Further analysis of the transient dynamic response of piezoelectric bodies subjected to electric impulses

International Journal of Solids and Structures 38 (2001) 2101±2114

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Further analysis of the transient dynamic response of piezoelectric bodies subjected to electric impulses Horacio Sosa *, Naum Khutoryansky Department of Mechanical Engineering and Mechanics, Drexel University, 32nd Chestnut Streets, Philadelphia, PA 19104, USA Received 12 August 1999; in revised form 10 December 1999

Abstract Electric and acoustic transient waves induced by an impulse point charge are represented and studied by means of dynamic Green's functions and their derivatives. Numerical results using such functions are generated to illustrate important features associated with the electroacoustic waves. In particular, the attenuation characteristics of the waves with distance from the source are analyzed for a class of material and electric load. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Dynamic response; Piezoelectric bodies; Electric impulses

1. Introduction Electromechanical vibration and wave phenomena in polarizable dielectrics, such as piezoelectric crystals and ceramics, is a fairly well-established ®eld of study, provided the excitations are of harmonic nature. In fact, such topics are treated in texts such as those of Tiersten (1969), Nelson (1979) and Ikeda (1990). Among many other important features, these treatises place emphasis on the fact that if an acoustic plane wave propagates in a piezoelectric solid, the acoustic wave drives the electric ®eld wave through the piezoelectric e€ect. This depolarizing electric ®eld (also known as the piezoelectrically induced ®eld) then a€ects the propagation of the acoustic wave that generates it. More speci®cally, by considering harmonic wave propagation of the form exp‰i…xt ÿ kn
x†Š (with x, k and n being the angular frequency, the propagation constant and unit vector in the direction of propagation, respectively) it can be shown that the induced ®eld is a longitudinal propagating wave, of magnitude proportional to the piezoelectric constants e and inversely proportional to the dielectric constant  in the n direction. Furthermore, it can be shown that such piezoelectric coupling leads to a sti€ening of the elasticity tensor (provided the elastic constant is measured at constant ®eld) and thus increases the velocity of propagation of acoustic waves. In other words, if C is the original elastic sti€ness of the material, the new sti€ness is given by C ˆ C ‡ e2 =.

*

Corresponding author. Tel.: +1-215-895-1542; fax: +1-215-895-1478. E-mail address: [email protected] (H. Sosa).

0020-7683/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 6 8 3 ( 0 0 ) 0 0 1 5 5 - 4

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Information of this nature is important because it allows designers of electromechanical devices to understand better and predict material response, thus, minimizing the risks of electrical and mechanical failure. A much less explored phenomenon, yet extremely important from a practical point of view, has to do with the transient dynamic response of piezo-active materials as a result of concentrated impulsive electric voltages or charges. Such loading event has been observed to arise accidentally in phase change transducers, pulse generators for igniters, high voltage transformers and insulators, high power lasers, and more recently, in spark plugs for combustion engines. While the design of such devices is based on known dielectric breakdown limits under nominal loading conditions, the nature of the applications of these devices tend to involve extremely high electrical peaks that take place randomly during short periods of time. For example, high-voltage transformers are designed to operate at approximately 400 kV. However, the voltages extracted from the power line can under certain circumstances reach peak values of more than 1000 kV. The actual situation is even more serious because the above numbers are based on the assumption of a perfect material. Most ceramics are not exempted from defects such as voids, cracks and impurities or discontinuities such as embedded electrodes (Furuta and Uchino, 1993); thus, in the neighborhood of such inhomogeneities, the ®elds are even much higher and the consequences far less controllable. Moreover, in some applications, the applied impulsive loads don't even need to be of large magnitude. This is particularly true in microelectronics and thin ®lm capacitors. Indeed, in piezoelectricity, the deformation c is proportional to the ®eld E, and if h is a characteristic length, both E and c are proportional to hÿ1 . Hence, since h  1 large strains and ®elds can be induced and initiate breakdown. In short, acoustic and electric transient disturbances are factors that should be considered in an electro-elastic model if the aim is to design increasingly robust devices. The present work is a sequel to our investigations on the subject of dynamic piezoelectricity (Khutoryansky and Sosa, 1995; Sosa and Khutoryansky, 1999). The article builds upon our previous results on Green's functions, reformulated here in a more convenient mathematical form not published before, which render a more direct interpretation of the results obtained with their use. These expressions are then applied to study the particular case of a concentrated charge that can vary very sharply with time in an arbitrary manner. The main feature of the article resides on the numerical results obtained by using the dynamic Green's functions and corresponding derivatives. Emphasis is placed on the attenuation characteristics of stress and electric ®eld waves as these move away from the impulsive source. It is shown numerically that if x is the distance from the source, such attenuation does not follow the 1=x2 pattern predicted by the static solution. In other words, we show that transient dynamic e€ects can still have a signi®cant impact on the material even at points in space remotely located from the source, and that a 1=x behavior is more representative of dynamic attenuation. The importance of these numerical results is reinforced by the fact that due to the complexity of the phenomena involved in electroelastic coupled problems, it is not possible to predict the aforementioned behavior by simply looking at the mathematical expressions derived in this article. Other characteristics such as in¯uence of the material constants and duration of the applied pulse are also considered. In this paper, the body under consideration is in®nite, homogeneous, electrically polarizable and nonconducting. Its material anisotropy is arbitrary, although numerical examples are provided for a transversely isotropic ceramic. Both direct and index notations are used, with the summation convention implied over repeated lower case Latin sub-indices. With a few exceptions lower and upper case Latin fonts will denote vectors and higher order tensors, respectively. With the choice of a Cartesian basis (say e1 , e2 , e3 ) these objects will also be referred to as column (or row) vectors and matrices, respectively. Subscript commas and dots over variables indicate di€erentiation with respect to the spatial variable x (or xi ) and the time variable t. Finally, r
and r refer to the operations of divergence and curl and to the tensor product of two vectors.

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2. The solution of the dynamic problem The study of elastic (acoustic) waves interacting with an electric ®eld in polarizable bodies, or conversely, the study of electromagnetic waves accompanied by mechanical deformations makes use of a combination of elastodynamics and electromagnetism. Speci®cally, we make use of the equation of motion for the stress T and of Maxwell's equations in the quasi-electrostatic approximation version (r
D ˆ q, r  E ˆ 0, with D, E and q the electric displacement, ®eld and charge, respectively). Such an approximation (which neglects the magnetic ®eld altogether) turns out to be quite reasonable for the wave problems mentioned in the Section 1. Additionally, making use of linear constitutive piezoelectric relations, the dynamic electromechanical problem, coupling displacement u with electric potential /, is described by ui ; Cijkl uk;lj ‡ ekij /;kj ‡ bi ˆ q

…1†

eikl uk;li ÿ ik /;ki ˆ q;

where Cijkl , eijk and ij are the elastic (measured at constant electric ®eld), piezoelectric and permittivity (measured at constant strain) constants, respectively. Eq. (1) indicates that the sound in a piezoelectric medium is accompanied by an electric ®eld through the piezoelectric e€ect and vice versa. If ekij is set equal to zero mechanical and electrical e€ects decouple and known results from elastodynamic and electrostatic theories are recovered. As mentioned in Section 1, harmonic solutions of Eq. (1) for particular cases in terms of spatial dimensions, boundary and initial conditions and material symmetry have been extensively discussed by many authors (refer, for example, to the already mentioned monographs). When ®elds switch sharply in time the solution to Eq. (1) becomes more dicult to obtain. To this end, a method based on Green's functions was developed by the authors (Khutoryansky and Sosa, 1995) which captures transient dynamic e€ects produced by point sources of mechanical and electrical nature that may also vary very sharply in time. Here we shall not repeat, the mathematical developments leading to the representation of the dynamic electroelastic solution. Instead, we will outline the relevant steps that yield the ®nal results, with an emphasis on the physical signi®cance of the methods involved in each representation. The results, however, are recast in this article in a more compact manner for the particular case of the electric charge. In passing, we note that Green's functions for piezoelectric materials have also been addressed by Chen (1993), Chen and Lin (1993), Lee and Jiang (1994) and Norris (1994). We assume that at the origin of a ®xed Cartesian coordinate system there exists a unit impulsive point charge q…x; t† ˆ d…t†d…x†

…2†

with d…x† ˆ d…x1 †d…x2 †d…x3 † being the three-dimensional delta function. Next, we introduce the notation     u 0 ; …3† Uˆ  ; F0 ˆ ÿ1 / where u and / are the fundamental solutions (or Green's functions) for the displacement and electric potential, respectively. Furthermore, we de®ne the di€erential operator



qo2 0 A a t

;

…4† ÿ L…r; ot † ˆ 0 0 aT ÿa where A…r†, a…r† and a…r† are tensors of rank two, one and zero, respectively, with components Aik …r† ˆ Cijkl

o o ; oxj oxl

ai …r† ˆ ekij

o o ; oxk oxj

a…r† ˆ ik

o o : oxi oxk

…5†

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In terms of these variables, Eq. (1) can be written as L…r; ot †U…x; t† ˆ d…t†d…x†F0 :

…6†

We have shown (Khutoryansky and Sosa, 1995) that the fundamental solution U…x; t† can be expressed in three di€erent manners, which we describe below: (1) First representation: This is based on the concept that any wave can be represented as a linear combination of plane waves. Hence, following Gelfand and Shilov (1964), U…x; t† and d…x† are represented in terms of integrals over the unit sphere jnj ˆ 1. Thus, in contrast to the alternative method of Fourier series, which uses frequency as the primary parameter, we include in this representation all possible directions of propagation of the plane wave via the use of the vector n. Hence, the fundamental solution is of the form. 1 Z  o n U…x; t† / H …t† aÿ1 Nÿ1 ad t ÿ x dS…n†; …7† ot jnjˆ1 k where H …t† is the Heaviside step function, k the wave speed of the material and N…n† ˆ qk2 I ÿ B…n†; B…n† ˆ A ‡ aÿ1 a a;

…8†

where I is the unit tensor. It is very important to note that in the sequel, the material characteristic tensors A, a and a (and consequently, B and N) are expressed as functions of the unit vector n (or in terms of the vector s, later introduced in Eq. (10)). For example, a…n† ˆ ni ik nk , with equivalent expressions for A and a. However, to economize notation, in many instances the argument n (or s) shall be omitted. Furthermore, in the n-space, these material tensors have the units of the elasticity, piezoelectric and dielectric tensors, that is (in the SI system) N mÿ2 , C mÿ2 and C2 Nÿ1 mÿ2 , respectively (with C ˆ Nm Vÿ1 ). Thus, B and N have units of N mÿ2 . We call N…n; k† the electro-acoustic tensor for a plane wave propagating in the direction of n with speed k. On the other hand, B plays the role of the ``e€ective'' elasticity tensor C mentioned in Section 1. However, here it is not clear whether the second term of Eq. (8) is positive or negative, thus, allowing for the possibility of larger or smaller wave velocities according to the values of the piezoelectric and dielectric constants. (2) Second representation: This results from a coordinate change or transformation that maps the unit sphere into the slowness surface of the material, represented by the equation Q…s† ˆ det N…s† ˆ 0; where s is the slowness vector, de®ned by n sˆ k

…9†

…10†

That is, there are three slowness vectors, one for each wave speed. The advantages of using slowness surfaces as a means of extracting information regarding the dynamic material characteristics of a body have been addressed, for example, by Musgrave (1970) and Abbudi and Barnett (1991) within the realm of anisotropic elastodynamics. Furthermore, graphical representations of slowness surfaces for three di€erent materials were given by Sosa and Khutoryansky (1999). On the slowness surface, the fundamental solution takes the form 1 We use the symbol of proportionality since here, we only want to emphasize the most relevant variables entering in the representation. The exact version can be found in Sosa and Khutoryansky (1999).

H. Sosa, N. Khutoryansky / International Journal of Solids and Structures 38 (2001) 2101±2114

U…x; t† / H …t†

o ot

Z Qˆ0

aÿ1 N ad…t ÿ x
s† dS…s†;

2105

…11†

where N is the cofactor matrix of N, satisfying the relation NN ˆ N N ˆ …det N†I ˆ Q…s†I:

…12†

(3) Third representation: This results from a new coordinate transformation, the Green's function is expressed as an integral over the line `…x; t† ˆ fQ…s† ˆ 0g \ fx
s ˆ tg;

…13†

which is the intersection of the slowness surface with a moving plane, regarded as the indicator of the direction of propagation of the acoustic or electric wave. We can show that in this ®nal and most appropriate representation, the fundamental solutions for the elastic displacement and electric potential reduce to Z H …t† o u…x; t† ˆ p…s† d`…s†; …14† 4p2 ot `…x;t†  t† ˆ H …t† o /…x; 4p2 ot

Z `…x;t†

aÿ1 a…s†
p…s† d`…s† ÿ

d…t† p ; 4p j…x†

…15†

where p…s† ˆ

sgn…s
rQ†N …s†a…s† q ; 2 2 2 a…s† jxj jrQj ÿ …x
rQ†

…16†

and j…x† ˆ ik xi xk with ik being the cofactor matrix of the dielectric tensor. Eqs. (14) and (15) have an inherent signi®cance from a computational point of view because they are not only one-dimensional representations, but also their integrands are free from the delta function. Moreover, there are a few features worthy of additional comments. (i) In view of the de®nitions for N and Q, the functional dependence of p is seen to be of the form ÿ1 ÿ1 p  a…aN† , with N given by Eq. (8). Such dependence reduces to a…aB† in the static case, as con®rmed later on by Eq. (25). Thus, inspection of Eqs. (14) and (15) allows one to predict whether electroacoustic waves will be magni®ed or not according to the individual values of the material constants. These features are later revealed through numerical examples. (ii) If piezoelectricity is neglected, which is accomplished by setting the vector a ˆ 0, then p ˆ 0, u ˆ 0 and the electric potential is given by the second term of Eq. (15), a result from electrostatics (Landau et al., 1984). (iii) The second term of Eq. (15) corresponds to the quasi-electrostatic solution. That is, the material's ability to transmit certain messages at in®nite speed. Important information regarding isotropic and anisotropic rigid dielectrics is obtained using this second term. (iv) The integrals in Eqs. (14) and (15) have singularities of order xÿ1 . Consequently, the integrals corresponding to the fundamental solutions for stress and electric ®eld (obtained via the constitutive relations) have singularities of order xÿ2 . These are common features with the corresponding static problem. However, as we shall see in Section 4, stress and electric ®eld components attenuate with distance from the source in a quite di€erent manner in the static and dynamic problems.

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3. On the electric source The fundamental solution U…x; t† of Section 2 has been obtained by assuming the sources to be delta functions in space and time. From a practical point of view, we are interested on more general and physically realistic loading conditions. These can be incorporated in our formulation by using properties of the convolution operation. To illustrate the procedure, consider the expression of u when represented over the slowness surface Q…s† ˆ 0 Z H …t† o u…x; t† ˆ v…s†d…t ÿ x
s† dS…s†; …17† 4p2 ot Qˆ0 where v…s† ˆ

sgn…s
rQ†N …s†a…s† : a…s†jrQj

…18†

Next, assume the impulsive charge to be represented by a continuous function w…t† with ®nite piecewise _ continuous ®rst derivative w…t†. Since any solution can be represented as the convolution of the fundamental solution and the source, we can write Z 1  u…x; t†  w…t† ˆ u…x; s†w…t ÿ s† ds …19† ÿ1

or using Eq. (17) u…x; t†  w…t† ˆ

Z

Z Qˆ0

v…s†

0

1

_ ÿ x
s† ds dS…s†: w…t ÿ s†d…s

…20†

Moreover, by virtue of the property Z 1 _ ÿ x
s† ds ˆ ÿd…x
s†w…t† ‡ H …x
s† w…t _ ÿ x
s†: w…t ÿ s†d…s ÿ1

Eq. (20) becomes u…x; t†  w…t† ˆ ÿw…t†

Z

Z Qˆ0

v…s†d…x
s† dS…s† ‡

Qˆ0

_ ÿ x
s† dS…s†; v…s†H …x
s†w…t

…21†

where the ®rst integral corresponds to the fundamental solution of the associated static problem, which we shall denote by ÿ uS …x†. Thus, the displacement becomes Z _ ÿ x
s† dS…s†: v…s†H …x
s†w…t …22† u…x; t† ˆ u…x; t†  w…t† ˆ w…t† uS …x† ‡ Qˆ0

Proceeding in a similar fashion, the electric potential becomes Z _ ÿ x
s† dS…s†;  t†  w…t† ˆ w…t†/S …x† ‡ aÿ1 a
v…s†H …x
s†w…t /…x; t† ˆ /…x; Qˆ0

…23†

where in the ®rst term, we have absorbed the quasi-static term of Eq. (15). One of the interesting features of Eq. (22) and (23) is that they conveniently separate the quasi-static solution (understood as the product of the function w…t† and the static solution) from the transient dynamic behavior. Furthermore, if the source is a delta function then the surface integrals of Eqs. (22) and (23) can be transformed into the line integral previously used for Eqs. (14) and (15). Otherwise, the best representation for U…x; t† is that over the slowness surface. One important exception, however, and which we employ to generate numerical results in Section 4, is a source with a time variation described by the triangle function. In this case, line integrals can

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 be used to represent the stress and electric ®eld because their second derivatives in time (i.e. w…t†) are delta functions. Finally, using the constitutive relations, we can obtain the stress tensor and electric displacement vector, namely o uk o/ …x; t†  w…t† ‡ ekij …s; t†  w…t†; …24a† Tij …x; t; w† ˆ Cijkl oxl oxk Di …x; t; w† ˆ eikl

o uk o/ …x; t†  w…t† ÿ ik …s; t†  w…t†; oxk oxl

…24b†

where the derivatives of the static fundamental solution can be calculated explicitly and are given by Z o uSi …x† xj o h ÿ1 al i ˆÿ nk Bil …n† …n† d`…n†; …25a† 3 oxk a 8p2 jxj `1 …x† onj o/S …x† xj ˆÿ 2 oxk 8p jxj3

Z `1 …x†

kj xj o h ÿ1 al ai i nk Bil …n† 2 …n† d`…n† ‡ ; a onj 4p‰ik xi xk Š3=2

…25b†

where `1 …x† is the intersection of the unit sphere jnj ˆ 1 with the stationary plane x
n ˆ 0. Note that since in the n space d`…n† is dimensionless, the units in which Eq. (25a) and (25b) are expressed as Cÿ1 and V Cÿ1 mÿ1 , respectively. Therefore, the left-hand sides of Eq. (24a) and (24b) result with units of N mÿ5 and C mÿ5 , respectively, before they are integrated over a given volume of material. 4. Numerical results We consider ceramics of the family PZT-4 (Berlincourt et al., 1964) that have been poled in the x3 direction. To gain an appreciation for the magnitudes of the variables involved in the calculations, we note that for most piezoceramics the elastic, piezoelectric and dielectric (ratio of material to vacuum permittivities) constants are of order 1011 N mÿ2 , 101 C mÿ2 and 102 ±103 , respectively. Furthermore, depoling of the ceramic can take place at ®elds of 1 MV mÿ1 , typical dielectric breakdown limits are around 100 MV mÿ1 and tensile and compressive strengths are approximately 80 and 600 MPa, respectively. The dependence on time of the point source is described by means of a symmetric triangular function like the one shown in Fig. 1, thus, q0 …26† w…t† ˆ ‰tH …t† ÿ 2…t ÿ s†H …t ÿ s† ‡ …t ÿ 2s†H …t ÿ 2s†Š: s Hence w…t† has the units of charge density. We point out that a function like Eq. (26) not only models in a realistic manner a disturbance of very short duration and high peak value, but can also be used as the building block of more complex electric signals. The intensity of the source is assumed to be q0 ˆ 10ÿ9 C mÿ3 and the duration of the electric peak is given by 2s ˆ 0:05 ls, unless otherwise speci®ed. We have shown before (Sosa and Khutoryansky, 1999) that the dynamic values of stress and electric ®eld at a ®xed location from the source are much larger than the values obtained in the static case. It was also shown that the ®eld induced in the x1 -direction is one order of magnitude larger than in the x3 -direction, and furthermore, normal stresses of signi®cant magnitude are induced only in the direction of polarization. In this article, while the applied point source has the same time-dependence characteristics, the calculations for the stress and electric ®eld components are done under di€erent conditions. Namely, we are interested on the values of these ®elds at ®xed instants. In this manner, it is possible to determine the location of the peaks induced by the load as well as the magnitude of such peaks. With such information, we

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Fig. 1. Electric impulsive charge.

can determine how the location where the maximum peak takes place relates to the magnitude of the peak. To put things in perspective, we recall that for the static solution stress and electric ®eld decrease as 1=x2 with x being the distance from the source. It is of interest to verify if the maximum values attained by the dynamic stress and electric ®eld obey the same law. The answer is in the armative if it is based on the spatial characteristics of the dynamic fundamental solution and its derivative. However, the time dependency in the integrand of these expressions have an important impact on the overall behavior of the ®elds, and thus, the attenuation of the maximum peaks does not obey such power law as can be inferred from the ®gures and tables given below. All ®gures are self-explanatory. In Figs. 2±5, we show the pro®le of the waves for some of the components of E and T in two mutually perpendicular directions at four di€erent instants (in other words, the ®gures correspond to the frozen images of the waves at four di€erent times, thus, capturing wave shape, maximum amplitude and location of the wave at the given time). These times plus three additional instants, as well as the maximum values achieved by the waves and the location, where such maximum takes place are tabulated in Tables 1 and 2. From these tables, it can be concluded that the attenuation of the maximum value of a dynamic ®eld does not follow the 1=x2 pattern of statics. Rather, a 1=x appears to be a better representation of such attenuation. Physically this means that at signi®cant distances from the source, the ®elds are still of relatively high magnitudes and thus transient dynamic e€ects pose greater risks of failure than static ones. In Figs. 6 and 7, we show the wave pro®les of the electric ®eld E1 and stress T13 for two di€erent materials (PZT-4 and PZT-5) at the instant t ˆ 0:2 ls when the pulse duration is still 2s ˆ 0:05 ls. While these two materials have very similar elastic and piezoelectric constants, the dielectric constants of a PZT-4 ceramic are approximately 25% smaller. On the other hand, Figs. 8 and 9 show the wave pro®les of E1 and T13 for a PZT-5 at the instant t ˆ 0:2 ls for two di€erent pulse durations, namely 2s ˆ 0:05 and 2s ˆ 0:10 ls. Finally, to understand the essential features of the pro®les of the waves shown in Figs. 2±9 in response to the impulse given by Eq. (26) it is enough to consider one of the terms entering in the calculation of such waves. First of all, we note that positive and negative values of the electroelastic ®elds are related to the positive and negative slopes of the impulsive source. Furthermore, the calculations of the ®elds are carried out over the lines resulting from the intersection of the slowness surface and plane representing the disturbance. Once again, due to the nature in time of the applied charge, there are three lines resulting from the aforementioned intersection (giving rise to three line integrals); thus, the convolution of the displacement gradient appearing in Eq. (24) is given by

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Fig. 2. Wave pro®les of E1 in x1 -direction for four ®xed values of t.

Fig. 3. Wave pro®les of T13 in x1 -direction for four ®xed values of t.

Z Z o uk o uS H …t† 2 H …t ÿ s† …x; t†  w…t† ˆ k …x†w…t† ÿ p…s† s d`…s† ‡ p…s† s d`…s† oxl oxl s s `…x;t† `…x;tÿs† Z H …t ÿ 2s† p…s† s d`…s† ÿ s `…x;tÿ2s†

…27†

2110

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Fig. 4. Wave pro®les of E3 in x3 -direction for four ®xed values of t.

Fig. 5. Wave pro®les of T33 in x3 -direction for four ®xed values of t.

with the derivative of the static solution given by Eq. (25a). On the other hand the expression for the  k  w…t† is similar to Eq. (27) with the integrand being replaced by aÿ1 …a
p†s. convolution o/=ox We can understand the meaning of each term in Eq. (27) better by looking at the convolution of the fundamental solution for the stress ®eld and the applied source. From the point of view of far ®elds (that is

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2111

Table 1 Maximum values and location of shear stress T13 and electric ®eld E1 in x1 -direction t (ls)

xmax (mm)

jTmax j (MPa)

jEmax j (kV mÿ1 )

0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.195 0.455 0.715 0.975 1.235 1.495 1.754

25.524 9.553 5.619 3.909 2.969 2.381 1.982

1291.8 484.4 286.0 199.5 151.9 122.1 101.7

Table 2 Maximum values and location of normal stress T33 and electric ®eld E3 in x3 -direction t (ls)

xmax (mm)

jTmax j (MPa)

jEmax j (kV mÿ1 )

0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.342 0.798 1.253 1.710 2.166 2.621 3.077

8.852 2.992 1.777 1.262 0.977 0.798 0.674

177.1 55.8 32.3 22.6 17.4 14.1 11.9

Fig. 6. Electric ®eld E1 in x1 -direction for two materials.

for times smaller or equal than the duration of the applied pulse) and when the slowness surface is convex, the leading part of such convolution is given by three terms of the form   jxj  _  w…t†; …28† Tk …x†d t ÿ ck

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Fig. 7. Stress T13 in x1 -direction for two materials.

Fig. 8. Electric ®eld E1 in x1 -direction for two pulses of di€erent duration.

where each term (k ˆ 1; 2; 3) has a Tk …x† associated with one slowness sheet, which varies as Tk  1=jxj, and a corresponding wave speed ck . Note that for convex slowness surfaces, in general, ck 6 kk (kk being the characteristic speed of the material), with the equality holding only along the coordinate axes. Since Tk is independent of time, it suces to analyze the convolution of the time derivative of the delta function and the applied pulse. Therefore, using the properties of convolution and delta function together with the time derivative of Eq. (26) we obtain         Z 1  _d t ÿ t0 ÿ jxj w…t0 † dt0 ˆ q0 H t ÿ jxj ÿ 2H t ÿ jxj ÿ s ‡ H t ÿ jxj ÿ 2s ; …29† s c c c c ÿ1

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Fig. 9. Stress T13 in x1 -direction for two pulses of di€erent duration.

which shows that the far ®eld of stress (or electric ®eld) has three jumps of di€erent magnitudes and signs according to each of the terms of Eq. (29). 5. Closure New results concerning the transient dynamic response of piezoelectric solids subjected to electric impulses have been presented. In the ®rst place, the dynamic Green's functions introduced by Sosa and Khutoryansky (1999) have been represented in a more convenient mathematical form, easier to visualize and interpret from a physical standpoint. Second, new information regarding the behavior of stress and electric ®eld has been obtained numerically for two di€erent directions of propagation, two di€erent materials and several instants. Particular attention has been placed to the attenuation characteristics of the waves. Finally, a brief analysis of the nature and pro®le of the jumps in the waves has been given using Eq. (27). We believe that the results presented in this article together with those previously published by the authors in this same journal provide an important tool to estimate and predict transient electroacoustic characteristics induced by rapidly varying concentrated sources. References Abbudi, M., Barnett, D., 1991. Three-dimensional computer visualization of the slowness surfaces of anisotropic crystals. In: Wu, J., Ting, T.C.T., Barnett, D. (Eds.), Modern Theory of Anisotropic Elasticity and Applications. SIAM, pp. 290±300. Berlincourt, D., Curran, D., Ja€e, H., 1964. Piezoelectric and piezoceramic materials and their function in transducers. In: Mason, W.P. (Ed.), Physical Acoustics, vol. I-A, Academic Press, New York, pp. 169±270. Chen, T., 1993. Green's functions and the nonuniform transformation problem in a piezoelectric medium. Mech. Res. Comm. 20, 271± 278. Chen, T., Lin, F., 1993. Numerical evaluation of derivatives of the anisotropic Green's functions. Mech. Res. Comm. 20, 501±506. Furuta, A., Uchino, K., 1993. Dynamic observation of crack propagation in piezoelectric multilayer actuators. J. Am. Ceram. Soc. 76, 1615±1617.

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Gelfand, I.M., Shilov, G.E., 1964. Generalized Functions, vol. 1. Academic Press, New York. Ikeda, T., 1990. Fundamentals of Piezoelectricity. Oxford University Press, Oxford. Khutoryansky, N., Sosa, H., 1995. Dynamic representation formulas and fundamental solutions for piezoelectricity. Int. J. Solids Struct. 32, 3307±3325. Landau, L.D., Lifshitz, E.M., Pitaevskii, L.P., 1984. Electrodynamics of Continuous Media, second ed., Pergamon Press, Oxford. Lee, J.S., Jiang, L.Z., 1994. A boundary integral formulation and 2-D fundamental solution for piezoelastic media. Mech. Res. Comm. 21, 47±54. Musgrave, M.J., 1970. Crystal Acoustics. Holden-Day, San Francisco. Nelson, D.F., 1979. Electric, Optic, and Acoustic Interactions in Dielectrics. Wiley, New York. Norris, A.N., 1994. Dynamic Green's functions in anisotropic piezoelectric, thermoelastic and poroelastic solids. Proc. R. Soc. Lond. A 447, 175±188. Sosa, H., Khutoryansky, N., 1999. Transient dynamic response of piezoelectric bodies subjected to internal electric impulses. Int. J. Solids Struct. 36, 5467±5484. Tiersten, H.F., 1969. Linear Piezoelectric Plate Vibrations. Plenum, New York.