An examination of the design of mathematical models incorporating both microstructural and surface effects in anti-plane deformations

• Author / Creator

  Sigaeva, Taisiya

• Micropolar theory and surface mechanics are rapidly becoming key tools in the development of more advanced models which can precisely describe the behavior of deformable elastic solids. Renewed interest in these areas has arisen due to the desire of researchers to generalize continuum-based models for applications in a wider class of materials, such as the micro-featured materials, and at smaller scales, such as the nano-scale. The analysis of such classes of materials, in which the effects of both the surface and microstructure are known to be significant, can be greatly benefited from micropolar theory and surface mechanics. However, the multidisciplinary study aimed to develop mathematically and physically adequate models based on both of these theories remains largely absent from the literature due to a number of difficulties. To fill this void in the literature, in this work we employ the theory of linear micropolar elasticity in conjunction with a new representation of micropolar surface mechanics to develop a comprehensive model for the deformations of a linearly micropolar elastic solid subjected to anti-plane shear loading. The proposed model represents the surface effect as a thin micropolar film of separate elasticity, perfectly bonded to the bulk. Hence, this model captures not only the micro-mechanical behavior of the bulk, which is known to be considerable in many real materials, but also the contribution of the surface effect which has been experimentally well-observed for bodies with significant size-dependency and large surface area to volume ratios. Our emphasis in this research is the rigorous mathematical treatment of this model, particularly its well-posedness analysis in the Hadamard's sense. Although challenging, the well-posedness analysis is vital in the development of brand-new models, since it can give a sufficient confidence to find numerically a uniquely existing solution to the problem. To perform this analysis, we apply boundary integral equation methods generalizing and utilizing them as necessary to account for strict requirements of the proposed model. The coupling of surface mechanics to bulk models gives rise to a highly non-standard boundary condition which has not been accommodated by classical studies in this area. Therefore, a portion of this work is devoted to the study of the surface effect in the classical linear elastic analogue of the proposed model. This supplementary model is thoroughly analyzed for well-posedness and an example demonstrating its efficiency is given. These investigations provided valuable insight on how to tackle the mathematical complexity of the general model, for which bulky micropolar governing equations are used in addition to the similar highly non-standard surface effect boundary condition. Accordingly, we supply a rigorous mathematical treatment of the mixed boundary-value problems in finite and infinite domains for the proposed model which combines both microstructural and surface effects. Boundary integral equation methods are employed to reduce these problems to systems of singular integro-differential equations using a representation of solutions in the form of modified single-layer potentials. Analysis of these systems demonstrates that the classical Noether's theorems reduce to Fredholm's theorems from which results on well-posedness are deduced. Finally, we provide a demonstration of the proposed model's contribution to fracture mechanics and argue that more sophisticated models produce higher accuracy in predicting material behavior.

• Subjects / Keywords

  • Anti-plane deformations
  • Surface effect
  • Microstructural effect
  • Boundary integral equation method

• Graduation date

  Spring 2015

• Type of Item

  Thesis

• Degree

  Doctor of Philosophy

• DOI

  https://doi.org/10.7939/R3GB1XP81

• 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.