PHYSICAL REVIEW E 91, 033208 (2015) Wave propagation in granular chains with local resonances Luca Bonanomi,1 Georgios Theocharis,2,3 and Chiara Daraio1,2 1Department of Mechanical and Process Engineering, Swiss Federal Institute of Technology (ETH), Zurich¨ 8092, Switzerland 2Graduate Aerospace Laboratories (GALCIT), California Institute of Technology, Pasadena, California 91125, USA 3LAUM, CNRS, Universite´ du Maine, Avenue O. Messiaen, 72085 Le Mans, France (Received 5 March 2014; revised manuscript received 14 October 2014; published 31 March 2015) We study wave propagation in a chain of spherical particles containing a local resonator. The resonant particles are made of an aluminum outer spherical shell and a steel inner mass connected by a polymeric plastic structure acting as a spring. We characterize the dynamic response of individual particles and the transmitted linear spectra of a chain of particles in contact. A wide band gap is observed both in theoretical and experimental results. We show the ability to tune the acoustic transmission by varying the contact interaction between particles. Higher driving amplitude leads to the generation of nonlinearities both in the response of a single particle and that of the whole chain. For a single resonant particle, we observe experimentally a resonant frequency downshift, which follows a complex nonlinear behavior. In the chain of particles, nonlinearity leads to the generation of nonlinear harmonics and the presence of localized modes inside the band gap. DOI: 10.1103/PhysRevE.91.033208 PACS number(s): 43.20.+g, 45.70.−n, 46.40.Cd, 43.35.+d I. INTRODUCTION ical metamaterials. We study experimentally the response of individual resonant particles and the wave propagation through Artificial materials with a designed geometrical structure a chain of resonant particles in contact, exploring the linear that supports localized resonances have been shown to present and nonlinear dynamic regimes. We support the experimental unusual behaviors with respect to electromagnetic [1] and findings in the linear regime with theoretical models. Similar acoustic wave propagation [2]. These media, referred to as granular chains with external resonant masses were described metamaterials, give rise to many novel phenomena, including earlier, but their analysis was limited to the linear dynamic subwavelength focusing and cloaking [3,4]. They also set the regime [23]. The fundamental characterization of granular basis for the design of next-generation devices with tunable and chains with internal resonators is useful for their potential switchable functionalities [5]. Many of these functionalities application in novel materials. Due to their simple geometry, can be realized due to nonlinear processes emerging when particles with internal resonators are suitable to be embedded metamaterials are excited by external stimuli [5]. in a matrix, for example, in a composite material [24]. In acoustics, periodic, locally resonant acoustic metamateri- The rest of the manuscript is organized as follows: We als [6] derive their unique properties from the presence of local describe the experimental setup in Sec. II. The theoretical resonators in the fundamental unit cells. Acoustic metamate- approaches used to describe the linear behavior of the rials have been demonstrated with periodic arrangements of system are summarized in Sec. III. Section IV presents and coated spheres or cylinders embedded in a linear homogeneous discusses the results, and it is divided in two subsections: host medium [7] and with arrays of Helmholtz resonators [8]. Sec. IV A discusses the results obtained in the linearized Currently, a great amount of work on acoustic metamaterials regime and Sec. IV B discusses results obtained at higher is devoted to the study of wave propagation in materi- driving amplitude, in the nonlinear regime. Section V includes als with near-zero density and zero-refractive index [9–11] a summary of the main findings and a concluding paragraph. and nonreciprocal acoustic devices [12,13]. Extraordinary transmission was obtained using density-near-zero ultranarrow channels [14] or through walls perforated with subwavelength II. EXPERIMENTAL APPROACH holes [15]. However, in contrast to the electromagnetic counterpart, which includes numerous studies on nonlinear We fabricate the resonant particles [Fig. 1(a)] combining an behavior, research on acoustics metamaterials has been mostly outer mass (a spherical shell), an inner mass (a solid sphere), limited to the linear dynamical regime. and an elastic structure between the two. The outer mass is an In the realm of nonlinear acoustic crystals, a great amount aluminum hollow shell (alloy 3003, diameter = 18.6mm,wall of work has been devoted to the study of granular chains, which thickness = 0.8 ± 0.1mm,massm1 = 2.3 ± 0.05 g, Young’s are closely packed, ordered arrangements of elastic particles modulus = 69 GPa, Poisson’s ratio = 0.33). The inner mass (spheres in most of the cases) in contact. Due to the Hertzian consists of a solid stainless steel bead (316 type, diameter = contact interaction between the particles [16], the dynamic 12.8mm, m2 = 8.6 ± 0.1 g, Young’s modulus = 193 GPa, vibrational response of these structures can be nonlinear and Poisson’s ratio = 0.3). The reported values of Young’s moduli tunable [17,18]. This makes granular chains a perfect example and Poisson’s ratios are standard specifications [25]. The for the study of fundamental phenomena [19] and engineering elastic structure (spring) connecting the two masses is made applications including tunable vibration filters [20], acoustic of a commercial plastic (Verowhite Plus, by Stratasys) having lenses [21], and acoustic rectifiers [22]. a Young’s modulus of 2500 MPa [26] and a mass of 0.2 g. In this work, we combine the concepts of elastic wave The shape of the elastic spring is shown in Fig. 1(a) and was propagation through tunable nonlinear contacts with that of fabricated by three-dimensional (3D) printing (in a Stratasys internal local resonators to design tunable, nonlinear, mechan- Object Connex 500). 1539-3755/2015/91(3)/033208(6) 033208-1 ©2015 American Physical Society LUCA BONANOMI, GEORGIOS THEOCHARIS, AND CHIARA DARAIO PHYSICAL REVIEW E 91, 033208 (2015) III. THEORETICAL APPROACH The nonlinear interaction law between two elastic particles is well described by the Hertzian contact law [16]. In Hertzian contacts, the contact force Fi,i+1 between two particles (i and i + 1) relates to the relative displacement i,i+1 of their particles’ centers, as FIG. 1. (Color online) (a) Parts and assembly of a resonant unit ni,i+1 F + = A + [ + ]+ . (1) cell. (b) Schematic of the experimental setup to test the dynamic i,i 1 i,i 1 i,i 1 response of a single particle. Here, Ai,i+1 and ni,i+1 depend on the geometry and material properties of the particles in contact. Values inside the bracket To test the dynamic response of a single particle and that of [s]+ only take positive values, which denotes the tensionless a chain of particles in contact, we align and hold the particle(s) characteristic of the system (i.e., there is no force between the in a support system consisting of four polycarbonate rods, held particles when they are separated). The contact interaction in place by polycarbonate guide plates [20]. An initial static between thin hollow spheres was studied in [27]. For the compression is applied on one side by using a translating steel range of wave amplitude considered in [27], a power-law type cube. The static force applied is measured with a piezoelectric relation (F = kδn) was also used to describe the contact. It static load cell [Fig. 1(b)]. We continuously drive the particle(s) was found, however, that the exponent n was smaller than the with a piezoelectric actuator mounted on a fixed steel cube, value 3/2 found in the classical Hertzian interaction between on the opposite side with respect to the static force sensor. solid spheres. The contact stiffness k and the exponent n We record the transmitted force-time history using a dynamic were found to be dependent on the thickness of the hollow force sensor. The input signal and the transmitted signal are sphere’s shell [27]. This dependence of the contact stiffness generated and measured using a lock-in amplifier. A heavy on the shell thickness can be used in the design of particles- steel cylinder is added to the system in order to measure based metamaterials to tune their resulting transmission simultaneously the dynamic force and the static load. spectrum. The linear regime is explored exciting the system with The characteristic dynamic response of a single resonant low amplitude (approximately 10 mN peak) swept sine signal particle compressed between two fixed walls (representing the from 0.5 to 20 kHz. We compute the power spectral density actuator and the dynamic force sensor in the experimental (PSD) of the signal collected by the dynamic force sensor. setup) is expected to present two resonances, corresponding Increasing the excitation amplitude (up to approximately to the in-phase (at lower frequency) and to the out-of-phase 0.5 N peak) it is possible to investigate the nonlinear (at higher frequency) motion of the outer and inner masses. regime, as shown in the Results and Discussion section. Ignoring dissipation, these eigenfrequencies are given by 2 1 2k0m2 + k2(m1 + m2) ± −8k0k2m1m2 + [2k0m2 + (m1 + m2)k2] f1,2 = . (2) 2π 2m1m2 Here k0 describes the linearized contact stiffness [28] between two masses, plays the role of a linear, damped spring with the outer shell and the steel wall, while k2 represents the an effective stiffness k2 and a viscous damping coefficient effective stiffness of the inner spring. To describe the dynamics η. We assume that most of the dissipation in our physical of the locally resonant granular chain considered in this work, system occurs inside each unit (introduced by the polymeric we adopt a lumped-element numerical approach.