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


Other title
Anti-plane deformations
Microstructural effect
Surface effect
Boundary integral equation method
Type of item
Degree grantor
University of Alberta
Author or creator
Sigaeva, Taisiya
Supervisor and department
Schiavone, Peter (Mechanical engineering)
Examining committee member and department
Federico, Salvatore (Mechanical and Manufacturing Engineering, University of Calgary)
Tang, Tian (Mechanical engineering)
Kim, Chun Il (Mechanical engineering)
Wang, Xiaodong (Mechanical engineering)
Schiavone, Peter (Mechanical engineering)
Department of Mechanical Engineering

Date accepted
Graduation date
Doctor of Philosophy
Degree level
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.
Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.
Citation for previous publication
Sigaeva, T., Schiavone, P.: Solvability of the Laplace equation in a solid with boundary reinforcement. Z. Angew. Math. Phys. 65, 809-815 (2014)Sigaeva, T., Schiavone, P.: Solvability of a theory of anti-plane shear with partially coated boundaries. Arch. Mech. 66, 113-125 (2014)Sigaeva, T., Schiavone, P.: Surface effects in anti-plane deformations of a micropolar elastic solid: Integral equation methods. Continuum Mech. Therm. DOI:10.1007/s00161-014-0404-3 (2014)Sigaeva, T., Schiavone, P.: The Effect of Surface Stress on an Interface Crack in Linearly Elastic Materials. Math. Mech. Solids, DOI:10.1177/1081286514534871 (2014)Sigaeva, T., Schiavone, P.: Influence of Boundary Elasticity on a Couple Stress Elastic Solid with a Mode-III Crack. Quart. J. Mech. Appl. Math. (Accepted)

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