AIRY STRESS FUNCTION FOR TWO DIMENSIONAL INCLUSION ...

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algorithm then in fact it may take longer than using a symbolic software byte-code compiler. At a technical level, Mathematica performs both symbolic and numeric calculations of cross-sectional properties such as areas, centroids, and moments of inertia. Symbolic softwares such as Mathematica can derive closed-form solutions for beams with circular, elliptical, equilateral-triangular, and rectangular cross sections [2]. Symbolic software also addresses the finite element method and is useful in finding shape functions, creating different types of meshes and can solve problems for both isotropic and anisotropic materials. They are also useful in the kinematic modeling of fully constrained systems[2]. This thesis will formulate a boundary value problem cast within a two- dimensional domain (subjected to far field loading) in the x-y plane using the Cartesian coordinates and then reformulating in the polar coordinates to allow further development and study in that coordinate system. The Airy stress function for specific two-dimensional plane conditions is computed and the stresses and displacements at a given point can be found using Mathematica. The thesis is divided into 4 chapters. Chapter 2 discusses the theory of two-dimensional elasticity behind the Airy stress function and the foundation of its formulation. It establishes a single governing equation for the plane stress and plane strain conditions by reducing the Navier equation to a form from which the Airy stress function can be derived. Chapter 3 contains the application of the Airy stress function to: 4
• A finite plate with a hole subjected to tensile loading • An infinite plate with a hole subjected to far field loading • A plate with a circular inclusion The formulations for all of the above problems were carried out using Mathematica and as a result the stresses and displacement given a certain point on the plate can be found. on the plate. Graphs will be shown to describe the variation of stresses at the various points Chapter 4 contains the conclusion and recommendation The Appendix will contain the description of all the formatted codes used to carry out the above. 5
Page 1 and 2: AIRY STRESS FUNCTION FOR TWO DIMENS
Page 3 and 4: Supervising Professor: Dr. Seiichi
Page 5 and 6: 3.2 Infinite Plate with a Hole Subj
Page 7 and 8: CHAPTER 1 INTRODUCTION 1.1 Overview
Page 9: our best knowledge. This thesis add
Page 13 and 14: epresentation It is observed that t
Page 15 and 16: 2 2 2 2 ⎛ ⎞ ⎛ ⎞ 4 ∂ 1 ∂
Page 17 and 18: where λ is called the Lame’s con
Page 19 and 20: σ + σ = σ + σ r θ x y 2iθ σ
Page 21 and 22: γ ( z) = − ψ ( z) = n ∑ k = 1
Page 23 and 24: CHAPTER 3 ADVANCED APPLICATIONS OF
Page 25 and 26: σ σ x y ( ± l, y) = S, ( x, ± c
Page 27 and 28: Sz b b χ ( z) = (log z) b . z z
Page 29 and 30: from which the stress components ca
Page 31 and 32: in σ = 2c + 2c r cosθ − d cos2
Page 33 and 34: 1 2a 4 4 4 2 ( a S + a S cos2θ + 2
Page 35 and 36: a b c 1 1 0 2 a d 2 a2 κ 1a2 − 2
Page 37 and 38: u out θ 4 2 2 4 S( a ( − µ + µ
Page 39 and 40: τ τ in xy out xy = 0, 6 10 6 2 6
Page 41 and 42: -50 -100 -150 0 τ 100 150 200 alon
Page 43 and 44: CHAPTER 4 CONCLUSIONS AND SUGGESTIO
Page 45 and 46: APPENDIX A MATHEMATICA CODE 39
Page 47 and 48: χ = Integrate[ϕ,z] χ1 = Integrat
Page 49 and 50: σθθ1 =FullSimplify@HD@φ1, 8r,2
Page 51 and 52: THE SIMULTANEOUS EQUATIONS THAT HAV
Page 53 and 54: aS aSκ1 aa1 − − 8 µ1 8 µ1 2
Page 55 and 56: PLATE τrθ1Final = ExpandAτrθ1In
Page 57 and 58: VERIFICATION OF THE EQUILIBRIUM EQU
Page 59: BIOGRAPHICAL INFORMATION Dharshini,

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