Floquet Theory for Discontinuously Supported Waveguides

Abstract

We apply Floquet theory of periodic coefficient second-order ODEs to an elastic waveguide. The waveguide is modeled as a uniform elastic string periodically supported by a discontinuous Winkler elastic foundation and, as a result, a Hill equation is found. The fundamental solutions, the stability regions, and the dispersion curves are determined and then plotted. An asymptotic approximation to the dispersion curve is also given. It is further shown that the end points of the band gap structure correspond to periodic and semiperiodic solutions of the Hill equation.

[ProQuest: [...] denotes non US-ASCII text; see PDF]

Academic Editor:Julius Kaplunov

Dipartimento di Scienze e Metodi dell'Ingegneria (DISMI), Universitá degli Studi di Modena e Reggio Emilia, Via G. Amendola 2, 42122 Reggio Emilia, Italy

Received 24 December 2015; Accepted 5 May 2016

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Periodic structures often appear in several mechanical systems, ranging from strings and beams [1, 2] to phononic crystals [3-5] to name just a few. Such systems exhibit a typical pass/block band structure when wave propagation is considered. Indeed, periodic structures are especially relevant when employed as waveguides [6, 7] or energy scavenging devices [8]. The analysis of the transmission property of waveguides is best carried out through Floquet theory of periodic coefficient ODEs, although this fact is seldom neglected in favor of a more direct approach by means of the Floquet-Bloch boundary conditions. In this paper, an analysis of the mechanical problem of a uniform string periodically supported on a Winkler foundation is presented form the standpoint of the stability theory of Hill's equation [9]. Besides, a high-frequency asymptotic homogenization procedure is presented, following [10, 11]. The discontinuous character of the support may be due to crack propagation [12-15] or debonding in composite materials [16, 17]. It could also be due to the tensionless character of the substrate [18]. This study follows upon a very vast body of literature on elastic periodic structures [2, 19-24]. The situation of wave propagation through a thin coating layer [25-27] could also be considered. Applications in the realm of civil engineering are also possible [28-32]. The paper is structured as follows: Section 2 sets up the mechanical model and the governing equations. Section 3 discusses stability of the solution of Hill's equation. The dispersion relation and its asymptotic approximation are presented in Section 4. Finally, conclusions are drawn in Section 5.

2. The Mechanical Model

Let us consider a homogeneous elastic string in uniform tension T, periodically supported on a Winkler elastic foundation (Figure 1). The governing equation for the transverse displacement w(x,t) in the absence of loading is periodic with period L: [figure omitted; refer to PDF] where ρA is the mass linear density of the string, assumed constant, β is the Winkler subgrade modulus (with physical dimension of stress), and H is Heaviside step function; that is, [figure omitted; refer to PDF] This equation may be rewritten in dimensionless form [figure omitted; refer to PDF] having introduced the dimensionless positive ratios: [figure omitted; refer to PDF] together with _x1 =x/L∈[0,1], the dimensionless axial coordinate, and τ=t/ρA/β, the dimensionless time. Here, prime denotes differentiation with respect to _x1 and dot differentiation with respect to τ and w(_x1 ,τ) and H(_x1 ) are assumed. We shall look for the harmonic behavior of w; that is, w(_x1 ,τ)=u(_x1 )exp[...](iΩτ), whence (3) becomes the constant coefficient ODE for u(_x1 ): [figure omitted; refer to PDF] We shift the unit period to range in the interval (-α/2,1-α/2) in order to consider an even/odd problem; namely, [figure omitted; refer to PDF] The general harmonic solution of (3) in the supported region _x1 ∈(-α/2,α/2) is given by [figure omitted; refer to PDF] while the solution in the free region _x1 ∈(α/2,1-α/2) is [figure omitted; refer to PDF] where _A1 , _A2 and _B1 , _B2 are integration constants to be determined through the boundary conditions (BCs). The BCs for the system require continuity at the supported/unsupported transition: [figure omitted; refer to PDF] where prime stands for _x1 differentiation. Furthermore, consideration of the Floquet-Bloch waves lends the periodicity conditions [figure omitted; refer to PDF] where ρ≠0 is the characteristic multiplier . For a second-order ODE, there are two nonnecessarily distinct characteristic multipliers, which are denoted by _ρ1 and _ρ2 . Besides, let _ρk =exp[...](_mk ) with k∈{1,2}; then _mk =iq is the characteristic exponent .

Figure 1: A homogeneous string periodically supported by an elastic foundation.

[figure omitted; refer to PDF]

3. Boundedness and Periodicity of the Solution

Equation (5) is known as Hill's equation, after Hill who studied it in 1886 in the context of lunar dynamics [33]. For the sake of clarity, we shall write (0,a) for the periodicity interval (-α/2,1-α/2). It is easy to find a fundamental system of solutions _[varphi]1 (_x1 ), _[varphi]2 (_x1 ) such that [figure omitted; refer to PDF] This is a set of two particular linearly independent solutions of the ODE (6); see [9] for further details. It is well known that, for any second-order ODE, [figure omitted; refer to PDF] where W is the Wronskian of the fundamental solutions [34]. In light of the problem's symmetry, _[varphi]1 is an even and _[varphi]2 an odd function. Let us introduce the discriminant [figure omitted; refer to PDF] being [figure omitted; refer to PDF] Hill's equation is said to be unstable if all nontrivial solutions are unbounded in R and conditionally stable if there is a nontrivial solution bounded in R and stable if all solutions are bounded in R.

By Theorem 1.3.1 of [9], the solution is stable if (D)<2 and unstable when (D)>2 and special consideration is required for the case (D)=2. Indeed, when (D)<2, the characteristic multipliers _ρ1 and _ρ2 are complex conjugated and have unit modulus, whereupon the characteristic exponents are opposite; that is, _q1 =-_q2 . Figure 2 plots the bounding curves D=2 (dashed) and D=-2 (solid) as a function of Ω and α for κ=0.5. Stable regions are shaded in gray. It is seen that Band Gaps (BGs) are bounded by dashed and solid curves in succession. When D=±2 it is _ρ1 =_ρ2 =±1, respectively.

Figure 2: Stability regions (shaded in gray) for Hill's equation (5) at κ=0.5. Solid and dashed curves represent the level lines D=±2, respectively.

[figure omitted; refer to PDF]

It is also well known [9, Section 1.2] that the bounding curve D=2 corresponds to solutions in the form [figure omitted; refer to PDF] where _p1 (_x1 ) and _p2 (_x1 ) are periodic functions with period 1, provided that the following condition which grants the availability of two eigenvectors for the repeated eigenvalue _ρ1,2 =1 holds [figure omitted; refer to PDF] Under the same condition, the bounding curve D=-2 corresponds to solutions in the form [figure omitted; refer to PDF] where _P1 (_x1 ) and _P2 (_x1 ) are semiperiodic function with period 1; that is, [figure omitted; refer to PDF] In Figure 3, the solution curves for the first and the second of the conditions (16) are plotted in dotted line style and they partly overlie the conditionally stable curves D=±2. However, it is observed that such conditions are satisfied only at some very special points. Indeed, we have [figure omitted; refer to PDF] whence crossing is possible either when [figure omitted; refer to PDF] or when α=1. The case α=1 relates to a fully supported string for which ^{[varphi]1[variant prime]} (a) and _[varphi]2 (a) are both proportional to [figure omitted; refer to PDF] so that they vanish at the isolated points Ω=1+^n2κ2π2 , n∈{0,1,2,...}. The first point, Ω=1, corresponds to the pivotal frequency for the 0th BG. Conversely, the case α=0 corresponds to a free string, which is a nondispersive situation. Indeed, BGs collapse into single points which, according to (20), are located at Ω=nκπ. Other (Ω,α,κ) combinations exist which satisfy (D)=2 and (16), such as (3.22313,0.512649,0.5). In all such points two periodic (or semiperiodic) solutions exist and conditional stability is reverted to stability. However, in the general case, (16) never hold together and thus conditional stability remains. Indeed, the second of the conditions (15) is replaced by [figure omitted; refer to PDF] which is obviously unstable. By the same token, the second of the conditions (17) becomes [figure omitted; refer to PDF] Thus, the coexistence problem for period 1 is generally answered in the negative; that is, a single periodic (or semiperiodic) solution exists which is accompanied by a nonperiodic one. Besides, from a mechanical standpoint, it is observed that, in the general case, a periodic solution p(x) is not acceptable on physical grounds, as it conveys a jump discontinuity at the boundary, unless [figure omitted; refer to PDF] Likewise, a semiperiodic solution P(x) is not acceptable unless [figure omitted; refer to PDF] We now prove that such conditions can always be met. To this aim we now take advantage of the problem's symmetry and recall that a nontrivial even solution exists if and only if [figure omitted; refer to PDF] respectively, are periodic and semiperiodic, whereas an odd solution stands if and only if [figure omitted; refer to PDF] again for periodic and semiperiodic [9, Theorem 1.3.4]. Now, when D=2 we have the nontrivial periodic solution _p1 (_x1 ) and let us define [figure omitted; refer to PDF] By the periodicity of _p1 , clearly _p1+ is proportional to the even part of _p1 and it is periodic. Besides, [figure omitted; refer to PDF] which are, respectively, (24) and the first of the conditions (26). With the choice for the denominator in (28), it is also _p1+ (0)=1 by which we conclude that _p1+ ≡_[varphi]1 extended in periodic fashion (it is also easy to show that no slope jump discontinuity is admitted by Hill's equation. Then, one can prove that ^{[varphi]1[variant prime]} (0)=^{[varphi]1[variant prime]} (a)=0 and ^{[varphi]2[variant prime]} (0)=±^{[varphi]2[variant prime]} (a), where the sign is given in the periodic and semiperiodic situation, resp.). Likewise, when D=-2, we have the nontrivial semiperiodic solution _P1 (_x1 ) and let us define [figure omitted; refer to PDF] which, in light of _P1 being semiperiodic, is again proportional to the even part of _P1 . It is the sum of two semiperiodic functions; _P1- (x) is semiperiodic; besides [figure omitted; refer to PDF] which are, respectively, (25) and the second of the conditions (27). With the choice for the denominator in (30), it is _P1- (0)=1 and we conclude that _P1- ≡_[varphi]1 extended in semiperiodic fashion. Through the analogous definitions of _p1- and _P1+ , one gets the odd function _[varphi]2 extended in periodic and semiperiodic fashion, respectively.

Figure 3: Periodic and semiperiodic solutions for Hill's equation (5) at κ=0.5. The red and the blue dotted curves represent the solution of the first and the second of the conditions (16), respectively.

[figure omitted; refer to PDF]

The role of κ is illustrated comparing Figure 2 with Figure 4. It is seen that the frequency axis roughly scales with κ. Besides, reducing κ brings wider BGs which tilt and tend to cluster together. It is noticed that no symmetry exists about α=0.5.

Figure 4: Stability regions (shaded in gray) for Hill's equation (5) at κ=0.25.

[figure omitted; refer to PDF]

4. Dispersion Relation

Imposing the BCs ((9), (10)) gives the dispersion relation: [figure omitted; refer to PDF] This relation can also be written as (see also [35]) [figure omitted; refer to PDF] and it is plotted in Figure 5 for κ=0.5 and α=0.25. Equation (32) conforms to the form of (4) in [19].

Figure 5: Dispersion relation for a string on a periodic support (κ=0.5, α=0.25).

[figure omitted; refer to PDF]

It is easy to prove that f(Ω) quickly asymptotes -1 from below for Ω>1, whence we can give a simple expression for the dispersion relation: [figure omitted; refer to PDF] Such approximation is superposed onto the dispersion curves in Figure 8 whereupon it is seen that it is very effective already at Ω close to 1, although it is unable to capture the BGs.

The relevant bounding values for each band gap (BG) are obtained considering the periodic and the semiperiodic eigenvalue problems, for which two sets of BCs need to be considered, respectively:

(1) the periodicity conditions u(0)=u(a) and ^{u[variant prime]} (0)=^{u[variant prime]} (a);

(2) the semiperiodicity conditions u(0)=-u(a) and ^{u[variant prime]} (0)=-^{u[variant prime]} (a).

The relevant eigenfrequencies are denoted by _λ0 ,_λ1 ,... and _μ0 ,_μ1 ,..., respectively, for the periodic and the semiperiodic eigenproblems. The eigenmode for the first periodic eigenfrequency _λ0 is shown in Figure 6 at =0.5, α=0.25. Likewise, the eigenmode for the first semiperiodic eigenfrequency _μ0 is shown in Figure 7. Such eigenfrequency is the lower boundary of the system's first BG.

Figure 6: The first eigenmode, relative to the eigenfrequency _λ0 =0.488445 at κ=0.5, α=0.25, and q=0, is a periodic function.

[figure omitted; refer to PDF]

Figure 7: The eigenmode at the lower end of the first BG, relative to the eigenfrequency _μ0 =1.57858 at κ=0.5, α=0.25, and q=π, is a semiperiodic function.

[figure omitted; refer to PDF]

Figure 8: Dispersion relation (solid) superposed onto its first approximation (34) (dashed).

[figure omitted; refer to PDF]

The dispersion curve intersection with the axis q=0 is given by _λm while the intersection with the axis q=π is given by _μm , where m∈N. BGs' size is obtained by _μ2m -_μ2m+1 and _λ2m+1 -_λ2m+2 .

5. Conclusions

In this paper, the Floquet theory of periodic coefficient ODEs is applied to describe the behavior of a mechanical waveguide. A homogeneous elastic string periodically supported by an elastic Winkler foundation is considered and it is found that the governing equation is given by a second-order Hill equation. Floquet-Bloch periodic boundary conditions are enforced. The stability regions together with the dispersion relation are found in terms of the Floquet theory through the discriminant. An asymptotic approximation to the lowest cutoff frequency is given. The fundamental eigensolutions of the periodic and of the semiperiodic problem are also determined. It is remarked that the present methodology can be extended to tackle functionally graded beams [36-39] or plates [40, 41]. Applications in the realm of civil engineering are also possible [28-32, 42]. A nice extension of the present theory could be related to temperature [43, 44] or viscoelastic effects in the fiber [45-48].

References

[1] D. J. Mead, "Wave propagation and natural modes in periodic systems: I. Mono-coupled systems," Journal of Sound and Vibration , vol. 40, no. 1, pp. 1-18, 1975.

[2] D. J. Mead, "Wave propagation in continuous periodic structures: research contributions from Southampton, 1964-1995," Journal of Sound and Vibration , vol. 190, no. 3, pp. 495-524, 1996.

[3] D. Yu, Y. Liu, H. Zhao, G. Wang, J. Qiu, "Flexural vibration band gaps in Euler-Bernoulli beams with locally resonant structures with two degrees of freedom," Physical Review B , vol. 73, no. 6, 2006.

[4] Y. Zhang, L. Han, L.-H. Jiang, "Transverse vibration bandgaps in phononic-crystal Euler beams on a Winkler foundation studied by a modified transfer matrix method," Physica Status Solidi (B) , vol. 250, no. 7, pp. 1439-1444, 2013.

[5] Y. Zhang, J. He, L.-H. Jiang, "Flexural vibration band gaps characteristics in phononic crystal Euler beams on two-parameter foundation," Advances in Mechanical Engineering , vol. 5, 2013.

[6] J. Miklowitz The Theory of Elastic Waves and Waveguides , Elsevier, New York, NY, USA, 2012.

[7] A. B. Movchan, L. I. Slepyan, "Resonant waves in elastic structured media: dynamic homogenisation versus Green's functions," International Journal of Solids and Structures , vol. 51, no. 13, pp. 2254-2260, 2014.

[8] A. Kudaibergenov, A. Nobili, L. Prikazchikova, "On low-frequency vibrations of a composite string with contrast properties for energy scavenging fabric devices," Journal of Mechanics of Materials and Structures , vol. 11, no. 3, pp. 231-243, 2016.

[9] M. S. Eastham The Spectral Theory of Periodic Differential Equations , Scottish Academic Press, Edinburgh, UK, 1973.

[10] R. V. Craster, J. Kaplunov, A. V. Pichugin, "High-frequency homogenization for periodic media," Proceedings of The Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences , vol. 466, no. 2120, pp. 2341-2362, 2010.

[11] T. Antonakakis, R. V. Craster, "High-frequency asymptotics for microstructured thin elastic plates and platonics," Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences , vol. 468, no. 2141, pp. 1408-1427, 2012.

[12] A. M. Tarantino, "On the finite motions generated by a mode I propagating crack," Journal of Elasticity , vol. 57, no. 2, pp. 85-103, 1999.

[13] A. M. Tarantino, "Crack propagation in finite elastodynamics," Mathematics and Mechanics of Solids , vol. 10, no. 6, pp. 435-456, 2005.

[14] R. Luciano, J. R. Willis, "Hashin-Shtrikman based FE analysis of the elastic behaviour of finite random composite bodies," International Journal of Fracture , vol. 137, no. 1-4, pp. 261-273, 2006.

[15] A. Nobili, E. Radi, L. Lanzoni, "A cracked infinite Kirchhoff plate supported by a two-parameter elastic foundation," Journal of the European Ceramic Society , vol. 34, no. 11, pp. 2737-2744, 2014.

[16] A. Caporale, R. Luciano, "Limit analysis of masonry arches with finite compressive strength and externally bonded reinforcement," Composites Part B: Engineering , vol. 43, no. 8, pp. 3131-3145, 2012.

[17] V. Salomoni, G. Mazzucco, C. Pellegrino, C. Majorana, "Three-dimensional modelling of bond behaviour between concrete and FRP reinforcement," Engineering Computations , vol. 28, no. 1, pp. 5-29, 2011.

[18] A. Nobili, "Superposition principle for the tensionless contact of a beam resting on a Winkler or a Pasternak foundation," Journal of Engineering Mechanics , vol. 139, no. 10, pp. 1470-1478, 2013.

[19] D. J. Mead, "Free wave propagation in periodically supported, infinite beams," Journal of Sound and Vibration , vol. 11, no. 2, pp. 181-197, 1970.

[20] M. Gei, A. B. Movchan, D. Bigoni, "Band-gap shift and defect-induced annihilation in prestressed elastic structures," Journal of Applied Physics , vol. 105, no. 6, 2009.

[21] G. Napoli, A. Nobili, "Mechanically induced Helfrich-Hurault effect in lamellar systems," Physical Review E--Statistical, Nonlinear, and Soft Matter Physics , vol. 80, no. 3, 2009.

[22] S. Nemat-Nasser, J. R. Willis, A. Srivastava, A. V. Amirkhizi, "Homogenization of periodic elastic composites and locally resonant sonic materials," Physical Review B , vol. 83, no. 10, 2011.

[23] R. V. Craster, J. Kaplunov, E. Nolde, S. Guenneau, "Bloch dispersion and high frequency homogenization for separable doubly-periodic structures," Wave Motion , vol. 49, no. 2, pp. 333-346, 2012.

[24] J. Kaplunov, A. Nobili, "The edge waves on a Kirchhoff plate bilaterally supported by a two-parameter elastic foundation," Journal of Vibration and Control , 2015.

[25] A. Nobili, A. M. Tarantino, "Magnetostriction of a hard ferromagnetic and elastic thin-film structure," Mathematics and Mechanics of Solids , vol. 13, no. 2, pp. 95-123, 2008.

[26] V. Guidi, L. Lanzoni, A. Mazzolari, "Patterning and modeling of mechanically bent silicon plates deformed through coactive stresses," Thin Solid Films , vol. 520, no. 3, pp. 1074-1079, 2011.

[27] N. Tullini, A. Tralli, L. Lanzoni, "Interfacial shear stress analysis of bar and thin film bonded to 2D elastic substrate using a coupled FE-BIE method," Finite Elements in Analysis and Design , vol. 55, pp. 42-51, 2012.

[28] A. M. Tarantino, "Homogeneous equilibrium configurations of a hyperelastic compressible cube under equitriaxial dead-load tractions," Journal of Elasticity , vol. 92, no. 3, pp. 227-254, 2008.

[29] V. A. Salomoni, C. E. Majorana, G. M. Giannuzzi, A. Miliozzi, "Thermal-fluid flow within innovative heat storage concrete systems for solar power plants," International Journal of Numerical Methods for Heat and Fluid Flow , vol. 18, no. 7-8, pp. 969-999, 2008.

[30] V. A. Salomoni, C. E. Majorana, B. Pomaro, G. Xotta, F. Gramegna, "Macroscale and mesoscale analysis of concrete as a multiphase material for biological shields against nuclear radiation," International Journal for Numerical and Analytical Methods in Geomechanics , vol. 38, no. 5, pp. 518-535, 2014.

[31] G. Dinelli, G. Belz, C. E. Majorana, B. A. Schrefler, "Experimental investigation on the use of fly ash for lightweight precast structural elements," Materials and Structures , vol. 29, no. 194, pp. 632-638, 1996.

[32] P. Bisegna, R. Luciano, "Bounds on the overall properties of composites with debonded frictionless interfaces," Mechanics of Materials , vol. 28, no. 1-4, pp. 23-32, 1998.

[33] G. W. Hill, "On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon," Acta Mathematica , vol. 8, no. 1, pp. 1-36, 1886.

[34] V. I. Smirnov A Course of Higher Mathematics: Volume 2, Advanced Calculus , Pergamon Press, 1964.

[35] J. Kaplunov, A. Nobili, "Multi-parametric analysis of strongly inhomogeneous periodic waveguideswith internal cutoff frequencies," Mathematical Methods in the Applied Sciences , 2016.

[36] R. Luciano, J. R. Willis, "Non-local constitutive equations for functionally graded materials," Mechanics of Materials , vol. 36, no. 12, pp. 1195-1206, 2004.

[37] R. Barretta, L. Feo, R. Luciano, "Some closed-form solutions of functionally graded beams undergoing nonuniform torsion," Composite Structures , vol. 123, pp. 132-136, 2015.

[38] R. Barretta, L. Feo, R. Luciano, F. Marotti de Sciarra, "A gradient Eringen model for functionally graded nanorods," Composite Structures , vol. 131, pp. 1124-1131, 2015.

[39] R. Barretta, L. Feo, R. Luciano, F. Marotti de Sciarra, "Variational formulations for functionally graded nonlocal Bernoulli-Euler nanobeams," Composite Structures , vol. 129, pp. 80-89, 2015.

[40] R. Barretta, R. Luciano, "Exact solutions of isotropic viscoelastic functionally graded Kirchhoff plates," Composite Structures , vol. 118, no. 1, pp. 448-454, 2014.

[41] A. Apuzzo, R. Barretta, R. Luciano, "Some analytical solutions of functionally graded Kirchhoff plates," Composites Part B: Engineering , vol. 68, pp. 266-269, 2015.

[42] A. M. Tarantino, "On extreme thinning at the notch tip of a neo-Hookean sheet," The Quarterly Journal of Mechanics and Applied Mathematics , vol. 51, no. 2, pp. 179-190, 1998.

[43] G. Xotta, G. Mazzucco, V. A. Salomoni, C. E. Majorana, K. J. Willam, "Composite behavior of concrete materials under high temperatures," International Journal of Solids and Structures , vol. 64, pp. 86-99, 2015.

[44] F. M. de Sciarra, M. Salerno, "On thermodynamic functions in thermoelasticity without energy dissipation," European Journal of Mechanics-A/Solids , vol. 46, pp. 84-95, 2014.

[45] R. Barretta, L. Feo, R. Luciano, "Torsion of functionally graded nonlocal viscoelastic circular nanobeams," Composites Part B: Engineering , vol. 72, pp. 217-222, 2015.

[46] L. Dezi, G. Menditto, A. M. Tarantino, "Homogeneous structures subjected to repeated structural system changes," Journal of Engineering Mechanics , vol. 116, no. 8, pp. 1723-1732, 1990.

[47] L. Dezi, A. M. Tarantino, "Time-dependent analysis of concrete structures with a variable structural system," ACI Materials Journal , vol. 88, no. 3, pp. 320-324, 1991.

[48] L. Dezi, G. Menditto, A. M. Tarantino, "Viscoelastic heterogeneous structures with variable structural system," Journal of Engineering Mechanics , vol. 119, no. 2, pp. 238-250, 1993.