UDC 539.3 BOUNDARY ELEMENT APPROACHES TO THE PROBLEM OF 2-D NON-STATIONARY ELASTIC VIBRATIONS

Two boundary element approaches are used to solve the problem on non-stationary vibrations of elastic solids. The first approach is based on the transition to the frequency domain by means of a Fourier series expansion. The second approach is associated with the direct solution of a system of time-dependent boundary integral equations, with a piecewise constant approximation of the dependence of the unknowns on time. In both cases, a collocation scheme is used to algebraize the integral equations, and the difficulties associated with the calculation of singular integrals are overcome by replacing the kernels with the initial segment of the Maclaurin series. After such a replacement, the kernels take the form of a sum, the first term of which is the corresponding fundamental solution of the statics problem while other terms are regular. Since integration of static kernels is not difficult the problem of calculating the diagonal coefficients of the SLAE turns out to be solved. The developed techniques are compared in the process of dynamics analysis solving of elastic media with two cylindrical cavities. The boundary of one of the cavities is subjected to a radial impulse load, which varies according to the parabolic law. Both approaches have shown the similar effectiveness and qualitative consistency.

The Boundary Element Method (BEM) is an effective tool for solving the problem of vibrations of elastic bodies with different cavities and inclusions. In this case, the solution of the problem can be obtained in two ways. In the first case, the Fourier transform is used to convert the problem into a frequency domain, while in the second case, a time-step procedure is used. In both cases, the algorithmic basis of the problem is the boundary analogue of the Somigliana identity for displacements. The identity links known displacements and stresses at the boundary points with those not unspecified by boundary conditions. The resolving equation describing two-dimensional harmonic vibrations of an elastic mass body, in the absence of mass loads, is as follows 1 ( , ) ( , ) ( , , ) ( , ) ( , , ) 2 j where ω is the circular frequency of vibrations; 1  the fundamental solution of the problem [1,2], which is given by the expression ;  -material density; λ and μ -Lame constants; , generalized derivative of the fundamental solution (stresses at the sites with normal n (y) j  that arise in elastic two-dimensional media due to the action in the point x  of concentrated unit force in the direction k x ; (1) m H is the Hankel function of the first kind.
In accordance with the collocation procedure of BEM the boundary is represented as a set of elements and a hypothesis defining how the unknown quantities change within each BE (i.e. basic function) is introduced. Then a set of poles l x  is assigned, agreed with the introduced hypothesis, and then equations (1) are written in each of the poles in turn. The result of this procedure is a system of linear algebraic equations with respect to unknown values of amplitudes of displacements and stresses at collocation points l x  . The system coefficients are integrals over boundary elements of product of kernel ( , , ) jk T x y    and the corresponding basic function.
When the distance r between the source x  point and the integration point y  tends to zero, i.e., if the pole is located at the same BE on which the integration is carried out, the Hankel functions, and with them the sub-integral expressions, take infinitely large values, which makes it impossible to calculate the diagonal system matrix coefficients using numerical integration. To overcome this obstacle, the Hankel functions are approximately replaced by the initial segments of the Maclaurin series. After such a replacement the kernels can be represented as follows [3] * * s jk T x y   are respectively the fundamental solution of the static 2D problem of and its generalized derivative.
Since integration of the kernels of the static problem does not cause difficulties, and regular additives * ( , , ) jk T x y    can be determined with any degree of accuracy, the problem of algebraization of the system of boundary integral equations (1) can be considered solved in the first approach.
The second approach requires solving a system of time-dependent boundary integral equations, which under zero initial conditions and the absence of mass forces can be written as follows where The peculiarity of this approach is that time is one of variables, on which the solution depends. Accordingly, it is necessary to divide the time interval into separate intervals and approximate the unknown within each interval. Assuming that the boundary stresses at each step remain constant, time integration of the both terms in the right side of equation (4) can be done analytically [4]. Moreover, the resulting expressions can also be presented in a form similar to that of representation (3), i.e., as a sum of the corresponding static kernel and some regular additive. Thus, even in this case, it is possible to overcome all the fundamental obstacles to the numerical solution of the problem [4].
Comparison of the developed algorithms was carried out using the problem about non-stationary vibrations of elastic space with two circular cylindrical cavities of radius R=3m. Boundary of the left cavity is exposed to radial influence of parabolic impulse, set by the formula Fig. 1). Time of the impulse action is  (4) solution. Figures 2 and 5 denote the graphs built within the first approach using 12 terms of the Fourier series, and figures 3 and 6 denote the graphs, for the construction of which 4 terms of the series were kept. At the same time, the diagrams 1, 2, 3 characterize the stress-strain state parameters at the point A, and the diagrams 4, 5, 6 correspond to the same parameters at the point B of the boundary (Fig. 1).
Extreme values of normalized parameters of stress-strain state, obtained with the help of two approaches, are contained in Tables    The data presented in the figures and tables show satisfactory qualitative consistency of the data obtained using different approaches. It should be noted that as the number of members of the Fourier series increases, the diagrams obtained in the first approach become more and more similar to the curves using the solution of the time-dependent boundary integral equations (4), i.e., using relations of the second approach.