# Dispersion equation of rayleigh waves in transversely isotropic nonlocal piezoelastic solids half-space

Tóm tắt Dispersion equation of rayleigh waves in transversely isotropic nonlocal piezoelastic solids half-space: ...n at all other points x′ in the domain. Recently, Ke et al. [7, 8] extended the nonlocal elasticity theory to the piezoelectric nanostructures- the nonlocal continuum theory of piezoelectricity (NLPE). Unlike the CPE continuum theory, the NLPE theory supposes that the stresses and the electrical ... surface of the half- space. In the present study, boundary conditions appropriate for particle motion in the x1x3 plane are considered at the plane surface x3 = 0. This surface is considered to be Dispersion equation of Rayleigh waves in transversely isotropic nonlocal piezoelastic solids half-s...eater the ones in the local theory case. It can be concluded that the nonlocality has significant effect on the velocities of propagation of plane waves. Next, the variations of the phase velocities with dimensionless parameter ep for θ = pi/3 are depicted in Fig. 2. Generally, this figure shows...

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been also illustrated. Keywords: dispersion equation, nonlocal, piezoelastic. 1. INTRODUCTION In recent years piezoelectric materials has drawn much attention towards applica- tion in surface acoustic wave (SAW) micro sensors, energy harvesting structure, health monitoring systems, transducers and actuators, etc. Both theoretical and experimental studies on wave propagation in piezoelectric materials have attracted the attention of scientists and engineers during last two decades. The survey of literature can be found in many related texts and books [1, 2]. We mention only a few such as: Zinchuk and Podlipenets [3] obtained dispersion equations for acousto-electric Rayleigh wave in a periodic layer piezoelectric half-space in a study for the 6 mm crystal class. Wave prop- agation in porous piezoelectric material (PPM), having crystal symmetry 6 mm, is stud- ied analytically by Vashishth et al. [4]. Sharma et al. [1] investigated the propagation of Rayleigh waves in a homogeneous, transversely isotropic, piezothermoelastic half-space subjected to stress free, electrically shorted/charge-free and thermally insulated/isother- mal boundary conditions. Secular equations for the half-space in closed form and isolated mathematical conditions in completely separate terms are derived. Recent development in science and technology requires that the high-performance electromechanical devices must have a higher sensitivity and larger storage capacity but c© 2019 Vietnam Academy of Science and Technology 364 Do Xuan Tung as maller size. Nano scale materials and structures have been introduced and devel- oped ever since. For these materials, the conventional continuum elasticity theory fails to represent the constitutive relationships properly [5]. A non-local model based on Erin- gens theory of non-local continuum mechanics has been proposed for the effects of the size dependency in very small structures. Particularly, Eringen’s nonlocal theory [6] has been extended to study the size dependent mechanical performances of the piezoelectric nanostructures by Ke et al. [7–9]. There are a few research works on the propagation of the guided elastic waves in nanoscale periodic piezoelectric structures. For examples, Chen et al. [10] studied the anti-plane transverse wave propagation in nanoscale periodic layered piezoelectric structures. Yan et al. [11,12] investigated the propagation of guided elastic waves in nanoscale layered periodic piezoelectric composites. However, only few researchs [11–13] on wave propagation in nanoscale periodic lay- ered piezoelectric structures have been reported in literature due to the complexity of the problem. In addition, many researches [1, 14] have been carried out on the propagation of Rayleigh waves in transversely isotropic piezoelastic materials solids. However, to the best of the authors knowledge, there are no researches analyzing the propagation of surface waves in transversely isotropic nonlocal piezoelastic half-space analytically avail- able in the literature. Therefore, the main purpose of this paper is to study the effect of nonlocality on the propagation of Rayleigh waves in transversely isotropic piezoelastic materials. The dispersion equation for the propagation of Rayleigh waves is derived for the boundary is stress-free, maintained at charge-free condition. 2. FORMULATION OF THE PROBLEM We consider homogeneous transversely isotropic, electrically conducting piezoelec- tric medium in the undeformed state at initial potential φ0. We assume that the medium is transversely isotropic in such a way that planes of isotropy are perpendicular to x3- axis. We take the origin of the coordinate system (x1, x2, x3) at any point on the plane surface and x3-axis pointing vertically downward into the half-space. Thus the consider- ing half-space is represented by x3 ≥ 0. For two-dimensional problem in which the plane wave is in the plane x1x3), the strains are related to the displacement field u1, u3 and the electric potential φ. The constitutive equations are given as [1, 11]: - Strain-displacement relations ε11 = u1,1, ε33 = u3,3, ε13 = ε31 = 1 2 (u1,3 + u3,1), (1) - Stress-strain and electric field relations σ11 = c11ε11 + c13ε33 − e31E3, σ33 = c13ε11 + c33ε33 − e33E3, σ13 = σ31 = 2c44ε13 − e15E1, D1 = 2e15ε13 + e11E1, D3 = e13ε11 + e33ε33 + e33E3, (2) Dispersion equation of Rayleigh waves in transversely isotropic nonlocal piezoelastic solids half-space 365 where Ei = −φ,i is the electric field and Di the electric displacement, ρ the mass density, σij the stress tensor, cij the elastic parameters tensor, eij the piezoelectric moduli, eij the electric permittivity (i, j = 1, 3). It is well known that in the classical piezoelectricity (CPE) theory, the stresses and the electric displacements at one point only dependend on the local strains and electric fields at the same point. But when the macro size reaches a few nanometers, the CPE continuum theory fails and we have usually to utilize other methods. The essence of the Eringen’s nonlocal elasticity theory [6,15] is that the stress at a point x in a body depends not only on the strain at that point but also on the strain at all other points x′ in the domain. Recently, Ke et al. [7, 8] extended the nonlocal elasticity theory to the piezoelectric nanostructures- the nonlocal continuum theory of piezoelectricity (NLPE). Unlike the CPE continuum theory, the NLPE theory supposes that the stresses and the electrical displacements at one point should be affected by the strains and electrical fields at all points of the whole body. Thus the relationship between the CPE stress and electrical displacement components and the NLPE stress and electrical displacement components can be written as [11, 12] tmn = (1 + e2∇2)σmn, dm = (1 + e2∇2)Dm, (3) where tmn and dm are the NLPE stress and electrical displacement components, respec- tively; σmn and Dm are the traditional stress and electrical displacement components, respectively. Constant e(= e0a) is the nonlocal parameter (e0 is the nonlocal constant and a is the internal characteristic length). For the wave propagation considered in this paper, the body forces, electric charge are ignored. Using the relations (3), the motion equations, Gauss equation are simplified as [11, 12] σ11,1 + σ13,3 = (1− e2∇2)ρu¨1, σ13,1 + σ33,3 = (1− e2∇2)ρu¨3, D1,1 + D3,3 = 0. (4) Substituiting (2) into (4) taking into account (1) we have c11u1,11 + c44u1,33 + (c13 + c44)u3,13 + (e15 + e31)φ,13 = (1− e2∇2)ρu¨1, (c13 + c44)u1,13 + c44u3,11 + c33u3,33 + e15φ,11 + e33φ,33 = (1− e2∇2)ρu¨3, (e15 + e31)u1,13 + e15u3,11 + e33u3,33 − e11φ,11 − e33φ,33 = 0. (5) For the waves propagating in the plane x3 = 0, we take the form of relevant components of displacement and the electric potential φ as [1, 16] u1 = a1e−ξyeik(x1−ct) u3 = a3e−ξyeik(x1−ct) with y = kx3, φ = A1e−ξyeik(x1−ct) (6) where a1, a3, A1 are polarization vectors, k is wavenumber, c is speed of wave propaga- tion, ξ is a complex coefficient whose imaginary part should be positive corresponding to the decay condition in the half-space x3 > 0. Substituting the expressions for displacement and electric potential from (6) into (5), we obtain the three homogeneous equations in three unknowns a1, a3, A1. For a nontrivial 366 Do Xuan Tung solution of these equations, we must have det(M) = 0, in which M = c11 − c44ξ2 − ρc2 − ρc2k2e2(1− ξ2) (c13 + c44)iξ (e15 + e31)iξ(c13 + c44)iξ c44 − c33ξ2 − ρc2 − ρc2k2e2(1− ξ2) e15 − e33ξ2 (e15 + e31)iξ e15 − e33ξ2 e33ξ2 − e11 . (7) This is the characteristics equation and it has the form h6p6 + h4p4 + h2p2 + h0 = 0, (8) where p = iξ and the coefficients hi, (i = 0, 2, 4, 6) are given in the Appendix. Eq. (8) is a cubic polynomial in p2. We order pn, n = 1, 2, . . . , 6, in such a way that p1, p2, p3 correspond to waves traveling in the positive x3 direction, and p4, p5, p6 correspond to the ones traveling in the negative x3, respectively. Since, we are interested in surface waves only so it is essential that motion is confined to free surface x3 = 0 of the half-space so that the characteristic roots p2i must satisfy the radiation condition Im(pi) ≥ 0. Then the general solution for displacements and electric potential are written as [1] u1 = 3 ∑ j=1 a1jepjyeik(x1−ct) u3 = 3 ∑ j=1 a3jepjyeik(x1−ct) φ = 3 ∑ j=1 A1jepjyeik(x1−ct) (9) where a1j, a3j and A1j are the amplitudes of the displacements and the electric potentials, respectively. Remark: For the propagation of plane waves with phase velocity c in the direction making an angle θ with the vertical axis, a surface wave of this form is expressed by u1 = a1eik(p1x1+p3x3−ct) u3 = a3eik(p1x1+p3x3−ct) φ = A1eik(p1x1+p3x3−ct) (10) where p1 = sin θ, p3 = cos θ are components of propagation unit vector. Substituting (10) into (5) and obtain a matrix similar to the matrix M in Eq. (7). By letting the determi- nant of this matrix equal zero, we have a quadratic equation in c2. Therefore, we obtain two real roots cj(j = 1, 2) corresponding the speeds of plane waves propagating in the medium. 3. BOUNDARY CONDITIONS AND DISPERSION EQUATIONS In this section, the Rayleigh wave equation for transversely isotropic nonlocal piezoe- lastic half-space can be derived using the boundary conditions at the surface of the half- space. In the present study, boundary conditions appropriate for particle motion in the x1x3 plane are considered at the plane surface x3 = 0. This surface is considered to be Dispersion equation of Rayleigh waves in transversely isotropic nonlocal piezoelastic solids half-space 367 stress-free (mechanical conditions), which requires the normal stress σ33 as well as the tangential stress σ13 to vanish at the surface x3 = 0. That means σ13 = σ33 = 0. (11) Another condition is required to represent that the surface of half-space is maintained at charge free condition (open circuit-surface), namely D3 = 0. (12) Substituting (9) into the boundary conditions (11), (12) and taking into account (2), we have a system of linear equations 3 ∑ j=1 [ c44(a3j + a1jpj) + e15A1j ] = 0, 3 ∑ j=1 [ c13a1j + c33a3jpj + e33A1jpj ] = 0, 3 ∑ j=1 [ e13a1j + e33a3jpj − e33A1jpj ] = 0. (13) For each pj(j = 1, 2, 3), the three corresponding unknowns a1j, a3j, A1j (7) are in a rela- tionship given by matrix M and we can express them as a1j = αjA1j, a3j = β jA1j where αj = (c13 + c44)pj(e15 + e33p2j )− ( c44 + c33p2j − ρc2 − ρc2k2e2(1 + p2j ) ) (e15 + e31)pj δj , β j = (c13 + c44)p2j (e15 + e31)− ( c11 + c44p2j − ρc2 − ρc2k2e2(1 + p2j ) ) (e15 + e33p2j ) δj , δj = [ c11 + c44p2j − ρc2 − ρc2k2e2(1 + p2j ) ][ c44 + c33p2j − ρc2 − ρc2k2e2(1 + p2j ) ] − (c13 + c44)2p2j , j = 1, 2, 3. Then we obtain a system of linear equations in amplitudes A11, A12, A13 only and it is in the form (c44β1 + c44α1p1 + e15)A11 + (c44β2 + c44α2p2 + e15)A12 + (c44β3 + c44α3p3 + e15)A13 = 0, (c13α1 + c33β1p1 + e33p1)A11 + (c13α2 + c33β2p2 + e33p2)A12 + (c13α3 + c33β3p3 + e33p3)A13 = 0, (e31α1 + e33β1p1 − e33p1)A11 + (e31α2 + e33β2p2 − e33p2)A12 + (e31α3 + e33β3p3 − e33p3)A13 = 0. (14) The dispersion equation of Rayleigh waves is obtained from det(CO) = 0 where matrix CO is the matrix of coefficients of the system of equation above as c44β1 + c44α1p1 + e15 c44β2 + c44α2p2 + e15 c44β3 + c44α3p3 + e15c13α1 + c33β1p1 + e33p1 c13α2 + c33β2p2 + e33p2 c13α3 + c33β3p3 + e33p3 e31α1 + e33β1p1 − e33p1 e31α2 + e33β2p2 − e33p2 e31α3 + e33β3p3 − e33p3 . (15) 368 Do Xuan Tung This dispersion equation is in implicit form and it shows the relation between the phase velocity c and the wave number k of the Rayleigh waves and the parameters of the medium. 4. NUMERICAL RESULTS AND DISCUSSION In order to illustrate theoretical results obtained in the preceding sections, the mate- rial chosen for the numerical calculations is CdSe (6 mm class) of hexagonal symmetry, which is transversely isotropic material. The physical data for a single crystal of CdSe material is given below [1, 14] c11 = 7.41× 1010 Nm−2, c13 = 3.93× 1010 Nm−2, c33 = 8.36× 1010 Nm−2, c44 = 1.32× 1010 Nm−2, ρ = 5504 kgm−3, e15 = −0.138 Cm−2, e31 = −0.16 Cm−2, e33 = 0.347 Cm−2, e11 = 8.26× 10−11 C2N−1m−2, e33 = 9.03× 10−11 C2N−1m−2, e0 = 0.39, a = 0.421× 10−9 m, e = e0a. (16) Denote ep = k2e2 the dimensionless frequency where k is the wavenumber. This is an important parameter that provides us the information of the wave-length of Rayleigh waves comparing to the nonlocal parameter of the medium. 0 10 20 30 40 50 60 70 80 90 Angle in degree 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 D im en si on le ss p ha se v el oc itie s c1 c2 Local Nonlocal ep=0.8 Fig. 1. Dimensionless velocities of plane waves c¯1 and c¯2 depending on the angle directions θ for local and nonlocal case First, we evaluate the effect of nonlocality to the speed of plane waves. Fig. 1 shows the dimentionless speed of plane waves c/bS, where bS = √ c44/ρ is the speed of SH- type wave, depending on the direction of propagation θ (angle between the direction Dispersion equation of Rayleigh waves in transversely isotropic nonlocal piezoelastic solids half-space 369 of propagation and vertical axis) in the piezoelectric medium for two case of local theory and nonlocal theory with the nonlocal parameter given by ep = 0.8. It can be seen that the phase velocities c1, c2 in the nonlocal theory case are greater the ones in the local theory case. It can be concluded that the nonlocality has significant effect on the velocities of propagation of plane waves. Next, the variations of the phase velocities with dimensionless parameter ep for θ = pi/3 are depicted in Fig. 2. Generally, this figure shows that the phase velocities c1, c2 are decreasing when ep is increasing. When ep < 1 these velocities decrease rapidly decrease while ep > 1 they are quite stable. 0 1 2 3 4 5 6 7 8 9 10 Dimensionless parameter ep=k2 2 0 0.5 1 1.5 2 2.5 D im en si on le ss p ha se v el oc itie s c1 c2 Fig. 2. The comparison of variations of the phase velocities with dimensionless parameter ep = k2e2 for θ = pi/3 Finally, the dimensionless speed of Rayleigh wave x = X/c44 with X = ρc2 depends upon the dimensionless parameter ep is illustrated by Fig. 3 for the boundary condition of open circuit surface (maintained charge free). The speed of Rayleigh wave is decreasing when the parameter ep is increasing. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ep 0 0.2 0.4 0.6 0.8 1 x Fig. 3. Effect of parameter ep on the speed of Rayleigh wave x for the open circuit surface 370 Do Xuan Tung 5. CONCLUSIONS In the present work, we have studied the propagation of Rayleigh waves in trans- versely isotropic piezoelastic nonlocal materials. Some important features are drawn be- low: (i) Under certain type of specific boundary condition: the surface x3 = 0 is consid- ered to be stress-free and maintained at charge-free condition, the dispersion equation of the Rayleigh waves is given. It is numerically concluded that the nonlocality has signifi- cant effect on the speed of Rayleigh wave. (ii) Phase velocities of plane waves are computed numerically and their variation on the incident angle θ, dimensionless frequency parameter ep, are presented graphically. The effect the nonlocality on the velocities of plane waves are also expressed through numerical example and the effect is also significant. ACKNOWLEDGMENT The work was supported by the Vietnam National Foundation for Science and Tech- nology Development (NAFOSTED) under Grant 107.02-2019.06. REFERENCES [1] J. N. Sharma, M. 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APPENDIX The coefficients of characteristic equation h4 = −b9b23 + 2b3b4b7 − b5b24 − b2b27 − 2b1b8b7 + b1b5b10 + b1b6b9 + b2b5b9, h2 = −b23b10 + 2b3b4b8 − b6b24 − b1b28 − 2b2b7b8 + b1b6b10 + b2b5b10 + b2b6b9, h0 = b2b6b10 − b2b28, h6 = b1b5b9 − b1b27, b1 = c44 − Xk2e2, b2 = c11 − X− Xk2e2, b3 = c13 + c44, b4 = e15 + e31, b5 = c33 − Xk2e2, b6 = c44 − X− Xk2e2, b7 = e33, b8 = e15, b9 = −e33, b10 = −e11,X = ρc2.

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