite limited and definitely needs to be expanded [1,3]. Wave propagation in multi-layered structures is unquestionably one of the most fun- damental problems of elastodynamics. Study of free guided waves in layered plates can be found in textbooks [8,9] and research papers [10, 11]. The dispersion equation of guided waves in fluid-solid bilayered plate is discussed in [12]. The phenomenon of osculation where two dispersion curves come near to each other was observed and care- fully studied by [13]. Wave motion generated by a loading is in general solved by the use of integral transform technique [8, 14–16] and by the reciprocity approach [17–27]. The integral transform approach is usually used for simple half-space problems. How- ever, it becomes more difficult for anisotropic solids, and impossible for inhomogeneous solids. The reciprocity approach is therefore suitable for guided wave motions in layered structures and composites. In this work, we present a model for ultrasonic guided waves in a trilayered sys- tem consisting of a bone-mimicking plate coupled with a soft-tissue layer and a marrow layer. In order to simplify the computation procedure, we consider the bone layer as an isotropic solid while the soft-tissue and the marrow layer as pure fluids. For modeling purpose, we also ignore the effect of viscosity. For this trilayered structure, boundary conditions are suitably applied at the free surfaces of fluids and the interfaces between the solid and the fluids to derive dispersion equations. Based on these equations, dis- persion curves are obtained by numerical computation. Explicit expressions of displace- ments and stresses in the trilayered structures are then introduced. The amplitudes of guided waves generated by a time-harmonic load applied in the solid plate are theoret- ically computed by using reciprocity in elastodynamics. As an example, the normalized amplitudes of the lowest wave modes are computed. 2. EXPRESSIONS FOR GUIDED WAVES IN A FLUID-SOLID-FLUID PLATE Let us consider a solid layer Ω representing the bone plate bonded in the middle of two fluid layers Ωˆ and Ω˜ which presents the upper and lower soft tissues, respectively. This forms a fluid-solid-fluid layered plate in Cartesian coordinate system (x, y, z), as shown in Fig.1. Propagation of free guided waves in the trilayered structure is of interest
  3. Theoretical model of guided waves in a bone-mimicking plate coupled with soft-tissue layers 93 in this section. The dispersion equation is derived resulting in the dispersion curves. In order to perform direct application of reciprocity theorem in the next section, new expressions of displacement and stress components are also introduced. Fig. 1. Coordinate system for trilayered plate Besides the amplitudes, guided waves in the fluid-solid-fluid plate are characterized by an angular frequency ω and a wavenumber k, where k = ω/c, c being the phase ve- locity, as well as Lame constants λˆ , µˆ, λ, µ and λ˜ , µ˜, and densities ρˆ, ρ and ρ˜ of the upper fluid, the solid and the lower fluid, respectively. Note that the non-viscous fluids consid- ered here do not sustain shear stresses and their equations of motion can be used as of a solid by assuming Lame constants µˆ = 0, µ˜ = 0. The upper fluid layer’s thickness is in- dicated by hˆ, the one of the solid is indicated by h and the lower fluid layer’s thickness is denoted by h˜. Wave solutions are expressed based on the partial wave theory discussed in detail, for example, in [9]. Since there are two partial waves in the upper fluid layer, the displacements may be written as  ikαˆ z −ikαˆ z ik(x−ct) uˆx = Aˆ 1e + Aˆ 2e e , (1)  ikαˆ z −ikαˆ z ik(x−ct) uˆz = αˆ Aˆ 1e − Aˆ 2e e . (2) For the solid layer, the number of partial waves is four. Thus, the displacement compo- nents are of the form   ikα1z ikα2z −ikα1z −ikα2z ik(x−ct) ux = A1e + A2e + A3e + A4e e , (3)  −  1 ikα1z ikα2z 1 −ikα1z −ikα2z ik(x−ct) uz = A1e + α2 A2e + A3e − α2 A4e e . (4) α1 α1 Since there are two partial waves in the lower fluid layer, the displacements may be written as  ikα˜ z −ikα˜ z ik(x−ct) u˜x = A˜ 1e + A˜ 2e e , (5)  ikα˜ z −ikα˜ z ik(x−ct) u˜z = α˜ A˜ 1e − A˜ 2e e . (6)
  4. 94 Hoai Nguyen, Ductho Le, Emmanuel Plan, Son Tung Dang, Haidang Phan Here, Aˆ j (j = 1, 2), Aj(j = 1, 2, 3, 4) and A˜ j(j = 1, 2) are constants to be determined. The dimensionless quantities are defined as follow q 2 2  αˆ = −1 + c /cˆL , (7) where q cˆL = λˆ /ρˆ, (8) are the longitudinal wave velocity of the fluid layer Ωˆ . Similarly, q q 2 2 2 2 α1 = −1 + c /cT, α2 = −1 + c /cL, (9) where p q cT = µ/ρ, cL = (λ + 2µ) /ρ, (10) are the transverse and longitudinal wave velocities, respectively, of the solid layer Ω. For the lower fluid layer q 2 2  α˜ = −1 + c /c˜L , (11) where q c˜L = λ˜ /ρ˜. (12) Note that the corresponding stress components τˆxx, τˆzz of the layer Ωˆ , τxx, τxz, τzz of the layer Ω and τ˜xx, τ˜zz of the layer Ω˜ can be easily calculated by the use of Hooke’s law. The boundary conditions are written as   τˆzz = 0 z = hˆ u = uˆ , τ = 0, τ = τˆ (z = 0) z z xz zz zz (13) uz = u˜z, τxz = 0, τzz = τ˜zz (z = −h)  τ˜zz = 0 z = − h + h˜ Eq. (13), after some manipulation, results in DA = 0, (14) where  T A = Aˆ 1 Aˆ 2 A1 A2 A3 A4 A˜ 1 A˜ 2 , (15) and D is the eight-by-eight matrix whose expression is given in Appendix A, see Eq. (A.1). In order to have nontrivial solutions, the determinant of the eight-by-eight matrix in Eq. (14) must be zero. This is referred to as the characteristic equation of guided waves in the fluid-solid-fluid plate with phase velocity as an unknown parameter when prop- erties of the three layers and frequency are given. Since the determinant appears with a frequency term via the wavenumber k, the phase velocity is dependent of frequency. Therefore, guided waves in the trilayered are dispersive. Note that the characteristic equation in general has infinite number of solutions, each of which corresponds to one
  5. Theoretical model of guided waves in a bone-mimicking plate coupled with soft-tissue layers 95 wave mode. In the examples shown in Fig.2, the relevant material properties are tab- ulated in Tab.1. The computation of the dispersion curves for a water-aluminum-water trilayered structure in Fig.2 coincides with the one obtained by Nguyen et al. [7]. Table 1. Material properties of aluminum and water Material ρ kg/m3 λ (GPa) µ (GPa) Water 1000 2.25 0 Aluminum 2700 55.25 25.94 (a) (b) Fig. 2. Dispersion curves of phase and group velocities: Lowest symmetric mode S0 (blue), Lowest antisymmetric mode A0 (red) and Higher modes (black) In Eq. (14), there are actually seven independent equations with eight unknowns Aˆ 1, Aˆ 2, A1, A2, A3, A4, and A˜ 1, A˜ 2. We find solution of Eq. (14) in the form of Aˆ1 = Adˆ1, Aˆ2 = Adˆ2, A1 = Ad1, A2 = Ad2, A3 = Ad3, A4 = Ad4, A˜1 = Ad˜1, A˜2 = Ad˜2 (16) where A is only unknown and dˆ1, dˆ2, d1, d2, d3, d4, d˜1, d˜2 are dimensionless quantities de- pending on material properties and thicknesses of the fluids and the solid, and the wave- number k. The expressions of dˆ1, dˆ2, d1, d2, d3, d4, d˜1, d˜2 are given in Appendix A, see Eq. (A.2). With the introduction of relative amplitude A, we may now rewrite the dis- placement and stress fields as follows: - For the upper fluid layer ik(x−ct) uˆx = AUˆ x (z) e , (17) ik(x−ct) uˆz = AUˆ z (z) e , (18) ik(x−ct) τˆxx = ikλATˆxx (z) e , (19)
  6. 96 Hoai Nguyen, Ductho Le, Emmanuel Plan, Son Tung Dang, Haidang Phan where ikαˆ z −ikαˆ z Uˆ x (z) = dˆ1e + dˆ2e , (20)  ikαˆ z −ikαˆ z Uˆ z (z) = α dˆ1e − dˆ2e , (21) 2  ikαˆ z −ikαˆ z Tˆxx (z) = 1 + αˆ dˆ1e + dˆ2e . (22) - For the solid layer ik(x−ct) ux = AUx (z) e , (23) ik(x−ct) uz = AUz (z) e , (24) ik(x−ct) τxx = ikµATxx (z) e , (25) ik(x−ct) τxz = ikµATxz (z) e , (26) where ikα1z ikα2z −ikα1z −ikα2z Ux (z) = d1e + d2e + d3e + d4e , (27) − 1 ikα1z ikα2z 1 −ikα1z −ikα2z Uz (z) = d1e + α2d2e + d3e − α2d4e , (28) α1 α1 ikα1z 2 2  ikα2z −ikα1z 2 2  −ikα2z Txx (z) = 2d1e + α1 − 2α2 + 1 d2e + 2d3e + α1 − 2α2 + 1 d4e , (29)     1 ikα1z ikα2z 1 −ikα1z −ikα2z Txz (z) = α1 − d1e + 2α2d2e − α1 − d3e − 2α2d4e . (30) α1 α1 - For the lower fluid layer ik(x−ct) u˜x = AU˜ x (z) e , (31) ik(x−ct) u˜z = AU˜ z (z) e , (32) ik(x−ct) τ˜xx = ikλAT˜xx (z) e , (33) where ikα˜ z −ikα˜ z U˜ x (z) = d˜1e + d˜2e , (34)  ikα˜ z −ikα˜ z U˜ z (z) = α d˜1e − d˜2e , (35) 2  ikα˜ z −ikα˜ z T˜xx (z) = 1 + α˜ d˜1e + d˜2e . (36) Instead of depending on eight unknowns as given in Eqs. (1)–(6), the guided wave fields now depend only on the amplitude A. Although guided waves in a plate are usu- ally separated into symmetric and antisymmetric modes, the displacement and stress fields given by Eqs. (17)–(36) already include both symmetric and antisymmetric modes. The main reason to introduce these expressions is to conveniently apply the reciprocity theorems and they are very important to the problem of guided wave motions subjected to time-harmonic loads in trilayered plates discussed in the next section.
  7. Theoretical model of guided waves in a bone-mimicking plate coupled with soft-tissue layers 97 3. COMPUTATION OF GUIDED WAVES DUE TO TIME-HARMONIC LOADING Considered in this section is the computation of guided wave motions in the trilay- ered structure (Fig.1) subjected to a time-harmonic line load. Suppose that we have a A vertical load fz at (x0, z0) where x0, z0 are the x-coordinate and the z-coordinate, respec- tively, of the point of application. The load demonstrated in Fig.3 may be written as A −ikct fz = Pδ (z − z0) δ (x − x0) e . (37) Fig. 3. Trilayered structure due to time-harmonic loading This load will generate various guided wave modes along the fluid-solid-fluid plate in both the positive x-direction and the negative x-direction with unknown scattered P+ P− amplitudes Am and Am (m = 0, 1, . . . , ∞), respectively. State A is called the actual state whose amplitudes will be determined using the reciprocity relation for a three-material body. Since many wave modes are generated in the trilayered structure, the far-field displacements of each layer of state A in the positive x-direction may be written in the form of ∞ ∞ m P+ ˆ m ikm(x−cmt) uˆx = ∑ uˆx = ∑ Am Ux (z) e , (38) m=0 m=0 ∞ ∞ m P+ ˆ m ikm(x−cmt) uˆz = ∑ uˆz = −i ∑ Am Uz (z) e , (39) m=0 m=0 ∞ ∞ m P+ m ikm(x−cmt) ux = ∑ ux = ∑ Am Ux (z) e , (40) m=0 m=0 ∞ ∞ m P+ m ikm(x−cmt) uz = ∑ uz = ∑ Am Uz (z) e , (41) m=0 m=0 ∞ ∞ m P+ ˜ m ikm(x−cmt) u˜x = ∑ u˜x = ∑ Am Ux (z) e , (42) m=0 m=0 ∞ ∞ m P+ ˜ m ikm(x−cmt) u˜z = ∑ u˜z = −i ∑ Am Uz (z) e . (43) m=0 m=0
  8. 98 Hoai Nguyen, Ductho Le, Emmanuel Plan, Son Tung Dang, Haidang Phan Reciprocity theorem in general provides a relation between displacements, tractions and body forces of two different loading states. Based on this relation of two states, so- lutions of scattered wave fields of the actual state can be derived. The original idea was discussed in [17], mostly for the cases of a solid half-space and a solid plate. The reci- procity approach was recently used for computation of guided wave motions in layered structures [19, 24]. The verification of this approach was reported in [18, 28, 29]. For a three-material body, the reciprocity relation follows from Eq. (3) of [22] Z   Z   Z   ˆA B ˆB A ˆ A B B A ˜A B ˜B A ˜ fj uˆj − fj uˆj dΩ + fj uj − fj uj dΩ + fj u˜j − fj u˜j dΩ Ωˆ Ω Ω˜ Z   Z   Z   (44) B A A B ˆ B A A B B A A B ˜ = τˆij uˆj − τˆij uˆj nˆ idS + τij uj − τij uj nidS + τ˜ij u˜j − τ˜ij u˜j n˜ idS, Sˆ S S˜ where fˆ, f and f˜ represent the forces applied on the upper fluid, the solid and the lower fluid, respectively, Sˆ, S and S˜ describes the contours around Ωˆ , Ω and Ω˜ without the interfaces with other bodies, respectively, while nˆ i, ni and n˜ i are normal vectors along Sˆ, S and S˜, respectively, see Fig.3. Also, superscripts A and B indicate two elastodynamic A states. The actual state A is the field generated by fz while the virtual state B is the guided wave field propagating in the trilayered plate. We should now choose an appropriate virtual state B based on the expressions of free guided wave fields in Eqs. (17)–(36). State B is set to include only a single wave mode with amplitude Bn. The negative x-direction state B is written as n ˆ n −ikn(x+cnt) uˆx = −BnUx (z) e , (45) n ˆ n −ikn(x+cnt) uˆz = −iBnUz (z) e , (46) n n −ikn(x+cnt) ux = −BnUx (z) e , (47) n n −ikn(x+cnt) uz = BnUz (z) e , (48) n ˜ n −ikn(x+cnt) u˜x = −BnUx (z) e , (49) n ˜ n −ikn(x+cnt) u˜z = −iBnUz (z) e . (50) The next step is to replace the expressions of states A and B into Eq. (44). The left- hand side of Eq. (44) can be simplified because the loading is applied only at (x0, z0). Note that the right-hand side of Eq. (44) vanishes when state A and state B are in same direction. Therefore, there is only contribution from the counter-propagating waves, see [17,19] for details. Since free boundary conditions are applied on the top and the bottom of the trilayered plate, there is no contribution of the integration. Moreover, using the orthogonality condition, derived in Appendix B, Eq. (B.10), the right-hand side of Eq. (44) cancels out for m 6= n. It should be noted that the time-harmonic load can be arbitrarily applied at any position in the structure. Without loss of generality, the load is applied in the solid layer Ω. We finally find, after some manipulation, the amplitude of guided
  9. Theoretical model of guided waves in a bone-mimicking plate coupled with soft-tissue layers 99 waves in the positive x-direction as n −ikn x0 P+ −iPUz (z0) e An = , (51) 2 µˆ Iˆn + λIn + µ˜ I˜n where hˆ Z ˆ  ˆ n ˆ n  In = ikn Txx (z) Ux (z) dz, (52) 0 Z0 n n n n In = ikn [Txx (z) Ux (z) + Txz (z) Uz (z)] dz, (53) −h −h Z ˜  ˜ n ˜ n  In = ikn Txx (z) Ux (z) dz. (54) −h+h˜ Note that Iˆn, In and I˜n are connected to the guided wave of mode n. They are ob- tained from Iˆmn, Imn and I˜mn expressed in Eqs. (B.7)–(B.9) of Appendix B, respectively, as m = n. If a virtual wave of mode n (state B) in the positive x-direction is chosen, we obtain n ikn x0 P− −iPUz (z0) e An = . (55) 2 µˆ Iˆn + λIn + µ˜ I˜n Similarly, for a horizontal load of the form A −ikct fx = Qδ (z − z0) δ (x − x0) e , (56) we find n −ikn x0 Q+ −QUx (z0) e An = , (57) 2 µˆ Iˆn + λIn + µ˜ I˜n n ikn x0 Q− QUx (z0) e An = . (58) 2 µˆ Iˆn + λIn + µ˜ I˜n Eqs. (51) and (55) represent the amplitudes of guided waves of mode n generated by the application of a vertical time-harmonic load of magnitude P at (x0, z0) obtained in closed-form solution. Similarly, Eqs. (57) and (58) express the amplitudes due to a horizontal force of magnitude Q at (x0, z0). As an example, we calculate these expressions for a trilayered model which includes a 3 mm-thick water layer, a 5 mm-thick aluminum layer and a 10 mm-thick water layer. The material properties are given in Tab.1. In this model, water is used to mimic human soft-tissue and marrow while the aluminum is used to mimic human cortical bone (see Fig. 2 of Ref. [7]). For this calculation, the vertical load is applied at the interface of the upper fluid layer and the solid layer with a magnitude chosen as P = µ/2. Also, low frequencies ranging from 5 to 40 kHz were used so only the lowest guided wave modes
  10. 100 Hoai Nguyen, Ductho Le, Emmanuel Plan, Son Tung Dang, Haidang Phan A0 and S0 will be generated. The normalized amplitudes of the lowest wave modes at the interface of the upper fluid and the solid are displayed in Fig.4. Fig. 4. Amplitudes of the lowest wave modes due to time-harmonic loading 4. CONCLUSIONS A theoretical approach for guided wave motions in an isotropic solid plate coupled with two fluid layers has been proposed in this article. We have derived the character- istic equation and obtained velocity dispersion curves for a three-layered plate. In order to perform reciprocity application, the expressions of free guided waves have been intro- duced. It has also analytically computed the amplitudes of guided wave modes subjected to a time-harmonic load applied in the solid layer. As an example, we have presented the results for normalized amplitudes of the lowest wave modes. The theoretical predic- tions obtained in the current work will be beneficial in building models for a cortical bone-mimicking plate coupled with soft-tissue layers and, in general, useful for further applications in bone quantitative ultrasound. ACKNOWLEDGMENT We would like to acknowledge the International Center of Physics for support of this research (Grant No. ICP.2020.12). REFERENCES [1] L. H. Le, Y. J. Gu, Y. Li, and C. Zhang. Probing long bones with ultrasonic body waves. Applied Physics Letters, 96, (11), (2010).
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  13. Theoretical model of guided waves in a bone-mimicking plate coupled with soft-tissue layers 103 APPENDIX A. EXPRESSIONS OF GUIDED WAVES IN THE TRILAYERED STRUCTURE Eight-by-eight matrix  ˆ ˆ  eikαˆ h e−ikαˆ h 0 0 0 0 0 0      αˆ −αˆ 1/α −α −1/α α 0 0   1 2 1 2     1 1   0 0 α1 − 2α2 − α1 −2α2 0 0   αˆ 1 α1             ˆ 2 ˆ 2 2 2   λ 1 + αˆ λ 1 + αˆ 2µ 1 − α1 µ 2µ 1 − α1 µ 0 0  =   D  −1 1  .  −ikα1h −ikα2h ikα1h − ikα2h − −ikα˜ h ikα˜ h   0 0 e α2e e α2e α˜ e α˜ e   α1 α1         1 −ikα h −ikα h 1 ikα h ikα h   0 0 α1 − e 1 2α2e 2 −α1 e 1 −2α2e 2 0 0   α1 α1             −ikα1h 2 −ikα2h ikα1h 2 ikα2h ˜ 2 −ikα˜ h ˜ 2 ikα˜ h   0 0 2µe µ 1−α1 e 2µe µ 1−α1 e λ 1+α˜ e λ 1 + α˜ e    ˜ ˜ 0 0 0 0 0 0 e−ikα˜ (h+h) eikα˜ (h+h) (A.1) Dimensionless quantities " #T h iT D∗ D∗ ∗ ∗ ∗ ∗ D∗ ˆ ˆ ˜ ˜ 1ˆ 2ˆ |D | |D | |D | |D | 1˜ d = d1 d2 d1 d2 d3 d4 d1 d2 = 1 2 3 4 1 , |D∗| |D∗| |D∗| |D∗| |D∗| |D∗| |D∗| (A.2) where  ˆ ˆ  eikαˆ h e−ikαˆ h 0 0 0 0 0      αˆ −αˆ 1/α1 −α2 −1/α1 α2 0     1 1   0 0 α1 − 2α2 − α1 −2α2 0   αˆ 1 α1    ∗  λˆ 1 + αˆ 2 λˆ 1 + αˆ 2 2µ 1 − α2 µ 2µ 1 − α2 µ 0  D =  1 1   −1 1   −ikα1h −ikα2h ikα1h ikα2h −ikα˜ h   0 0 e α2e e −α2e −α˜ e   α1 α1    1   1    − e−ikα1h e−ikα2h − eikα1h − eikα2h   0 0 α1 2α2 α1 2α2 0   α1 α1  −ikα1h 2 −ikα2h ikα1h 2 ikα2h ˜ 2 −ikα˜ h 0 0 2µe µ 1 − α1 e 2µe µ 1 − α1 e λ 1 + α˜ e   0      0       0      and k =  0  ,    ikα˜ h   −α˜ e       0    −λ˜ 1 + α˜ 2 eikα˜ h (A.3) ∗ ∗ with Di formed by replacing the column i of matrix D by k. APPENDIX B. ORTHOGONALITY CONDITION We derive here an orthogonality condition for counter-propagating guided waves in the trilayered plate using the reciprocity relation given in Eq. (44). The condition is related to two free guided waves of mode m with wavenumber km (state A) and mode n with wavenumber kn (state B). The domain is defined by a ≤ x ≤ b, −(h + h˜) ≤ z ≤ hˆ as demonstrated in Fig.3. With the absence of a force term, the left-hand side of Eq. (44)
  14. 104 Hoai Nguyen, Ductho Le, Emmanuel Plan, Son Tung Dang, Haidang Phan vanishes. This leads to Z   Z   Z   B A A B ˆ B A A B B A A B ˜ τˆij uˆj − τˆij uˆj nˆ idS + τij uj − τij uj nidS + τ˜ij u˜j − τ˜ij u˜j n˜ idS = 0. (B.1) Sˆ S S˜ Applying free boundary conditions on the top and the bottom of the trilayered plate, Eq. (B.1) becomes hˆ 0 −h hˆ 0 −h Z Z Z Z Z Z ˆmn + mn + ˜mn = ˆmn + mn + ˜mn FAB x=a dz FAB |x=a dz FAB x=a dz FAB x=b dz FAB |x=b dz FAB x=b dz, 0 −h −(h+h˜) 0 −h −(h+h˜) (B.2) ˆmn mn ˜mn where FAB , FAB and FAB are expressed as ˆmn Bn Am Am Bn FAB = τˆxx uˆx − τˆxx uˆx , (B.3) mn Bn Am Bn Am Am Bn Am Bn FAB = τxx ux + τxz uz − τxx ux − τxz uz , (B.4) ˜mn Bn Am Am Bn FAB = τ˜xx ux − τ˜xx u˜x . (B.5) Using the displacement and stress expressions of the states A and B into Eq. (B.2), after some manipulation, yields   i(km−kn)a i(km−kn)b  e − e µˆ Iˆmn + µImn + µ˜ I˜mn = 0, (B.6) where hˆ 1 Z Iˆ = i k Tˆ n (z)Uˆ m(z + k Tˆ m (z)Uˆ n(z) dz, (B.7) mn 2 n xx x m xx x 0 0 1 Z I = i [k (Tn (z)Um(z) − Tn (z)Um(z)) + k (Tm (z)Un(z) − Tm (z)Un(z))] dz, (B.8) mn 2 n xx x xz z m xx x xz z −h −h 1 Z I˜ = i k T˜ n (z)U˜ m(z) + k T˜ m (z)U˜ n(z) dz. (B.9) mn 2 n xx x m xx x −(h+h˜) Note that Eq. (B.6) must be satisfied for arbitrary values of a and b. Clearly, it is satisfied if m = n. When m 6= n, it can be satisfied only if µˆ Iˆmn + µImn + µ˜ I˜mn = 0. For m 6= n, thus, we obtain the orthogonality condition of guided waves in a trilayered plate as µˆ Iˆmn + µImn + µ˜ I˜mn = 0. (B.10) It should be noted that as m = n, quantities Iˆn, In and I˜n expressed in Eqs. (52)–(54) are Iˆmn, Imn and I˜mn given in Eqs. (B.7)–(B.9).