Dynamics of a Monolayer of Microspheres on an Elastic Substrate

We present a model for wave propagation in a monolayer of spheres on an elastic substrate. The model, which considers sagittally polarized waves, includes: horizontal, vertical, and rotational degrees of freedom; normal and shear coupling between the spheres and substrate, as well as between adjacent spheres; and the effects of wave propagation in the elastic substrate. For a monolayer of interacting spheres, we find three contact resonances, whose frequencies are given by simple closed-form expressions. For a monolayer of isolated spheres, only two resonances are present. The contact resonances couple to surface acoustic waves in the substrate, leading to mode hybridization and "avoided crossing" phenomena. We present dispersion curves for a monolayer of silica microspheres on a silica substrate, assuming adhesive, Hertzian interactions, and compare calculations using an effective medium approximation to a discrete model of a monolayer on a rigid substrate. While the effective medium model does not account for discrete lattice effects at short wavelengths, we find that it is well suited for describing the interaction between the monolayer and substrate in the long wavelength limit. We suggest that a complete picture of the dynamics of a discrete monolayer adhered to an elastic substrate can be found using a combination of the results presented for the discrete and effective medium descriptions. This model is potentially scalable for use with both micro- and macroscale systems, and offers the prospect of experimentally extracting contact stiffnesses from measurements of acoustic dispersion

Wrinkles Riding Waves in Soft Layered Materials

The formation of periodic wrinkles in soft layered materials due to mechanical instabilities is prevalent in nature and has been proposed for use in multiple applications. However, such phenomena have been explored predominantly in quasi-static settings. In this work, we measure the dynamics of soft elastomeric blocks with stiff surface films subjected to high-speed impact, and observe wrinkles forming along with, and riding upon, waves propagating through the system. We analyze our measurements with large-deformation, nonlinear visco-hyperelastic Finite Element simulations coupled to an analytical wrinkling model. The comparison between the measured and simulated dynamics shows good agreement, and suggests that inertia and viscoelasticity play an important role. This work encourages future studies of the dynamics of surface instabilities in soft materials, including large-deformation, highly nonlinear morphologies, and may have applications to areas including impact mitigation, soft electronics, and the dynamics of soft sandwich composites

Longitudinal Eigenvibration of Multilayer Colloidal Crystals and the Effect of Nanoscale Contact Bridges

Longitudinal contact-based vibrations of colloidal crystals with a controlled layer thickness are studied. These crystals consist of 390 nm diameter polystyrene spheres arranged into close packed, ordered lattices with a thickness of one to twelve layers. Using laser ultrasonics, eigenmodes of the crystals that have out-of-plane motion are excited. The particle-substrate and effective interlayer contact stiffnesses in the colloidal crystals are extracted using a discrete, coupled oscillator model. Extracted stiffnesses are correlated with scanning electron microscope images of the contacts and atomic force microscope characterization of the substrate surface topography after removal of the spheres. Solid bridges of nanometric thickness are found to drastically alter the stiffness of the contacts, and their presence is found to be dependent on the self-assembly process. Measurements of the eigenmode quality factors suggest that energy leakage into the substrate plays a role for low frequency modes but is overcome by disorder- or material-induced losses at higher frequencies. These findings help further the understanding of the contact mechanics, and the effects of disorder in three-dimensional micro- and nano-particulate systems, and open new avenues to engineer new types of micro- and nanostructured materials with wave tailoring functionalities via control of the adhesive contact properties

Granular Media-Based Tunable Passive Vibration Suppressor

and vibration suppression device is composed of statically compressed chains of spherical particles. The device superimposes a combination of dissipative damping and dispersive effects. The dissipative damping resulting from the elastic wave attenuation properties of the bulk material selected for the granular media is independent of particle geometry and periodicity, and can be accordingly designed based on the dissipative (or viscoelastic) properties of the material. For instance, a viscoelastic polymer might be selected where broadband damping is desired. In contrast, the dispersive effects result from the periodic arrangement and geometry of particles composing a linear granular chain. A uniform (monatomic) chain of statically compressed spherical particles will have a low-pass filter effect, with a cutoff frequency tunable as a function of particle mass, elastic modulus, Poisson fs ratio, radius, and static compression. Elastic waves with frequency content above this cutoff frequency will exhibit an exponential decay in amplitude as a function of propagation distance. System design targeting a specific application is conducted using a combination of theoretical, computational, and experimental techniques to appropriately select the particle radii, material (and thus elastic modulus and Poisson fs ratio), and static compression to satisfy estimated requirements derived for shock and/or vibration protection needs under particular operational conditions. The selection of a chain of polymer spheres with an elastic modulus .3 provided the appropriate dispersive filtering effect for that exercise; however, different operational scenarios may require the use of other polymers, metals, ceramics, or a combination thereof, configured as an array of spherical particles. The device is a linear array of spherical particles compressed in a container with a mechanism for attachment to the shock and/or vibration source, and a mechanism for attachment to the article requiring isolation (Figure 1). This configuration is referred to as a single-axis vibration suppressor. This invention also includes further designs for the integration of the single-axis vibration suppressor into a six-degree-of-freedom hexapod "Stewart"mounting configuration (Figure 2). By integrating each singleaxis vibration suppressor into a hexapod formation, a payload will be protected in all six degrees of freedom from shock and/or vibration. Additionally, to further enable the application of this device to multiple operational scenarios, particularly in the case of high loads, the vibration suppressor devices can be used in parallel in any array configuration

Nonlinear Waves in Disordered Diatomic Granular Chains

We investigate the propagation and scattering of highly nonlinear waves in disordered granular chains composed of diatomic (two-mass) units of spheres that interact via Hertzian contact. Using ideas from statistical mechanics, we consider each diatomic unit to be a "spin", so that a granular chain can be viewed as a spin chain composed of units that are each oriented in one of two possible ways. Experiments and numerical simulations both reveal the existence of two different mechanisms of wave propagation: In low-disorder chains, we observe the propagation of a solitary pulse with exponentially decaying amplitude. Beyond a critical level of disorder, the wave amplitude instead decays as a power law, and the wave transmission becomes insensitive to the level of disorder. We characterize the spatio-temporal structure of the wave in both propagation regimes and propose a simple theoretical interpretation for such a transition. Our investigation suggests that an elastic spin chain can be used as a model system to investigate the role of heterogeneities in the propagation of highly nonlinear waves.Comment: 10 pages, 8 figures (some with multiple parts), to appear in Physical Review E; summary of changes: new title, one new figure, additional discussion of several points (including both background and results

Synthetically Non-Hermitian Nonlinear Wave-like Behavior in a Topological Mechanical Metamaterial

Topological mechanical metamaterials have enabled new ways to control stress and deformation propagation. Exemplified by Maxwell lattices, they have been studied extensively using a linearized formalism. Herein, we study a two-dimensional topological Maxwell lattice by exploring its large deformation quasi-static response using geometric numerical simulations and experiments. We observe spatial nonlinear wave-like phenomena such as harmonic generation, localized domain switching, amplification-enhanced frequency conversion, and solitary waves. We further map our linearized, homogenized system to a non-Hermitian, non-reciprocal, one-dimensional wave equation, revealing an equivalence between the deformation fields of two-dimensional topological Maxwell lattices and nonlinear dynamical phenomena in one-dimensional active systems. Our study opens a new regime for topological mechanical metamaterials and expands their application potential in areas including adaptive and smart materials, and mechanical logic, wherein concepts from nonlinear dynamics may be used to create intricate, tailored spatial deformation and stress fields greatly exceeding conventional elasticity.Comment: 12 pages, 7 figure

Topology optimization of nonlinear periodically microstructured materials for tailored homogenized constitutive properties

A topology optimization method is presented for the design of periodic microstructured materials with prescribed homogenized nonlinear constitutive properties over finite strain ranges. The mechanical model assumes linear elastic isotropic materials, geometric nonlinearity at finite strain, and a quasi-static response. The optimization problem is solved by a nonlinear programming method and the sensitivities computed via the adjoint method. Two-dimensional structures identified using this optimization method are additively manufactured and their uniaxial tensile strain response compared with the numerically predicted behavior. The optimization approach herein enables the design and development of lattice-like materials with prescribed nonlinear effective properties, for use in myriad potential applications, ranging from stress wave and vibration mitigation to soft robotics

Dependence of the kinetic energy absorption capacity of bistable mechanical metamaterials on impactor mass and velocity

Using an alternative mechanism to dissipation or scattering, bistable structures and mechanical metamaterials have shown promise for mitigating the detrimental effects of impact by reversibly locking energy into strained material. Herein, we extend prior works on impact absorption via bistable metamaterials to computationally explore the dependence of kinetic energy transmission on the velocity and mass of the impactor, with strain rates exceeding $10^2$ s$^{-1}$. We observe a large dependence on both impactor parameters, ranging from significantly better to worse performance than a comparative linear material. We then correlate the variability in performance to solitary wave formation in the system and give analytical estimates of idealized energy absorption capacity under dynamic loading. In addition, we find a significant dependence on damping accompanied by a qualitative difference in solitary wave propagation within the system. The complex dynamics revealed in this study offer potential future guidance for the application of bistable metamaterials to applications including human and engineered system shock and impact protection devices