Quasistationary temperature stresses in multiply connected plates in the process of heating
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QUASISTATIONARY TEMPERATURE STRESSES IN MULTIPLY CONNECTED PLATES IN THE PROCESS OF HEATING R. M. Kushnir and T. Ya. Solyar
UDC 539.3
We present an algorithm for the evaluation of quasistationary temperature stresses in multiply connected plates with heat transfer heated by heat sources. For the solution of the problem, we use the Laplace transformation, improved formulas of its numerical inversion, and the method of integral equations. We present several examples of the numerical evaluation of nonstationary temperatures and temperature-induced stresses in plates of various shapes.
Statement of the Problem We study the problem of thermoelasticity for a thin plate whose median surface occupies a domain D bounded by curves L0 , L 1 , … , LN ( L0 is the outer contour; the domain D encloses all other contours). The plate is heated by internal heat sources. The heat exchange with the medium is realized through the cylindrical boundaries of the plate according to the Newton law. The initial temperature of the plate is denoted by T0 ( x, y ). Its boundary surfaces are unloaded. The temperature is constant across the thickness of the plate. Determination of the Temperature Field of the Plate The thermal state of the plate in the case where its heating is symmetric about the median surface is determined by the temperature field T obtained as a solution of the following boundary-value problem of heat conduction [1]: ΔT − χ 2 T =
λ
1 ∂T − q, a ∂τ
∂T + α(T − Tc ) = 0, ∂n
( x, y ) ∈ D,
( x, y ) ∈ L,
τ > 0,
τ > 0, T | τ = 0 = T0 ,
(1)
(2)
where Tc is the temperature of the medium outside the cylindrical boundary surface, q =
w , 2λh
Δ =
∂2 ∂2 + , ∂x 2 ∂y 2
χ2 =
α+ , λh
L = L 0 + L1 + … + LN ,
α + = α– are the heat transfer coefficients for the planes, z = ± h is the density of heat sources per unit area of Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, Ukrainian Academy of Sciences, Lviv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 42, No. 6, pp. 27–33, November–December, 2006. Original article submitted September 20, 2006. 1068–820X/06/4206–0739
© 2006
Springer Science+Business Media, Inc.
739
740
R. M. KUSHNIR
AND
T. YA. S OLYAR
the median surface of the plate, α is the heat transfer coefficient for the cylindrical surface, λ is the heat conduction coefficient, and h is the half thickness of the plate. We now apply the integral Laplace transformation with respect to the variable t = a τ to Eq. (1) and conditions (2). As a result, we get the following boundary-value problem for the Laplace transform of temperature T˜ ( x, y) : 2 ΔT˜ − χ1 T˜ = − T0 − q˜
( x, y ) ∈ D,
λ
∂T˜ + α (T˜ − T˜c ) = 0 ∂n
( x, y ) ∈ L,
(3)
where χ12 = χ 2 + s
T˜ =
and
∞
∫ T˜ ( x, y, τ) e
− sτ
dτ
0
If the Laplace transform of temperature T˜ ( x, y, s) is known and T ( x, y, t ) → T∞ ( x, y, t ) as t → ∞, then the temperature T ( x, y, t ) can be represented in the form of the following rapidly convergent series [2, 3]: T ( x, y, t ) =
⎡ ⎛t ⎞⎤ 1 ∞ ˆ 1 1 Tn exp (sn t ) + T0
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