Graduation Semester and Year
Spring 2025
Language
English
Document Type
Dissertation
Degree Name
Doctor of Philosophy in Mechanical Engineering
Department
Mechanical and Aerospace Engineering
First Advisor
Seiichi Nomura
Abstract
This research presents an analytical framework for determining stress fields in an elastic medium containing a circular anisotropic inclusion embedded in an infinitely extended isotropic matrix subjected to far-field uniform stresses. Stress distributions within both the inclusion and the matrix are described using classical stress functions, widely employed in two-dimensional elasticity theory.
To manage the complexity of the mathematical derivations, symbolic computation software (Mathematica) is used to streamline the analysis and obtain closed-form solutions. This approach overcomes traditional computational barriers that have limited the practical application of complex variable methods (CVM) in elasticity.
The methodology builds upon the foundational work of Kolosov and Muskhelishvili, who introduced complex variable techniques for solving plane elasticity problems in isotropic materials through Airy stress functions. These were later extended by Lekhnitskii to anisotropic materials, where stress functions are expressed as the real parts of two analytic functions of independent complex variables – derived from the roots of a characteristic equation based on the material’s elastic constants.
In this study, the Kolosov–Muskhelishvili complex variable method is applied to the isotropic matrix, while Lekhnitskii’s formalism is used to model the anisotropic inclusion. The stress function in the matrix is expanded in a Laurent series, and a Taylor series is used for the inclusion. Continuity of displacement and traction across the matrix–inclusion interface, along with far-field boundary conditions, are used to determine the unknown complex coefficients in these expansions.
The resulting stress field solutions are novel, exact, and presented in closed form. While the primary focus is on stress analysis of anisotropic inclusion problems, the developed methodology is broadly applicable to a wide range of physical problems – including anisotropic structural and thermal analyses, as well as their coupled-field phenomena – demonstrating its versatility in tackling advanced mechanics and materials science challenges.
Keywords
Plane Problems, Airy stress function, Lekhnitskii formulation, Symbolic computation, Closed-form solution, Displacement and Traction continuity, Complex variables, Laurent series and Taylor series, Anisotropic inclusion, Isotropic matrix
Disciplines
Applied Mechanics
License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License
Recommended Citation
Chen, Liming, "STRESS ANALYSIS OF ANISOTROPIC INCLUSION PROBLEMS USING COMPLEX VARIABLES" (2025). Mechanical and Aerospace Engineering Dissertations. 426.
https://mavmatrix.uta.edu/mechaerospace_dissertations/426