Theoretical transient analysis and wave propagation
of piezoelectric bi-materials
Chien-Ching Ma
a,*, Xi-Hong Chen
a, Yi-Shyong Ing
ba
Department of Mechanical Engineering, National Taiwan University, Taipei 106, Taiwan, ROC b
Department of Aerospace Engineering, Tamkang University, Tamsui 251, Taiwan, ROC Received 12 September 2006; received in revised form 5 March 2007
Available online 4 April 2007
Abstract
The transient response of piezoelectric bi-materials subjected to a dynamic anti-plane concentrated force or electric charge with perfectly bonded interface is examined in the present study. The problem is solved by using the Laplace trans-form method and the inverse Laplace transtrans-form is evaluated by means of Cagniard’s method. Exact transient full-field solutions of the contribution for each wave are expressed in explicit closed forms. The transient behavior of field quantities is examined in detail by numerical calculations. The existence condition of a propagating surface wave along the interface is discussed in detail. A surface wave can be guided by the interface of two semi-infinite materials in contact if one, at least, of these two materials is piezoelectric. The propagation velocity of the surface wave is explicitly expressed and is found to be less than the lower shear wave velocity of the two materials. The existence of the surface wave for piezoelectric–piezo-electric bi-materials is restricted to the situation that the shear waves of the two piezopiezoelectric–piezo-electric materials are very close. The possibility for the existence of the surface wave for piezoelectric–elastic bi-materials is much greater than that of the pie-zoelectric–piezoelectric bi-materials.
2007 Elsevier Ltd. All rights reserved.
Keywords: Transient response; Dynamic loading; Piezoelectric bi-materials; Surface wave; Laplace transform method
1. Introduction
The propagation of stress waves through an unbounded material is not a difficult subject. A half-space bounded by one plane surface is the simplest model for observing elastic waves in solid. Many applica-tions of electrodynamics begin with the model of a half-space. The classical analysis in this area was first proposed by Lamb (1904); he considered the elastic half-space subjected to point and line loads on the surface of a semi-infinite half-space. Since this early analysis of Lamb, a great many contributions have appeared, pertaining to what is commonly referred to as Lamb’s problem. de Hoop (1960) and Cagnard (1962) proposed a general method to evaluate the inverse Laplace transforms, which made the solving of
0020-7683/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2007.03.027
* Corresponding author. Tel.: +886 2 23659996; fax: +886 2 23631755. E-mail address:ccma@ntu.edu.tw(C.-C. Ma).
elastic wave propagation problem become possible. The generalized ray theory was developed since 1939 when Cagniard studied the transient waves in two homogeneous half-spaces in contact. In his monumental work, he had shown that by going through a sequence of contour deformations and changes of integra-tion variables, one is able to find the inverse Laplace transforms of the expressions for each ray. A review of this theory was given by Pao and Gajewski (1977). Spencer (1960) used the generalized ray method to investigate the surface response of a stratified half-space to the radiation from a localized source. The method leads to an infinite series of the generalized ray integral constructed in the Laplace transform domain by assembling the source function, reflection and transmission coefficient, the receiver function, and the phase function. The method therefore obviates the necessity for solving a tedious boundary value problem. The time function associated with each ray integral is obtained by using the Cagniard method.
Ma and Huang (1993) have constructed the exact transient solution of buried dynamic forces for elastic bi-material problem. Buried source problems develop considerable interests in seismology and have been studied by many investigators. All the research works mentioned above are concerned with isotropic elas-tic materials.
Piezoelectric materials possess the important property of linear coupling between mechanical and electrical fields, which renders them useful in many areas of modern technology. These materials have thus been widely used for a long time as electromechanical transducers, filters, sensors and actuators, to mention only a few. In recent years, they are finding new applications in non-destructive evaluation, ultrasonic medical imaging, smart structures and active control of sound and vibration. In this endeavor, composite materials, consisting of combinations of two or more different piezoelectric and non-piezoelectric material phases, have been designed to meet specific technical needs. Such composites permit the tailoring of special properties, unavail-able in homogeneous phases, and therefore they are becoming increasingly important in diverse areas of mod-ern technology. Quite recently, a new class of highly advanced composites, where the material properties vary continuously in a particular direction, has been fabricated and introduced for aerospace applications. They have been termed functionally gradient materials and they are expected to play even a more important role (Yamanouchi, 1990).
The study of wave propagation in piezoelectric materials is a rather involved problem. The situation is even more formidable when non-homogeneity has to be taken into account. It is, therefore, not sur-prising that only scant information regarding transient wave propagation problems has been available. As in the case of the Stoneley wave, whose mechanical displacements are in the sagittal plane, the ampli-tude of this wave decreases with distance away from the interface into both media (Stoneley, 1924).
Bleustein (1968) and Gulayev (1969) simultaneously discovered that there exists a shear horizontal (SH) electro-acoustic surface mode in a class of transversely isotropic piezoelectric media, which is known today as the BG wave. The BG wave is a unique result in the repertoire of surface acoustic wave (SAW) theory, because it has no counterpart in purely elastic solids. As a matter of fact, since then, the BG wave theory has become one of the cornerstones for the modern electro-acoustic technology. It is shown that BG wave can exist in cubic crystals of 43m and 23 classes, along [1 1 0] direction on the ð110Þ plane and their equivalent orientations. The velocity equations for piezoelectric surface wave and elastic surface wave were derived and their characteristics were discussed by Tseng (1970). A pure shear elastic surface wave (MT wave) can propagate along the interface of two identical crystals, in class 6 mm, when the z-axis of these crystals, both parallel to the interface and perpendicular to the propa-gation direction, are in opposite directions (Maerfeld and Tournois, 1971). The general equations and the fundamental piezoelectric matrix were derived for the anti-plane wave motion and Floquet theory was applied to obtain the passing and stopping bands in a periodically layered infinite space by Honein and Herrman (1992). Taking into account both optical effect as well as the contribution from the rota-tional part of electric field, the solutions obtained were not only valid for any wave speed range, but also provide accurate formulas to evaluate the acousto-optic interaction due to piezoelectricity. As the wave speed is much less than the speed of light, the solution degenerates to the well-known BG wave or MT wave (Li, 1996).
The surface acoustic wave (SAW) can be excited and detected efficiently by using an interdigital trans-ducer (IDT) placed on a piezoelectric substrate and a vast amount of effort was invested in the research and development of SAW devices for military and communication applications, such as delay lines and
filters for radar. The propagation mode in most devices is the Rayleigh wave on a free surface of a pie-zoelectric substrate. Since SAWs concentrate their energy near the substrate free surface, their propagation surface should be sufficiently flat and free of contamination. This means that devices based on SAWs must be put into a package for protection against surface disturbance. The packages are usually much larger than the substrate and account for most of the cost of the device. Hence it is difficult to satisfy the requirement for smaller device and reliability.
With the fast evolution of passive high frequency filtering requirements, much effort has been devoted to the improvement of classical SAW. Among possible new devices, the use of interface surface waves (ISWs) in place of SAWs has been proposed. As one of the families of such ISWs, Stoneley waves, which basically consist of longitudinal and shear-vertical displacement components, have been most extensively discussed in the past. Because of the very restricted range of existence of Stoneley waves and this acoustic wave can not be excited by interdigital transducers in such nonpiezoelectric media, it has been rather difficult to find practical device applications. Hence a piezoelectric material must be chosen as one or both media. Recently, some technologies appeared that allow one to create a direct contact between two previously grown piezocrystals of different symmetry (Dvoesherstov et al., 2002). Accordingly, interest in the development of interface surface waves has grown considerably. Camou and Laude (2003) used a interdigital transducer at the interface for the excitation of the ISW. The application of interface surface waves in acousto-electronic devices can give a number of important advantages over the common SAW. One of the most cited advantages of ISW devices as opposed to SAW devices is the natural protection of the excitation interface, which is isolated from, and hence insensitive to, external disturbances, such as dust or wetness. This should lead to a simplification of packaging requirements, especially at high frequencies. Furthermore, in some cases, the phase velocity of transverse ISWs can be higher than that of SAWs. By choosing specific piezoelectric media, one can also improve the parameters of the wave, namely, raise the electromechanical coupling coefficient and improve the thermal stability of the wave simultaneously (Irino et al., 1988; Irino and Shimizu, 1989). It was pointed out by Yamashita et al. (1997) that the penetration depth of shear-horizontal type of ISWs into the bulk is small. Hence the substrate thickness needed for practical applications can be reduced to several wavelengths. This suggests that the direct wafer bonding technique could be applied to realize the structure.
A lot of engineering applications mentioned above are involved with wave propagations through the pie-zoelectric components, and hence their dynamic or transient behaviors are the primary concern in design as well as in performance. In this paper, an analysis is presented to study the transient behaviors of a transversely isotropic piezoelectric material under anti-plane mechanical and in-plane electrical line sources, which is con-sidered as one of the fundamental important problems in electro-elastodynamics. This problem exhibits the distinct feature of piezoelectricity, which is different from the behaviors of ordinary elasto-dielectric solids. The analysis presented in this study provides a sound theoretical ground and interesting physics for a better understanding of the transient behaviors of piezoelectric materials. The results obtained in this study will be useful for the design and application of ISWs devices. This article is divided into five sections. Following this brief introduction, Section2outlines the basic equations needed to formulate the problem. The exact full-field transient solutions for piezoelectric bi-materials are presented in explicit forms. The possibility that there exists a piezoelectrically-induced electromagnetic surface wave propagating along the interface between two materi-als is discussed in detail in Section3. Then, a number of numerical results of transient behaviors for interesting cases and the surface wave velocity along the interface of bi-materials are examined in Section4. Finally we conclude the paper in Section5.
2. The transient solutions for piezoelectric bi-materials
In a piezoelectric material, the interdependence of electric and mechanical fields implies coupling elastic and electromagnetic waves. Because elastic waves in a typical material are five orders of magnitude slower than electromagnetic waves, so the piezoelectrically coupled electric field is assumed to be quasi-static. Maxwell’s equations therefore reduce to (Hayt and Buck, 2001)
Di;i¼ qs; ð1Þ
Ei¼ U;i ð2Þ
where Di, Ei, U and qsare electric displacement, electric field, electrostatic potential and free charge density,
respectively, and the comma denotes differentiation in the usual tensor notation. The piezoelectric material is assumed to be a perfect insulator so that qsis zero, then(1) reduces to the Laplace’s equation
U;ii¼ 0 ð3Þ
Consequently, the propagation of elastic and electromagnetic waves can be treated separately. The mechanical field quantities must satisfy Newton’s stress equation of motion which, in absence of internal body forces, becomes
sij;j¼ q€ui ð4Þ
where sij, uiand q are stress tensor, mechanical displacement and density, respectively, and the dot denotes the
differentiation with respect to time. Infinitesimal strain tensor Sijis defined by
Sij ¼
1
2ðui;jþ uj;iÞ: ð5Þ
The above relationships are coupled through the piezoelectric equations of state
sij¼ cijklSkl ekijEk; ð6Þ
Di¼ eiklSklþ eikEk; ð7Þ
where cijkl, ekijand eikare the elastic, piezoelectric and permittivity tensors of the material, respectively.
Substi-tuting(2)–(5)into(6) and (7), the second-order coupled differential equations of the system for uland U are
obtained (Royer and Dieulesaint, 2000)
cijklul;jkþ ekijU;kj¼ q€ui; ð8Þ
ejklul;jk ejkU;jk¼ 0: ð9Þ
In what follows, the wave polarized in the z-direction, propagating in the x–y plane of a hexagonal crystal in class 6mm will be analyzed. The x, y and z axes are aligned with the crystal axes X, Y and Z, respectively. In this configuration, the boundary value problem simplifies considerably because there exists the anti-plane dis-placement uz, which couples only with the in-plane electric fields Exand Ey, such that
uz¼ wðx; y; tÞ; Ex¼ Exðx; y; tÞ; Ey ¼ Eyðx; y; tÞ: ð10Þ
This is the so-called anti-plane problem, upon substituting these relations into(8) and (9), we are led to
c44r2wþ e15r2U¼ q€w; ð11Þ
e15r2w e11r2U¼ 0; ð12Þ
where $2is the two-dimensional Laplacian operator andr2 o2 ox2þ o
2
oy2. The constitutive equations can be
sim-plified as syz¼ c44 ow oy þ e15 oU oy; ð13Þ Dy¼ e15 ow oy e11 oU oy; ð14Þ sxz¼ c44 ow oxþ e15 oU ox; ð15Þ Dx¼ e15 ow ox e11 oU ox: ð16Þ We define a function w by
w U e15 e11
w: ð17Þ
The solutions of(11) and (12)can be obtained from the solutions of the following two uncoupled equations r2w¼ q c44 € w; ð18Þ r2 w¼ 0; ð19Þ where c44 c44þ e2 15
e11is the piezoelectrically stiffened elastic constant. The constitutive equations are reduced to
the form of syz¼ c44 ow oyþ e15 ow oy; ð20Þ Dy ¼ e11 ow oy; ð21Þ sxz¼ c44 ow oxþ e15 ow ox; ð22Þ Dx ¼ e11 ow ox: ð23Þ
2.1. The transient solutions for applying a dynamic anti-plane concentrated force
In this problem, the system of coordinates (x, y, z) is chosen so that perfectly conducting plane is described by the equation y = 0, as shown inFig. 1. This is also an interface between two half-spaces of piezoelectric materials. The piezoelectric bi-material is initially undisturbed. Material (1) is subjected to a dynamic anti-plane concentrated force at x = 0, y = d with magnitude p at time t = 0. The jump condition for the shear stress is
sð1yzþÞjy¼d sð1Þ
yz jy¼d ¼ pdðxÞHðtÞ; ð24Þ
where d(x) is the delta function of x and H(t) is the Heaviside function of t. The continuous conditions are Dð1yþÞjy¼d Dð1Þ y jy¼d ¼ 0; ð25Þ wð1þÞjy¼d wð1Þjy¼d ¼ 0; ð26Þ Uð1þÞjy¼d Uð1Þ jy¼d ¼ 0; ð27Þ sð1yzÞjy¼0¼ sð2Þyz jy¼0; ð28Þ
Dð1yÞjy¼0¼ Dð2Þ
y jy¼0; ð29Þ
wð1Þjy¼0¼ wð2Þjy¼0; ð30Þ
Uð1Þjy¼0¼ Uð2Þj
y¼0: ð31Þ
Note that, the superscripts(1) and (2)indicate materials(1) and (2), respectively. This problem can be solved by the application of integral transforms. The one-sided Laplace transform over time t and the bilateral La-place transform on the spatial variable x for (18) and (19)can be represented of the form
d2w dy2 ðb 2 k2Þs2w¼ 0; ð32Þ d2w dy2 ðe 2 k2 Þs2w¼ 0; ð33Þ
where s which is the Laplace transform parameter is a positive real number, large enough to ensure the con-vergence of the integral and k is a complex variable. Where b¼qffiffiffiffifficq44 is the slowness of the shear wave and e! 0+. The overbar symbol is used denoting the transform on time t and the star symbol is used denoting the transform on the spatial variable x. The general solutions of(32) and (33)represented in the matrix form are wð1þÞ wð1þÞ " # ¼ Y1 A1 C1 þ Y B1 D1 ; ð34Þ wð1Þ wð1Þ " # ¼ Y1 E1 G1 þ Y F1 H1 ; ð35Þ wð2Þ wð2Þ " # ¼ Y1 I1 K1 þ Y J1 L1 ; ð36Þ where Y¼ e say 0 0 esby ; Y1¼ e say 0 0 esby ; Y¼ e say 0 0 esby ; Y1¼ e say 0 0 esby ; aðkÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b21 k2 q ; bðkÞ ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffie2 k2; aðkÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 2 k 2 q ; b1¼ ffiffiffiffiffiffiffi qð1Þ cð1Þ44 s ; b2¼ ffiffiffiffiffiffiffi qð2Þ cð2Þ44 s :
The coefficients A1, B1, C1, D1, E1, F1, G1, H1, I1, J1, K1and L1are determined by satisfying the jump and
continuous conditions. The solutions of the mechanical and electric fields presented in the Laplace transform domain are wð1Þ wð1Þ " # ¼1 2s2ðY 1RD1M1Zþ BM1ZÞ; ð37Þ wð2Þ wð2Þ " # ¼1 2s2Y QTD1M1Z; ð38Þ sð1Þ yz Dð1Þy " # ¼ 1 2sðM1UY 1RD1M1Zþ M 1UBM1ZÞ; ð39Þ sð1Þ xz Dð1Þ x " # ¼1 2s ðM1kaY 1RD1M1Zþ M 1kbBM1ZÞ; ð40Þ
sð2Þ yz Dð2Þ y " # ¼1 2sN1VY QTD1M1Z; ð41Þ sð2Þ xz Dð2Þx " # ¼1 2sN1kcY QTD1M1Z; ð42Þ where M¼ c ð1Þ 44a e ð1Þ 15b 0 eð1Þ11b " # ; N¼ c ð2Þ 44a e ð2Þ 15b 0 eð2Þ11b " # ; M1¼ cð1Þ44 eð1Þ15 0 eð1Þ11 " # ; N1¼ cð2Þ44 eð2Þ15 0 eð2Þ11 " # ; R¼ ðM þ NQÞ1ðM NQÞ; T¼ ðM þ NQÞ1ðM NQÞ þ I; ka¼ kþ1 0 0 kþ3 " # ; kb¼ kþ2 0 0 kþ4 " # ; kc¼ kþ5 0 0 kþ6 " # ; U¼ a 0 0 b ; V¼ a 0 0 b ; D1¼ e sad 0 0 esbd ; B¼ e sajydj 0 0 esbjydj " # ; I¼ 1 0 0 1 ; Q¼ 1 0 eð1Þ15 eð1Þ11 eð2Þ15 eð2Þ11 1 2 4 3 5; Z ¼ p 0 :
The next step consists in evaluating the inverse Laplace transform of (37)–(42) by means of the Cagn-iard-de Hoop scheme. The CagnCagn-iard-de Hoop inversion method is used to perform the two integrations in one single operation leaving only the convolution to be done. We have to include the integral around the branch cut whatever different slowness combines. This additional integral path represents the head wave. There are two situations, b1jcosh1j > b2and b1jcosh1j < b2, to be investigated and the k-contours are shown
in Fig. 2. As the Cagniard-de Hoop inversion method is employed, we introduce Cagniard contours by setting aðy þ dÞ kx ¼ t; ð43Þ aðy dÞ kx ¼ t; ð44Þ byþ ad kx ¼ t; ð45Þ bðy dÞ kx ¼ t; ð46Þ ayþ ad kx ¼ t; ð47Þ by þ ad kx ¼ t: ð48Þ
Note that, vales of k1, k2, k3, k4, k5 and k6 are the roots of(43)–(47)and(48), respectively. When the imag-inary part of root in(45) and (48)vanishes, the correspondent arrive times are denoted by t3and t6,
respec-tively. The additional integral path represents the head wave in(43) and (47), where the wave fronts of head wave arrive at time t = t1HDand t = t5HD, respectively.
Finally, the transient solutions in matrix form for displacement, shear stresses and electric displacement for piezoelectric materials(1) and (2)are explicitly presented and are summarized as follows
wð1Þ wð1Þ " # ¼1 2p Z t 0 HaIm½k0aRM 1Z dt þZ t 0 HbIm½k0bM 1Z dt ; ð49Þ wð2Þ wð2Þ " # ¼1 2p Z t 0 HcIm½k0cQTM 1Z dt; ð50Þ sð1Þ yz Dð1Þ y " # ¼ 1 2pðIm½Hak 0 aM1URM1Z þ Im½Hbk0bM1UM1ZÞ; ð51Þ
sð1Þ xz Dð1Þ x " # ¼1 2pðIm½Hak 0 aM1kaRM1Z þ Im½Hbk0bM1kbM1ZÞ; ð52Þ sð2Þ yz Dð2Þy " # ¼1 2pIm½Hck 0 cN1VQTM1Z; ð53Þ sð2Þ xz Dð2Þx " # ¼1 2pIm½Hck 0 cN1kcQTM1Z; ð54Þ where
k0a¼ okþ1 ot 0 0 ok þ 3 ot 2 4 3 5; k0 b¼ okþ 2 ot 0 0 ok þ 4 ot 2 4 3 5; k0 c¼ okþ5 ot 0 0 ok þ 6 ot 2 4 3 5; Ha¼ Hðt t1HDÞ 0 0 Hðt t3Þ Hb¼ Hðt b1r2Þ 0 0 Hðt er2Þ ; Hc¼ Hðt t5HDÞ 0 0 Hðt t6Þ ; r1¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2þ ðy þ dÞ2 q ; r2¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2þ ðy dÞ2 q ; kþ1 ¼ t r1 cos h1þ ij sin h1j ffiffiffiffiffiffiffiffiffiffiffiffiffiffi t2 r2 1 b2 1 s ; kþ2 ¼ t r2 cos h2þ ij sin h2j ffiffiffiffiffiffiffiffiffiffiffiffiffiffi t2 r2 2 b2 1 s ; t1HD¼ b1ðr2sin h2þ dÞ; t5HD¼ b2ðr2sin h2þ dÞ þ b1d; cos h1¼ x r1 ; sin h1¼ ðy þ dÞ r1 ; cos h2¼ x r2 ; sin h2¼ ðy dÞ r2 :
The transient solutions in matrix form presented in(51)–(54)can also be completely expressed in explicit form as indicated inAppendix A. If the piezoelectric material (2) is reduced to an elastic material, which implies that the piezoelectric effect is neglected. Then we have
cð2Þ44 ¼ l; ð55Þ
eð2Þ15 ¼ 0; ð56Þ
eð2Þ11 ! 1; ð57Þ
where l is shear modulus. Substitution of(55)–(57)into(49)–(54)yields the transient solutions in matrix form for displacement, stress and electric displacement for piezoelectric and elastic bi-material.
2.2. The transient solutions for applying a dynamic electric charge
In this problem, the material (1) is subjected to a dynamic electric charge (qd(x)H(t)) at time t = 0, as shown inFig. 1. The jump condition is represented as
Dð1yþÞjy¼d Dð1yÞjy¼d ¼ qdðxÞHðtÞ: ð58Þ
The continuous conditions are the same as that presented in(26)–(31)and the jump of shear stress syzin(24)
equal zero. The Cagniard-de Hoop inversion method is applied and Cagniard contours are introduced by set-ting(43)–(48)and
ayþ bd kx ¼ t; ð59Þ
bðy þ dÞ kx ¼ t; ð60Þ
ayþ bd kx ¼ t; ð61Þ
bðy dÞ kx ¼ t: ð62Þ
Note that vales of k7, k8, k9 and k10are the roots of(59)–(62), respectively. When the imaginary part of root in(59) and (61)vanishes, the correspondent arrive times are denoted by t7and t9, respectively. Accordingly,
from the similar procedure that we have used for applying a dynamic concentrated force, the transient solu-tions for displacement, shear stresses and electric displacements for piezoelectric materials(1) and (2)are sum-marized as follows wð1Þ wð1Þ " # ¼1 2p Z t 0 HaIm½k0aRM 1G dt þZ t 0 HbIm½k0bM 1G dt ; ð63Þ wð2Þ wð2Þ " # ¼1 2p Z t 0 HcIm½k0cQTM 1G dt; ð64Þ
sð1Þ yz Dð1Þy " # ¼ 1 2pðIm½M1UHdRk 0 dM 1G þ Im½M 1UHek0eM 1GÞ; ð65Þ sð1Þ xz Dð1Þx " # ¼1 2pðIm½M1kdHdRk 0 dM 1G þ Im½M 1keHek0eM 1GÞ; ð66Þ sð2Þ yz Dð2Þy " # ¼1 2pIm½N1VHfQTk 0 fM 1G; ð67Þ sð2Þ xz Dð2Þx " # ¼1 2pIm½N1kfHfQTk 0M1G; ð68Þ where G¼ 0 q ; k0d¼ k 0 d1 0 0 k0d3 " # ; k0e¼ k 0 e2 0 0 k0e4 " # ; k0f¼ k 0 f5 0 0 k0f6 " # ; kd¼ kd1 0 0 kd3 ; ke¼ ke2 0 0 ke4 ; kf ¼ kf5 0 0 kf6 ; Hd¼ Hd1 0 0 Hd3 ; He¼ He2 0 0 He4 ; Hf ¼ Hf5 0 0 Hf6 ; k0e2He2¼ okþ2 ot Hðt b1r2Þ; k 0 e4He4¼ okþ4 ot Hðt er2Þ; k 0 d1Hd1¼ okþ1 ot Hðt t1HDÞ; k0d3Hd1¼ okþ7 ot Hðt t7Þ; k0d1Hd3¼ okþ3 ot Hðt t3Þ; k 0 d3Hd3¼ okþ8 ot Hðt er1Þ; k 0 f5Hf5¼ okþ5 ot Hðt t5HDÞ; k0f6Hf5¼ okþ9 ot Hðt t9Þ; k0f5Hf6¼ okþ6 ot Hðt t6Þ; k 0 f6Hf6¼ okþ10 ot Hðt er2Þ; ke2k 0 e2He2¼ kþ2 okþ2 ot Hðt b1r2Þ; ke4k0e4He4¼ kþ4 okþ4 ot Hðt er2Þ; kd1k 0 d1Hd1¼ kþ1 okþ1 ot Hðt t1HDÞ; kd1k 0 d3Hd1¼ kþ7 okþ7 ot Hðt t7Þ; kd3k0d1Hd3¼ kþ3 okþ3 ot Hðt t3Þ; kd3k 0 d3Hd3¼ kþ8 okþ8 ot Hðt er1Þ; kf5k 0 f5Hf5¼ kþ5 okþ5 ot Hðt t5HDÞ; kf5k0f6Hf5¼ kþ9 okþ9 ot Hðt t9Þ; kf6k 0 f5Hf6¼ kþ6 okþ6 ot Hðt t6Þ; kf6k 0 f6Hf6¼ kþ10 okþ10 ot Hðt er2Þ:
3. The existence of surface wave at the piezoelectric bi-material interface
It is important to examine the possibility of surface waves propagating along the interface between piezo-electric bi-materials. We will discuss the existence condition of the surface wave in this section. An explicit expression of the surface wave velocity will be given if the surface wave does exist. The second term of
(A.1), which represents the reflected wave, is used to study the surface wave at the interface. The reflected wave is expressed as follows p 2pIm R2 R1 okþ1 ot Hðt t1HDÞ; ð69Þ where
R1¼ ðeð2Þ15e ð1Þ 11 e ð1Þ 15e ð2Þ 11Þ 2 bðkþ 1Þ þ c ð1Þ 44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þaðk þ 1Þ þ c ð2Þ 44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þa ðkþ 1Þ; R2¼ ðeð2Þ15e ð1Þ 11 e ð1Þ 15e ð2Þ 11Þ 2 bðkþ 1Þ þ c ð1Þ 44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þaðk þ 1Þ c ð2Þ 44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þa ðkþ 1Þ; bðkþ1Þ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2 ðkþ 1Þ 2 q ; aðkþ1Þ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b21 ðkþ1Þ2 q ; aðkþ1Þ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b22 ðkþ1Þ2 q :
The surface wave at the piezoelectric bi-material interface can be constructed by setting the denominator of
(69)equal to zero R1¼ ðeð2Þ15e ð1Þ 11 e ð1Þ 15e ð2Þ 11Þ 2 bðkþ1Þ þ c ð1Þ 44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þaðk þ 1Þ þ c ð2Þ 44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þaðk þ 1Þ ¼ 0: ð70Þ
Note that,(70)is the same as the well-known velocity equation for the MT wave obtained byMaerfeld and Tournois (1971). The number of roots for (70) is determined by means of the principle of the argument (Achenbach, 1976). Let Gk(z) be analytic everywhere inside and on a simple closed curve Ck, except for a finite
number of poles inside Ckand Gk(z) has no zeros on Ck. Then
1 2pi Z Ck dGk dz dz GkðzÞ ¼ Zk Pk; ð71Þ
where z is a complex variable. Zkis the number of zeros inside Ckand Pkis the number of poles. The numbers
Zkand Pkinclude the orders of poles and zeros. It is convenient to rewrite(70)in the form
RðkÞ ¼ ðeð2Þ15eð1Þ11 eð1Þ15e11ð2ÞÞ2pffiffiffiffiffiffiffiffiffiffiffiffiffiffie2 k2þ cð1Þ 44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b21 k2 q þ cð2Þ44eð1Þ11eð2Þ11ðeð1Þ11 þ eð2Þ11Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b22 k2 q ¼ 0: ð72Þ In the complex k-plane, the function R(k) is rendered single-valued by introducing branch cuts. Now con-sider the contour Ckconsisting of Ct, Cland Cras indicated inFig. 3. Since the function R(k) clearly does not
have poles in the complex k-plane, the number of zeros within the contour Ck= Ct+ Cl+ Cris given by
Zk¼ 1 2pi Z Ck dR dk dk RðkÞ: ð73Þ
The counting of the number of zeros is carried out by mapping the k-plane on the v-plane through the relation v¼ RðkÞ; dv¼dRðkÞ
dk dk: ð74Þ
If Cvis the mapping of Ckin the v-plane, the integral(73)in the v-plane becomes
1 2pi Z Cv dv v ¼ Zk: ð75Þ
The integral in(75)has a simple pole at v = 0 and thus the value of Zkis simply the number of times the
image contour Cvencircles the origin in the v-plane in the counter-clockwise direction. To determine the
num-ber of zeros in the k-plane, we thus carefully trace the mapping of the contour Ckinto the v-plane. Since
R(k) = R(k), the images of Crand Clare the same and only one of them, say Cr, needs to be considered.
There are two cases to be discussed, that are b1> b2and b2> b1, as follows
Case (a): b1> b2(seeFig. 3).
At O point: RðeÞ ¼ cð1Þ44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi b21 e2 q þ cð2Þ44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi b22 e2 q : ð76Þ Along OA: RðkÞ ¼ cð1Þ44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b21 k2 q þ cð2Þ44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b22 k2 q ðeð2Þ15e ð1Þ 11 e ð1Þ 15e ð2Þ 11Þ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 e2 p h i i; ð77Þ
where the minus sign applies above the cut and the plus sign applies below the cut. Also, at A point: Rðb2Þ ¼ cð1Þ44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b21 b22 q ðeð2Þ15e ð1Þ 11 e ð1Þ 15e ð2Þ 11Þ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi b22 e2 q i: ð78Þ Along AB: RðkÞ ¼ cð1Þ44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b21 k2 q ðeð2Þ15e ð1Þ 11 e ð1Þ 15e ð2Þ 11Þ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 e2 p þ cð2Þ44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 b2 2 q i: ð79Þ At B point: Rðb1Þ ¼ ðe ð2Þ 15e ð1Þ 11 e ð1Þ 15e ð2Þ 11Þ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi b21 e2 q þcð2Þ44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b21 b2 2 q i: ð80Þ Forjkj is large, we find RðkÞ ¼ ½ðeð2Þ15e ð1Þ 11 e ð1Þ 15e ð2Þ 11Þ 2 þ eð1Þ11e ð2Þ 11ðc ð1Þ 44 þ c ð2Þ 44Þðe ð1Þ 11 þ e ð2Þ 11Þ ffiffiffiffiffiffiffiffi k2 p : ð81Þ
Finally, we find that the number of zeros for the function R(k) is mainly controlled by the positive or negative value of the bracket in(80). For the case that
ðeð2Þ15e ð1Þ 11 e ð1Þ 15e ð2Þ 11Þ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi b21 e2 q þ cð2Þ44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b21 b2 2 q >0; ð82Þ
as indicated inFig. 4(a), the contours Ct, Crand Clin the k-plane are mapped to the contours C0t, C 0 rand C
0 lin
the v-plane. The contour C0t encircles the origin (pole) counterclockwise in the v-plane and C0land C0rencircle the origin clockwise in the v-plane.
The contour of C0t is C0! D0! E0and E00! F00! C00, and Z
k¼ 2 12¼ 1.
The contour of C0ris B0! A0! O0! A00! B00, and Zk¼ 12.
The contour of C0lis the same as the route of C0r, and Zk¼ 12.
Hence the number of zeros is zero, i.e. Zk¼ 2 12 2 12¼ 0. For the case that
ðeð2Þ15e ð1Þ 11 e ð1Þ 15e ð2Þ 11Þ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi b21 e2 q þ cð2Þ44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b21 b22 q <0; ð83Þ
the contours C0t, C0rand C0lin the v-plane is indicated inFig. 4(b). The contour C0tencircles the origin counter-clockwise in the v-plane while C0l and C0rencircle the origin counterclockwise in the v-plane.
The contour of C0t is C0! D0! E0and E00! F00! C00, and Z
k¼ 2 12¼ 1.
The contour of C0r is B0! A0! O0! A00! B00, and Z k¼12.
The contour of C0lis the same as the route of C0r, and Zk¼12.
Hence the number of zeros for the function R(k) is two, i.e. Zk¼ 2 12þ 2 12¼ 2.
Case (b): b2> b1. At O point: RðeÞ ¼ cð1Þ44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi b21 e2 q þ cð2Þ44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi b22 e2 q : ð84Þ
Along OA: RðkÞ ¼ cð1Þ44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b21 k2 q þ cð2Þ44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b22 k2 q ðeð2Þ15e ð1Þ 11 e ð1Þ 15e ð2Þ 11Þ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 e2 p h i i; ð85Þ
where the minus sign applies above the cut and the plus sign applies below the cut. Also, at A point: Rðb1Þ ¼ cð2Þ44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b22 b2 1 q ðeð2Þ15eð1Þ11 eð1Þ15eð2Þ11Þ2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi b21 e2 q i: ð86Þ Along AB: RðkÞ ¼ cð2Þ44eð1Þ11eð2Þ11ðeð1Þ11 þ eð2Þ11Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b22 k2 q ðeð2Þ15e ð1Þ 11 e ð1Þ 15e ð2Þ 11Þ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 e2 p þ cð1Þ44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 b21 q i: ð87Þ At B point: Rðb2Þ ¼ ðe ð2Þ 15e ð1Þ 11 e ð1Þ 15e ð2Þ 11Þ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi b22 e2 q þ cð1Þ44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b22 b2 1 q i: ð88Þ
Forjkj is large, the result is the same as that indicated in(81). Finally, follow the similar discussion as given in case (a), we find that
ðeð2Þ15e ð1Þ 11 e ð1Þ 15e ð2Þ 11Þ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi b22 e2 q þ cð1Þ44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b22 b2 1 q >0; Zk¼ 0; ð89Þ and ðeð2Þ15e ð1Þ 11 e ð1Þ 15e ð2Þ 11Þ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi b22 e2 q þ cð1Þ44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b22 b2 1 q <0; Zk¼ 2: ð90Þ
We have shown that the condition for the existence of surface waves is indicated in (83) (or(90)). We will determine the velocity of the surface wave if it exists. Consider the problem that at time t = 0, a dynamic anti-plane loading is applied at the interface (x, y) = (0, 0) and the receiver is located at (x, y) = (x,0), then sin h1= 0 and cos h1= 1. We have
kþ1 ¼ t x
1 vs
; ð91Þ
where vsis the surface wave velocity. The solution of vsin(70)can be obtained and the surface wave velocity is
explicitly expressed as vs¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2D2 E2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E22 4D2F2 q v u u t ; ð92Þ where D2¼ ½A22 B 2 2; E2¼ ½B22ðb 2 1þ b 2 2Þ 2A2C2; F2¼ ½C22 B 2 2b 2 1b 2 2; A2¼ ½eð1Þ11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þ 2 ½ðcð1Þ44Þ 2 þ ðcð2Þ44Þ 2 ½eð2Þ15e ð1Þ 11 e ð1Þ 15e ð2Þ 11 4 ; B2¼ 2cð1Þ44c ð2Þ 44½e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þ 2 ; C2¼ ½eð1Þ11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þ 2 ½ðcð1Þ44Þ 2 b21þ ðcð2Þ44Þ 2 b22:
If the piezoelectric material (2) is reduced to an elastic material, which implies that the piezoelectric effect is neglected and the interface between piezoelectric and elastic bi-material is metallized. The surface wave at the interface between piezoelectric and elastic bi-material is also studied and the result is presented in
Appendix B.
If the surface of piezoelectric material (1) is covered with an infinitesimally thin perfect conducting film and the mechanical effect in material (2) is neglected. Then the scalar potential of the electric field is set to be zero on the surface. We have
cð2Þ44 ¼ 0; eð2Þ15 ¼ 0; eð2Þ11 ! 1: ð93Þ Substitution of(93)into(70)yields
aðkþ1Þ k2 ebðk þ 1Þ ¼ 0; ð94Þ where k2e¼ðe ð1Þ 15Þ 2
cð1Þ44eð1Þ11. By substituting(91)into(94), a simple velocity equation for the electromagneto-acoustic
sur-face wave is obtained as
vs¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cð1Þ44 qð1Þð1 k 4 eÞ s : ð95Þ
This surface wave velocity recovers the classic BG wave solution. If the surface of piezoelectric material (1) is a free surface, which is in contact with a vacuum half-space on the top, we have
cð2Þ44 ¼ 0; eð2Þ15 ¼ 0; eð2Þ11 ¼ e0: ð96Þ
Substitution of(96)into(70)yields aðkþ1Þ k2 vbðk þ 1Þ ¼ 0; ð97Þ where k2v ¼ ðe ð1Þ 15Þ 2e 0 cð1Þ44eð1Þ11ðeð1Þ11þe0Þ
. Then the electromagneto-acoustic surface wave has the velocity as
vs¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cð1Þ44 qð1Þð1 k 4 vÞ s : ð98Þ
Once again, we recover the classic result of the classic BG wave solution.
4. Numerical results
With the explicit transient solutions at hand, the numerical calculation of transient response for pie-zoelectric bi-material subjected to a dynamic anti-plane concentrated force or electric charge is investi-gated in detail. The correspondent static solutions of the same problem are expressed in Appendix C. At time t/b1r2= 0, the loading is applied suddenly at (x, y) = (0, d) in piezoelectric material (1) as shown
in Fig. 1. Table 1lists the commonly used piezoelectric material properties and correspondent shear wave velocities. There are two cases of piezoelectric bi-material to be considered in this study. One has faster shear wave velocity in material (2) and material (1)–material (2) is PZT4–ZnO. The pattern of wave fronts is shown in Fig. 5 (points A and B are receivers). The transient responses of shear stresses and
Table 1
The material properties and shear wave speeds of piezoelectric materials
c44(1010N/m2) e15(C/m) e11(109F/m) q(kg/m3) 1/b (m/s)
ZnO 4.25 0.48 0.0757 5676 2832.65
PZT4 2.56 12.70 6.4634 7500 2596.26
CdS 1.49 0.21 0.0799 4824 1789.73
electric displacements for two receivers are presented in Figs. 6–9. The other case has slower shear wave velocity in material (2) and material (1)–material (2) is PZT4–CdS. Similarly, the pattern of wave fronts is shown in Fig. 10 and the transient responses of shear stresses and electric displacements are presented in Figs. 11–14. The waves shown in Figs. 5–14 are composed of incident wave, reflected wave, refracted wave, head wave, elastic wave and electromagnetic wave and are denoted by i, r, f, h, e and p, respec-tively. For instance, symbol of rpe represents the elastic wave reflected from the interface by the
electro-magnetic incident wave and symbol of hfee represents the elastic head wave refracted from the interface by the elastic incident wave. It is shown in Figs. 5 and 10 that the wave fronts of head wave are inclined ðhr
eeÞ and horizontal ðh r ep, h
f
epÞ straight lines for PZT4–ZnO but there is no head wave h r
ee for PZT4–CdS.
The time presented in the transient response curves has been normalized by dividing by b1r2. The arrival
time for each wave front and the corresponding static values are also indicated in these figures. It is found that the transient responses of shear stresses and electric displacements tend to static value after the arrival of the last wave.
InFig. 6(a), a small illustration window on the right hand side presents the transient response of sð1Þ yz at time
t/b1r2= 0–1. The repwave is the first wave arrival at the receiver A at the normalized time equal to 0.5. This
wave propagates toward the interface with the elastic wave speed and then travels with the electromagnetic wave velocity after reflected from the interface. It is shown clearly inFig. 6(a) that the stress field behaves with a square root singularity at the incident ieand reflected reewave fronts. However, the contribution of shear
stress from repand hrepwaves is relatively small. InFig. 6(b), we can see that the only contribution of the
elec-tric displacement is due to the repwave which is indicated in(A.2). InFig. 7, the receiver B is located in
mate-rial (2) and the arrival time of feeis less than unity because ZnO has faster shear wave velocity than PZT4. In Fig. 5. The pattern of wave fronts for PZT4–ZnO subjected to a dynamic anti-plane concentrated force.
order to see the transient response induced by electric loading, a dynamic electric charge in piezoelectric bi-material is considered. InFig. 8, the transient behavior of shear stress and electric displacement in material (1) are presented and we can see that more waves are generated as compared with Fig. 6.Fig. 9 represent the transient response for material (2). Consequently, it is worthy to note that all kinds of elastic and electro-magnetic waves in piezoelectric bi-material can be generated by applying a dynamic electric charge. For the piezoelectric bi-material is PZT4–CdS, the wave fronts (Fig. 10) for ieand reeare circular curves, while feewave
front is a smooth curve which is constructed by numerical calculations. The transient phenomena inFigs. 11– 14have similar features as that indicated inFigs. 6–9.
Fig. 6. The transient responses of (a) sð1Þ
yz and (b) D ð1Þ
The transient solution for applying a dynamic anti-plane concentrated force in a piezoelectric and elastic bi-material can be reduced from the solution for piezoelectric bi-bi-materials. The transient responses of shear stres-ses for PZT4–Aluminum alloy 2014-T6 and PZT4–Bronze are indicated inFig. 15 and 16, respectively. Sim-ilarly, the transient solutions are shown to approach the correspondent static solutions as time increases. Compared with piezoelectric bi-material, the main distinction is that sð2Þ
yz does not have the wave of fepowing
to non-piezoelectricity in material (2). Basically, the transient phenomena as presented in these figures have similar characteristics to those shown inFigs. 6–9.
The existence of surface wave for piezoelectric bi-material for the commonly used material indicated in
Table 1is examined. From the existence condition indicated in(83)(or(90)), it is found that there is no surface
Fig. 7. The transient responses of (a) sð2Þ
yz and (b) D ð2Þ
wave for any combination of piezoelectric materials listed inTable 1. However, we can not rule out the pos-sibility that there is a surface wave propagating along the interface between piezoelectric bi-material. Hence, we try to investigate if there exist ranges of piezoelectric properties inTable 1that a surface wave can prop-agate along the piezoelectric bi-material interface. The virtual piezoelectric material properties are obtained by changing the elastic constant of c44to satisfy(83)(or(90)) and the positive definite conditions. For case (a)
(b1> b2), the bi-material is PZT4–virtual ZnO, all the material properties are set to be fixed except c44 for Fig. 8. The transient responses of (a) sð1Þ
yz and (b) D ð1Þ
ZnO, we find that the existence condition (83) is satisfied only in the range 3:53 1010 N=m26cð2Þ 44 63:58 10 10 N=m2(i.e., c 44for ZnO is 4.25· 10 10
N/m2). The correspondent shear wave speed for virtual ZnO is 2599.11 m/s 6 1/b262616 m/s. The velocity of the surface wave can be obtained
from(92)and the result is 2593 m/s 6 vs62596.19 m/s. The shear wave speed for PZT4 is 2596.26 m/s as
indi-cated inTable 1. We can see that the existence of the surface wave only at the situation that the shear wave speed of two materials is close. Furthermore, the surface wave velocity is close to and less than the slower shear wave speed (i.e., PZT4) of the two materials. For case (b) (b2> b1), the bi-material is virtual BaTiO3–
Fig. 9. The transient responses of (a) sð2Þ
ZnO and we change the value of c44for BaTiO3. It is found that the existence condition of surface wave is
satisfied only in the range 3:26 1010N=m26cð1Þ
44 63:29 10 10
N=m2 (i.e., c
44 for BaTiO3is 4.4· 1010N/
m2) and the correspondent shear wave speed for virtual BaTiO3 is 2833.52 m/s 6 1/b162842.79 m/s. The
velocity of the surface wave is 2830.04 m/s 6 vs62832.58 m/s. Similar phenomenon in case (a) is found in
case (b). The surface wave velocity is close to and less than the slow shear wave speed (i.e., 2832.65 m/s for ZnO).
Table 2 lists the surface wave velocity vs (if it exists) of the piezoelectric–elastic bi-material which is
composed of PZT4 (material (1)) and elastic materials (material (2)). It is interesting to note that the surface wave does exist in the interface of ZnO and nine elastic materials. For example, the surface wave velocity of the bi-material PZT4–Glass is 2595.51 m/s which is slightly less than the shear wave speed of PZT4 (2596.26 m/s). If we change only the elastic constant c44 of PZT4 and to find the range of c44 such
that the surface wave exists, then, we find that there will have a surface wave along the interface of PZT4–Glass bi-material for 2:12 1010
N=m26cð1Þ
44 66:87 10 10
N=m2 and the surface wave velocity is
2480.71 m/s 6 vs63467.1 m/s. For the other case that the elastic material is Bronze, the shear wave
speed (2316.43 m/s) is less than that of PZT4. The surface wave velocity is found to be 2313.96 m/s which is slightly less than the shear wave speed of Bronze. The surface wave of PZT4–Bronze exists if c44 of PZT4 is in the range of 1:53 1010N=m26cð1Þ44 62:72 10
10
N=m2 and the correspondent
sur-face wave velocity is 2212.64 m/s 6 vs62316.43 m/s. We can see from both cases that the range of cð1Þ44
for the existence of the surface wave is much larger than that of piezoelectric bi-materials discussed previously.
For the case that the shear wave speed of the elastic material is larger than that of the piezoelectric material, i.e., b1> b02, the existence condition of the surface wave can be expressed as
1 ðe ð1Þ 15Þ 4 ðleð1Þ11Þ 2 " # lqð1Þ qð2Þ ðeð1Þ15Þ 2 eð1Þ11 < c ð1Þ 44 < lqð1Þ qð2Þ ðeð1Þ15Þ 2 eð1Þ11 and c ð1Þ 44 >0: ð99Þ
However, if the shear wave speed of the elastic material is less than that of the piezoelectric material, i.e., b1< b02, then the condition for the existence of the surface wave becomes
Fig. 11. The transient responses of (a) sð1Þ
lqð1Þ qð2Þ ðeð1Þ15Þ 2 eð1Þ11 < c ð1Þ 44 < lqð1Þ 2qð2Þþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lqð1Þ 2qð2Þ 2 þ ðe ð1Þ 15Þ 2 eð1Þ11 !2 v u u t ðeð1Þ15Þ 2 eð1Þ11 : ð100Þ
After the detailed numerical investigation of the existence of the surface wave for piezoelectric–piezoelectric bi-materials and piezoelectric–elastic bi-materials, it can be concluded that the surface wave velocity is always less than the slower shear wave speed of the two materials. Furthermore, the existence of the surface wave for piezoelectric–piezoelectric bi-materials is restricted to the situation that the shear waves of the two
piezoelec-Fig. 12. The transient responses of (a) sð2Þ
yz and (b) D ð2Þ
tric materials are very close. However, the possibility for the existence of the surface wave for piezoelectric– elastic bi-materials is much greater than that of the piezoelectric–piezoelectric bi-materials. In order to present the contribution of surface wave, a material point near the interface between PZT4 and Aluminum alloy 2014-T6 is chosen for investigation. The transient response of a short time interval for the arrival of the surface
Fig. 13. The transient responses of (a) sð1Þ
wave is shown inFig. 17. Due to the influence of the surface wave, a large variation of shear stress is found in
Fig. 17.
5. Conclusion
In this paper, a general methodology is proposed to construct the full-field transient solutions of pie-zoelectric bi-materials subjected to a dynamic anti-plane concentrated force and a dynamic electric charge.
Fig. 14. The transient responses of (a) sð2Þ
The Cagniard-de Hoop method is used to construct the transient solutions in time domain. The analytical results obtained in this study are exact and are expressed in explicit closed forms, each term representing a physical transient wave. The corresponding static solutions are also obtained in this study. The dynamic response of shear stress in the transient period is much larger than that of the static value. In the transient
Fig. 15. The transient responses of (a) sð1Þ
yz and (b) sð2Þyz for PZT4–Aluminum alloy 2014-T6 subjected to a dynamic anti-plane concentrated force.
period, the stress will change radically when the reflected or the refracted wave arrives. The transient value of shear stress will tend toward the static value after the last wave has passed. The existence condition of a surface wave propagating along piezoelectric bi-material interface is established in this study. The veloc-ity of surface wave for the fully-coupled SH electromagneto-acoustic surface wave is obtained in a simple, closed form. It is found in this study that the surface wave velocity is always close to and less than the
Fig. 16. The transient responses of (a) sð1Þ
slower shear wave speed of the two materials. Furthermore, the existence of the surface wave for electric–piezoelectric bi-materials is restricted to the situation that the shear wave speed of the two piezo-electric bi-materials is close.
Acknowledgement
The financial support of the authors from the National Science Council, Republic of China, through Grant NSC 92-2212-E002-074 to National Taiwan University is gratefully acknowledged.
Table 2
The material properties of elastic materials and the correspondent surface wave velocities
l(1010N/m2) q(kg/m3) 1/b (m/s) v s(m/s) Aluminum alloy 2014-T6 2.8 2800 3162.28 2559.25 Aluminum alloy 7075-T6 2.7 2800 3105.30 2549.59 Aluminum alloy 6061-T6 2.6 2700 3103.16 2544.27 Brass 4.1 8400 2209.29 – Bronze 4.4 8200 2316.43 2313.96 Cast iron 6.9 7200 3095.70 –
Copper and copper alloys 4.7 8900 2298.02 2297.00
Glass 3.5 2800 3535.53 2595.51 Magnesium alloys 1.7 1830 3047.89 2478.38 Monel 6.6 8800 2738.61 2590.00 Nickel 8.0 8800 3015.11 – Rubber 0.0001 1300 27.74 – Steel 7.5 7850 3090.98 – Titanium alloys 3.9 4500 2943.92 2577.25 Tungsten 14.0 1900 8583.95 –
Fig. 17. The transient response of sð1Þ
yz at the interface for PZT4–Aluminum alloy 2014-T6 subjected to a dynamic anti-plane concentrated force.
Appendix A sð1Þyz ¼ p 2pIm okþ2 ot Hðt b1r2Þ þ p 2pIm ðeð2Þ15e ð1Þ 11 e ð1Þ 15e ð2Þ 11Þ 2 bðkþ1Þ þ c ð1Þ 44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þaðk þ 1Þ cð2Þ44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þaðk þ 1Þ ! ðeð2Þ15e ð1Þ 11 e ð1Þ 15e ð2Þ 11Þ 2 bðkþ1Þ þ c ð1Þ 44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þaðk þ 1Þ þcð2Þ44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þaðk þ 1Þ !ok þ 1 ot 2 6 6 6 6 4 3 7 7 7 7 5Hðt t1HDÞ þ p 2pIm 2eð1Þ15eð2Þ11ðeð2Þ15eð1Þ11 eð1Þ15eð2Þ11Þbðkþ 3Þ
ðeð2Þ15eð1Þ11 eð1Þ15eð2Þ11Þ2bðkþ3Þ þ cð1Þ44eð1Þ11eð2Þ11ðeð1Þ11 þ eð2Þ11Þaðkþ3Þ þcð2Þ44eð1Þ11eð2Þ11ðeð1Þ11 þ eð2Þ11Þaðkþ 3Þ !ok þ 3 ot 2 6 6 6 6 4 3 7 7 7 7 5Hðt t3Þ; ðA:1Þ Dð1Þy ¼p 2pIm 2eð1Þ11e ð2Þ 11ðe ð2Þ 15e ð1Þ 11 e ð1Þ 15e ð2Þ 11Þbðk þ 3Þ ðeð2Þ15e ð1Þ 11 e ð1Þ 15e ð2Þ 11Þ 2 bðkþ3Þ þ c ð1Þ 44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þaðk þ 3Þ þcð2Þ44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þaðk þ 3Þ !ok þ 3 ot 2 6 6 6 6 4 3 7 7 7 7 5Hðt t3Þ; ðA:2Þ sð1Þ xz ¼ p 2pIm kþ2 aðkþ2Þ okþ2 ot Hðt b1r2Þ þ p 2pIm kþ1 ðe ð2Þ 15e ð1Þ 11 e ð1Þ 15e ð2Þ 11Þ 2 bðkþ1Þ þ c ð1Þ 44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þaðk þ 1Þ cð2Þ44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þaðk þ 1Þ ! aðkþ1Þ ðeð2Þ15e ð1Þ 11 e ð1Þ 15e ð2Þ 11Þ 2 bðkþ1Þ þ c ð1Þ 44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þaðk þ 1Þ þcð2Þ44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þaðk þ 1Þ !ok þ 1 ot 2 6 6 6 6 4 3 7 7 7 7 5Hðt t1HDÞ þ p 2pIm 2eð1Þ15eð2Þ11ðeð2Þ15e ð1Þ 11 e ð1Þ 15e ð2Þ 11Þk þ 3 ðeð2Þ15e ð1Þ 11 e ð1Þ 15e ð2Þ 11Þ 2 bðkþ3Þ þ c ð1Þ 44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þaðk þ 3Þ þcð2Þ44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þaðk þ 3Þ !ok þ 3 ot 2 6 6 6 6 4 3 7 7 7 7 5Hðt t3Þ; ðA:3Þ Dð1Þx ¼ p 2pIm 2eð1Þ11e ð2Þ 11ðe ð2Þ 15e ð1Þ 11 e ð1Þ 15e ð2Þ 11Þk þ 3 ðeð2Þ15e ð1Þ 11 e ð1Þ 15e ð2Þ 11Þ 2 bðkþ3Þ þ c ð1Þ 44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þaðk þ 3Þ þcð2Þ44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þaðk þ 3Þ !ok þ 3 ot 2 6 6 6 6 4 3 7 7 7 7 5Hðt t3Þ; ðA:4Þ sð2Þyz ¼p 2pIm 2cð2Þ44eð1Þ11eð2Þ11ðeð1Þ11 þ e ð2Þ 11Þaðk þ 5Þ ðeð2Þ15e ð1Þ 11 e ð1Þ 15e ð2Þ 11Þ 2 bðkþ5Þ þ c ð1Þ 44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þaðk þ 5Þ þcð2Þ44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þaðk þ 5Þ !ok þ 5 ot 2 6 6 6 6 4 3 7 7 7 7 5Hðt t5HDÞ p 2pIm 2eð2Þ15eð1Þ11ðeð1Þ15e ð2Þ 11 e ð2Þ 15e ð1Þ 11Þbðk þ 6Þ
ðeð2Þ15eð1Þ11 eð1Þ15eð2Þ11Þ2bðkþ6Þ þ cð1Þ44eð1Þ11eð2Þ11ðeð1Þ11 þ eð2Þ11Þaðkþ6Þ þcð2Þ44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þaðk þ 6Þ !ok þ 6 ot 2 6 6 6 6 4 3 7 7 7 7 5Hðt t6Þ; ðA:5Þ
Dð2Þy ¼p 2pIm 2eð1Þ11e ð2Þ 11ðe ð2Þ 15e ð1Þ 11 e ð1Þ 15e ð2Þ 11Þbðk þ 6Þ ðeð2Þ15e ð1Þ 11 e ð1Þ 15e ð2Þ 11Þ 2 bðkþ6Þ þ c ð1Þ 44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þaðk þ 6Þ þcð2Þ44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þaðk þ 6Þ !ok þ 6 ot 2 6 6 6 6 4 3 7 7 7 7 5Hðt t6Þ; ðA:6Þ sð2Þ xz ¼ p 2pIm 2cð2Þ44eð1Þ11e11ð2Þðeð1Þ11 þ eð2Þ11Þkþ 5
ðeð2Þ15eð1Þ11 eð1Þ15eð2Þ11Þ2bðkþ5Þ þ cð1Þ44eð1Þ11eð2Þ11ðeð1Þ11 þ eð2Þ11Þaðkþ5Þ þcð2Þ44eð1Þ11eð2Þ11ðeð1Þ11 þ eð2Þ11Þaðkþ 5Þ !ok þ 5 ot 2 6 6 6 6 4 3 7 7 7 7 5Hðt t5HDÞ p 2pIm 2eð2Þ15eð1Þ11ðeð1Þ15eð2Þ11 eð2Þ15eð1Þ11Þkþ6 ðeð2Þ15e ð1Þ 11 e ð1Þ 15e ð2Þ 11Þ 2 bðkþ6Þ þ cð1Þ44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þaðk þ 6Þ þcð2Þ44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þaðk þ 6Þ !ok þ 6 ot 2 6 6 6 6 4 3 7 7 7 7 5Hðt t6Þ; ðA:7Þ Dð2Þx ¼p 2pIm 2eð1Þ11eð2Þ11ðeð2Þ15e ð1Þ 11 e ð1Þ 15e ð2Þ 11Þk þ 6 ðeð2Þ15e ð1Þ 11 e ð1Þ 15e ð2Þ 11Þ 2 bðkþ6Þ þ c ð1Þ 44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þaðk þ 6Þ þcð2Þ44e ð1Þ 11e ð2Þ 11ðe ð1Þ 11 þ e ð2Þ 11Þaðk þ 6Þ !ok þ 6 ot 2 6 6 6 6 4 3 7 7 7 7 5Hðt t6Þ: ðA:8Þ Appendix B
The existence condition of a surface wave propagating along the piezoelectric–elastic bi-material interface is analyzed in this appendix. Substitution of(55)–(57)into(A.1)yields the reflected wave subjected to a dynamic anti-plane concentrated force as follows
p 2pIm R4 R3 okþ1 ot Hðt t1HDÞ; ðB:1Þ where R3¼ ðeð1Þ15Þ 2 bðkþ1Þ þ c ð1Þ 44e ð1Þ 11aðk þ 1Þ þ le ð1Þ 11aðk þ 1Þ; R4¼ ðeð1Þ15Þ 2 bðkþ1Þ þ c ð1Þ 44e ð1Þ 11aðk þ 1Þ le ð1Þ 11aðk þ 1Þ; b02¼ ffiffiffiffiffiffiffi qð2Þ l s ; aðkþ1Þ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b022 ðkþ1Þ 2 q :
Considering the denominator of(B.1), let R3¼ ðeð1Þ15Þ 2 bðkþ1Þ þ cð1Þ44e ð1Þ 11aðk þ 1Þ þ le ð1Þ 11aðk þ 1Þ ¼ 0: ðB:2Þ
The principle of argument in Section III is applied to (B.2)and rewrite (B.2)in the form R0ðkÞ ¼ ðeð1Þ15Þ2pffiffiffiffiffiffiffiffiffiffiffiffiffiffie2 k2þ cð1Þ 44e ð1Þ 11 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b21 k2 q þ leð1Þ11 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b022 k2 q ¼ 0: ðB:3Þ
Two cases, i.e., b1> b02and b 0
2> b1, are discussed as follows Case (I): b1> b02. We find that if the condition
ðeð1Þ15Þ2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi b21 e2 q þ leð1Þ11 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b21 b02 2 q >0 ðB:4Þ
is satisfied, then the contours C0
t, C0rand C0lin the v-plane is the same as that indicated inFig. 4(a). The number
of zeros for the function R0(k) is zero, i.e., Z
k¼ 2 12 2 12¼ 0. For the condition
ðeð1Þ15Þ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi b21 e2 q þ leð1Þ11 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b21 b022 q <0; ðB:5Þ
the contours C0
t, C0rand C0lin the v-plane is indicated inFig. 4(b) and the number of zeros for the function R0(k)
is two, i.e. Zk¼ 2 12þ 2 12¼ 2. Case (II): b02> b1. We find that
ðeð1Þ15Þ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b022 e2 q þ cð1Þ44e ð1Þ 11 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b022 b2 1 q >0; Zk¼ 0; ðB:6Þ and ðeð1Þ15Þ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b022 e2 q þ cð1Þ44e ð1Þ 11 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b022 b2 1 q <0; Zk¼ 2: ðB:7Þ
We have shown that the constraint of(B.5)(or(B.7)) is the existence condition of the surface wave. The sur-face wave velocity can be explicitly expressed as
vs¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2A4 B4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B2 4 4A4C4 q v u u t ; ðB:8Þ where A4¼ ð1 þ g2 k4eÞ 2 4g2; B4¼ 2ðb21þ g 2b2 2Þð1 þ g 2 k4 eÞ þ 4g 2ðb2 1þ b 2 2Þ; C4¼ b41 2g 2b2 1b 2 2þ g 4b4 2; g¼ l cð1Þ44 :
Appendix C. The static solutions for piezoelectric bi-materials
C.1. The static solutions for applying an anti-plane concentrated force
As indicated inFig. 1, the material (1) is subjected to a static anti-plane concentrated force with magnitude p applied at x = 0 and y = d. The governing equations for the static problem are
r2w¼ 0; ðC:1Þ
r2U¼ 0: ðC:2Þ
The constitutive equations are the same as that expressed in(13)–(16). The jump condition is sð1yzþÞjy¼d sð1Þ
yz jy¼d ¼ pdðxÞ: ðC:3Þ
The continuous conditions are presented in(25)–(31). This problem can be solved by the application of the Fourier transform. The Fourier transform on the spatial variable x for(C.1) and (C.2)can be represented of the form d2w~ dy2 x 2w~ ¼ 0; ðC:4Þ d2U~ dy2 x 2~ U¼ 0; ðC:5Þ
where x is the Fourier transform parameter and the overwave symbol is used to denote the transform on the spatial variable x. The general solutions of(C.4) and (C.5)are
~ wð1þÞ ~ Uð1þÞ " # ¼ exy A3 C3 þ exy B3 D3 ; ðC:6Þ
~ wð1Þ ~ Uð1Þ " # ¼ exy E3 G3 þ exy F3 H3 ; ðC:7Þ ~ wð2Þ ~ Uð2Þ " # ¼ exy I3 K3 þ exy J3 L3 ; ðC:8Þ
The static solutions in the transform domain are ~ wð1Þ ~ Uð1Þ " # ¼ 1 2x2ðe xjydjIþ exðyþdÞRÞ ^^ M1Z; ðC:9Þ ~ wð2Þ ~ Uð2Þ " # ¼ 1 2x2e xðydÞT ^^M1Z; ðC:10Þ where ^ R¼ ð ^Mþ ^NÞ1ð ^M ^NÞ; T^ ¼ ð ^Mþ ^NÞ1ð ^M ^NÞ þ I; M^ ¼ c ð1Þ 44 e ð1Þ 15 eð1Þ15 eð1Þ11 " # ; N^ ¼ c ð2Þ 44 e ð2Þ 15 eð2Þ15 eð2Þ11 " # :
After the Fourier inversion transform is employed, the static solutions for displacement, shear stresses and electric displacements are
wsð1Þ Usð1Þ " # ¼ 1 2pð ^R ^M 1Z ln r 1þ ^M1Z ln r2Þ; ðC:11Þ wsð2Þ Usð2Þ " # ¼ 1 2p ^ T ^M1Z ln r2; ðC:12Þ ssð1Þ yz Dsð1Þy " # ¼ 1 2p ^ M ^R ^M1Zyþ d r2 1 þ Zy d r2 2 ; ðC:13Þ ssð1Þ xz Dsð1Þx " # ¼ 1 2p ^ M ^R ^M1Zx r2 1 þ Zx r2 2 ; ðC:14Þ ssð2Þ yz Dsð2Þy " # ¼ 1 2pN^^T ^M 1Zy d r2 2 ; ðC:15Þ ssð2Þ xz Dsð2Þx " # ¼ 1 2p ^ N^T ^M1Zx r2 2 : ðC:16Þ
C.2. The static solutions for applying an electric charge The jump condition is
Dð1yþÞjy¼d Dð1Þ
y jy¼d ¼ qdðxÞ: ðC:17Þ
The static solutions for displacement, shear stresses and electric displacements in piezoelectric materials (1) and (2) are presented as follows
wsð1Þ Usð1Þ " # ¼ 1 2pð ^R ^M 1G ln r 1þ ^M1G ln r2Þ; ðC:18Þ
wsð2Þ Usð2Þ " # ¼ 1 2pT ^^M 1G ln r 2; ðC:19Þ ssð1Þ yz Dsð1Þy " # ¼ 1 2p ^ M ^R ^M1Gyþ d r2 1 þ Gy d r2 2 ; ðC:20Þ ssð1Þ xz Dsð1Þx " # ¼ 1 2p ^ M ^R ^M1Gx r2 1 þ G x r2 2 ; ðC:21Þ ssð2Þ yz Dsð2Þy " # ¼ 1 2p ^ N^T ^M1Gy d r2 2 ; ðC:22Þ ssð2Þ xz Dsð2Þ x " # ¼ 1 2p ^ N^T ^M1Gx r2 2 : ðC:23Þ References
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![Fig. 4. The v-plane for (a) Im[R(b 1 )] > 0 and (b) Im[R(b 1 )] < 0.](https://thumb-ap.123doks.com/thumbv2/9libinfo/8844530.240046/13.816.272.556.92.788/fig-the-plane-im-r-gt-and-im.webp)
