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Two vector Wiener-Hopf equations with 2x2 kernels containing oscillatory terms

Presented by: 
Pavlos Livasov
Friday 16th August 2019 - 12:00 to 12:30
INI Seminar Room 1
In the first part we discuss a steady-state problem for an interface crack between two dissimilar elastic materials. We consider a model of the process zone described by imperfect transmission conditions that reflect the bridging effect along a finite part of the interface in front of the crack. By means of Fourier transform, the problem is reduced into a Wiener-Hopf equation with a 2x2 matrix, containing oscillatory terms. We factorize the kernel following an existing numerical method and analyse its performance for various parameters of the problem. We show that the model under consideration leads to the classic stress singularity at the crack tip. Finally, we derive conditions for the existence of an equilibrium state and compute admissible length of the process zone.  
For the second part of the talk, we consider propagation of a dynamic crack in a periodic structure with internal energy. The structural interface is formed by a discrete set of uniformly distributed alternating compressed and stretched bonds. In such a structure, the fracture of the initially stretched bonds is followed by that of the compressed ones with an unspecified time-lag. That, in turn, reflects the impact of both the internal energy accumulated inside the pre-stressed interface and the energy brought into the system by external loading. The application to the original problem of continuous (with respect to time) and  selective discrete (with respect to spatial coordinate) Fourier transforms yields another vector Wiener-Hopf equation with a kernel containing oscillating terms. We use a perturbation technique to factorise the matrix.   Finally, we show similarities and differences of the matrix-valued kernels mentioned above and discuss the chosen approaches for their factorisation.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons