Visualization of Contact Interaction of Nanobeams M.V. Zhigalov 1 , V.A. Apryskin 1 , V.A. Krysko 1 zhigalovm@yandex.ru|wwooow@yandex.ru 1 Yuri Gagarin Saratov State Technical University, Saratov, Russia The paper presents a visualization of the contact interaction of two Bernoulli-Euler nanobeams connected through boundary conditions. Mathematical models of beams are based on the gradient deformation theory and the theory of contact interaction of B. Y. Cantor. The visualization is based on Fourier transform and wavelet transform, phase portrait. Keywords: Bernoulli-Euler nanobeam, gradient deformation theory, contact problem, Fourier transform, wavelet. 1. Introduction The designs of modern devices are complex multi-layer packages with small gaps between the elements, so an important issue is to take into account the contact interaction of layers, which in turn leads to a strong nonlinearity - constructive. Such nano electro-mechanical systems (NEMS) are widely used in various electronic devices, in particular in gyroscopes (layered flat micromechanical accelerometers (MMA). Note that the account of contact interaction leads to the chaotic state of the system already at small amplitudes of oscillations. The presence of a gap between the elements, such as beams, already at small deflections, commensurate with the gap between the elements, can lead the object under study in a state of chaotic oscillations. Therefore, the need for research with visualization of the contact interaction of nano-mechanical systems in the form of beams is undoubtedly relevant and requires attention. 2. Statement of the problem Micro-and nano-sized beams are widely used in various micro-and nano-Electromechanical systems (vibration sensors [1], micro-drives [2], micro-switches [3]). The dependence of elastic behavior on the body size at the micron scale was observed experimentally in different substances (metals [4, 5] and alloys [6], polymers [7], crystals [8]). The dependences of elastic behavior on size can be explained using molecular dynamics (MD) simulations or higher order continuum mechanics. Although the molecular dynamics approach can provide more accurate approximations to real objects, it is too expensive from a computational point of view. Therefore, the higher order continuum mechanics approach has been widely used in the modeling of small-scale structures. The development of a higher order continuum theory can be traced back to the earliest work of Piola in the 19th century, as shown in [9], and the work of the Kosser brothers [10] in 1909. However, the ideas of the Kosser brothers received considerable attention from researchers only since the 1960s, when a large number of higher-order continuum theories were developed. In general, these theories can be divided into three different classes, namely, the family of strain gradient theories, the microcontinuum, and nonlocal elasticity theories. Based on the higher order stress theory of Mindlin [11] and Lam et al. [12] proposed the theory of elasticity of the deformation gradient, in which, in addition to the classical equations of equilibrium of forces and moments, a new additional equilibrium equation is introduced, which determines the behavior of stresses of higher orders and the equation of equilibrium of moments. Three parameters of the material length scale are introduced for isotropic linear elastic materials 0 1 2 (, , ) l l l . According to this theory, the total strain energy density is a function of the symmetric strain tensor, the dilation gradient vector, the deviator tension gradient tensor, and the symmetric rotation gradient tensor. In this paper, a mathematical model of Bernoulli-Euler nanobeams connected through boundary conditions under the action of transverse load is constructed. Three material length scale parameters are introduced to account for dimensional effects 0 1 2 (, , ) l l l . To account for the contact between the beams, a Winkler coupling between the compression and the contact pressure between the two beams is used [13]:   1 2 1 1 2 k sign w h w        , (1) where 1,  if 1 2 w k w h   то that is, there is contact between the plate and the beam, else 0  , 1 2 w, , k w h - deflections of the first and second beams and the gap between them, respectively. The mathematical model of contact between two nanobeams, based on the kinematic Bernoulli-Euler hypothesis, is described by a system of resolving equations:   4 3 2 2 2 0 2 1 4 96 2 12 225 m w bh l bh l bh l bh x                   6 3 3 2 2 0 1 6 2 1 2 2 14 2 12 225 12 ( 1) ( ) , m m m w bh bh l l q x w w Kw w h t t                         (2) where m – beam number (m=1,2), h - the gap between the beams. Boundary conditions are: 0; 0 m m w w x     . (3) Initial conditions are:   ,0 ( ,0) 0; 0 m m w x w x t     . (4) The system (2-4) was reduced to the Cauchy problem using the finite difference method O(h 2 ). The Cauchy problem was solved by the Runge-Kutta method of 4 orders. A study of convergence by the method of finite differences, on the basis of which the optimal number of partitions was chosen, was carried out. The partitioning step for the Runge-Kutta method was determined according to the Runge principle. 3. The results of the study of the influence of length scale factors on the nature of oscillations For the considered tasks the following parameter values were used: / 30 a h  ,   1 10sin 5.3 , q t   2 0 q  , 0.1,   0.01. h  The research results for two of the nine considered combinations of coefficients 0 1 2 (, , ) l l l are shown in Figure 1, 2. For the first case (Fig. 1), all coefficients are zero, i.e. considered beams on the classical theory. The second case (Fig. Copyright © 2019 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).