Journal of the Mechanics and Physics of Solids 105 (2017) 179–198 Contents lists available at ScienceDirect Journal of the Mechanics and Physics of Solids journal homepage: www.elsevier.com/locate/jmps A hybrid elastic metamaterial with negative mass density and tunable bending stiffness ∗ Yangyang Chen a, Gengkai Hu b, Guoliang Huang a, a Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211, USA b School of Aerospace Engineering, Beijing Institute of Technology, Beijing 10 0 081, China a r t i c l e i n f o a b s t r a c t Article history: Achieving vibration and/or wave attenuation with locally resonant metamaterials has at- Received 16 February 2017 tracted a great deal of attention due to their frequency dependent negative effective mass Revised 29 April 2017 density. Moreover, adaptive phononic crystals with shunted piezoelectric patches have also Accepted 15 May 2017 demonstrated a tunable wave attenuation mechanism by controlling electric circuits to Available online 17 May 2017 achieve a negative effective stiffness. In this paper, we propose an adaptive hybrid meta- Keywords: material that possesses both a negative mass density as well as an extremely tunable Elastic metamaterial stiffness by properly utilizing both the mechanical and electric elements. A multi-physical Tunable bending stiffness analytical model is first developed to investigate and reveal the tunable wave manipula- Programmable wave control tion abilities in terms of both the effective negative mass density and/or bending stiff- ness of the hybrid metamaterial. The programmed flexural wave manipulations, broad- band negative refraction and waveguiding are then illustrated through three-dimensional (3D) multi-physical numerical simulations in hybrid metamaterial plates. Our numerical results demonstrate that the flexural wave propagation can essentially be switched be- tween “ON/OFF” states by connecting different shunting circuits. ©2017 Elsevier Ltd. All rights reserved. 1. Introduction Acoustic/elastic metamaterials are artificially engineered materials composed of subwavelength microstructures that be- have like a continuous medium with unusual effective properties. For example, the assembly of dipolar locally resonant meta-structures into a bulk medium exhibited a peculiar negative effective mass density ( Yu et al., 2006; Oudich et al., 2010; Chen et al., 2014a ). Furthermore, a negative effective modulus can also be realized by embedding monopolar (ex- pansion/compression) resonators inside a host medium ( Fang et al., 2006; Lee et al., 2009 ). Based on this concept, many acoustic/elastic metamaterials are designed and viewed as promising candidates for dynamic applications including low- frequency vibration shielding, guided wave attenuation, seismic wave mitigation, mechanical wave guiding, absorption and energy harvesting ( Ma and Sheng, 2016; Cummer et al., 2016; Haberman and Guild, 2016; Carrara et al., 2013; Miniaci et al., 2016; Chen et al., 2016a ). Because of their enhanced range of dynamic material properties, acoustic/elastic metamaterials possessing double-negative properties have been realized by designing microstructural units with built-in monopolar and dipolar resonators, respectively, for emerging applications such as elastic wave superlenses, subwavelength imaging and acoustic cloaking ( Liu et al., 2011; Zhu et al., 2014; Wu et al., 2011 ). Unfortunately, the microstructural designs required to achieve double negative properties are notably more complicated than their single negative-property counterparts. The main ∗ Corresponding author. E-mail address: [email protected] (G. Huang). http://dx.doi.org/10.1016/j.jmps.2017.05.009 0022-5096/© 2017 Elsevier Ltd. All rights reserved. 180 Y. Chen et al. / Journal of the Mechanics and Physics of Solids 105 (2017) 179–198 challenge consists of identifying a single building-block that exhibits multiple types of resonances simultaneously. The major limitation of the current acoustic/elastic metamaterial designs with mechanically resonant mechanisms is that the frequency range of the single and/or double negative properties is fixed and limited to a narrow frequency band. This inherent limita- tion makes their use in practical engineering applications difficult. One of the most significant challenges in acoustic/elastic metamaterial development is the ability to tune their performance without requiring physical microstructural modifications. There has been a growing effort to explore adaptive or active acoustic/elastic metamaterial designs capable of overcoming the challenges described above and increase their effectiveness in relevant applications. The terms ‘adaptive’ or ‘active’ are used quite generally to denote a meta-structure that can actively report the energy characteristics of an impinging wave and supply feedback to the elastic/acoustic control system. By integrating adaptive or smart materials into the meta-structure, the effective properties of the adaptive metamaterial can be tuned or adapted electronically for increasingly complex sys- tems. Among currently available adaptive materials, shunted piezoelectric materials provide an ideal platform to tune and control elastic/acoustic properties of a metamaterial in a compact way. This is because of their strong response to electrical signals which can be controlled with relatively simple electronics ( Collet et al., 2012; Airoldi and Ruzzene, 2011; Chen et al., 2013 ). The piezoelectric effect allows energy to be exchanged between the mechanical and electrical domains so that, when a piezoelectric element is shunted through an electrical circuit, it behaves as a variable stiffness element that can be used to convert mechanical energy. In a more general representation, this coupled electro-mechanical mechanism can be employed to alter the dynamic acoustic and mechanical behavior of the structure. The piezoelectric shunting technique was first intro- duced by Forward (1979) to damp mechanical vibrations in optical systems, while Hagood and von Flotow (1991) provided the first analytical formulation for the passive electrical shunts. Since their pioneering works, an array of electrical shunting designs have been proposed to improve the acoustic and dynamic mitigation performance of structures. In these works, piezoelectric components connected to inductor-resistor (LR) shunt circuits appear as a promising passive technique for noise and vibration control in dynamic structures ( Airoldi and Ruzzene, 2011; Chen et al., 2013; Forward, 1979; Hagood and von Flotow, 1991; Bergamini et al., 2014; Casadei et al., 2012 ). This type of shunt generates an electrical resonance in the piezoelectric patch. If this electrical resonance is tuned to one of the structural modal frequencies, a considerable damping effect of the corresponding mode can be achieved. Generally, the resonant shunts with simple LR components suffer from a drawback because their mitigation performance is limited to a relatively narrow frequency range and therefore very sensitive to parameter variations in the system. To overcome this problem, many efforts on piezoelectric metastructures with resonant shunts have been made to enlarge the wave attenuation region by introducing electric transmission line ( Bergamini et al., 2015 ), implementing amplifier-resonator feedback circuits ( Wang and Chen, 2016 ) and networks of different resonant shunts ( Cardell et al., 2016; Thorp et al., 2001 ). As another type of shunting strategy to enlarge the wave attenuation region, nega- tive capacitance shunts have been proposed and investigated ( Collet et al., 2012; Beck et al., 2011; Tateo et al., 2014 ). Theo- retically, the negative capacitance of the shunting circuit eliminates the piezoelectric patch’s inherent capacitance such that the remaining resistance can optimally convert acoustic and/or elastic wave energy in a broadband frequency region. The applications of periodic piezoelectric patches with negative capacitance shunts have also been proposed for various spatial wave control, which include the flexural wave focusing ( Yi et al., 2016 ), adaptive wave filtering with tunable Kirigami lat- tices ( Ouisse et al., 2016 ), and wave directionality manipulations ( Celli and Gonella, 2015; Zhu and Deng, 2016 ). In addition, Chen et al. (2014b) implemented the negative capacitance piezoelectric shunting method into meta-structures to actively tune the locally resonant frequencies of mechanical resonators by modifying the stiffness of the resonant microstructure. By controlling the resonance point of the mechanical resonators, the resulting band gap frequency region of the elastic meta- material can then be actively controlled. Based on this concept, Zhu et al. (2016) designed an adaptive elastic metamaterial where harmonic vibration as well as transient wave propagation tests were conducted on the fabricated metamaterial sam- ples to illustrate their tunable vibration suppression and elastic wave attenuation abilities. More recently, by integrating hybrid shunted piezoelectric stacks with self-adaptive/tunable frequency-dependent stiffness into the resonant microstruc- tures, extremely high- and low-pass filtering capabilities of an adaptive metamaterial were demonstrated both analytically and numerically ( Chen et al., 2016b ). However, in the previous active/adaptive metamaterial design, attention is focused primarily on the frequency tunability of the negative mass density by introducing variable stiffness components into the resonators. In