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Surface Effects in Plane Deformations of Micropolar Elastic Solids

  • Author / Creator
    Gharahi, Alireza
  • The predictive modeling of the mechanics of materials at small scales has attracted increasing attention in the recent literature mainly due to the scientific and technological demand for models which accommodate the influence of both material internal structure and surface effects from micro- to nano-scales. Materials whose deformation is significantly influenced by their internal structure and surface effects include the class of micro/nano composites as well as polycrystalline, granular, and fibrous materials used increasingly in a wide variety of advanced technological applications. Micropolar theory and surface mechanics are developed to bestow upon the continuum-based mathematical models the capability of analyzing such advanced materials. Notwithstanding the fact that the internal micro/nano-constituents and the surface effects are incorporated into the model as two different enhancing strategies, the simultaneous use of them remains rare in the literature. In particular, a systematic design and examination of such a model which describes plane deformations in micro/nano-materials is missing. In this thesis, we present two new linear micropolar surface mechanics models coupled with the plane deformations of a micropolar elastic bulk. The surface models are proposed as elastic micropolar shells capable of incorporating flexural resistance. In that sense, the adopted surfaces are the micropolar counterpart of the classical linear Steigmann-Ogden surface model. In conjunction with the micropolar bulk, the proposed surface models capture both the effects of surface and the effects of internal structures which are known to be dominant in most of the materials from micro- to nano-meter scale. Among the two, we apply and demonstrate the feasibility of the higher order model using three popular micro/nano-mechanics problems. In addition, we discuss the improvements that follow from such a comprehensive mathematical model. We evaluate the problems of stress concentration around a cavity, effective elastic properties of nano-composites and a dislocation near a surface. In each case, we obtain meaningful results which are in corroboration with the existing literature. In the next step, we perform a rigorous mathematical analysis to investigate \\wellposedness" of the models. First, we establish the general fundamental boundary value problems associated with the models and discuss the uniqueness of solutions. We then proceed with the analysis by applying the boundary integral equation method to examine the existence of solutions of the corresponding boundary value problems. The incorporation of surface effects as a separate micropolar structure gives rise to a set of highly nonstandard boundary conditions that require special treatments. The boundary integral equations method allows us to reduce the boundary value problems to systems of singular integro-differential equations. However, a series of carefully chosen transforms are required to rearrange the system of singular integro-differential equations in a form accommodated by the well-established theory of singular integral equations. Consequently, we establish the solvability of the boundary value problems and validate the corresponding mathematical models.

  • Subjects / Keywords
  • Graduation date
    Fall 2021
  • Type of Item
  • Degree
    Doctor of Philosophy
  • DOI
  • License
    This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.