Stéphane Junca

INTERACTION BETWEEN PERIODIC ELASTIC WAVES AND TWO CONTACT NONLINEARITIES

Propagation of elastic waves is studied in a 1D medium containing two cracks. The latter are modeled by smooth nonlinear jump conditions accounting for the finite, non-null compressibility of real cracks. The evolution equations are written in the form of a system of two nonlinear neutral delay differential equations, leading to a well-posed Cauchy problem. Perturbation analysis indicates that, under periodic excitation, the periodic solutions oscillate around positive mean values, which increase with the forcing level. This typically nonlinear phenomenon offers non-destructive means to evaluate the cracks. Existence, uniqueness and attractivity of periodic solutions is then examined. At some particular values of the ratio between the wave travel time and the period of the source, results are obtained whatever the forcing level. With a much larger set of ratios but at small forcing levels, results are obtained under a Diophantine condition. Lastly, numerical experiments are proposed to illustrate the behavior of the periodic diffracted waves.

Dilatation of a One-Dimensional Nonlinear Crack Impacted by a Periodic Elastic Wave

The interactions between linear elastic waves and a nonlinear crack with finite compressibility are studied in the one-dimensional context. Numerical studies on a hyperbolic model of contact with sinusoidal forcing have shown that the mean values of the scattered elastic displacements are discontinuous across the crack. The mean dilatation of the crack also increases with the amplitude of the forcing levels. The aim of the present theoretical study is to analyze these nonlinear processes under a larger range of nonlinear jump conditions. For this purpose, the problem is reduced to a nonlinear differential equation. The dependence of the periodic solution on the forcing amplitude is quantified under sinusoidal forcing conditions. Bounds for the mean, maximum, and minimum values of the solution are presented. Lastly, periodic forcing with a null mean value is addressed. In that case, a result about the mean dilatation of the crack is obtained.

Nonsmooth modal analysis: Investigation of a 2-dof spring-mass system subject to an elastic impact law

The well-known concept of normal mode for linear systems has been extended to the framework of nonlinear dynamics over the course of the 20th century, initially by Lyapunov, and later by Rosenberg and a growing community of researchers in modal and vibration analysis. This effort has mainly targeted nonlinear smooth systems — the velocity is continuous and differentiable in time — even though systems presenting nonsmooth occurrences have been increasingly studied in the last decades to face the growing industrial need of unilateral contact and friction simulations. Yet, these systems have nearly never been explored from the standpoint of modal analysis.This contribution addresses the notion of modal analysis of nonsmooth systems. Developments are illustrated on a seemingly simple 2-dof autonomous system, subject to unilateral constraints reflected by a perfectly elastic impact law. Even though friction is ignored, its dynamics appears to be extremely rich. Periodic solutions are sought for given numbers of impacts per period and nonsmooth modes are illustrated for one and two impacts per period in the form of two-dimensional manifolds in the phase space. Also, an unexpected bridge between these modes in the frequency-energy plots is observed.Copyright

Vanishing pressure in gas dynamics equations

Abstract. This work is devoted to the analysis of the behaviour of solutions of gas dynamics equations as the pressure goes to 0 in the context of regular solutions. We obtain in this way a first justification of the connection to pressureless gases model.

A Continuous Model For Ratings

We study rating systems, such as the famous Elo system, applied to a large number of players. We assume that each player is characterized by an intrinsic inner strength and follow the evolution of their rating evaluations by deriving a new continuous model, a kinetic-like equation. We then investigate the validity of the rating systems by looking at their large time behavior as one would ideally expect the rating of each player to converge to their actual strength. The simplistic case when all players interact indeed yields an exponential convergence of the ratings. However, the behavior in the more realistic cases with only local interactions is more complex with several possible equilibria depending on the exact initial distribution of initial ratings and possibly very slow convergence.

Reflexions d'oscillations monodimensionnelles

We prove the propagation of oscillations with an asymptotic development for an oscillating initial boundary value problem of semilinear hyperbolic systems in the spirit of J.L.Joly, G.Metivier and J.Rauch. In particular we simplify the Joly-metivier-Rauchs proof for the Cauchy problem. Then, we show the new phenomenon of localised oscillations.