Thermal Stresses in Spherical Bodies
In this chapter the thermal stresses in spherical bodies are presented. First, one-dimensional problems for a solid and a hollow sphere are discussed. Next, two-dimensional axisymmetric problems are treated by Goodier’s thermoelastic potential and the Bou
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Thermal Stresses in Spherical Bodies
In this chapter the thermal stresses in spherical bodies are presented. First, one-dimensional problems for a solid and a hollow sphere are discussed. Next, twodimensional axisymmetric problems are treated by Goodier’s thermoelastic potential and the Boussinesq harmonic functions. Problems and solutions for thermal stresses in a solid and a hollow cylinder subjected to the steady and the transient temperature field are presented. [See also Chap. 24.]
19.1 One-Dimensional Problems in Spherical Bodies The equilibrium equation without body force for a one-dimensional problem in a spherical coordinate system is obtained from Eq. (16.52) 2 dσrr + (σrr − σθθ ) = 0 dr r
(19.1)
Hooke’s law is from Eq. (16.59) σrr = 2μrr + λe − βτ σθθ = σφφ = 2μθθ + λe − βτ
(19.2)
where e = rr + 2θθ . The strain-displacement relations are rr =
du r , dr
θθ = φφ =
ur r
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_19, © Springer Science+Business Media Dordrecht 2013
(19.3)
475
476
19 Thermal Stresses in Spherical Bodies
The stress components in terms of the displacement component u r are du r ur E (1 − ν) + 2ν − (1 + ν)ατ (1 + ν)(1 − 2ν) dr r du u E r r ν + − (1 + ν)ατ = σφφ = (1 + ν)(1 − 2ν) dr r
σrr = σθθ
(19.4)
The equilibrium equation in terms of the displacement component u r is d 1 d 2 1 + ν dτ r ur = α dr r 2 dr 1 − ν dr
(19.5)
The general solution of Eq. (19.5) is ur =
1+ν 1 α 1 − ν r2
τr 2 dr + C1r + C2
1 r2
(19.6)
where C1 and C2 are constants. The stresses are expressed by σrr
(1 + ν)(1 − 2ν) 1 E −2 α 3 = (1 + ν)(1 − 2ν) 1−ν r
τr 2 dr
+ (1 + ν)C1 − 2(1 − 2ν)C2 σθθ = σφφ
(1 + ν)(1 − 2ν) 1 E = α 3 (1 + ν)(1 − 2ν) 1−ν r
1 r3
+ (1 + ν)C1 + (1 − 2ν)C2
τr 2 dr − τ
(19.7)
1 r3
The displacement and the thermal stresses in a solid sphere of radius a with free traction are a 1 r 2 α r ur = (1 + ν) 2 τr dr + 2(1 − 2ν) 3 τr 2 dr 1−ν r 0 a 0 a r αE 2 2 τr 2 dr − 3 τr 2 dr (19.8) σrr = 1 − ν a3 0 r 0 a αE 2 1 r 2 2 σθθ = σφφ = τr dr + τr dr − τ 3 3 1−ν a 0 r 0 The displacement and the thermal stresses in a hollow sphere of inner radius a and outer radius b with free traction are
19.1 One-Dimensional Problems in Spherical Bodies
ur =
1+ν 1 α 1 − ν r2
r
477
τr 2 dr
a
b r 1 b 2 2(1 − 2ν) a3 2 τr dr + τr dr 1 + ν b3 − a 3 a b3 − a 3 r 2 a αE 2(r 3 − a 3 ) b 2 2 r 2 = τr dr − 3 τr dr (19.9) 1 − ν r 3 (b3 − a 3 ) a r a b αE 2r 3 + a 3 1 r 2 2 = σφφ = τr dr + τr dr − τ 1 − ν r 3 (b3 − a 3 ) a r3 a +
σrr σθθ
The displacement and the thermal stresses in an infinite space with a spherical cavity of radius a with free traction are 1+ν 1 r 2 α τr dr ur = 1 − ν r2 a r 2αE 1 τr 2 dr (19.10) σrr = − 1 − ν r3 a αE 1 r 2 τr dr − τ σθθ = σφφ = 3 1−ν r a
19.2 Two-Dimensional Axisymmetric Problems We now consider two-dimensional axisynetric problems of a spherical body. The equilibrium equa
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