Mechanics Research Communications 103 (2020) 103464 Contents lists available at ScienceDirect Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom On nonlinear modeling of an acoustic metamaterial ∗ 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 acoustic metamaterial. 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 metamaterials 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 frequency 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 amplitude 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]. 2 2 β ( + η) + β 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 β2 ω ∗ = + , 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- β1 − 4 ω β2 ω = + 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 + .