Plane Thermoelastic Problems
In this chapter the basic treatment of plane thermoelastic problems in a state of plane strain and a plane stress are recalled. Typical three methods for the solution of plane problems are presented: the thermal stress function method for both simply conn
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Plane Thermoelastic Problems
In this chapter the basic treatment of plane thermoelastic problems in a state of plane strain and a plane stress are recalled. Typical three methods for the solution of plane problems are presented: the thermal stress function method for both simply connected and multiply connected bodies, the complex variable method with use of the conformal mapping technique, and potential method for Navier’s equations [See also Chap. 7].
17.1 Plane Strain and Plane Stress The unified systems of the governing equations for both plane strain and plane stress are as follows: The generalized Hooke’s law is 1 σx x − ν ∗ σ yy + α∗ τ − c∗ E∗ 1 = ∗ σ yy − ν ∗ σx x + α∗ τ − c∗ E 1 σx y = 2G
x x = yy x y
(17.1)
An alternative form σx x = (λ∗ + 2μ)x x + λ∗ yy − β ∗ τ σ yy = (λ∗ + 2μ) yy + λ∗ x x − β ∗ τ σx y = 2μx y where ∗
E =
⎧ ⎨
E =
⎩E
(17.1 )
E for plane strain 1 − ν2 for plane stress
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_17, © Springer Science+Business Media Dordrecht 2013
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424
17 Plane Thermoelastic Problems
∗
ν = α∗ = λ∗ =
ν = ν
ν 1−ν
for plane strain for plane stress
α = (1 + ν)α for plane strain α for plane stress
⎧ ⎨λ
for plane strain
2μλ ⎩λ = λ + 2μ ⎧ ⎨β 2μβ β∗ = ⎩ β = λ + 2μ ν0 ∗ c = 0
for plane stress
(17.2)
for plane strain for plane stress for plane strain for plane stress
The equilibrium equations in the absence of body forces are ∂σ yx ∂σx x + = 0, ∂x ∂y
∂σx y ∂σ yy + =0 ∂x ∂y
(17.3)
The compatibility equation is ∂ 2 yy ∂ 2 x y ∂ 2 x x + =2 2 2 ∂y ∂x ∂x∂ y
(17.4)
Navier’s equations are from Eqs. (7.25) and (7.35) ∂τ ∂e − β∗ =0 ∂x ∂x ∂τ ∂e − β∗ =0 μ∇ 2 u y + (λ∗ + μ) ∂y ∂y
μ∇ 2 u x + (λ∗ + μ)
(17.5)
where e = x x + yy + c∗ . The boundary conditions are σx x l + σ yx m = pnx , σx y l + σ yy m = pny
(17.6)
Next, we show typical three analytical methods for the plane problem. Thermal stress function method We introduce a thermal stress function χ related to the components of stress as follows ∂2χ ∂2χ ∂2χ (17.7) σx x = , σ yy = , σx y = − 2 2 ∂y ∂x ∂x∂ y
17.1 Plane Strain and Plane Stress
425
The governing equation for the thermal stress function χ is ∇ 4 χ = −α∗ E ∗ ∇ 2 τ
(17.8)
where ∇4 = ∇2∇2 =
∂2 ∂2 + 2 2 ∂x ∂y
∂2 ∂2 + 2 2 ∂x ∂y
=
∂4 ∂4 ∂4 + 2 2 2 + 4 (17.9) 4 ∂x ∂x ∂ y ∂y
⎧ ⎨ αE for plane strain α∗ E ∗ = 1 − ν ⎩ αE for plane stress
(17.10)
The components of displacement can be expressed in the form ∂χ 1 1 ∂ψ − − c∗ x + 2G ∂x 1 + ν∗ ∂ y ∂χ 1 ∂ψ 1 − + − c∗ y uy = 2G ∂y 1 + ν ∗ ∂x
ux =
(17.11)
where c∗ is a constant and the function ψ satisfies the equation σx x + σ yy + α∗ E ∗ τ = ∇ 2 χ + α∗ E ∗ τ ≡ in which
∂2ψ ∂x∂ y
∂2 ∇2ψ = 0 ∂x∂ y
(17.12)
(17.13)
When the external force does not apply to the body, the boundary conditions of pure thermal stress problems are χ(P) = C1 x + C2 y + C3 ∂χ(P) = C1 cos(n , x) + C2 cos(n , y) ∂n
(17.14)
where n denotes some direction which does
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