Mechanics of Deformable Solids
Loktev А.А., Zaletdinov А.V., Gridasova E.A., Zapol’nova E.V.
ALEXEY A. LOKTEV, Doctor of Physical-Mathematical Sciences, Professor, Dean of the Faculty of Informatization, Economics and Management, Head of the Department of Structural Mechanics, Machinery and Equipment, Moscow State University of Railway Engineering, Moscow. 2, Minaevsky Lane, Moscow, Russia, 127055, e-mail: prtlokt@yandex.ru
ARTUR V. ZALETDINOV, Candidate of Technics, Consultant, LLC “Accenture”, Moscow. 2, Paveletskaya square, bild. 2, Moscow, Russia, 115054, e-mail: azaletdinov@gmail.com
EKATERINA A. GRIDASOVA, Candidate of Technics, Associate Professor of Welding Engineering, School of Engineering, Far Eastern Federal University, Vladivostok. 8 Sukhanova St., Vladivostok, Russia, 690050, е-mail:olvin@list.ru
EVGENIYA V. ZAPOL’NOVA, Junior Engineer SAPR, Codest International SRL, Moscow. 7 Soimonovskii Av., Moscow, Russia, 19034,
e-mail: jenzapolnova@ya.ru
The algorithm to determine the spots to reinforce the flat element of orthotropic composite material when shock is applied
The paper presents a modified method of approach to the dynamic effects of two rigid bodies in the context of the wave theory of impact. The approach is based on undertaking a joint task: that of the impact of two bodies as an individual solution of the contact task and the wave one as well as their subsequent combination through the use of boundary conditions on the contour of the loading area. The devised technique may be applied when dealing with a wide range of tasks concerning wave generation and transmission in rigid bodies after a dynamic impact of a tool having various mechanical and geometrical properties. One of the tasks is that of determining the spots to reinforce the elements made of composite materials by high strength threads or fittings. In the present paper, it is proposed to determine the spots of the eventual reinforcement of anisotropic composite as points of interaction of elastic waves reflecting from the upper and lower boundaries of the flat element. Wave surfaces appear in the target due to the impact upon its rigid body, whose rheological properties cover the directly affected contact zone. The advance of wave surfaces having terminal velocity has been made possible owing to the wave equation of the Uflyand – Mindlin – Reissner model taking into consideration both transverse shift deformation and rotary inertia of cross-sections. The equations determining the execution behaviour of the points of the flat element enable one to suggest that the latter’s deformation outside the area of the interaction of the impact tool and the target may be caused also by the advance of elastic waves with terminal velocities.
The paper presents also a model for considering wave surfaces in colliding bodies, an algorithm of considering various rheological properties of interactive bodies based on the analytical method which represents unknowns like decompositions in space and time, starting and boundary conditions, and numerical methods to solve finite integro-differential equations. The need for a more accurate notion of the behaviour of structures under loading makes mathematical models of processes and objects more complicated. Relevant is also to derive a rather simple computational model and algorithms to compute the tasks of impact interaction with regard to the advancement of surface waves having terminal velocities as well as their actuialisation in the form of a software application of the algorithm to determine the points of interaction of direct and reflected waves. The developed computational algorithm and software system have been tested, which allowed determining the points of location of the reinforcing filaments or rods in the planar composite in order to provide the necessary stiffness of the whole structure.
Key words: dynamic impact, non-classical plate, elastic waves, orthotropic properties, reflected waves, Legendre polynomials, computational algorithm, software package.
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