Coupled Thermoelasticity
When a structure is under the thermal shock load, the governing equation of thermoelasticity and the first law of thermodynamics are coupled. This thermal shock may be applied to the surface of a body or may be caused through the body heating.
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Coupled Thermoelasticity
When a structure is under the thermal shock load, the governing equation of thermoelasticity and the first law of thermodynamics are coupled. This thermal shock may be applied to the surface of a body or may be caused through the body heating. When the period of applied thermal shock is considerably smaller than the time period of the first natural frequency of the structure, then coupled equations may be justified to be employed to obtain the stress and deformation of the structure. In this chapter the governing equations for the classical coupled thermoelasticity theory are given. Some basic problems, such as solid sphere and one-dimensional rod, are considered and their behavior under thermal shock loads are discussed.
26.1 Governing Equations, Classical Theory Returning to displacement formulations, and introducing the displacement vector U = ui + vj + wk, the vectorial form of governing equations of the classical coupled thermoelasticity are
and
˙ = −R k∇ 2 T − ρc T˙ − αT0 (3λ + 2μ) div U
(26.1)
¨ μ∇ 2 U + (λ + μ) grad divU − (3λ + 2μ)α grad T = ρU
(26.2)
The displacement vector can now be written as the sum of an irrotational and a potential part as given in the form U = grad ψ + curl
(26.3)
where ψ is a scalar potential, and is a vector potential. We may substitute Eq. (26.3) into Eqs. (26.2) and (26.1) to arrive at
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_26, © Springer Science+Business Media Dordrecht 2013
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26 Coupled Thermoelasticity
1 ¨ (3λ + 2μ) α(T − T0 ) ψ= 2 λ + 2μ c1 1 ¨i = 0 ∇ 2 i − 2 i = 1, 2, 3 c2 k∇ 2 T − ρc T˙ − αT0 (3λ + 2μ)∇ 2 ψ˙ = −R
∇2ψ −
(26.4)
where c1 and c2 are the speed of propagation of the elastic longitudinal wave and the speed of the shear wave, respectively. Elimination of T between the first and the last of Eq. (26.4) results in a single equation for ψ, namely 1 ∂ 1 ∂2 m1 R β 2 T0 2 2 ∇ 2 ψ˙ = − ∇ − 2 2 ψ− ∇ − κ ∂t (λ + 2μ)k k c1 ∂t with m1 =
β λ + 2μ
β = (3λ + 2μ)α
κ=
k ρc
(26.5)
(26.6)
and the equation for the components of vector remains as ∇ 2 i −
1 ¨i = 0 c22
i = 1, 2, 3
(26.7)
For quasi-steady problems, when the variation of temperature with respect to time is slow and the inertia effects are neglected, the system of equations reduces to 1 ˙ αT0 (3λ + 2μ) 2 ˙ R ∇ ψ=− T− κ k k α(3λ + 2μ) (T − T0 ) ∇2ψ = λ + 2μ
∇2T −
(26.8)
The function ψ can be eliminated from Eq. (26.8) and the equation for heat conduction takes the form R (26.9) ∇ 2 T − m T˙ = − k where m=
1 α2 T0 (3λ + 2μ)2 + κ k(λ + 2μ)
which is the uncoupled heat conduction equation in a solid body.
(26.10)
26.2 Problems and Solutions of Coupled Thermoelasticity
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26.2 Problems and Solutions of Coupled Thermoelasticity Problem 26.1. Solve Eq. (26.5) for a solid spherical domain with the radial thermal flow when R = 0 Solution: When the heat generation, R, thought the elastic medium is neglected, Eq. (26.5) reduces to 1 ∂ 1 ∂2
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