Solutions to Particular Two-Dimensional Boundary Value Problems of Elastostatics
In this chapter a number of two-dimensional boundary value problems for a body under plane strain conditions or under generalized plane stress conditions are solved. The problems include: (i) a semispace subject to an internal concentrated body force, (ii
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Solutions to Particular Two-Dimensional Boundary Value Problems of Elastostatics
In this chapter a number of two-dimensional boundary value problems for a body under plane strain conditions or under generalized plane stress conditions are solved. The problems include: (i) a semispace subject to an internal concentrated body force, (ii) an elastic wedge subject to a concentrated load at its tip, and (iii) an infinite elastic strip subject to a discontinuous temperature field. To solve the problems a two-dimensional version of the Boussinesq-Papkovitch-Neuber solution as well as an Airy stress function method, are used.
9.1 The Two-Dimensional Version of Boussinesq-Papkovitch-Neuber Solution for a Body Under Plane Strain Conditions An elastic state s = [u, E, S] corresponding to a body under plane strain conditions is described by the equations [see Eqs. 7.70 and 7.71 in Problem 7.1.] u α = ψα −
1 (xγ ψγ + ϕ),α 4(1 − ν)
(9.1)
bα μ
(9.2)
xγ bγ μ
(9.3)
where ψα,γ γ = − and ϕ,γ γ =
The strains E αβ and stresses Sαβ , associated with u α , are given, respectively, by
M. Reza Eslami et al., Theory of Elasticity and Thermal Stresses, Solid Mechanics and Its Applications 197, DOI: 10.1007/978-94-007-6356-2_9, © Springer Science+Business Media Dordrecht 2013
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9 Solutions to Particular Two-Dimensional Boundary Value Problems
E αβ =
1 [2(1 − 2ν) ψ(α,β) − xγ ψγ ,αβ − ϕ,αβ ] 4(1 − ν)
(9.4)
and Sαβ =
μ [2(1 − 2ν) ψ(α,β) − xγ ψγ ,αβ + 2ν ψγ ,γ δαβ − ϕ,αβ ] 4(1 − ν)
(9.5)
If a concentrated force P0 normal to the boundary of a semispace |x1 | < ∞, x2 ≥ 0 is applied at the point (x1 , x2 ) = (0, 0), and suitable asymptotic conditions are imposed on s = [u, E, S] at infinity, then a suitable choice of the pair (ϕ, ψα ) leads to the stress tensor Sαβ in the form S11 = −
2P0 2 2P0 2P0 x x2 , S22 = − 4 x23 , S12 = − 4 x1 x22 π r4 1 πr πr
where r = |x| =
x12 + x22
(9.6)
(9.7)
In polar coordinates (r, ϕ) related to the Cartesian coordinates (x1 , x2 ) by x1 = r cos ϕ, x2 = r sin ϕ we obtain Srr = −
2P0 sin ϕ, Sϕϕ = Sr ϕ = 0 πr
(9.8)
(9.9)
Clearly, it follows from (9.6) and (9.9) that |S| → 0 as r → ∞
(9.10)
Similarly, if a concentrated force T0 tangent to the boundary of a semispace |x1 | < ∞, x2 ≥ 0 is applied at the point (x1 , x2 ) = (0, 0), and suitable asymptotic conditions are imposed on s = [u, E, S] at infinity, then a suitable choice of the pair (ϕ, ψα ) leads to the stress tensor Sαβ in the form S11 = −
2T0 3 2T0 2T0 x1 , S22 = − 4 x1 x22 , S12 = − 4 x12 x2 4 πr πr πr
(9.11)
In polar coordinates (r, ϕ) we obtain Srr = −
2T0 cos ϕ, Sϕϕ = Sr ϕ = 0 πr
(9.12)
and it follows from Eqs. (9.11) and (9.12) that |S| → 0 as r → ∞
(9.13)
9.2 Problems and Solutions Related to Particular Two-Dimensional Boundary
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9.2 Problems and Solutions Related to Particular Two-Dimensional Boundary Value Problems of Elastostatics Problem 9.1. Find an elastic state s = [u, E, S] corresponding to a concentrated body force in an interior of a homogeneous and isotropic semispace |x1 | < ∞, x2 ≥ 0, under
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