AIRY STRESS FUNCTION FOR TWO DIMENSIONAL INCLUSION ...

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formulation and solution. Because all real elastic structures are three-dimensional, the theories set forth here will be approximate models. The nature and accuracy of the approximation depend on the problem and loading geometry. Two basic theories, plane stress and plane strain represent the fundamental plane problem in elasticity. These two theories apply to significantly different types of two-dimensional bodies although their formulations yield very similar field equations. Numerous solutions to plane stress and plane strain problems can be obtained through the use of a particular stress functions technique. The method employs the Airy stress function [1] and will reduce the general formulation to a single governing equation in terms of a single unknown. The resulting governing equation is then solvable by several methods of applied mathematics, and thus many analytical solutions to problems of interest can be generated. The stress function formulation is based on the general idea of developing a representation for the stress field that satisfies equilibrium and yields a single governing equation from the compatibility statement. This thesis is a successful attempt to apply the above-mentioned method to a plate of infinite length and width with a central hole and a disc separately. The problem of a circular hole in an infinite plate has been studied for many years with various approaches [1] including the Airy stress function approach. This problem has many applications in engineering as it can reveal stress singularity around the hole. However, due to the complexity of algebra involved, there has been no work about a two-dimensional plate with a circular inclusion (disc) using the Airy stress function to 2
our best knowledge. This thesis addresses such a problem using symbolic algebra software 1.2 Use of Symbolic Software The development of hardware and software of computers has made available symbolic algebra software packages such as MATLAB, MAPLE, MATHEMATICA [2]. Older packages such as Macsyma- one of the very first general-purpose symbolic computations systems were written in LISP whereas new ones such as Mathematica are written in the C language and its variations and is one of the most widely available symbolic systems. Using symbolic algebra systems, one can evaluate mathematical expressions analytically without any approximation. Differentiations, integrations, expansions and solving equations exactly are the major features of symbolic algebra systems. Most of the symbolic algebra systems have been used by mathematicians and theoretical physicists [8]. The ability to deal with symbolic formulae, as well as with numbers, is one of its most powerful features. This is what makes it possible to do algebra and calculus. It has been demonstrated that in certain circumstances the widely held view that one can always dramatically improve on the CPU time required for lengthy computations by using compiled C or Fortran code instead of advanced quantitative programming environments such as Mathematica, MATLAB etc… is wrong. A well written C program can be expected to outperform Mathematica, R, S-Plus or MATLAB [7] but, if the C program is not efficiently programmed using the best possible 3
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: CHAPTER 1 INTRODUCTION 1.1 Overview
Page 11 and 12: • A finite plate with a hole subj
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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AIRY STRESS FUNCTION FOR TWO DIMENSIONAL INCLUSION ...

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