įigure 2a shows the change in the deflection W (x, y, t) in time (dashed line) along the axis OX and y = 0.2, and in Fig. Note that, when solving boundary value problems, an orthonormal SКF is used for the biharmonic and unit operator, respectively, and then a numerical method based on the use of quadrature formulas is used to solve the autonomous systems of integral and integro-differential equations. logical conjunction operator of zero order. The normalized equation of the geometry of the region for the plate shown in Fig. Ф 1 ≠ 0 и Ф 2=0, but can not Ф 2 ≠ 0 и Ф 1=0, since this circumstance leads to the appearance of an «extra» boundary condition. We take one of the undefined functions equal to zero. The presence of two or more numbers of undefined functions in the structure creates difficulties in solving boundary value problems. Ф s =, - unknown components to be determined, - complete linearly independent system of functions D 2, T 2- differential operators of R-functions, ω - normalized boundary equation of the domain. In the above structural formulas, Ф1 and Ф2 are undefined components of the structure, which are represented in the form In order to solve the problem numerically, we use the structure The nucleus of shear relaxation is the nucleus R(t)=ε. Let the plate be subjected to a load (q = 1). Note that SКF exactly satisfy all the boundary conditions that are constructed using the Rvachev R-functions method įirst, let us consider the problems of quasistatic bending of the freely supported viscoelastic plates shown in Fig. Where - systems of coordinate functions (Chebyshev polynomials, power, trigonometric, Schoenberg splines, etc.) SКF. Solutions of equations (1) and (4) are sought in the form Where - differential operators that depend on the boundary conditions Г - boundary of the region и - initial values. Where ρh - the mass of the slab per unit surface.Įquations (4) are solved for the corresponding boundary and initial conditions K=E/3(1- 2µ) - bulk modulus of elasticity h - is the thickness of the plate.Īs is known, the equation of an oscillating thin viscoelastic plate has the form Where G=E/2(1+µ) - shear modulus E - elastic modulus - integral operator with shear relaxation kernels - integral operator, i.e. If, on the other hand, we use the hypothesis of the elasticity of volume deformations, then for bending and twisting moments it is calculated by the following formulas Μ - Poisson's ratio q (x, y, t) is the intensity of the external load. Where D is the stiffness of viscoelastic plates integral operator with relaxation kernels R(t), т. Note that if we use the hypothesis of the constancy of the Poisson's ratio when formulating the basic physical relationships, then the bending and twisting moments are calculated by the following formulas: The paper is devoted to numerical modeling and automation of the solution of boundary value problems for viscoelastic plates of arbitrary configuration for various models of viscosity in a Maple system environment.Īs is known, the mathematical model of this problem has the form: Therefore, it is necessary to create reliable methods for calculating the load-bearing elements of thin-walled structures. Only a reasonable combination of analytical and numerical methods is a necessary condition for success in solving practical problems. The value of the computational experiment can not be overestimated, especially if the field experiment is dangerous, expensive or simply impossible.
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