AIRY STRESS FUNCTION FOR TWO DIMENSIONAL INCLUSION ...

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2.1.1 Polar Coordinate Formulation The polar coordinates play an important part in solving many plane problems in elasticity and the above developed equation can be worked out by converting it into the polar coordinates thus giving us the governing equations in this curvilinear system. For the polar coordinate system, the solution to plane stress and plane strain basic vector differential problems involves the determination of the in-plane displacement and stresses { u , u , ε , ε , γ , σ , σ , τ } in region R subjected to prescribed boundary condition on S. r θ r θ rθ r θ rθ The two-dimensional in-plane stresses from the Cartesian to the polar coordinates will transform as follows: 2 2 σ = σ cos θ + σ sin θ + 2τ sin θ cos θ, r x y xy 2 2 σ = σ sin θ + σ cos θ − 2 τ sin θ cos θ, (2.1.6) θ x y xy 2 2 τ = − σ sin θ cosθ + σ sin θ cos θ + τ (cos θ − sin θ). rθ x y xy The relations (2.1.3) between the stress components and the Airy stress function can be transformed to the polar form to yield σ σ τ r θ rθ 2 1 ∂φ 1 ∂ φ = + 2 2 , r ∂r r ∂ θ 2 ∂ φ = 2 , ∂r ∂ ⎛ 1 ∂φ ⎞ = − ⎜ ⎟ . ∂r ⎝ r ∂θ ⎠ 8 (2.1.7) In the absence of a body force, the biharmonic equation (2.1.5) changes to the polar coordinates as
2 2 2 2 ⎛ ⎞ ⎛ ⎞ 4 ∂ 1 ∂ 1 ∂ ∂ 1 ∂ 1 ∂ ∇ φ = ⎜ 2 + + 2 2 ⎟ ⎜ 2 + + 2 2 ⎟ φ = 0. (2.1.8) ⎝ ∂r r ∂r r ∂ θ ⎠ ⎝ ∂r r ∂r r ∂ θ ⎠ The plane problem again is formulated in term of the Airy stress function, φ( r , θ) , with a single governing biharmonic equation as required. 2.2 Complex Variable Methods A complex variable z is defined by two real variables x and y This definition can also be expressed in polar form by z = x + iy. (2.2.1) Figure 2-2 Complex Plane. z r i re i = (cosθ + sin θ) = 2 2 where r = x + y known as the modulus of z and θ = −1 tan ( y / x) the argument 9 θ (2.2.2) z x iy re i − θ = − = . (2.2.3) z is the complex conjugate of the variable z
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: epresentation It is observed that t
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,

AIRY STRESS FUNCTION FOR TWO DIMENSIONAL INCLUSION ...

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