Interaction between periodic elastic waves and two contact nonlinearities (Q2905298)

The propagation of elastic waves in a \(1D\) medium containing two cracks modeled by linear elastodynamics is studied. The physical problem is formulated in terms of the linear hyperbolic partial differential equations with nonlinear jump conditions. NEWLINE\[NEWLINE \rho \frac{\partial \upsilon}{\partial t} = \frac{\partial \sigma}{\partial x}, \quad \frac{\partial \sigma}{\partial x} = \rho c^2 \frac{\partial \sigma}{\partial x} + S(t) \delta (x-x_s), NEWLINE\]NEWLINE where \( \upsilon = \frac{\partial u}{\partial t} \) is the elastic velocity, \( u \) is the elastic displacement, and \( \sigma \) is the elastic stress. The source \( S(t) \) is causal, \( T\)-periodic, and oscillates around a null mean value. The stated problem is reduced into a system of nonlinear neutral delay differential equations \((NDDE) \) which can be studied more easily. Existence and uniqueness of global solutions is proved. A perturbation analysis is performed, and two numerical methods are introduced. Interesting numerical examples are given for the test of the introduced numerical methods.