AIRY STRESS FUNCTION FOR TWO DIMENSIONAL INCLUSION ...

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ur1Ini = FullSimplify@ur1D êêTrigReduce 1 8r 3 µ1 H−r4 S+r 4 Sκ1+2r 4 SCos@2θD−4r 2 f1−4rCos@θDf2−4Cos@2θDf3+ 4r 2 Cos@2θDm1+4r 2 κ1Cos@2θDm1+8rCos@3θDm2+4rκ1Cos@3θDm2L ur = HFullSimplify@urIni êêTrigReduceDL ê.r →a ur1 = HFullSimplify@ur1Ini êêTrigReduceDL ê.r →a aH−1+κLa1+Cos@θDHκa0+a 2 H−2+κLa2−b0L−aCos@2θDb1−a 2 Cos@3θDb2 2µ 1 8a 3 µ1 Ha4 SH−1+κ1+2Cos@2θDL−4Cos@2θDf3+ 4aH−af1−Cos@θDf2+aH1+κ1LCos@2θDm1+H2+κ1LCos@3θDm2LL uθ1Ini = HComplexExpand@ui1DêêTrigReduceLê.Cos@θD^2 +Sin@θD^2 →1 1 4r 3 µ1 H−r4 SSin@2θD−2rSin@θDf2−2Sin@2θDf3+ 2r 2 Sin@2θDm1−2r 2 κ1Sin@2θDm1+4rSin@3θDm2−2rκ1Sin@3θDm2L uθ = HSimplify@uθIniD êêTrigReduceL ê.r →a uθ1 = HSimplify@uθ1IniD êêTrigReduceL ê.r →a −κSin@θDa0+2a 2 Sin@θDa2+a 2 κSin@θDa2+Sin@θDb0+aSin@2θDb1+a 2 Sin@3θDb2 2µ 1 4a 3 µ1 H−a4 SSin@2θD−2aSin@θDf2−2Sin@2θDf3+ 2a 2 Sin@2θDm1−2a 2 κ1Sin@2θDm1 +4aSin@3 θDm2 −2aκ1Sin@3 θDm2L 44
THE SIMULTANEOUS EQUATIONS THAT HAVE TO BE SOLVED TO FIND THE CONSTANTS Eq1 =FullSimplify@ Coefficient@σrr êêExpand,Cos@θDD − Coefficient@σrr1 êêExpand, Cos@θDDD 2aa2 − 2f2 a 3 Eq2 = FullSimplify@Coefficient@σrr êê Expand, Cos@2 θDD − Coefficient@σrr1 êêExpand, Cos@2 θDDD − S 2 3f3 4m1 −b1− + a4 a2 Eq3 = FullSimplify@Coefficient@σrr êêExpand, Cos@3 θDD − Coefficient@σrr1 êêExpand, Cos@3 θDDD −2ab2+ 10m2 a 3 Eq4 =FullSimplify@Coefficient@τrθ êêExpand, Sin@θDD − Coefficient@τrθ1 êêExpand, Sin@θDDD 2aa2 − 2f2 a 3 Eq5 = FullSimplify@Coefficient@τrθ êêExpand, Sin@2 θDD − Coefficient@τrθ1 êêExpand, Sin@2 θDDD S 2 3f3 2m1 +b1− + a4 a2 Eq6 =FullSimplify@Coefficient@τrθ êêExpand, Sin@3 θDD − Coefficient@τrθ1 êêExpand, Sin@3 θDDD 2ab2 + 6m2 a 3 45
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 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: σθθ1 =FullSimplify@HD@φ1, 8r,2
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,

AIRY STRESS FUNCTION FOR TWO DIMENSIONAL INCLUSION ...

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