# Optimal control of the length of a straight crack for a model describing an equilibrium of a two-dimensional body with two intersecting cracks

• Lazarev Nyurgun P., nyurgun@ngs.ru M. K. Ammosov North-Eastern Federal University, 48 Kulakovsky Street, Yakutsk 677000, Russia; Lavrentiev Institute of Hydrodynamics, 15 Lavrentiev Avenue, Novosibirsk 630090, Russia
• Rudoy Evgeny M., rem@hydro.nsc.ru Lavrentyev Institute of Hydrodynamics, 15 Akad. Lavrentyev Avenue, Novosibirsk 630090, Russia
• Popova Tatiana S., ptsokt@mail.ru M. K. Ammosov North-Eastern Federal University, 48 Kulakovsky Street, Yakutsk 677000, Russia
Keywords: variational inequality, optimal control problem, nonpenetration, non-linear boundary conditions, crack

### Abstract

A mathematical model describing an equilibrium of cracked two-dimensional bodies with two mutually intersecting cracks is considered. One of these cracks is assumed to be straight, and the second one is described with the use of a smooth curve. Inequality type boundary conditions are imposed at the both cracks faces providing mutual non-penetration between crack faces. On the external boundary, homogeneous Dirichlet boundary conditions are imposed. We study a family of corresponding variational problems which depends on the parameter describing the length of the straight crack and analyze the dependence of solutions on this parameter. Existence of the solution to the optimal control problem is proved. For this problem, the cost functional is defined by a Griffith-type functional, which characterizes a possibility of curvilinear crack propagation along the prescribed path. Meanwhile, the length parameter of the straight crack is chosen as a control parameter.

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How to Cite
Lazarev, N., Rudoy, E. and Popova, T. (&nbsp;) “Optimal control of the length of a straight crack for a model describing an equilibrium of a two-dimensional body with two intersecting cracks”, Mathematical notes of NEFU, 25(3), pp. 43-53. doi: https://doi.org/10.25587/SVFU.2018.99.16950.
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Mathematics