behavior) is that the ratio of the wave amplitude to the initial precompression is not a small ... with the finite width of 5 particle diameters and ratio of solitary wave ...
TUNABLITY OF SOLITARY WAVE PROPERTIES IN ONE DIMENSIONAL STRONGLY NONLINEAR PHONONIC CRYSTALS
C. Daraio1, V. F. Nesterenko1,2*, E. B. Herbold2, S. Jin1,2
Materials Science and Engineering Program1 Department of Mechanical and Aerospace Engineering2 University of California at San Diego, La Jolla CA 92093-0418 USA
Abstract.
One dimensional strongly nonlinear phononic crystals were assembled from
chains of PTFE (polytetrafluoroethylene) and stainless steel spheres with gauges installed inside the beads. Trains of strongly nonlinear solitary waves were excited by an impact. A significant modification of the signal shape and an increase of solitary wave speed up to two times (at the same amplitude of dynamic contact force) were achieved through a noncontact magnetically induced precompression of the chains.
Data for PTFE based chains are
presented for the first time and data for stainless steel based chains were extended into a smaller range by more than one order of magnitude than previously reported. Experimental results were found to be in reasonable agreement with the long wave approximation and with numerical calculations based on Hertz interaction law for discrete chains.
PACS numbers: 05.45.Yv, 46.40.Cd, 43.25.+y, 45.70.-n
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INTRODUCTION One dimensional strongly nonlinear systems composed by chains of different granular materials are presently a very active area of research because they represent a natural step from weakly nonlinear to strongly nonlinear wave dynamics [1-39]. These systems permit the unique possibility of tuning the wave propagation behavior from linear, to weakly nonlinear and further to strongly nonlinear regimes [19]. In this paper we demonstrate that through simple noncontact magnetically induced precompression it is possible to tune the wave propagation response of the system from the strongly nonlinear to the weakly nonlinear regime. This allows a fine control over the propagating signal shape and speed with an adjustable precompressive force. Novel applications in different areas may arise from understanding the basic physics of these tunable strongly nonlinear 1-D systems, especially at a low amplitude range of stresses corresponding to signals used in ultrasonic diagnostics or in the audible range. For example, tunable sound focusing devices (acoustic lenses), tunable acoustic impedance materials, sound absorption layers and sound scramblers are among the most promising engineering applications [37]. The non-classical, strongly nonlinear wave behavior appears in granular materials if the system is “weakly” compressed with a force F0 [1,2,19]. The term “weakly” is used when the precompression is very small with respect to the wave amplitude. The principal difference between this case and the "strongly" compressed chain (approaching linear wave
2
behavior) is that the ratio of the wave amplitude to the initial precompression is not a small parameter as it was in the latter case. A supersonic solitary wave that propagates with a speed Vs in a “weakly” compressed chain with an amplitude much higher than the initial precompression can be closely approximated by one hump of a periodic solution corresponding to a zero prestress (ξ0=0). This exact solution has a finite length equal to only five particle diameters for a Hertzian type of contact interaction [1,17,19]. In the continuum approximation the speed of this solitary wave Vs has a nonlinear dependence on maximum strain ξm, which translates to a nonlinear dependence on maximum force Fm between particles in discrete chains. When static precompression (ξ0) is applied the speed of a solitary wave Vs has a nonlinear dependence on normalized maximum strain ξr=ξm /ξ0 in continuum approximation or on normalized force fr=Fm/F0 in discrete chain with beads of diameter a, bulk density ρ, Poisson's ratio ν and Young’s modulus E (Eq. (1), [19,37]): 4 5 Vs = c0 3 + 2 ξ r 2 − 5ξ r (ξ r − 1) 15 1
1
2
4 E 2 F0 = 0 . 9314 a 2ρ 3 1 −ν
(
)
2 2
1
6
1 f 2 3 r
4 5 2 3 + 2 f r 3 − 5 f r 3 − 1 15
1
2
.
(1)
The sound speed c0 in a chain precompressed with a force F0 can be deduced from Eq. (1) at the limit for fr=1: 3 c0 = 2
1
2
1
c ξ0 4
2E = 0.9314 3 aρ 2 1 − ν 2
(
)
1
3 1
F0 6 ,
(2)
where the constant c is
3
2E
c =
πρ (1 − ν
2
)
.
(3)
Equation (1) also allows the calculation of the speed of weakly nonlinear solitary waves which is the solution of the Korteweg-de Vries equation [19]. When fr or ξr are very large (i.e. F0 (ξ0) is approaching 0) Eq. (1) reduces to the Eq. (4) for solitary wave speed Vs in “sonic vacuum” [3,6] for continuum approximation and discrete chains respectively
Vs =
2
1
c ξm4
5
2E = 0 .6802 3 a ρ 2 1 −ν
(
2
)
1
3 1
Fm 6 .
(4)
For simplicity only the leading approximation was used to connect strains in continuum limit and forces in discrete chain in Eqs. (1) and (4). The similarity between Eq. (2) and Eq. (4) is striking (and can even be misleading) though these equations describe qualitatively different types of wave disturbances. Equation (2) represents the speed of a sound wave with an infinitely small amplitude in comparison with the initial precompression and with a long wave length. This wave is the solution of the classical linear d’Alambert wave equation. Equation (4) corresponds to the speed of a strongly nonlinear solitary wave with the finite width of 5 particle diameters and ratio of solitary wave amplitude to initial precompression equal infinity. This solitary wave is the solution of the strongly nonlinear wave equation first derived in [1]. A strongly nonlinear compression solitary waves exist for any general power law interaction between particles with an exponent in the force dependence on displacement n>1 (Hertz law is only a partial case with n=3/2) [3,5,10,18,19].
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The exponent n determines the width of the solitary wave and the dependence of its speed on the maximum strain. The corresponding equations for the speed and width of these solitary waves in continuum approximation were first derived in [3], the results of numerical calculations for discrete chains can be found in [31]. A general type of strongly nonlinear interaction law also supports the strongly nonlinear compression or rarefaction solitary waves depending on elastic “hardening” or “softening” behavior [10,19]. It is possible to find from Eqs. (1)-(4) that the speed Vs of a solitary wave propagating in a one dimensional granular media can be significantly smaller than the sound speed in the material composing the beads and can be considered approximately constant at any narrow interval of its relative amplitude fr due to the small exponent in the power law dependence. The signal speed of strongly nonlinear and linear compression waves in this condensed “soft” matter can be below the level of sound speed in gases at normal conditions as was experimentally demonstrated for PTFE (polytetrafluoroethylene) based sonic vacuum [37]. The described properties of strongly nonlinear waves may permit the use of these materials as effective delay lines with an exceptionally low speed of the signal. In stainless steel based phononic crystals, previous experimental and numerical data for the dependence of the solitary wave speed on amplitude exists only at relatively large amplitudes of dynamic forces (20 N – 1200 N) [11,15] and 2 N – 100 N [36] and for the large diameter of stainless steel beads – 8 mm and 26 mm respectively. In the present study we extended the range of experimental data to a lower amplitude range down to 0.1 N to characterize the behavior of stainless steel based strongly nonlinear systems at amplitudes
5
closer to the amplitude of the signals used in ultrasonic diagnostics and for the diameter of beads 4.76 mm. Scaling down of the size of the beads is important for the assembling of practical devices.
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RESULTS AND DISCUSSION
(a)
Al2O3 striker
(b)
Caps
Ring Magnet PTFE cylinder m = 0.5g, ø5 mm, Steel PTFE or Stainless Steel Brass cover-plate
Piezo-sensor Epoxy layer
Piezo-gauge, RC~103µs Wave-guide
FIG. 1. (a) Experimental set-up for testing of 1-D phononic materials, the magnetic particle on the top is used for magnetic tuning of “sonic vacuum”; (b) Schematic drawing of a particle with embedded piezo-sensor. Soliton parameters (speed, duration and force amplitude) and reflected pulse from the wall were measured using the experimental set-up presented in Fig. 1. One dimensional PTFE based phononic crystals were assembled in a PTFE cylinder, with inner diameter 5 mm and outer diameter 10 mm, vertically filled with chains of 20 and 52 PTFE balls (McMasterCarr catalogue) with diameter a=4.76 mm and mass 0.1226 g (standard deviation 0.0008 g) (Fig. 1(a)). PTFE has a strong strain rate sensitivity [40] and an exceptionally low elastic
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modulus in comparison with metals [41,42]. This property can be very attractive to ensure very low speed of soliton propagation and tunability of interfacial properties in laminar composite systems made from chains of different materials. These composite systems exhibit unusual properties with respect to wave reflection [8,19,26] and were proposed as shock protectors [35]. It was demonstrated that chains composed of PTFE beads do support strongly nonlinear solitary waves [37]. Also, chains made from different linear elastic materials like stainless steel, brass, glass [2,11,15,36], and viscoelastic polymeric material like Homalite 100 and nylon [7,9,15] support this type of wave. For comparison of tunability of signal speed, stainless steel based chains were assembled in the same holder from 20 stainless steel beads (316 steel, McMaster-Carr catalogue) with diameter a=4.76 mm and mass 0.4501 g (standard deviation 0.0008 g). A magnetic steel ball, with diameter a=5 mm and mass 0.5 g, was then placed on top of the PTFE or stainless steel chains to ensure magnetically induced precompression equal to the weight of the magnet (Fig.1(a)). Three calibrated piezo-sensors (RC ~103µs) were connected to a Tektronix oscilloscope to detect force-time curves. Two piezo-gauges made from lead zirconate titanate (square plates with thickness 0.5 mm and 3 mm side) with Nickel plated electrodes and
custom microminiature wiring supplied by Piezo Systems, Inc. were embedded inside two of the PTFE and two steel particles in the chains similar to [8,19,36,37]. The wired piezoelements were glued between two top parts cut from original beads (Fig. 1(b)). This design ensures a calculation of the speed of the signal propagation based on the time interval
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between maxima detected by different gauges separated by a known distance (usually it was equal to 5 particle diameters) simultaneously with the measurement of the force acting inside the particles. The speed of the pulse was related to the averaged amplitude between the two gauges. In PTFE, a typical particle with an embedded sensor consisted of two similar caps (total mass 2M=0.093 g) and a sensor with a mass m=0.023 g glued to them. The total mass of these particles was equal to 0.116 g, which is very close to the mass of the original PTFE particle (0.123 g). In theory, the introduction of a foreign element in a chain of particles of equal masses results in weak wave reflections, as observed in [12,18,20].
Numerical
calculations with single particle in the chain with a mass 0.116 g embedded into the chain of particles with mass 0.123 g created negligible effects of wave reflections [37]. Therefore such small deviation of the sensor mass from the particle mass (
