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UBC Theses and Dissertations

Analytical and finite element solutions for an elastic matrix with two-dimensional nanoscale inhomogeneities Tian, Lian


A new frontier of research at the interface of material science and mechanics of solids has emerged with the current focus on the development of nanomaterials such as nanotubes, nanowires and nanoparticles. As the dimensions of a structure approach the nanoscale, its properties can be size-dependent. The classical continuum theory, however, does not admit an intrinsic size, and is not applicable to the analysis of nanoscale materials and structures. Mechanics of nanomaterials-based composites can be understood by incorporating the effects of surface and interfacial energy. A fundamental problem in the study of behaviour of such materials is the examination of size-dependent elastic field of an elastic matrix with nanoscale inhomogeneities. Classical inhomogeneity problems have a rich history since the celebrated work of Eshelby. However, the classical solutions cannot be applied to study nanoscale inhomogeneity problems and new solutions accounting for surface/interface energy have to be derived. This thesis therefore presents an analytical scheme and a finite element formulation to study the size-dependent elastic field of an elastic matrix containing two-dimensional nanoscale inhomogeneities. The Gurtin-Murdoch surface/interface elasticity model is applied to incorporate the surface/interface energy effects. By using the complex potential technique of Muskhelishvili, a closed-form analytical solution is obtained for the elastic field of a nanoscale circular inhomogeneity in an infinite matrix under arbitrary remote loading and a uniform eigenstrain. In the case of an elliptical inhomogeneity, the analytic potential functions are obtained approximately. A new finite element formulation that takes into account the surface stress effects is presented. Elastic field is found to depend on the characteristic dimensions of the inhomogeneity, surface elastic constants and surface residual stress. A striking feature of the new solutions is the existence of singular elastic fields below some dimensions of the inhomogeneity. This phenomenon requires careful further investigation. Eshelby tensor of a nanoscale circular inhomogeneity in an infinite matrix due to a uniform eigenstrain is uniform but becomes size-dependent; however, the tensor is size-dependent and non-uniform in the case of an elliptical inhomogeneity.

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