Граничные интегральные уравнения первого рода с гладкими ядрами в динамических задачах ортотропной теории упругости

Steady-state harmonic vibrations of an orthotropic elastic body are considered in a plane formulation. To solve boundary-value problems, a method of alternative boundary integral equations (ABIE) of the first kind with smooth kernels is proposed. The method enforces the solvability conditions directly on the characteristic set of the differential operator instead of constructing an explicit fundamental solution. The boundary formulation reduces the dimension of the problem by one and is especially effective for problems in which boundary displacements and stresses are unknown. In the zero-frequency limit, the scheme formally turns into the static ABIE formulation with the same set of boundary unknowns. A procedure for reducing the boundary equations to an overdetermined system of linear algebraic equations is proposed. To find its regularized solution, the truncated singular value decomposition (TSVD) method is applied. Numerical verification of the method is carried out by considering four model problems: for an elliptical domain with a prescribed stress vector on the boundary; for a rectangular domain clamped on one side; for the same rectangular domain with homogeneous mixed boundary conditions in a spectral formulation; and for an elliptical domain, in which the resonance frequencies are being determined. Comparison with the results of finite-element computations shows that, in the first problem, the discrepancy in the boundary displacements of the ellipse is 3–5%, while in the second problem the maximum discrepancy in components is 2.7%; in the problems of determining resonant frequencies, the maximum relative discrepancy between the computed frequencies does not exceed 0.07% for the rectangular domain and 10-3% for the elliptical domain. The obtained results confirm the computational stability and applicability of the proposed ABIE scheme.