International audiencePropagation of elastic waves is studied in a 1D medium containing N cracks modeled by nonlinear jump conditions. The case N = 1 is fully understood. When N > 1, the evolution equations are written as a system of nonlinear neutral delay differential equations, leading to a well-posed Cauchy problem. In the case N = 2, some mathematical results about the existence, uniqueness and attractivity of periodic solutions have been obtained in 2012 by the authors, under the assumption of small sources. The difficulty of analysis follows from the fact that the spectrum of the linear operator is asymptotically closed to the imaginary axis. Here we propose a new result of stability in the homogeneous case, based on an energy method. One deduces the asymptotic stability of the zero steady-state. Extension to N = 3 cracks is also considered, leading to new results in particular configurations. 1. Introduction. Understanding the interactions between ultrasonic waves and contact defects have crucial applications in the field of mechanics, especially as far as the non-destructive testing of materials is concerned. When the cracks are much smaller than the wavelengths, they are usually replaced by interfaces with appropriate jump conditions. Here we consider realistic models describing cracks with finite compressibility, in a 1D geometry (section 2). The case of N = 1 crack, which involves a nonlinear ordinary differential equation, has been completely analysed in [7]. When tackling with N > 1 cracks, the analysis becomes much more intricate. The successive reflections of waves between the cracks are described mathematically by a system of N nonlinear neutral-delay differential equations (NDDE) with forcing [5]. The main features of such systems are already contained in the following scalar NDDE:
