An Introduction to Magnetohydrodynamics

Solitary waves in the granular chain

Phononic crystals (PCs) and metamaterials (MMs) can exhibit abnormal properties, even far beyond those found in nature, through artificial design of the topology or ordered structure of unit cells. This emerging class of materials has diverse application potentials in many fields. Recently, the concept of tunable PCs or MMs has been proposed to manipulate a variety of wave functions on demand. In this review, we survey recent developments in tunable and active PCs and MMs, including bandgap and bandgap engineering, anomalous behaviors of wave propagation, as well as tunable manipulation of waves based on different regulation mechanisms: tunable mechanical reconfiguration and materials with multifield coupling. We conclude by outlining future directions in the emerging field.

Tunable and Active Phononic Crystals and Metamaterials

Acoustic lenses are employed in a variety of applications, from biomedical imaging and surgery to defense systems and damage detection in materials. Focused acoustic signals, for example, enable ultrasonic transducers to image the interior of the human body. Currently however the performance of acoustic devices is limited by their linear operational envelope, which implies relatively inaccurate focusing and low focal power. Here we show a dramatic focusing effect and the generation of compact acoustic pulses (sound bullets) in solid and fluid media, with energies orders of magnitude greater than previously achievable. This focusing is made possible by a tunable, nonlinear acoustic lens, which consists of ordered arrays of granular chains. The amplitude, size, and location of the sound bullets can be controlled by varying the static precompression of the chains. Theory and numerical simulations demonstrate the focusing effect, and photoelasticity experiments corroborate it. Our nonlinear lens permits a qualitatively new way of generating high-energy acoustic pulses, which may improve imaging capabilities through increased accuracy and signal-to-noise ratios and may lead to more effective nonintrusive scalpels, for example, for cancer treatment.

https://www.pnas.org/content/pnas/107/16/7230.full.pdf

Generation and control of sound bullets with a nonlinear acoustic lens

Electrical flow control devices are fundamental components in electrical appliances and computers; similarly, optical switches are essential in a number of communication, computation and quantum information-processing applications. An acoustic counterpart would use an acoustic (mechanical) signal to control the mechanical energy flow through a solid material. Although earlier research has demonstrated acoustic diodes or circulators, no acoustic switches with wide operational frequency ranges and controllability have been realized. Here we propose and demonstrate an acoustic switch based on a driven chain of spherical particles with a nonlinear contact force. We experimentally and numerically verify that this switching mechanism stems from a combination of nonlinearity and bandgap effects. We also realize the OR and AND acoustic logic elements by exploiting the nonlinear dynamical effects of the granular chain. We anticipate these results to enable the creation of novel acoustic devices for the control of mechanical energy flow in high-performance ultrasonic devices.

/pdf/granular-acoustic-switches-and-logic-elements-15zygvtvg9.pdf

Granular acoustic switches and logic elements

It has been seen that inertial mismatches in 1D granular chains lead to remarkable energy absorption which increases with the number of spheres, $N$, and tapering, $q$. Short chains, however, are limited in that regard, and we therefore present one solution which greatly improves performance for any size chain. These strongly nonlinear and scalable systems feature surprisingly complicated dynamics and are inadequately represented by a hard-sphere approximation. Additionally, such systems have shock absorption capacities that vary as a function of position along the chain. In this Letter, we present results in the form of normalized kinetic energy diagrams to illustrate the impressive mitigation capability of both original and improved tapered chains.

Decorated, tapered, and highly nonlinear granular chain.

We present an analytical and numerical study of the problem of mechanical impulse propagation through a horizontal alignment of progressively shrinking (tapered) elastic spheres that are placed between two rigid end walls. The studies are confined to cases where initial loading between the spheres is zero (i.e., in the ``sonic vacuum'' region). The spheres are assumed to interact via the Hertz potential. Force and energy as a function of time for selected grains that comprise the solitary wave are provided and shed light on the system's behavior. Propagation of energy is analytically studied in the hard-sphere approximation and phase diagrams plotting normalized kinetic energy of the smallest grain at the tapered end are developed for various chain lengths and tapering factors. These details are then compared to kinetic energy phase diagrams obtained via extensive dynamical simulations. Our figures indicate that the ratios of the kinetic energies of the smallest to largest grains possess a Gaussian dependence on tapering and an exponential decay when the number of grains increases. The conclusions are independent of system size, thus being applicable to tapered alignments of micron-sized spheres as well as those that are macroscopic and more easily realizable in the laboratory. Results demonstrate the capabililty of these chains to thermalize propagating impulses and thereby act as potential shock absorbing devices.

Impulse absorption by tapered horizontal alignments of elastic spheres.

Rapid absorption of impulses using light-weight, small, reusable systems is a challenging problem. An axially aligned set of progressively shrinking elastic spheres, a “tapered chain,” has been shown to be a versatile and scalable shock absorber in earlier simulational, theoretical, and experimental works by several authors. We have recently shown (see R. L. Doney and S. Sen, Phys. Rev. Lett. 97, 155502 (2006)) that the shock absorption ability of a tapered chain can be dramatically enhanced by placing small interstitial grains between the regular grains in the tapered chain systems. Here we focus on a detailed study of the problem introduced in the above mentioned letter, present extensive dynamical simulations using parameters for a titanium-aluminum-vanadium alloy Ti6Al4V, derive attendant hard-sphere analyses based formulae to describe energy dispersion, and finally discuss some preliminary experimental results using systems with chrome spheres and small Nitinol interstitial grains to present the underlying nonlinear dynamics of this so-called decorated tapered granular alignment. We are specifically interested in small systems, comprised of several grains. This is because in real applications, mass and volume occupied must inevitably be minimized. Our conclusion is that the decorated tapered chain offers enhanced energy dispersion by locking in much of the input energy in the grains of the tapered chain rather than in the small interstitial grains. Thus, the present study offers insights into how the shock absorption capabilities of these systems can be pushed even further by improving energy absorption capabilities of the larger grains in the tapered chains. We envision that these scalable, decorated tapered chains may be used as shock absorbing components in body armor, armored vehicles, building applications and in perhaps even in applications in rehabilitation science.

/pdf/energy-partitioning-and-impulse-dispersion-in-the-decorated-18vnn4gcfx.pdf

Energy partitioning and impulse dispersion in the decorated, tapered, strongly nonlinear granular alignment: A system with many potential applications

Making shock proof layers is an outstanding challenge. Elastic spheres are known to repel softer than springs when gently squeezed but develop strong repulsion upon compression and the forces between adjacent spheres lead to ballistic-like energy transfer between them. Here we demonstrate that a small alignment of progressively shrinking spheres of a strong, light-mass material, placed horizontally in an appropriate casing, can absorb ∼80% (∼90%) of the incident force (energy) pulse. The system can be scaled down in size. Effects of varying the size, radius shrinkage and restitutive losses are shown via computed “dynamical phase diagrams.”

Robert L. Doney

Papers

Solitary waves in the granular chain

Decorated, tapered, and highly nonlinear granular chain.

Impulse absorption by tapered horizontal alignments of elastic spheres.

Energy partitioning and impulse dispersion in the decorated, tapered, strongly nonlinear granular alignment: A system with many potential applications

Absorption of short duration pulses by small, scalable, tapered granular chains