AIRY STRESS FUNCTION FOR TWO DIMENSIONAL INCLUSION ...

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y R ( a ) ( b ) ( c ) Figure 2-3 Typical Domains of Interest : (a) Finite Simply Connected, (b) Finite Multiply Connected, (c) Infinite Multiply Connected. 2.3.2 Finite Multiply Connected Domain For a general region surrounded with a defined external boundary assumed to have n internal boundaries as shown in Figure 2-3(b), the potential need not be single valued. This can be demonstrated by considering the behavior of the stresses and displacements around each of the n contours Ck in region R. Assuming that the displacement and stresses are single valued and continuous everywhere, we get 14 C1 x x y R C3 C2 C0
γ ( z) = − ψ ( z) = n ∑ k = 1 n ∑ k = 1 Fk * log( z − zk ) + γ ( z), 2π ( 1 + κ ) κFk * log( z − zk ) + ψ ( z), 2π ( 1 + κ ) 15 (2.3.2) where Fk is the resultant force on each contour Ck, zk is a point within the contour Ck, γ * (z) and ψ * (z) are arbitrary analytic functions in R, and κ is the material constant 2.3.3 Infinite Domain For an externally unbounded region having m internal boundaries as shown in Figure 2-3©, the potentials can be determined by taking into consideration that the stresses remain bounded at infinity. Taking the requirement into consideration we get ∞ ∞ ∞ where σ σ τ m ∑ Fk ∞ ∞ σ σ k x + = 1 y ** γ ( z) = − log z + z + γ ( z), 2π ( 1 + κ ) 4 m ∑ κ Fk ∞ ∞ ∞ σ σ iτ k y − x + 2 = 1 xy ** ψ ( z) = log z + z + ψ ( z). 2π ( 1 + k) 2 x , y , xy are the stresses at infinity and γ ** ( z) and ψ ** ( ) (2.3.4) z are arbitrary analytic functions outside the region enclosing all n contours. Using power series theory these analytic functions can be expressed as: γ ψ ** ** ∞ ∑ −n ( z) = a z , n= 1 ∞ ∑ −n ( z) = b z . n= 1 n n (2.3.5)
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 and 10: our best knowledge. This thesis add
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: σ + σ = σ + σ r θ x y 2iθ σ
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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