Static and harmonic BEM solutions of gradient elasticity problems with axisymmetry

Abstract

An advanced boundary element method/fast Fourier transform (BEM/FFT) methodology for treating static and time harmonic axisymmetric problems in linear elastic structures exhibiting microstructure effects, is presented. These microstructure effects are taken into account with the aid of a simple strain gradient elastic theory proposed by Aifantis and co-workers [Aifantis (1992), Altan and Aifantis (1992), Ru and Aifantis (1993)]. Boundary integral representations of both static and dynamic gradient elastic problems are employed. Boundary quantities, classical and non-classical (due to gradient terms) boundary conditions are expanded in complex Fourier series in the circumferential direction and the problem is decomposed into a series of problems, which are solved by the BEM by discretizing only the surface generator of the axisymmetric body. The BEM integrations are performed by FFT in the circumferential directions simultaneously for all Fourier coefficients and by Gauss quadrature in the generator direction. All the strongly singular integrals are computed directly by employing highly accurate three-dimensional integration techniques. The Fourier transform solution is numerically inverted by the FFT to provide the final solution. The accuracy of the proposed boundary element methodology is demonstrated by means of representative numerical examples.