One problem of torsion of piecewise homogeneous elastic bodies
By means of method of hybrid integral transform of Legendre-Fourier-Fourier type integral representation of exact analytical solution of the problem of torsion of semi-bounded piecewise homogeneous elastic cylinder is obtained. Методом гібридного інтегрального перетворення типу Лежандра-Фур’є-Фур’є...
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Інститут кібернетики ім. В.М. Глушкова НАН України
2014
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| Zitieren: | One problem of torsion of piecewise homogeneous elastic bodies / T.M. Pylypiuk // Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2014. — Вип. 10. — С. 160-169. — Бібліогр.: 6 назв. — англ. |
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| citation_txt | One problem of torsion of piecewise homogeneous elastic bodies / T.M. Pylypiuk // Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2014. — Вип. 10. — С. 160-169. — Бібліогр.: 6 назв. — англ. |
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| container_title | Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки |
| description | By means of method of hybrid integral transform of Legendre-Fourier-Fourier type integral representation of exact analytical solution of the problem of torsion of semi-bounded piecewise homogeneous elastic cylinder is obtained.
Методом гібридного інтегрального перетворення типу Лежандра-Фур’є-Фур’є одержано інтегральне зображення точного аналітичного розв’язку задачі кручення напівобмеженого кусково-однорідного пружного циліндра.
|
| first_indexed | 2025-12-01T04:31:46Z |
| format | Article |
| fulltext |
Математичне та комп’ютерне моделювання
160
UDC 517.946
T. M. Pylypiuk, assistant
Kamianets-Podilsky Ivan Ohienko National University,
Kamianets-Podilsky
ONE PROBLEM OF TORSION OF PIECEWISE
HOMOGENEOUS ELASTIC BODIES
By means of method of hybrid integral transform of Legendre-
Fourier-Fourier type integral representation of exact analytical so-
lution of the problem of torsion of semi-bounded piecewise homo-
geneous elastic cylinder is obtained.
Key words: Legendre equation, Fourier equation, Sturm-
Liouville problem, hybrid integral transform, hybrid differential
operator, the main solutions.
Introduction. The problems of the theory of torsion of elastic bodies
with different geometric structure are of considerable theoretical and practi-
cal interest [1–3]. One of the effective methods for solving such problems in
the case of piecewise-homogeneous environments is a method of hybrid
integral transforms. The hybrid integral transform of Legendre-Fourier-
Fourier type is constructed in this paper, and this transform is applied for
solving the problem of torsion of semi-bounded piecewise homogeneous
elastic cylinder with different physical and mechanical characteristics.
Formulation of the problem. Let’s consider a semi-bounded piece-
wise homogeneous elastic cylinder with radius R , which is composed of
different materials. Physical and mechanical properties of this cylinder are
changed according to the law
1 1 2 1 2 3 2( ) ( ) ( ) ( ) ( ) ( ),G z G shz z l z G z l l z G z l
;jG const 1,3j ,
here ( )x is the Heaviside step function.
We consider inhomogeneous areas of cylinder be soldered together, and
the bottom end 0z is free from stress. We consider that the movement is
limited if z , and lateral surface of the cylinder is loaded efforts ( )f z .
The problem of torsion of such cylinder mathematically is reduced to
a construction bounded on the set
1 1 2 2( , ) : (0, ); (0, ) ( , ) ( , )D r z r R z l l l l
solution of differential separate system of partial differential equations [1]
1 0 1 1 1
1
( , ) ( , ), (0, ),
4
B u r z F r z z l
© T. M. Pylypiuk, 2014
Серія: Фізико-математичні науки. Випуск 10
161
2
1 2 2 1 22
( , ) ( , ), ( , ),B u r z F r z z l l
z
(1)
2
1 3 3 22
( , ) ( , ), ( , ),B u r z F r z z l
z
with boundary conditions
1
0
0,
z
u
z
0
0,j
r
u
r
1 ( )
,
( )
j
j
jr R
u f z
u
r r G z
1,3j , (2)
and conditions of mechanical contact
1
1
1 2
1 2
1 2
0,
0,
z l
z l
u u
u u
G shz G
z z
2
2
2 3
32
2 3
0,
0,
z l
z l
u u
uu
G G
z z
(3)
here
2
1 2 2
1 1
B
r rr r
is Bessel operator,
2
0 2
1
4
cthz
zz
is
Legendre operator.
The main part. Let’s construct the exact analytical solution of the
boundary value problem of conjugation (1)–(3) by the method of hybrid
integral transform of Legendre-Fourier-Fourier type.
1. The hybrid integral transform of Legendre-Fourier-Fourier
type. Let’s consider the singular spectral Sturm-Liouville problem of the
structure of solution, which is limited on the set
2 1 1 2 2: (0, ) ( , ) ( , )I r r R R R R
of separate system of ordinary differential Legendre and Fourier equations
of the 2-nd order
2 2
1 1 1 1 1 1[ ] ( ) 0, (0, ),L V b a V r r R
(4)
2
2 2
1 32
[ ] ( ) 0, ( , ); 2, 3;m m m m m m m
d
L V b a V r r R R m R
dr
with the conjugate conditions
1 1 2 2 1( ) ( ) 0; , 1, 2,
k
k k k k
j j k j j k
r R
d d
V r V r j k
dr dr
(5)
here 0;ja 0;m
jk 0;m
jk
1
22 2 ;j jb 2 0;j 2 1
k k
jk j jc
1 2 0;k k
j j
2 2
2 2
1
;
4
d d
cthr
drdr sh r
1
;
2
is Legendre
operator [4].
Математичне та комп’ютерне моделювання
162
The fundamental system of solutions for the equation 1 1[ ] 0L V is
formed by attached Legendre functions
1
1
2
( )
iq
P chr
and
1
1
2
( )
iq
chr
L [4],
and for equation [ ] 0m mL V — by trigonometric functions cos mq r and
sin mq r [5]; 1 2( )j j jq a b .
It is directly verify that functions
1
,1 21 22 2 3 1
2
( , ) ( ) ( ) ( )
iq
V r c c q q P chr
,
1
1
11, 1
,2 22 3 1 22 2 1 21
;11
2
11, 1
1 12 2 1 21
;21
2
( , ) ( ) ( ) ( , )
( ) ( , ) ,
iq
iq
V r c q Z chR q R q r
Z chR q R q r
(6)
,3 ,2 3 ,1 3( , ) ( ) cos ( )sinV r q r q r
are the solution of the boundary value problem (4), (5).
We use such denotation in equalities (6):
1
2 11,
, 3 2 1 11 2 1 2 2 21 2 1 2 222 1
;21
2
( ) ( ) ( ) ( , ) ( , )j
j
iq
v q R Z chR q R q R q R q R
1 1
211, 11,
1 3 2 12 2 1 2 2 1121 1
;11 ;21
2 2
( ) ( ) ( , ) ( )j
iq iq
Z chR v q R q R q R Z chR
(7)
1
11,
22 2 1 2 2 11
;11
2
( , ) ( ) ,
iq
q R q R Z chR
1, 2;j
11 22 12 21
2 1 2 1( , ) ( ) ( ) ( ) ( ),jk j k j kx y v x v y v x v y , 1, 2;j k
1 ( ) sin sin ;k k k
mj s k mj s s k jm s kv q R q q R q R
2 ( ) sin sin ;k k k
mj s k mj s s k jm s kv q R q q R q R
1, 1 1
1 1 1 1 1 1 11, 1 1 1( ) ( ) ( ),j jjZ chq R shR P chR P chR
1 1
1
;
2
iq
bar means the derivative of the argument.
Let's define values and functions:
2
11 12 1
1
21 22 1
,
c c a
c c shR
212
2 2
22
,
c
a
c
2
3 3 ;a
Серія: Фізико-математичні науки. Випуск 10
163
,3 2( , ) ( );V r r R 1 2 2 1
3 ,1 ,2( ) ([ ( )] [ ( )] ) ,q
1 1 2 1 2 3 2( ) ( ) ( ) ( ) ( ) ( ).r shr r R r r R R r r R (8)
Theorem 1. If the function
1 1 2 2( ) ( )[ ( ) ( ) ( ) ( ) ( )]g r f r shr r R r r R R r r R
is piecewise continuous, absolutely summable and has bounded variation
in the interval (0; ) , then for 2r I integral representation is true
0 0
1 2
( 0) ( 0) ( , ) ( ) ( ) ( , ) ( ) .
2
f r f r V r d f V d
(9)
Proof. Functions , ( , )jV r and , ( , )jV r are the solutions of dif-
ferential equations
2 2 2
1 1 ,1
2 2 2
1 1 ,1
( , ) 0,
( , ) 0;
a V r
a V r
(10)-(11)
2
2 2 2
,2
2
2 2 2
,2
( , ) 0,
( , ) 0, 2,3.
j j j
j j j
d
a V r
dr
d
a V r j
dr
(12)-(13)
Let’s multiply the equality (10) on the function ,1( , )V r shr , and
equality (11) — on the function ,1( , )V r shr and subtract second from
the first. We obtain:
,1 ,1
2
,1 ,11
,1 ,12 2
( , ) ( , )
( , ) ( , )
( , ) ( , ) .
V r V r shr
dV r dV ra d
shr V r V r
dr dr dr
(14)
Let’s multiply the equality (12) on the function , ( , )jV r , and equality
(13) — on the function , ( , )jV r and subtract second from first. We obtain:
2
, , 2 2
, ,
, ,
( , ) ( , )
( , ) ( , )
( , ) ( , ) .
j
j j
j j
j j
a
V r V r
dV r dV rd
V r V r
dr dr dr
(15)
Let’s set a fairly large number 2.R R Let’s multiply the equality
(14) on 1dr and integrate from 0 to 1R , and equality (15) let’s multiply
Математичне та комп’ютерне моделювання
164
on jdr and integrate from jR to 1 3( 1, 2; )jR j R . At the re-
sult of adding the integrals we have, that
, ,32 2
0
,3 ,
1
( , ) ( , ) ( ) ( , ) ( , )
( , ) ( , ) .
R
j
j
r R
d
V r V r r V r V r
dr
d
V r V r
dr
(16)
Let’s calculate the double integral
0
2
( ) ( , ) ( ) ( , ) ( )
d
c
I g V r d V r r dr
(17)
for arbitrary positive numbers c and d ( )c d and arbitrary finite func-
tion ( )g , which is defined on the segment ,c d .
Due to the equation (16) double integral (17) can be rewritten as:
,3 ,32 2
,3 ,3
2 ( )
( , ) ( , )
( , ) ( , ) ( ) .
lim
d
R
c
g d
V R V R
dr
d
V R V R d
dr
I
(18)
As a result of elementary transformations we obtain that
,3 ,3 ,3 ,3 3 32 ( , ) ( , ) ( , ) ( , ) ( ) ( )
d d
V R V R V R V R q q
dr dr
,2 ,2 ,1 ,1 3 3 3 3( ) ( ) ( ) ( ) sin ( ) ( ) ( ) ( )R q q q q
,1 ,1 ,2 ,2 3 3( ) ( ) ( ) ( ) sin ( ) ( )R q q (19)
3 3 ,1 ,2 ,1 ,2 3 3( ) ( ) ( ) ( ) ( ) ( ) cos ( ) ( )q q R q q
3 3 ,1 ,2 ,1 ,2 3 3( ) ( ) ( ) ( ) ( ) ( ) cos ( ) ( ) .q q R q q
If to assume that the function ( )g is continuous, absolutely integrable
and has bounded variation on [ , ]c d , then substituting (19) into (18), with
further using Dirichlet and Riemann lemmas [6] leads to the equality
0
( ), [ , ];2
( ) ( , ) ( ) ( , ) ( )
0, [ , ].
d
c
g c d
I g V r d V r r dr
c d
(20)
If the function ( )g has properties on the interval (0, ) , which
discussed above, then we obtain that
Серія: Фізико-математичні науки. Випуск 10
165
0 0
( ), [ , ];2
( ) ( , ) ( ) ( , ) ( )
0, [ , ].
g c d
g V r d V r r dr
c d
(21)
Let now the function
0
2
( ) ( ) ( , ) ( ) .f r g V r d
(22)
Let’s multiply the equality (22) on ( , ) ( )V r r dr , where is arbi-
trary positive number and integrate by r from 0r to r . Due to
equation (21) we have that
0
( ) ( , ) ( ) ( ).f r V r r dr g
(23)
Let’s substitute the function
0
( ) ( ) ( , ) ( )g f V d
to
equality (22). We obtain the integral representation
0 0
2
( ) ( , ) ( ) ( ) ( , ) ( ) .f r V r d f V d
(24)
Rejection from continuity of the function ( )f r in the point r leads
to the integral representation (9). The theorem is proved.
The integral representation (9) defines the direct
;2
0
[ ( )] ( ) ( , ) ( ) ( )H f r f r V r r dr f
(25)
and inverse
1
;2
0
2 1
[ ( )] ( ) ( , ) ( ) ( 0) ( 0)
2
H f f V r d f r f r
(26)
hybrid integral transform of Legendre-Fourier-Fourier type.
Algebra of hybrid differential operator
2
2 2 2
1 1 2 1 2 3 22 2
( ) ( ) ( ) ( ) ( )
d d
M a r R r a r R R r a r R
dr dr
can be constructed due to the main identity.
Theorem 2. If the function ( )f r is a twice continuously differenti-
able on the set 2I , satisfies the conjugation conditions and conditions of
the limited
,1 ,1( , ) ( ) ( , ) 0,lim
r
df d
shr V r f r V r
dr dr
Математичне та комп’ютерне моделювання
166
,3
,3 0,lim
r
dVdf
V f
dr dr
(27)
then the basic identity of integral transform of hybrid differential operator
M is true:
1
3
2 2
;2 ,
1
[ ( )] ( ) ( ) ( , ) ( ) ,
j
j
R
j j j j
j R
H M f r f f r V r r dr
(28)
0 0,R 3 ;R 1( ) ;r shr 2 3( ) ( ) 1.r r
Proof. Let's define the values:
0
( ) lim ( ),
k
k
r R
f R f r
0
( ) lim ( );
k
k
r R
f R f r
11 11 22 21 12 ,k k k k k 12 11 22 21 12 ,k k k k k
21 11 22 21 12 ,k k k k k 22 11 22 12 21.k k k k k
From the conjugate conditions we find the relations:
21 12
1
11 22
1
( ) 1
( ) ( )
1
( ) ( ) ( ) , 1, 2
j j j
j j
j
j j
j j j
j
df R df
R f R
dr c dr
df
f R R f R j
c dr
(29)
The components , ( , )jV r of the spectral function ( , )V r have the
same connections:
, 1
, , 111 12
1
, , 1
, 121 22
1
( , )1
( , ) ( , ) ,
( , ) ( , )1
( , ) .
j jj j
j j j j
j
j j j jj j
j j
j
dV R
V R V R
c dr
dV R dV R
V R
dr c dr
(30)
From equations (29) and (30) the identity follows
,
,
2 , 1
, 1
1
( ) ( , )
( , ) ( )
( ) ( , )
( , ) ( ) , 1, 2.
j j j
j j j
j j j j
j j j
j
df R dV R
V R f R
dr dr
c df R dV R
V R f R j
c dr dr
(31)
The proof of the theorem is obtained by integration by parts under
the integral with following using of the limited conditions (27), identity
(31), the properties of functions ,1 ,2 ,3, , , ( )V V V f r and structures of
1 2 3, , . The theorem is proved.
Серія: Фізико-математичні науки. Випуск 10
167
The identity (28) makes it possible to apply the introduced hybrid in-
tegral transform of Legendre-Fourier-Fourier type to the solving of singu-
lar problems of mathematical physics of inhomogeneous structures.
2. The solution of the problem (1)–(3). Let's write the system (1)
and boundary conditions (2) in matrix form:
1 0 1
12
1 2 22
32
1 32
1
( ) ( , )
4
( , )
( ) ( , ) ( , ) ,
( , )
( ) ( , )
B u r z
F r z
B u r z F r z
z
F r z
B u r z
z
(32)
1
2
3 0
0
0 ,
0
r
u
u
r
u
1
1 11
1
2 2 2
1
3 2 3
( )( )
1
( )
( )r R
f z G shzu
u f z G
r r
u f z G
. (33)
Listed by equations (6)–(8) values and functions for this case
11 21 12 22( 0,k k k k 11 12 1,k k 1, 2;k 1
21 1 1,G shl 1
22
2
2 21,G 2
22 3,G 0) we denote by 1 2 3, , , 11( , ),V z
21( , )V z and 31( , ).V z In this case 11 1 1,c G shl 12 21 2 ,c c G
22 3 ,c G .iG const Spectral density for this case we denote by 0 ( ) .
Let’s represent the integral operator 0;2H , which acts by the formula
(25) as an operator matrix-row
1 2
1 2
0;2 11 1 21 2 31
0
[...] ... ( , ) ... ( , ) ... ( , ) .
l l
l l
H V z shzdz V z dz V z dz
(34)
Let's apply the operator matrix-row (34) to the problem (32), (33) ac-
cording to matrices multiplication rule. As a result of main identity (28)
(when 2 2 2
1 2 3 1,a a a 2
1 0, 2 2
2 3
1
)
4
we get a boundary value
problem: to construct a limited in the interval (0, )R solution of Bessel
equation for modified functions
2
2
2 2
1 1
( , ) ( , );
d d
q u r F r
r drdr r
2 2 1
4
q (35)
with boundary conditions
0
0,
r
du
dr
1
( ).
r R
d
u f
dr r
(36)
Математичне та комп’ютерне моделювання
168
It is possible to verify that the desired solution of the boundary value
problem (35), (36) is a function
0
( , ) ( , ) ( ) ( , , ) ( , ) .
R
u r W r f E r F d (37)
In the formula (37) there are the Green's function
1 1
1 0 0 1
1
( )
( , ) ( )( ( ) 2 ( ))
( )
RI qr
W r RI qr q RI qR I qR
and fundamental function
1 2 1 1 1
1 2 1 1 11
( )[ ( ) ( ) ( ) ( )], 0 ;1
( , , )
( )[ ( ) ( ) ( ) ( )], 0 ,( )
I qr I q K q r R
E r
I qr I qr K qr r R
here 2 0 1( ) ( ) 2 ( );qRK qR K qR ( ),I x ( )K x are modified Bessel
functions of the first and second kind.
For resuming the function 1 2 3( , ) ( , ); ( , ); ( , )u r z u r z u r z u r z by its
image ( , )u r let’s apply the operator matrix column to the matrix-
element ( , )u r (function ( , )u r is defined by the formula (37)) accord-
ing to matrices multiplication rule
11 0
0
0;2 21 0
0
31 0
0
2
... ( , ) ( )
2
... ... ( , ) ( )
2
... ( , ) ( )
V z d
H V z d
V z d
,
as the inverse operator of the operator which is defined by (34).
As a result of elementary transformations we obtain unique solution
of the conjugate boundary value problem (1)–(3):
1
3
1
( , ) ( , , ) ( ) ( )
m
m
l
j jm m m
m l
u r z W r z f d
10
( , , , ) ( , ) ( ) ,
m
m
lR
jm m m m
l
H r z F d d
here 0 0,l 3l , 1( )z shz , 2 3( ) ( ) 1z z , 1
1 3 1 1G G shl ,
1
2 3 2G G , 3 1 , Green's functions
Серія: Фізико-математичні науки. Випуск 10
169
( , , )jmW r z 1 1 0
0
( , ) ( , ) ( , ) ( )j mW r V z V d
and the influence functions
1 1 0
0
( , , , ) ( , , ) ( , ) ( , ) ( )jm j mH r z E r V z V d
of the boundary value problem (1)–(3).
If ( )jf z and ( , )jF r z are given then the position of cylinder which
is discussed becomes known.
Conclusion. By means of method of hybrid integral transform of
Legendre-Fourier-Fourier type integral representation of solution of the
problem of torsion of semi-bounded piecewise homogeneous elastic cylin-
der is obtained.
References:
1. Arutyunyan N. Torsion of Elastic Bodies / N. Arutyunyan, B. Abramyan. —
Physmatgis, 1963. — 688 p.
2. Grilitsky D. Torsion of two-layer elastic medium / D. Grilitsky // Appl.
Mech. — 1961. — Vol. 7, № 1. — P. 89–95.
3. Protsenko V. Hybrid integral Fourier-Hankel transforms and some torsion
problem of piecewise-homogeneous media / V. Protsenko, Т. Кashavel // Dy-
namics of systems with the mobile distributed load : col. of scien. p. —
Kharkov, 1978. — № 1. — P. 120–124.
4. Konet I. Integral Mehler — Fock transforms / І. Коnеt, М. Leniuk. —
Chernivtsi : Prut, 2002. — 248 p.
5. Stepanov V. The course of differential equations / V. Stepanov. — M. : Phys-
matgis, 1959. — 468 p.
6. Fikhtengol'ts G. Course of differential and integral calculus : in 3 volumes /
G. Fikhtengol'ts. — M. : Nauka, 1969. — Vol. 3. — 656 p.
Методом гібридного інтегрального перетворення типу Лежандра-
Фур’є-Фур’є одержано інтегральне зображення точного аналітичного
розв’язку задачі кручення напівобмеженого кусково-однорідного
пружного циліндра.
Ключові слова: рівняння Лежандра, рівняння Фур’є, задача
Штурма-Ліувілля, гібридне інтегральне перетворення, гібридний ди-
ференціальний оператор, головні розв’язки.
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/HRV (Za stvaranje Adobe PDF dokumenata pogodnih za pouzdani prikaz i ispis poslovnih dokumenata koristite ove postavke. Stvoreni PDF dokumenti mogu se otvoriti Acrobat i Adobe Reader 5.0 i kasnijim verzijama.)
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/ITA (Utilizzare queste impostazioni per creare documenti Adobe PDF adatti per visualizzare e stampare documenti aziendali in modo affidabile. I documenti PDF creati possono essere aperti con Acrobat e Adobe Reader 5.0 e versioni successive.)
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| id | nasplib_isofts_kiev_ua-123456789-86549 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 2308-5878 |
| language | English |
| last_indexed | 2025-12-01T04:31:46Z |
| publishDate | 2014 |
| publisher | Інститут кібернетики ім. В.М. Глушкова НАН України |
| record_format | dspace |
| spelling | Pylypiuk, T.M. 2015-09-21T14:18:48Z 2015-09-21T14:18:48Z 2014 One problem of torsion of piecewise homogeneous elastic bodies / T.M. Pylypiuk // Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2014. — Вип. 10. — С. 160-169. — Бібліогр.: 6 назв. — англ. 2308-5878 https://nasplib.isofts.kiev.ua/handle/123456789/86549 517.946 By means of method of hybrid integral transform of Legendre-Fourier-Fourier type integral representation of exact analytical solution of the problem of torsion of semi-bounded piecewise homogeneous elastic cylinder is obtained. Методом гібридного інтегрального перетворення типу Лежандра-Фур’є-Фур’є одержано інтегральне зображення точного аналітичного розв’язку задачі кручення напівобмеженого кусково-однорідного пружного циліндра. en Інститут кібернетики ім. В.М. Глушкова НАН України Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки One problem of torsion of piecewise homogeneous elastic bodies Article published earlier |
| spellingShingle | One problem of torsion of piecewise homogeneous elastic bodies Pylypiuk, T.M. |
| title | One problem of torsion of piecewise homogeneous elastic bodies |
| title_full | One problem of torsion of piecewise homogeneous elastic bodies |
| title_fullStr | One problem of torsion of piecewise homogeneous elastic bodies |
| title_full_unstemmed | One problem of torsion of piecewise homogeneous elastic bodies |
| title_short | One problem of torsion of piecewise homogeneous elastic bodies |
| title_sort | one problem of torsion of piecewise homogeneous elastic bodies |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/86549 |
| work_keys_str_mv | AT pylypiuktm oneproblemoftorsionofpiecewisehomogeneouselasticbodies |