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Section 3. Mechanic engineering
DOI:10.29013/AJT-24-5.6-69-71
ANALYSIS OF THE OPERATION OF A HOLLOW THICK-WALLED CYLINDER IN THE CASE OF A STATIONARY AXISYMMETRIC TEMPERATURE LOAD
Kalmova Maria1
1 Department of Structural Mechanics, Engineering Geology, Foundations and Foundations, Samara State Technical University
Cite: Kalmova M. (2024). Analysis of the Operation of a Hollow Thick-Walled Cylinder in The Case of a Stationary Axisymmetric Temperature Load. Austrian Journal of Technical and Natural Sciences 2024, No 3 - 4. https://doi.org/10.29013/AJT-24-5.6-69-71
Abstract
The work of a hollow thick-walled cylinder made of anisotropic material is analyzed in the case of a stationary axisymmetric load acting on it. The law of change in the increment of the temperature load in the cylinder is known. The design scheme is represented by a thick-walled cylinder, the end surfaces of which are rigidly fixed in the axial plane, and there are no fasteners in the radial plane, the cylindrical surfaces are stress-free. It is concluded that only radial and circumferential deformations can be taken into account in the calculations, and relative deformations along the height of the cylinder can be neglected.
Keywords: axisymmetric problem, thermoelectroelasticity, finite integral transformations, stationary action
Introduction
Consider a hollow thick-walled cylinder made of an anisotropic material under the action of a stationary temperature load. The mathematical formulation of the problem includes differential equations of equilibrium of the components of the displacement vector and temperature increment, as well as boundary conditions (Grinchenko V. T., Ulitko A. F., Shulga N. A., 1989; Senitsky Yu. E. 2011):
d d2U d2W
—VU + a —t- + a
dr
1 dz2
dr dz
se
dr
(1)
dW d2W du de
a1V--h a3 —— + a 2 V-= a 4 —
dr dz dz dz
dW dU
z = 0,h W = 0, + = o , (2)
dr dz
D1 dU U dW f }
r = ~ + a5 — + a6-T— = {®1,®2) ,
dr r dz
dW+dU=o, (3)
dr dz
This system of equations is presented in a
Section 3. Mechanic engineering
The system (5) is reduced to a resolving
dimensionless form. We investigate it by us- equation with respect to the function Ws:
ing Fourier transforms:
Uc (r ,n ) = \h0U (r, z )C0S (jnz )dz ,
Ws (r,n) = J W(r,z)sin(jnz)dz, U (r, z ) = Yi^~1Uc (r ,n )cos (jnz),
n=1 2 ^
W(r'z) = 2TsW (rn)sin(jnz),
h n=1
d 2 dW
-VUC -aifUc + «2jn ~rJ~ = Fl, dr dr
d^ 2 a-a3)nWs -a2JnVUc = a4F2;
V d V W - HV W - = F,, (7)
dr dr dr
the right-hand side of which admits the following factorization into commutative multipliers:
d
V
V —- A2 dr
d
W - FH, (8)
(4)
(5)
r = R ,1
dr dU
U
+ a5 + a6 jnWs ={fflic ^c] >
dr r
V--B
JV dr J
The general solution of the differential equation (9) has the form:
W5 (r ,n ) = DJ o (Anr) + D2nKo (Anr ) + +D3nIo (Bj) + D4nKo (Bj) +
idW^(r(9)
The expression for the function Uc (r,n) is obtained by reducing the system (5) to (7) and has the form:
dW
, . 1 d V7dW5 Uc (r ,n ) =—r—^ +
dr
- jnUc = 0,
(6)
a2 J3 dr dr
where F1 (r,n) = — £ 0(r,z)cos(jnz)dz .
(a2a2 - a3) dW s
F - a 4 dF2
•2 1 „ „ -3
(10)
dr
F2 (r ^ ) = J£
dz sin(Jnz)dz,
©1C (R,n),©2C (1,n)} = d£ |©1 (R,z),©2 (1,z)}cos(jnz)dz .
a.a2jn dr aj2 axa2jn dr The final expressions for determining displacements are obtained as a result of substituting (9), (10) into (4).
Analyzing the numerical dependences given, graphs of changes in height displacements of a piezoceramic cylinder with finite dimensions, without taking into account the electrical load, were obtained.
Figure 1. Graphs of changes in cylinder height movements
a)
U (l,z )
UUA( 1, z) UUB(1, z) ■ 10 UUC(1, z) ■ 10
1.5x10
1x10
5x10
- 5x10
- 1x10
3 /
/
2 1
10
20
30
40
50
5
z
0
z
W (l,z )
WWA( 1, z) WWB(1, z) 10 WWC(1, z) -10
1x10
5x10
- 5x10
1x10
b)
Section 3. Mechanic engineering
/ 3
1 2 „.j***"""
10
20
30
40
50
3
4
0
4
z
3
0
z
When the temperature load changes, the radius of the cylinder changes, associated at the first stage with a decrease in it, and then with an increase. When determining the movements along the height of the cylinder, we come to the conclusion that the values decrease (Fig. 1, b, line 1, 2). When exposed to a constant temperature load along the height of the piezocermal cylinder, a radial component arises that practically does not change
in height, this follows from Graph 3, Fig. 1 b, and small the values determine the vertical movements.
It follows from the above that when studying a thick-walled cylinder with finite dimensions under the influence of a constant temperature load, the problem can be described using thermal conductivity equations that take into account radial and circumferential deformations (Kalmova M. A., 2023).
References:
Bardzokas D. I. Mathematical modeling in problems of mechanics of coupled fields. Vol. II: Static and dynamic problems of electroelasticity for composite multi-connected bodies.-M.: Komkniga, 2005.- 376 p.
Kovalenko A. D. Introduction to thermoelasticity.- Kiev: Nauk. Dumka, 1965.- 204 p.
Grinchenko V. T., Ulitko A. F., Shulga N. A. Mechanics of coupled fields in structural elements.- Kiev: Nauk. Dumka, 1989.- 279 p.
Senitsky Yu. E. The method of finite integral transformations - a generalization of the classical decomposition procedure by eigenvector functions // Izv. Saratov University. A new series. Math., mechanics., computer science,2011.- No. 3(1).- P. 61-89.
Kalmova M. A. Unsteady mechanics of radial axisymmetric thermoelectroelastic fields in a long piezoceramic cylinder: dis. ... candidate of Technical Sciences: 1.1.8 / Kalmova Maria Alexandrovna.- Samara, 2023.- 142 p.
submitted 11.06.2024;
accepted for publication 25.06.2024;
published 30.07.2024
© Kalmova M.
Contact: kalmova@inbox.ru
