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Kinematic hypotheses of various approximations in coupled problems of thermodynamics taking into account the contact interaction


E.Yu. Krylova – Ph.D.(Phys.-Math.), Associate Professor, Department of Mathematical and Computer Modeling, Saratov State University named after N.G. Chernyshevsky
I.V. Papkova – Ph.D.(Phys.-Math.), Associate Professor, Department «Mathematics and Modeling», Yuri Gagarin State Technical University of Saratov
O.A. Saltykova – Ph.D.(Phys.-Math.), Associate Professor, Department «Mathematics and Modeling», Yuri Gagarin State Technical University of Saratov
V.A. Krysko – Dr.Sc.(Eng.), Head of Department «Mathematics and Modeling», Yuri Gagarin State Technical University of Saratov

In the article mathematical models of the contact interaction of two geometrically nonlinear beams with a small gap, taking into account the kinematic hypotheses of different approximations (first approximation – Euler-Bernoulli, second approximation -Timoshenko, third approximation - Pelekh-Sheremetyev) were constructed. The beam structure is in the temperature field. In the general case, the temperature field is different for each beam of the packet. There are no restrictions imposed on the temperature distribution over the thickness of the packet. Two-dimensional heat conduction equations are considered taking into account the coupling of the temperature and deformation fields. The resolving equations of motion, the boundary and initial conditions were obtained from the Ostrogradsky-Hamilton energy principle. The transverse distributed alternating stress given by the harmonic law is applied to the upper beam of the packet. Contact interaction is accounted for by the Cantor model. The tasks are highly nonlinear. The geometric nonlinearity of beams according to the Karman model and constructive nonlinearity (contact between beams) are taken into account in constructing a mathematical model. The system of partial differential equations reduces to the ODE system by the method of finite differences of the second order. The resulting system is solved by several methods of Runge-Kutta type of various orders. Two cases are considered. In each of the cases various kinematic hypotheses are adopted for different beams of the packet. For the first beam, the hypothesis of Bernoulli-Euler was adopted in the first case, in the second - Timoshenko. For the beam - the «substrate», which is not affected by the external normal load, the hypotheses of Pelekh-Sheremetev are accepted in both cases. A study is made of the convergence of the solution methods with respect to the spatial coordinate, depending on the choice of the number of fission segments. As well as convergence, depending on the method of solving and step on the time coordinate. The truth of the chaotic oscillations of the system under consideration is established on the basis of the chaos definition by Gulik. For this purpose, the spectra of Lyapunov exponents were constructed using three different methods (Wolff, Kants, and Resenstein). On the basis of the study, we can speak of the convergence of the results at 400 points of the interval along the spatial coordinate obtained by the Runge-Kutta method of the eighth order (Prince Dormand). The investigation of nonlinear dynamics and contact interaction of beam structures of dependence on selected hypotheses is carried out. It was revealed that when the beams of a packet come into contact, the nature of the oscillations of the beams becomes chaotic, regardless of the selected hypotheses. In the conducted experiments there is a phenomenon of chaotic frequency synchronization.

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May 29, 2020

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