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Dynamics Of Heterogeneous Materials

The study of granular crystals, which are nonlinear metamaterials that consist of closely packed arrays of particles that interact elastically, is a vibrant area of research that combines ideas from disciplines such as materials science, nonlinear dynamics, and condensed-matter physics. Granular crystals exploit geometrical nonlinearities in their constitutive microstructure to produce properties (such as tunability and energy localization) that are not conventional to engineering materials and linear devices. In this topical review, we focus on recent experimental, computational, and theoretical results on nonlinear coherent structures in granular crystals. Such structures—which include traveling solitary waves, dispersive shock waves, and discrete breathers—have fascinating dynamics, including a diversity of both transient features and robust, long-lived patterns that emerge from broad classes of initial data. In our review, we primarily discuss phenomena in one-dimensional crystals, as most research to date has focused on such scenarios, but we also present some extensions to two-dimensional settings. Throughout the review, we highlight open problems and discuss a variety of potential engineering applications that arise from the rich dynamic response of granular crystals.

/pdf/nonlinear-coherent-structures-in-granular-crystals-2lq75wrayo.pdf

Nonlinear coherent structures in granular crystals

The study of granular crystals, metamaterials that consist of closely packed arrays of particles that interact elastically, is a vibrant area of research that combines ideas from disciplines such as materials science, nonlinear dynamics, and condensed-matter physics. Granular crystals, a type of nonlinear metamaterial, exploit geometrical nonlinearities in their constitutive microstructure to produce properties (such as tunability and energy localization) that are not conventional to engineering materials and linear devices. In this topical review, we focus on recent experimental, computational, and theoretical results on nonlinear coherent structures in granular crystals. Such structures --- which include traveling solitary waves, dispersive shock waves, and discrete breathers --- have fascinating dynamics, including a diversity of both transient features and robust, long-lived patterns that emerge from broad classes of initial data. In our review, we primarily discuss phenomena in one-dimensional crystals, as most research has focused on such scenarios, but we also present some extensions to two-dimensional settings. Throughout the review, we highlight open problems and discuss a variety of potential engineering applications that arise from the rich dynamic response of granular crystals.

/pdf/nonlinear-coherent-structures-in-granular-crystals-49qubouhjj.pdf

Nonlinear Coherent Structures in Granular Crystals

We consider a reaction-diffusion equation ut = uxx + f(u), where f has exactly three zeros 0, α and 1 (0 α having two interfaces at a large distance, u(x, t) approaches a pair of travelling wave solutions φ(x − p1(t)) + φ(−x + p2(t)) for a long time, and then the travelling fronts eventually disappear by colliding with each other. While our results establish this process, they show that there is a (backward) global solution ψ(x, t) and that the annihilation process is approximated by a solution ψ(x−x0, t−t0).

/pdf/backward-global-solutions-characterizing-annihilation-4a7kley5j5.pdf

Backward global solutions characterizing annihilation dynamics of travelling fronts (Nonlinear Diffusive Systems and Related Topics)

In the present work, motivated by generalized forms of the Hertzian dynamics associated with granular crystals, we consider the possibility of such models to give rise to both dispersive shock and rarefaction waves. Depending on the value p of the nonlinearity exponent, we find that both of these possibilities are realizable. We use a quasicontinuum approximation of a generalized inviscid Burgers model in order to predict the solution profile up to times near the formation of the dispersive shock, as well as to estimate when it will occur. Beyond that time threshold, oscillations associated with the highly dispersive nature of the underlying model emerge, which cannot be captured by the quasicontinuum approximation. Our analytical characterization of the above features is complemented by systematic numerical computations.

/pdf/emergence-of-dispersive-shocks-and-rarefaction-waves-in-18lb0a6fp3.pdf

Emergence of dispersive shocks and rarefaction waves in power-law contact models.

We study traveling waves for reaction diffusion equations on the spatially discrete domain ${\mathbb Z^2}$. The phenomenon of crystallographic pinning occurs when traveling waves become pinned in certain directions despite moving with non-zero wave speed in nearby directions. In [19] it was shown that crystallographic pinning occurs for all rational directions, so long as the nonlinearity is close to the sawtooth, which itself was considered in [6]. In this paper we show that crystallographic pinning holds in the horizontal and vertical directions for bistable nonlinearities which satisfy a specific computable generic condition. The proof is based on dynamical systems. In particular, it relies on an examination of the heteroclinic chains which occur as singular limits of wave profiles on the boundary of the pinning region.

Universality of Crystallographic Pinning

We study the interaction of small amplitude, long wavelength solitary waves in the Fermi-Pasta-Ulam model with general nearest-neighbor interaction potential. We establish global-in-time existence and stability of counter-propagating solitary wave solutions. These solutions are close to the linear superposition of two solitary waves for large positive and negative values of time; for intemediate values of time these solutions describe the interaction of two counterpropagating pulses. These solutions are stable with respect to perturbations in l 2 and asymptotically stable with respect to perturbations which decay exponentially at spatial ±∞.

/pdf/counterpropagating-two-soliton-solutions-in-the-fpu-lattice-1zhbs9uvrq.pdf

Counterpropagating two-soliton solutions in the FPU lattice

We prove the existence of asymptotic two-soliton states in the Fermi-Pasta-Ulam model with general interaction potential. That is, we exhibit solutions whose difference in $\ell^2$ from the linear superposition of two solitary waves goes to zero as time goes to infinity.

Asymptotic two-soliton solutions in the Fermi-Pasta-Ulam model

We consider general reaction diusion systems posed on rectangular lattices in two or more spatial dimensions. We show that travelling wave solutions to such systems that propagate in rational directions are nonlinearly stable under small perturbations. We employ recently developed techniques involving point-wise Green’s functions estimates for functional dierential equations of mixed type (MFDEs), allowing our results

/pdf/multi-dimensional-stability-of-waves-travelling-through-238aw20zf3.pdf

Multi-dimensional stability of waves travelling through rectangular lattices in rational directions

We consider scalar lattice dierential equations posed on square lattices in two space dimensions. Under certain natural conditions we show that wave-like solutions exist when obstacles (characterized by \holes") are present in the lattice. Our work generalizes to the discrete spatial setting the results obtained in [9] for the propagation of waves around obstacles in continuous spatial domains. The analysis hinges upon the development of sub and super-solutions for a class of discrete bistable reaction-diusion problems and on a generalization of a classical result due to Aronson and Weinberger that concerns the spreading of localized disturbances.

/pdf/entire-solutions-for-bistable-lattice-differential-equations-2pu9rvd8ih.pdf

Entire Solutions for Bistable Lattice Differential Equations With Obstacles

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