Mechanics Research Communications 103 (2020) 103464

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Mechanics Research Communications

journal homepage: www.elsevier.com/locate/mechrescom

On nonlinear modeling of an acoustic

∗ A.V. Porubov a,b, , E.F. Grekova a a Institute for Problems in Mechanical Engineering, Bolshoy 61, V.O., Saint-Petersburg, Russia b Peter the Great St. Petersburg Polytechnic University (SPbPU), Polytechnicheskaya st., 29, Saint Petersburg, Russia

a r t i c l e i n f o a b s t r a c t

Article history: Two kinds of nonlinearity are studied in the framework of a model of a discrete diatomic model of an Received 30 August 2019 . It is shown that the continuum limit of a discrete model is similar to the equa-

Revised 21 September 2019 tions obtained using a model of a reduced continuum with a microstructure. An asymptotic approach is Accepted 29 November 2019 developed to obtain a modulation nonlinear governing equation for dynamical processes in an acoustic Available online 6 December 2019 metamaterial. New metamaterial features caused by nonlinearity are found.

Keywords: © 2019 Elsevier Ltd. All rights reserved. Metamaterial Nonlinear wave Nonlinear elasticity

1. Introduction be spatial Kirchhoff rods whose mechanical response is controlled by a deformation field and a rotation field. This leads naturally The interest to the study of the acoustic has to a model based on the Cosserat elasticity. Rigidity constraints grown considerably in the recent years, see [1–6] and references are introduced that effectively reduce the model to a variant of a therein. The linear acoustic metamaterials draw more attention second-gradient elasticity theory [3,4] . A metamaterial is modeled [2,3,7,8] , however, more and more works devoted to the nonlin- in [7] where a homogenization approach based on the asymptotic ear acoustic metamaterials appear in the last years [2,9–13] . An analysis establishes a connection between the different character- influence of nonlinearity on the band gap, effective istics at micro- and macroscales like in composite materials with negative modulus and density allows to describe new features of the inclusion embedded in the matrix. A metamaterial model stud- metamaterials. The nonlinearities shift the double negative pass- ied in [8] , consists of periodic cylindrical cavities carried out from band into the adjacent modulus single negative forbidden band an elastic matrix. In [14] it is shown that some classes of reduced and transform the metamaterial from an acoustic insulator into continua , e.g. media, whose strain energy depends on the gener- an acoustic conductor [12] . Novel nonlinear phenomena affecting alized co-ordinates, but does not depend on the gradient of part the bandwidth are discovered in [9,10] . The dependent of them, are the acoustic metamaterials. Continuum modeling of frequency band gaps are noted in [1,2,12] . media with a microstructure [14,15] gives rise to the linear model Metamaterials are usually studied on the basis of discrete lat- equations whose solutions allow frequency bandgaps (single neg- tice models. Theoretical and experimental description of acous- ative acoustic metamaterials) and/or decreasing parts of the dis- tic metamaterials on the basis of mas-in mass discrete system is persion curves (double negative acoustic metamaterials). Contin- shown in [2] where nonlinearity is introduced using a nonlinear uum models are usually linear. At the same time nonlinear con- rigidity of the internal spring in the model. The Helmholtz res- tinuum generalization of the Cosserat model, see, e.g., [16,17] , re- onators and membranes periodically distributed along a pipe are sults in derivation of the governing nonlinear dispersion equations considered in [12] . In [13] an acoustic metamaterial model contains which model an influence of a microstructure and nonlinearity on a pure Duffing oscillator shell with the internal oscillator. Various the dispersion properties of a material. A similarity between the metamaterial lattice models are considered in [9,10] . description of metamaterials and composites allows to employ the Continuum modeling have been developed in [3,7,8,14,15] . A approaches used for nonlinear description of the composite mate- three-dimensional continuum theory for fibrous mechanical meta- rials, see, e.g., [19] materials is proposed in [3] , in which the fibers are assumed to In this paper we study a nonlinear metamaterial model fol- lowing both from a discrete consideration [9,10] and a continuum modeling of a microstructure [14,15] . Two kinds of nonlinearity are

∗ introduced in the model. An asymptotic transformation of the orig- Corresponding author at: Institute for Problems in Mechanical Engineering, Bol- shoy 61, V.O., Saint-Petersburg, Russia. inal equations results in a nonlinear governing modulation equa- E-mail address: [email protected] (A.V. Porubov). tion for a nonlinear acoustic metamaterial. https://doi.org/10.1016/j.mechrescom.2019.103464 0093-6413/© 2019 Elsevier Ltd. All rights reserved. 2 A.V. Porubov and E.F. Grekova / Mechanics Research Communications 103 (2020) 103464

2. Statement of the problem where ( ∗) is a complex conjugate. Substitution of Eq. (7) into Eqs. (5) and (6) allows us to resolve A from Eq. (6) , Consider a discrete problem of a di-atomic chain studied pre- β − ω 2 ) = 1 . viously in [9,10]. Let us modify this model be an inclusion of an A β B (8) additional nonlinearity caused by the non-Hookean interactions of 1 the springs between masses m in the chain. The additional masses Eq. (7) with the use of Eq. (8) gives rise to the dispersion relation, m1 are attached by springs to each mass m in the chain, see [9,10]. β ( + η) + β 2 2 2 1 1 0 p h Masses m1 do not interact directly between themselves, thus ω = ± corresponding on the continuum level to the distributed dynamic 2 absorber, while interacting masses m, after continualization, will  1 correspond to the bearing continuum [14] . The displacement of (β (1 + η) + β p2 h 2 ) 2 − 4 β β p2 h 2. (9) 2 1 0 0 1 the mass, m , with the number n is denoted by x n , while that of

ε2 m1 is denoted by yn . Then the discrete equations of motion are Next order equations at are

2 = β 2 + ηβ ( − ) − x¨ n = β0 (x n −1 − 2 x n + x n +1 ) + ηβ1 (y n − x n ) + ηβ2 (y n − x n ) + u1 ,tt 0 h u1 ,xx 1 v1 u1

2 2 2 2 2 u , + 2 β h u , + ηβ ( − u ) + 2 β hu , u , , (10) β3 ((x n +1 − x n ) − (x n −1 − x n ) ) , (1) 0 tT 0 0 xX 2 v0 0 3 0 x 0 xx

2 2 v 1 ,tt = −β1 (v 1 − u 1 ) − 2 v 0 ,tT − β2 (v 0 − u 0 ) . (11) y¨ n = −β1 (y n − x n ) − β2 (y n − x n ) . (2) The solution to Eqs. (10) and (11) is sought in the form, Here η = m 1 /m, while the linear stiffness of the spring of the β β chain is 0 m, the nonlinear stiffness is 3 m. Corresponding linear u 1 = A 1 (X, T , τ ) exp (ı (px − ωt)) + Q 1 (X, T , τ ) exp (2 ı (px − ωt))+ , β β and nonlinear stiffnesses of the attached spring are 1 m1 , 2 m1 respectively. We exclude dissipative parts in the model of [9,10] for ∗ ( ) + ( , , τ ) , simplicity. F1 X T (12) It is known that discrete nonlinear models are difficult for an analysis, and we proceed with a continuum limit of Eqs. (1) and v 1 = B 1 (X, T , τ ) exp (ı (px − ωt)) + Q 2 (X, T , τ ) exp (2 ı (px − ωt))+ (2) . Following the standard procedure we introduce the continuum functions u ( x, t ), v ( x, t ) for description of the displacements of the ∗ ( ) + ( , , τ ) . masses m, m1 with the number n while the continuum displace- F2 X T (13) ments of the neighboring masses are sought using the Taylor series Substituting Eqs. (12) and (13) into Eqs. (10) and (11) and equat- around u . Retaining only the first nonzero term in the expansion ing to zero the terms at corresponding exponents, we obtain the we obtain

coupled equations for finding A1 , B1 .F1 , F2 , Q1 , Q2 . We obtain using 2 2 utt = β0 h u xx + ηβ1 (v − u ) + ηβ2 (v − u ) + 2 β3 hu x u xx , (3) Eq. (8) 4 β ω ∗ = + 2 , F1 F2 BB (14) = −β ( − ) − β ( − ) 2, β3 vtt 1 v u 2 v u (4) 1

2 4 where h is a distance between the masses m in the chain. The lin- β − 4 ω β ω = 1 + 2 2. Q1 Q2 B (15) earized version of Eqs. (3), (4) is similar to that of the equations for β β3 1 1 a reduced Cosserat model for metamaterials obtained in [14,15] .   4 3 2 2 2 ηβ2 ω − 2 ıβ3 hp (ω − β1 ) B Q = . (16) 3. Derivation of governing nonlinear modulation equation 2 4 2 2 2 2 2 β1 (16 ω − 4[ β1 (1 + η) + 4 β0 h p ] ω + 4 β0 β1 h p )

( ( − ω )) We consider a weakly nonlinear case. For this purpose a small However, equating terms at exp ı px t to zero in parameter ε is introduced, and the solution is sought in the form Eqs. (10) and (11) does not result in the solutions for A1 , B1 due to Eq. (9) . Indeed, it follows from Eq. (11) that the terms at

= ε + ε 2 + ε 3 + . . . , u u0 u1 u2 exp (ı (px − ωt)) being equating to zero, give rise to the equation

2 (β1 − ω ) B 1 − β1 A 1 − 2 ıωB T = 0 , (17) 2 3 v = ε v + ε v + ε v + . . . . 0 1 2 whose solution is = Moreover we introduce fast and slow variables, so as ui β − ω 2 2 ıω 1 ( , , , , τ ) , = ( , , , , τ ) , = ε , = ε , τ = ε 2 A 1 = B 1 − B T . (18) ui x t T X vi vi x t T X T t X x t. Then β1 β1 we obtain coupled linear equations from Eqs. (3) and (4) at order ( ( − ω )) ε, Substitution of Eq. (18) into equating to zero exp ı px t part of Eq. (10) , 2 u , = β h u , + ηβ ( − u ) , (5) 0 tt 0 0 xx 1 v0 0 2 2 2 2 (β0 h p + ηβ1 − ω ) A 1 − β1 ηB 1 = 2 ı (ωA T + β0 h pA X ) , (19)

results in the equation for B. The part proportional to B1 disappears v , = −β (v − u ) . (6) 0 tt 1 0 0 because of Eq. (9) and we obtain from Eq. (19) ,

The solution to Eqs. (5) and (6) is sught in the form, 2 2 2 2 2 ω(β1 (1 + η) − 2 ω + β0 h p ) B T + (β1 − ω ) β0 h pB X = 0 . (20) ∗ = ( , , τ ) ( ( − ω )) + ( ) , u0 A X T exp ı px t Then B = B (θ, τ ) where θ = X − W T ,

(β − ω 2 ) β h 2 p ∗ W = 1 0 , (21) v 0 = B (X, T , τ ) exp (ı (px − ωt)) + ( ) , (7) 2 2 2 ω(β1 (1 + η) − 2 ω + β0 h p ) A.V. Porubov and E.F. Grekova / Mechanics Research Communications 103 (2020) 103464 3 and the solution at this order does not contain secular or growing For the nonlinear Schr odinger¯ equation, α2 = 0 , or only one of β β terms. nonlinearities is taken into account in the model, either 2 or 3 To complete the solution and define B , we consider at order ε3 is zero. In this case bright or dark solitary wave solution exists de- ( ( − α γ only the problem of suppression the secular terms at exp ı px pending on the sign of the product 1 , ωt)) . Assume that   2 α ∗ 1 = ( , , τ ) ( ( − ω )) + ( ) + . . . , P = a sech − a (θ − 2 γ qτ ) , ϕ = q (θ − 2 γ qτ ) , u 2 A2 X T exp ı px t γ   ∗ = ( , , τ ) ( ( − ω )) + ( ) + . . . 2 α v2 B2 X T exp ı px t = − 1 (θ − γ τ ) , ϕ = (θ − γ τ ) . P a tanh γ a 2 q q 2 q Then the same as before algebraic manipulations with the terms at exp (ı (px − ωt)) result in the equation for B The solution describes a dependence between the amplitude and

2 ∗ the wave number, also the shape of modulation of the harmonic ıB τ + (α + ıα ) B B + γ B θθ = 0 , (22) 1 2 wave profile is defined by the sign of the equation coefficients. This where is not covered within the linearized consideration. These solutions  − α 4 2 2 2 1 do not exist in the case of nonzero 2 corresponding to a degen- α = β ω[ β (η − 2) + 3 ω − β h p 1 1 1 2 erate case of the Ginzburg- Landau equation (GLE) see [18] and α references therein. Nonzero 2 appears due to a mutual influence  − 2 2 2 2 2 2 1 of nonlinearities of the chain, β  = 0, and of the internal oscilla- ω [ β (1 + η) + 4(β h p β − ω )] − β h p β 3 1 0 1 0 1 β  = tor, 2 0. The solutions similar to the known localized solutions  to the GLE require complex coefficient at the dispersion term and

β2β2 2 6 (β − ω 2 )(β − ω 2 ) − β2ω 6( β η + ω 2 − β 2 2 ) the presence of the linear term proportional to B in Eq. (22) . This 2 1 3 h p 1 4 1 2 2 2 1 0 h p may be achieved by further modification of the original model by  including the dissipative factors in it. 4 2 2 2 The approach developed for obtaining governing nonlinear [12 ω − β1 ω (3 + 4 η) + 3 β0 h p (β1 − 4 ω 2 )] , equations in the paper may be used for a modeling of more com-   plicated metamaterials like those based on a pantographic model −1 α = β4ω β (η − ) + ω 2 − β 2 2 [20,21] . 2 1 [ 1 2 3 2 h p

 − Declaration of Competing Interest 2 2 2 2 2 2 1 ω [ β1 (1 + η) + 4(β0 h p β1 − ω )] − β0 h p β1 There is no any conflict of interest. 

β β β 3 ω 4(β − ω 2 ) ω 2 − β2η − β ω+ 1 2 3 hp 1 [20 7 1 8 1 Acknowledgment

 2 2 2 The work has been supported by the Russian Foundation for Ba- 4 β h p (2 β − 5 ω )] , 0 1 sic Researches, grant No 17-01-00230-a.

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