Perturbational solution for quasi-axial propagation in a piezoelectric, continuously twisted, structurally chiral medium

Applied Acoustics 62 (2001) 1019±1023 www.elsevier.com/locate/apacoust

Technical note

Perturbational solution for quasi-axial propagation in a piezoelectric, continuously twisted, structurally chiral medium Akhlesh Lakhtakia * Department of Engineering Science and Mechanics, 212 Earth-Engineering Sciences Building, The Pennsylvania State University, University Park, PA 16802-6812, USA Received 25 September 2000; accepted 1 December 2000

Abstract A simple formula is provided for quasi-axial propagation in a so-called piezoelectric, continuously twisted, structurally chiral medium (PCTSCM), which is a reciprocal medium whose constitutive properties vary helicoidally along its axis of spirality. # 2001 Elsevier Science Ltd. All rights reserved.

1. Introduction The piezoelectric, continuously twisted, structurally chiral medium (PCTSCM) was introduced in this journal a few years ago [1]. The stress tensor in a PCTSCM is symmetric, while the sti€ness, the permittivity and the piezoelectric stress tensors vary helicoidally about the axis of spirality. Sculptured thin ®lm technology has made tremendous strides during the last three years [2], and the fabrication of useful PCTSCMs with this new technology appears feasible so much so that reports on the mechanical characterization of structurally chiral mediums have now begun to appear in print [3±5]. The equations of acoustic wave propagation along the axis of spirality can be solved exactly and easily [1]. For oblique propagation too, an exact solution in terms of a uniformly convergent matrix polynomial is available [6]. But the slow convergence of that solution makes it error-prone for thick PCTSCM layers. A simple * Tel.: +1-814-863-4319; fax: +1-814-863-7967. E-mail address: [email protected] (A. Lakhtakia). 0003-682X/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0003-682X(00)00100-6

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and adequate solution for quasi-axial propagation is desirable, as the relevant conditions are likely to be invoked in order to separate the incident and the re¯ected plane waves for experimental characterization of PCTSCMs. To that end, this communication presents a perturbational solution. The plan of this communication is as follows: Linear constitutive equations are presented and a key matrix di€erential equation for oblique propagation is reproduced in the following section. The perturbational solution for quasi-axial propagation is then presented in the last section. An exp(-i!t) time-dependence is implicit in this work; the z axis is taken as the axis of spirality and oblique propagation (i.e. @ @ @ @xik, @y 0, @z 60) in a PCTSCM is handled without loss of generality with respect to the constitutive properties. The interested reader is referred to Refs. [1] and [5] for other relevant details. 2. Constitutive relations and matrix di€erential equation The Kelvin notation [7] is employed to denote stress and strain as the column 6vectors  t ) ‰T…r†Š ˆ Txx …r†; Tyy …r†; Tzz …r†; Tyz …r†; Txz …r†; Txy …r†  t ; …1† ‰S…r†Š ˆ Sxx …r†; Syy …r†; Szz …r†; 2Syz …r†; 2Sxz …r†; 2Sxy …r† while the particle displacement, the electric ®eld, the magnetic ®eld, and the electric displacement are respectively denoted by the column 3-vectors  t  t ) ‰u…r†Š ˆ ux …r†; uy …r†; uz …r† ‰E…r†Š ˆ Ex …r†; Ey …r†; Ez …r†  t  t …2† ‰H…r†Š ˆ Hx …r†; Hy …r†; Hz …r† ‰D…r†Š ˆ Dx …r†; Dy …r†; Dz …r† where the superscript t denotes the transpose. The constitutive equations of a PCTSCM may be stated in matrix form as [1]  ‰D…r†Š ˆ ‰"…z†Š‰E…r†Š ‡ ‰e…z†Š‰S…r†Š ; …3† ‰T…r†Š ˆ ‰e…z†Št ‰E…r†Š ‡ ‰c…z†Š‰S…r†Š where the 33 permittivity matrix ["(z)], the 66 sti€ness matrix [c(z)] and the 36 piezoelectric stress matrix [e(z)] are helicoidally dependent on z. Thus, 9 ‰"…z†Š ˆ ‰A…z†Š‰"0 Š‰A…z†Št = …4† ‰c…z†Š ˆ ‰R…z†Š‰c0 Š‰R…z†Št ; ; ‰e…z†Š ˆ ‰A…z†Š‰e0 Š‰R…z†Št where ["0 ], [c0 ] and [e0 ] may be called the reference constitutive matrixes. In Eq. (4), the unitary matrix [1]

A. Lakhtakia / Applied Acoustics 62 (2001) 1019±1023

0

1 z z sin 0



B C z z C ‰A…z†Š ˆ B @ sin cos 0A



0 0 1

1021

cos

…5†

denotes a clockwise rotation about the z axis by an angle pz/ , while the corresponding Bond matrix is given as [1] 0 B B B B B B ‰R…z†Š ˆ B B B B B B @

z

2 z sin

0 0

cos2

z

2 z cos

0 0 sin2

0

0

0

0 1 0

0 0 z cos

z sin

0

0 0 z sin

z cos

0

1 2z sin

C 2z C C sin

C C C 0 C: C 0 C C C 0 C A 2z cos

…6†

0 0 0 1 2z 1 2z sin sin 0 2

2

The structural period of a PCTSCM equals 2 . Without loss of generality, all spatially varying column vectors in Eqs. (1) and (2) can be taken independent of y and the Fourier transform with respect to x implemented on them. Thus,  ‰T…r†Š ‡ eix ‰T…z; †Š ; …7† ‰E…r†Š ‡ eix ‰E…z; †Š etc.,  being the transversal or the horizontal wavenumber. Following the developments in Refs. [1] and [6] in order to arrive at a 1010 matrix di€erential equation, the following transformation is e€ected: 9 ‰E0 …z; †Š ˆ ‰A…z†Št ‰E…z; †Š ‰H0 …z; †Š ˆ ‰A…z†Št ‰H…z; †Š = …8† ‰D0 …z; †Š ˆ ‰A…z†Št ‰D…z; †Š ‰T0 …z; †Š ˆ ‰R… z†Š‰T…z; †Š : ; ‰u0 …z; †Š ˆ ‰A…z†Št ‰u…z; †Š ‰S0 …z; †Š ˆ ‰R…z†Št ‰S…z; †Š Eventually, the matrix di€erential equation d 0 ‰f …z; †Š ˆ i‰Q0 …z; †Š‰f0 …z; †Š; d z

…9†

emerges, where the column 10-vector [f0 (z,)] is given by h ‰f…z; †Š ˆ E0x …z; †; E0y …z; †; H0x …z; †; H0y …z; †; u0x …z; †; u0y …z; †; u0z …z; †; it T0xz …z; †; T0yz …z; †; T0zz …z; † …10†

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A. Lakhtakia / Applied Acoustics 62 (2001) 1019±1023

The 1010 matrix [Q0 (z,)] is far too huge for reproduction here but can be obtained using symbolic manipulation programs. More importantly [6], ‰Q0 …z; †Š ˆ ‰PŠ ‡  eiz= ‰W1 Š ‡ e

iz=


‰W2 Š ‡ 2 ‰W3 Š ‡ ei2z= ‰W4 Š ‡ e

i2z=

‰W5 Š




…11† for any PCTSCM, where all six 1010 matrixes [P] and [Wl], (14l45), are independent of both z and , but not necessarily of . 3. Quasi-axial propagation In a typical problem involving a structurally chiral medium, the latter's thickness (i.e. the dimension in the z direction) is assumed to be ®nite [8]. Quintessentially therefore, one is interested in the response of a ®nite-thickness PCTSCM layer to an incident plane wave with wavenumber kinc. Therefore, arcsin(/kinc) is an angle whose value is ®xed by the incidence conditions; for instance, arcsin(/kinc)=0 for a normally incident plane wave. Quasi-axial propagation will occur in a PCTSCM when the ratio =kinc <<1. Under this condition, the  and the 2 terms on the right side of Eq. (11) may be considered as perturbations with respect to [P]. A procedure to solve di€erential equations analogous to Eq. (10) has been discussed in great detail by Yakubovich and Starzhinskii [9]. As [Q0 (z,)]=[Q0 (z+2 ,)], the solution of Eq. (10) in general must be of the form  ‰f 0 …z; †Š ˆ ‰F…z; †Šexp ‰K…†Šz ‰f 0 …0; †Š; …12† according to the Floquet±Lyapunov theorem; here, [F(z,)]=[F(z+2 ,)] is a periodic matrix while the matrix [K()] is independent of z. The Yakubovich±Starzhinskii procedure yields the perturbative expressions
) ‰F…z; †Š  ‰IŠ ‡ ‰hF1 …iz†Š ‡ 2 ‰F2 …z†Š ‡ O 3
; …13† ‰K…†Š  i‰PŠ ‡ 2 K~ ‡ O 4 where [I] is the 1010 identity matrix. The other matrixes appearing in Eq. (13),  9 ‰F1 …z†Š ˆ i eiz= R1 …‰W1 Š† ‡ e iz= R 1 …‰W2 Š† > = i2z=

R2 …‰G1 Š† ‡ ei2z= R 2 …‰G2 Š† …14† h‰F2i…z†Š ˆ e > K~ ˆ i‰W3 Š ‰W2 ŠR1 …‰W1 Š† ‰W1 ŠR 1 …‰W2 Š† ; make use of the matrixes ‰G1 Š ˆ i‰W4 Š ‰G2 Š ˆ i‰W5 Š

‰W1 ŠR1 …‰W1 Š† ‰W2 ŠR 1 …‰W2 Š†

 …15†

as well as the matrix function Rm([V]) of the matrix [V] de®ned via the linear relation

A. Lakhtakia / Applied Acoustics 62 (2001) 1019±1023

m Rm …‰VŠ† ˆ ‰PŠRm …‰VŠ†

Rm …‰VŠ†‰PŠ

i‰VŠ:

1023

…16†

The obtained perturbational solution is exact for axial propagation (i.e. =0). It is easy to implement numerically as well as symbolically on a computer for any PCTSCM, so long as =kinc <<1. It also yields a zero-th order approximation 
‰f0 …z; †Š  exp i ‰PŠ ‡ 2 ‰W3 Š z ‰f0 …0; †Š; …17† which indicates that the eigenvalues and eigenvectors of the matrix [P]+2 [W3] may be useful in developing analytical formulas for quasi-axial propagation in PCTSCMs of speci®c types. References [1] Lakhtakia A. Wave propagation in a piezoelectric, continuously twisted, structurally chiral medium along the axis of spirality. Applied Acoustics 1995;44:25-37; errata: 1995;44:385. [2] See several papers in: Lakhtakia A, Messier RF, editors. Engineered nanostructural ®lms and materials. SPIE Optical Engineering Press, Bellingham, WA, 1999. [3] Seto MW, Robbie K, Vick D, Brett MJ, Kuhn L. Mechanical response of thin ®lms with helical microstructures. J Vac Sci Technol 1999;B 17:2172±7. [4] Lakhtakia A, Sherwin JA. Displacement in a continuously twisted structurally chiral medium due to axial loading. J Acoust Soc Am 2000;107:3549±51. [5] Knepper R, Messier R. Morphology and mechanical properties of oblique angle columnar thin ®lms. Proc SPIE 2000;4097:291±8. [6] Lakhtakia A. Exact analytic solution for oblique propagation in a piezoelectric, continuously twisted, structurally chiral medium. Applied Acoustics 1996;49:225±36. [7] Auld BA. Acoustic ®elds and waves in solids, vol. I. Malabar, FL: Krieger, 1990. [8] Venugopal VC, Lakhtakia A. Sculptured thin ®lms: conception, optical properties, and applications. In: Singh ON, Lakhtakia A, editors. Electromagnetic Fields in Unconventional Materials and Structures. Wiley, New York, 2000. [9] Yakubovich VA, Starzhinskii VM. Linear di€erential equations with periodic coecients. Chichester, UK (chapter IV-6): Wiley, 1975.