Jesus Cuevas (Nonlinear Physics, Sevilla Univ.), Panayotis Kevrekidis (Mathematics and Statistics, UMass), Anna Vainchtein and Lifeng Liu (Mathematics, Univ. of Pittsburgh), Yuli Starosvetsky (Mechanical Engineering, Technion IIT), Caishan Liu (State Key Lab. for Turbulence and Complex Systems, Peking University), Michael Herrmann (Mathematics and Computer Sciences, Münster University)

-impact propagation in granular metamaterials, leading to the formation of solitary waves

-vibrational energy localization (breathers) in oscillator chains

For physical systems with discrete spatial structures, naive continuum limits provide only a poor account of the dynamics, which is better captured by nonlinear lattice differential equations involving typically an infinite number of particles. For example, solitary waves and breathers in granular chains under precompression can be modeled using the celebrated Fermi-Pasta-Ulam lattice. In such spatially discrete models, the analysis of traveling wave solutions leads to challenging mathematical questions related to advance-delay differential equations. Classical phenomena induced by spatial discreteness include robust energy trapping in the form of structurally stable breathers, propagation failure for localized structures (fronts or pulses) and lattice-induced anisotropy.

Our research primarily focuses on two classes of nonlinear systems, either nonsmooth or strongly nonlinear :

-Nonsmooth discrete systems can be characterized by several types of nonsmooth interaction laws, leading to differential inclusions (as in the case of set-valued Coulomb friction law), differential equations with impulsive forces (rigid impacts), or differential equations with piecewise-smooth nonlinearities (Hertz contact law). Such nonlinear systems can be reformulated as complementarity systems.

-Strongly (or essentially) nonlinear discrete systems are characterized by interaction potentials with a vanishing second derivative at the ground state. Classical examples concern the Hertzian contact force between two touching beads (or more general smooth non-conforming surfaces) and geometric nonlinearities implemented with mechanical springs.

In such systems, classical linear and weakly nonlinear wave theory do not apply and unusual physical phenomena can show up. This is one of the essential ideas behind strongly nonlinear metamaterials, in which the main mode of energy propagation corresponds to (essentially) compactly supported solitary waves. The formation of waves localized on a few discrete elements opens up the possibility of controlling stress wave propagation much more efficiently than in classical linear (or weakly-nonlinear) systems dominated by dispersion. For example, experimental applications of granular metamaterials include the non-destructive testing of elastic media interfaced with granular chains, the use of two-dimensional granular crystals for localized wave redirection or the design of acoustic lenses.

Original publication available at royalsocietypublishing.org. Preprint available on arXiv: pdf file.

Original publication available at link.springer.com. Preprint available on arXiv: pdf file.

Original publication available at www.worldscientific.com. Preprint available on arXiv: pdf file

Original publication available at link.springer.com. Preprint available on arXiv: pdf file.

Original publication available at epubs.siam.org. Preprint available on arXiv: pdf file.

Original publication available at www.sciencedirect.com. pdf file.

Preprint available on arXiv: pdf file.

Original publication available at royalsocietypublishing.org : (part I), (part II). Preliminary version

Original publication available at aps.org. pdf file

Original publication available at springer.com. Preprint available on HAL.

Monography available at springer.com.

ITN proposal submitted to H2020 program (2014).

IXXI project (Rhône-Alpes Institute for Complex Systems) accepted in November 2012.

This project concerns the study of nonsmooth mechanical systems with a particular focus on granular chains, nonlinear waves and nonlinear modes.

Project funded by the French National Research Agency (ANR) and National Natural Science Foundation of China (NSFC).

Summary and Project description.

B. Brogliato, Multiple impacts and Painlevé paradox, Aussois summer school (Sept 2012). Slides

G. James, Nonlinear Waves in Granular Chains, GdR 3437 DYNOLIN, ENSAM Lille (Oct 2013). Slides

V. Acary, G. James and F. Pérignon, Periodic motions of coupled impact oscillators, minisymposium Non-Smooth Dynamics, 8th European Nonlinear Dynamics Conference (ENOC), Vienna, July 6-11, 2014.

B. Bidégaray-Fesquet, E. Dumas and G. James, An asymptotic model for small amplitude solutions to Newton's cradle, SIAM Conference on Nonlinear Waves and Coherent Structures, Cambridge, UK, August 11-14, 2014.

G. James, Localized waves in fully nonlinear media, SIAM Conference on Nonlinear Waves and Coherent Structures, Cambridge, UK, August 11-14, 2014.