Strongly Nonlinear Waves in 3d Phononic Crystals

Three dimensional phononic crystal ("sonic vacuum" without prestress) was assembled from 137 vertical cavities arranged in hexagonal pattern in Silicone matrix filled with stainless steel spheres. This system has unique strongly nonlinear properties with respect to wave propagation inherited from nonlinear Hertz type elastic contact interaction. Trains of strongly nonlinear solitary waves excited by short duration impact were investigated. Solitary wave with speed below sound speed in the air and reflection from the boundary of two "sonic vacuums" were detected. INTRODUCTION Linear elastic phononic crystals are materials with a periodic structure causing acoustic band gap [1,2]. An approach for modeling of compressional waves in weakly nonlinear phononic materials “the phononic lattice solid with fluids (PLSF)” at the microscopic scale was proposed in [3]. This paper presents the results on wave dynamics in strongly nonlinear [4] phononic crystals based on granular chains in a Silicone elastomer or Teflon matrix. STRONGLY NONLINEAR WAVES Non-classical wave behavior appears if a chain of grains is “weakly” compressed [4]. The principal difference between this case and the "strongly" compressed chain is due to a lack of a small parameter with respect to a wave amplitude in the former case. Long wave equation for displacement u in this case is: (1) Here c is not a sound speed, instead c0 is a sound speed corresponding to initial strain ξ0 . This equation has no characteristic wave speed independent on amplitude (equation for general interaction law can be found in [4]). Despite its complex nature the equation has simple stationary solutions with unique properties. For example, supersonic solitary wave propagates with a speed Vs depending on the ratio ξr of initial ξo and maximum ξm strains: (2) This strongly nonlinear solitary wave is of a fundamental interest because Eq. 1 is more general than weakly nonlineaar KdV equation. In a system moving with a speed Vp, its periodic solution is represented by a sequence of humps (ξo=0) [4]: (3) Solitary shape can be taken as one hump of periodic solution (it has only two harmonics) with finite length equal five particle diameters. This unique wave was observed in numerical calculations and detected in experiments [4]. Solitary wave can be considered as a quasiparticle with mass equal about 1.4 mass of grain in the chain and its speed Vs has a nonlinear dependence on maximum strain ξm (or particle velocity υm): Downloaded 14 Aug 2007 to 132.239.202.236. Redistribution subject to AIP license or copyright, see http://proceedings.aip.org/proceedings/cpcr.jsp (4) We may see that the speed of this wave can be infinitely small if the amplitude is small! It means that using this material as a matrix in NTPC (Nonlinear, Tunable Phononic Crystals) we can ensure infinite elastic contrast of components, important for monitoring of band gaps. At the same time speed of solitary waves can be considered as constant at any relatively narrow interval of amplitudes due to power law dependence with small exponent. These properties allow using NTPCs as effective delay lines with exceptionally low speed of signal. Simple estimation based on Eq. 4 shows that it is possible to create materials with impulse speed in the interval 10 – 100 m/s corresponding to the amplitude of audible signal. EXPERIMENTAL RESULTS AND DISCUSSION We processed a 3-D phononic crystal (Fig. 1) based on a Silicone elastomer matrix filled with one-dimensional chains of steel spheres. We will present a results describing how these chains support waves of different amplitude and duration. In experiments we measured the force between the bottom plate and the last particle in the chain resting on this plate (Fig. 2). Piezoelectric gauges were placed under the plates of different diameters allowing support one or seven chains. They were connected with a wave guide a long steel rod with a length about 20 cm embedded into the massive steel block. Typical time of the electric circuit of the gauge was RC=10 μs which was enough to ensure a good quality of signals with characteristic period up to 100 μs. The gauges were calibrated using impact with the parallel detection of acceleration. One dimensional testing was performed using chains of balls placed in Teflon or Silicone elastomer matrixes to investigate how it may influence wave propagation in the chains (Fig. 3). From Fig. 3 a remarkable feature of "sonic vacuum" is evident – very rapid decomposition of initial impulse on the distances comparable with the soliton width. In fact, the impulse is split after traveling only through 20 particles. This example also demonstrates that “short” duration impact on highly nonlinear ordered periodic systems (lattices) with weak dissipation may result in a chain of solitary waves instead of intuitively expected shock wave. Increase of the duration of impact results in shock wave with oscillatory structure where the leading pulse can be KdV-type for weakly nonlinear chain or compacton-like for strongly nonlinear case [4]. This property of strongly nonlinear phononic crystal can be used for controlled impulse transformation in relatively short transmission lines. If chains of grains are placed into a polymer matrix the nonlinear elastic behavior is accompanied by strong dependence of electrical resistivity on local pressure [5]. This behavior can result in a new phenomena like train of locally conductive solitary waves. Single solitary wave can be generated in the strongly nonlinear chains under impact of particles (pistons) with a mass equal or smaller than mass of particle in the chain [4]. Figure 1. 3-D phononic crystal Figure 2. Set-up for testing of 1-D chain. m = 0.03g, steel