New boundary element methods for group of elastostatic problems in advanced materials

Abstract

Nowadays, more and more advanced materials appear to meet the needs of various requirements in engineering and science. These materials are generally characterized by heterogeneous and anisotropic materials. Their complex material properties have significant influence on the elastic behaviors.

This research aims to develop the new Boundary Element Methods (BEMs) to solve groups of elastostatic problems in advanced materials. Firstly, four fundamental solutions are applied to the BEMs and two of them are used to develop the BEMs for the first time. They are for the multilayered homogeneous and isotropic fullspace and the transversely isotropic bi-material fullspace under a point load or a circular ring load. Hence, in analyzing layered and graded materials, the boundary surfaces need to be discretized while the discretizations on the material interfaces can be avoided. This feature significantly decreases the calculation amounts and makes it possible to analyze the elastostatic problems of heterogeneous and anisotropic materials efficiently and accurately.

Secondly, different types of boundary elements satisfy different requirements in discretizing the boundary surfaces. Discontinuous elements are used to avoid the complex calculations at the strike lines of material interfaces and capture the crack-tip behavior. Infinite elements are applied to consider the influence of the far-field areas. Discontinuous infinite elements are developed to combine the characteristics of discontinuous elements and infinite elements together. Thirdly, many types of numerical integral methods are summarized and applied to solve the regular, nearly singular and singular integrals in the discretized boundary integral equations. Particularly, Kutt’s numerical quadrature together with the coordinate transform is applied to solve strongly singular integrals. Numerical verifications for benchmark problems are conducted, which gives the excellent agreement with previously results.

Fourthly, applying the proposed BEMs, this thesis makes breakthroughs in analyzing the complicated elastic responses of isotropic and transversely isotropic layered halfspaces. The first application is to analyze the non-horizontally layered homogeneous and isotropic layered halfspace or transversely isotropic bi-material halfspace under external loadings. Numerical results show that the angle values between the non-horizontal material interface and the horizontal boundary surface can have significant influence on the elastic responses under external loadings. The displacements are non-smoothly continuous at the material interfaces and the stresses are non-smoothly continuous or discontinuities at the material interfaces. The second application is to study the axisymmetric problems of layered homogeneous and isotropic halfspace or transversely isotropic bi-material halfspace without or with a cavity under external or internal loadings, respectively. It is shown that the influence of material properties and cavity shapes on elastic displacements and stresses are remarkable. The third application is to solve crack problems in layered and graded halfspace. Results show that the material heterogeneity and crack shapes can have a profound effect on the stress intensity factors.

This research also provides fundamental cornerstones to future works. These future works include the extensions of new BEMs for non-horizontally layered halfspace problems under complex loading conditions, complex axisymmetric problems with cavity, and the preparation of a handbook for stress intensity factors and corresponding software products.