ISSN 1066-369X, Russian Mathematics, 2018, Vol. 62, No. 2, pp. 19–27.
c
Allerton Press, Inc., 2018.
Original Russian Text
c
N.G. Gur’yanov, O.N. Tyuleneva, 2018, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2018, No. 2, pp. 23–31.
Cylindrical Tank Filled With a Liquid
in a Three-Dimensional Temperature Field
N. G. Gur’yanov
*
and O.N.Tyuleneva
**
Kazan Federal University
ul. Kremlyovskaya 18, Kazan, 420008 Russia
Received November 8, 2016
Abstract—In the cylindrical coordinate system, we construct an exact solution of the three-
dimensional thermoelasticity problem for a tank filled with a liquid. After determining the tem-
perature field from the heat conduction equation, we solve the equations of the asymmetrical
problem of the theory of elasticity. In doing so, the system of resolving equations is reduced to four
separate equations with respect to the displacements of the construction. Several exact solutions of
boundary-value problems are found. The results are presented in the form of rather simple formulas.
DOI: 10.3103/S1066369X18020032
Keywords: temperature field, thermoelasticity, integrable combinations, boundary-value
problems, analytical solutions.
In the cylindrical coordinate system, the most compact form of the equations of the three-dimensional
elasticity theory, which are written in terms of the displacements and stresses, is given in the monographs
[1, 2]. Analogous equations with allowance for the temperature terms, which were constructed using
the Duhamel–Neumann relations, are presented in the monograph [3]. All these equations have a
common disadvantage, namely, it is not possible to obtain their integrable combinations, which leads
to insurmountable difficulties in constructing exact solutions. In the monograph [4], this problem was
solved for equations that do not contain temperature terms by means of introducing into the system
an additional equation with respect to the volume deformation. An exact solution to this equation was
already published in the work [1], but the authors of the work [4] succeeded in using the equation to solve
the entire system of resolving relations.
The aim of this work is to construct the system of equations of the elasticity theory, which are most
convenient for integration, and to find the solution of these equations.
As initial equations, we take those for a circular cylinder [3]. Without taking into account the mass
forces, they can be written with respect to the displacements as follows:
Δw +
R
(1 − 2υ)
1
ε
∂θ
∂γ
− 2(1+υ) α
T
1
ε
∂T
∂γ
=0,
Δ −
1
α
2
u +
R
(1 − 2υ)
∂θ
∂α
− 2(1+υ) α
T
∂T
∂α
−
2
α
2
∂v
∂β
=0,
Δ −
1
α
2
v +
R
(1 − 2υ)
1
α
∂θ
∂β
− 2(1+υ) α
T
1
α
∂T
∂β
+
2
α
2
∂u
∂β
=0,
θ =
1
R
1
α
∂ (αu)
∂α
+
1
α
∂v
∂β
+
1
ε
∂w
∂γ
.
(1)
Here α, β,andγ are the dimensionless cylindrical coordinates (α is scaled by the outer radius R of
the cylinder, γ is scaled by the height H of the cylinder, β is the angular coordinate along the guide);
r is the radius of the inner lateral surface; u, v,andw are the displacements along the coordinate
*
E-mail: gng.ggb@mail.ru.
**
E-mail: tdv.ton@mail.ru.
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