Thermal-magnetic-elastic effect analysis of a thin current-carrying conical frustum shell
A thermal-magnetic-elastic problem for a thin current-carrying conical frustum shell in a magnetic field is studied. The normal Cauchy form of nonlinear differential equations, which include in total eight basic unknown variables, are obtained by the variable replacement method. Using the Newmark’s...
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irk-123456789-1882202023-02-17T01:26:28Z Thermal-magnetic-elastic effect analysis of a thin current-carrying conical frustum shell Bian, Y.H. Zhang, C. A thermal-magnetic-elastic problem for a thin current-carrying conical frustum shell in a magnetic field is studied. The normal Cauchy form of nonlinear differential equations, which include in total eight basic unknown variables, are obtained by the variable replacement method. Using the Newmark’s stable finite equidifferent formulas and the quasi-linearization method, the nonlinear partial differential equations are reduced to a sequence of quasi-linear differential equations, which can be solved by the discrete-orthogonalization method. The temperature field in a thin conical frustum shell and the integral eigenvalues are derived after considering Joule’s heat effect in an electromagnetic field and the thermal equilibrium equation. The change of stresses, displacements, and temperatures in the thin current-carrying conical frustum shell with variation of the electromagnetic parameters is discussed. It is proved that the stresses, strains, and temperatures in thin shells can be controlled by changing the electromagnetic and mechanical parameters by considering a specific example. These results are expected to be a theoretical reference for further analysis of this case. Вивчено термомагнітопружну задачу для тонкої конічної зрізаної оболонки, що проводить струм і перебуває у магнітному полі. Отримано методом заміни змінних нелінійні диференціальні рівняння нормального за Коші типу, які включають всього вісім невідомих змінних. За допомогою стійких скінченних формул Ньюмарка і методу квазілінеаризації нелінійні диференціальні рівняння з частинними похідними зведені до послідовності квазілінійних диференціальних рівнянь, які далі можуть розв'язуватися методом дискретної ортогоналізації. З розгляду теплового ефекту Джоуля в електромагнітному полі і рівняння теплової рівноваги визначено температурне поле в тонкій конічній зрізаній оболонці і інтегральні власні значення. На спеціальному прикладі проаналізовано зміну напружень, зміщень та температури зі зміною електромагнітних параметрів. Отримані результати матимуть теоретичне продовження в дослідженні поставленої задачі 2020 Article Thermal-magnetic-elastic effect analysis of a thin current-carrying conical frustum shell / Y.H. Bian, C. Zhang // Прикладная механика. — 2020. — Т. 56, № 1. — С. 128-143. — Бібліогр.: 23 назв. — англ. 0032-8243 http://dspace.nbuv.gov.ua/handle/123456789/188220 en Прикладная механика Інститут механіки ім. С.П. Тимошенка НАН України |
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A thermal-magnetic-elastic problem for a thin current-carrying conical frustum shell in a magnetic field is studied. The normal Cauchy form of nonlinear differential equations, which include in total eight basic unknown variables, are obtained by the variable replacement method. Using the Newmark’s stable finite equidifferent formulas and the quasi-linearization method, the nonlinear partial differential equations are reduced to a sequence of quasi-linear differential equations, which can be solved by the discrete-orthogonalization method. The temperature field in a thin conical frustum shell and the integral eigenvalues are derived after considering Joule’s heat effect in an electromagnetic field and the thermal equilibrium equation. The change of stresses, displacements, and temperatures in the thin current-carrying conical frustum shell with variation of the electromagnetic parameters is discussed. It is proved that the stresses, strains, and temperatures in thin shells can be controlled by changing the electromagnetic and mechanical parameters by considering a specific example. These results are expected to be a theoretical reference for further analysis of this case. |
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Bian, Y.H. Zhang, C. |
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Bian, Y.H. Zhang, C. Thermal-magnetic-elastic effect analysis of a thin current-carrying conical frustum shell Прикладная механика |
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Bian, Y.H. Zhang, C. |
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Bian, Y.H. |
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Thermal-magnetic-elastic effect analysis of a thin current-carrying conical frustum shell |
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Thermal-magnetic-elastic effect analysis of a thin current-carrying conical frustum shell |
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Thermal-magnetic-elastic effect analysis of a thin current-carrying conical frustum shell |
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Thermal-magnetic-elastic effect analysis of a thin current-carrying conical frustum shell |
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Thermal-magnetic-elastic effect analysis of a thin current-carrying conical frustum shell |
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thermal-magnetic-elastic effect analysis of a thin current-carrying conical frustum shell |
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Інститут механіки ім. С.П. Тимошенка НАН України |
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Thermal-magnetic-elastic effect analysis of a thin current-carrying conical frustum shell / Y.H. Bian, C. Zhang // Прикладная механика. — 2020. — Т. 56, № 1. — С. 128-143. — Бібліогр.: 23 назв. — англ. |
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Прикладная механика |
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AT bianyh thermalmagneticelasticeffectanalysisofathincurrentcarryingconicalfrustumshell AT zhangc thermalmagneticelasticeffectanalysisofathincurrentcarryingconicalfrustumshell |
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2020 П Р И К Л А Д Н А Я М Е Х А Н И К А Том 56, № 1
128 ISSN0032–8243. Прикл. механика, 2020, 56, № 1
Y . H . B i a n , C . Z h a n g
THERMAL-MAGNETIC-ELASTIC EFFECT ANALYSIS OF A THIN
CURRENT-CARRYING CONICAL FRUSTUM SHELL
Key Laboratory of Mechanical Reliability for Heavy Equipments and Large Structures
of Hebei Province, College of Civil Engineering and Mechanics, Yanshan University,
Qinhuangdao, China; e-mail: yh_bian@sina.com
Abstract. A thermal-magnetic-elastic problem for a thin current-carrying conical frus-
tum shell in a magnetic field is studied. The normal Cauchy form of nonlinear differential
equations, which include in total eight basic unknown variables, are obtained by the variable
replacement method. Using the Newmark’s stable finite equidifferent formulas and the qua-
si-linearization method, the nonlinear partial differential equations are reduced to a sequence
of quasi-linear differential equations, which can be solved by the discrete-orthogonalization
method. The temperature field in a thin conical frustum shell and the integral eigenvalues
are derived after considering Joule’s heat effect in an electromagnetic field and the thermal
equilibrium equation. The change of stresses, displacements, and temperatures in the thin
current-carrying conical frustum shell with variation of the electromagnetic parameters is
discussed. It is proved that the stresses, strains, and temperatures in thin shells can be con-
trolled by changing the electromagnetic and mechanical parameters by considering a specif-
ic example. These results are expected to be a theoretical reference for further analysis of
this case.
Key words: thermal-magnetic-elasticity, thin conical frustum shell, coupling effect, Lo-
rentz force, quasi-linearization method, discrete-orthogonalization method.
1. Introduction.
Applications of electromagnetic elastic structures are broad in modern technological
fields, such as aerospace industry, electromagnetic emission, power supply equipment,
magnetoacoustic processing, etc. The objects moving in the applied magnetic field will
come under the interaction of an electromagnetic field, a temperature field, and a mechani-
cal field. The more complex mechanics behaviors have been shown, and the operations of
the system are influenced. Therefore, the researches on the thermal-magnetic-elastic prob-
lem have both theoretically and practically important significance. In recent years, electro-
magnetic elastic mechanics has developed very quickly. Many important achievements have
been obtained, such as Pao and Yeh [1] (1973); Ambartsumyan [2] (1977); Moon [3]
(1984); Van de Ven [4] (1986); Ulitko, Mol’chenko, Kovalchuk [5] (1994); Mol’chenko,
Grigorenko [6] (2010); Mol’chenko [7 – 12] (1989, 2012, 2013, 2014, 2015, 2016); Bian [13]
(2015); Zheng, Zhang, Zhou [14] (2005); Qin [15, 16] (2003, 2009); Pratiher [17] (2011);
Ootao, Ishihara [18] (2013); Kuang [19] (2014); Soni, Jain, Joshi [20] (2017); Moham-
madimehr, Rostami [21] (2018), and others. These research achievements laid a good foun-
dation for studies on electromagnetic elastic mechanics and its applications. However, these
researches fasten mostly on the problems about the vibration and stability of magnetoelastic
bodies, studies on magnetoelastic stress-strain problems of current-carrying plates and
shells, especially, studies on a thermal-magnetic-elastic problem with considering tempera-
ture fields have rarely been seen. Therefore, the thermal-magnetic-elastic stress-strain analy-
sis for current-carrying plates and shells is recently one of the interested topics studied.
129
In this paper, a thermal-magnetic-elastic problem for a thin current-carrying conical
frustum shell in a magnetic field is studied. Based on the nonlinear magnetoelastic kinetic
equations, geometric equations, physical equations, and electrodynamics equations of a thin
shell under the interaction of an electromagnetic field, a temperature field, and a mechanical
field, the fundamental equations for the nonlinear stress-strain problem under the action of
the coupling field are given. The temperature field in a thin conical frustum shell and the
integral eigenvalues are derived after considering Joule’s heat effect in an electromagnetic
field and the thermal equilibrium equation. The stresses, displacements, and temperatures of
the thin conical frustum shell in a magnetic field are computed. The effect of the side cur-
rent, electromagnetic induction density, etc. on the stresses, displacements, and temperatures
in the thin conical frustum shell is analyzed.
2. Fundamental Equations.
Under the precondition of symmetrical loads, the thin conical frustum shell can be re-
garded axisymmetric. An orthogonal curvilinear coordinate system , ,s is shown in
Fig. 1. By satisfying the magnetoelastic supposition of the thin shell [7] and using elastic
mechanics theories, Ohm’s law and Maxwell equations in electromagnetic basic theories,
the fundamental equations for the thin conical frustum shell can be derived as follows:
The magnetoelastic kinetic equations are given by
2
2
( )
cos ( ) ;s
s s
rN u
N r F f r h
s t
(1)
2
2
( )
sin ( ) ;srQ w
N r F f r h
s t
(2)
23
2
( ) sin
cos .
12
s s
s s s
rM r h
M r N M r Q
s r t
(3)
The electrodynamics equations are given by
( )1
;
B rE
t r s
(4)
1
( )
2
s s
s s
B B Bw u
E B B B
t t s rh
. (5)
The relations between the displacements and the strains are given by
21
;
2s s
u
s
cos sin ;
u w
r r (6a, b)
;s
s s
cos
;sr
.s
w
s
(7a, b, c)
The relations between the internal forces and the strains are given by
[ (1 ) ];s N s TN D [ (1 ) ];N s TN D (8a, b)
[ (1 ) ];s M s TM D [ (1 ) ],M s TM D (9a, b)
where is the electrical conductivity of the material; is the permeability of the material;
h is the thickness of the thin shell; t is the time variable; r is the radius of the section
round; is the angle between the normal of the neutral surface and the rotation axis; sB
130
and sB are the values of sB on the outer and inner surfaces of the thin conical frustum
shell, respectively; s and are the strains in the corresponding directions; s and
are the bending strains; u , w , and s are the displacements and the angle of rotation in the
corresponding directions; E is the electric field intensity in the -direction; B is the
magnetic induction intensity in the -direction; sF and F are the surface forces in the
corresponding directions on the thin shell; sf and f are the Lorentz forces in the corre-
sponding directions; is the mass density of the medium; sN , N , sQ , sM , and M are
the internal forces and moments in the corresponding directions in the thin conical frustum
shell; 2( / (1 ))ND Eh and 3 2( / [12(1 )])MD Eh are the tensile and bending rigidi-
ties of the thin conical frustum shell, respectively; E is elastic modulus; is Poisson’s
ratio; T and T are the integral eigenvalues of the temperature field T along the thickness
of the thin conical frustum shell, that is
2
2
1
( , , , )d
h
T T
h
T s t
h
;
2
3
2
12
( , , , ) d
h
T T
h
T s t
h
, (10a, b)
where T is the linear expansion coefficient of the material; ( , , , )T s t is the tempera-
ture distributing function in the thin shell.
For obtaining normal Cauchy form nonlinear partial differential equations, let u , w ,
s , sN , sQ , sM , B , and E be the basic unknown variables. We obtain:
2
2(1 ) sin cos 1
(1 ) ;
2s s T
u v v v
N w u v
s Eh r r
(11)
;s
w
s
(12)
2
3
12(1 ) cos
(1 ) ;s
s s T
v v
M v
s rEh
(13)
2
2
cos cos sin cos
( 1) ( ) ;s
s T s s
N u
v N Eh u w Eh F f h
s r r r r t
(14)
2
2
sin cos sin cos
( ) ;s
s T s
Q w
vN Eh u w Eh Q F f h
s r r r r t
(15)
3 3cos cos sin cos
( 1)
12 12
s
s s T s s T s
M Eh Eh
v M v M
s r r r r
23
2
;
12
s
s s s
h
N Q
t
(16)
1
( ) ;
2
s s
s s
B B Bw u
E B B B
s t t rh
(17)
cos
.
BE
E
s t r
(18)
131
3. Computational Method.
Equations (11) – (18) can be written as boundary-value problems:
0( , ) ( );Ns s s s
s
N
F N (19)
1 0 1( ) ;D s N d 2 2( ) ,ND s N d (20a, b)
where T{ , , , , , , , }s s s su w N Q M B E N is eight-dimensional vector; 1D and 2D are
given orthogonal matrixes that the ranks are 8k and (8 ) 8 ( 8)k k ; 1d and 2d are
given vectors.
Eq. (19) is a set of nonlinear partial differential equations with eight basic unknown var-
iables, the difficulties are how to solve directly the equations. First, Newmark’s stable finite
equidifferent formulas [7] are used to find the derivatives with respect to time in Eqs. (11) –
(18) for a time step length:
2
1
0,5 ;
( )
t t t t
t t tu u u
u u
tt
0,5 ( ),t t t t t tu u t u u (21a, b)
where t is the time difference step length; is the parameter of the scheme, when
0,25, Newmark’s arithmetic operators for linear dynamic system are unconditionally
stable [22].
After differencing, Eqs. (11) – (18) can be expressed as
0
d
( , ) ( );
d Ns s s s
s
N
F N (22)
1 0 1( ) ;D s N d 2 2( ) .ND s N d (23a, b)
In order to solve the nonlinear problem described by Eq. (22), with an iterative method,
thus, nonlinear problems can be turned into a series of linear problems. Taken iterative
equations are [7]
( 1)
( ) ( ) ( 1) ( )( , ) ( , )( );
k
k k k kd
s s
ds
N
F N Γ N N N (24)
( 1)
1 0 1( ) ;kD s N d ( 1)
2 2( )k
ND s N d ( 0, 1, 2 . . .),k (25a, b)
where ( )( , )ksΓ N is Jacobi’s matrix.
By using Eqs. (11) – (18), we have the linear iterative equations:
( 1)kdu
ds
2
( 1) ( ) 2 ( 1) ( ) ( 1) ( 1) ( 1)1 1 sin cos
( ) (1 ) ;
2
k k k k k k k
s s s s TN w u
Eh r r
(26)
( 1)
( 1) ;
k
k
s
dw
ds
(27)
( 1) 2
( 1) ( 1) ( 1)
3
12(1 ) cos
(1 ) ;
k
k k ks
s s T
d
M
ds rEh
(28)
132
( 1)
( 1)( 1) ( 1) ( 1)d cos cos sin
( 1)
d
k
kk k ks
s T
N
v N Eh Eh u w
s r r r
2 ( 1)
( 1)
2
( ) ;
k
k
s s
u
F f h
t
(29)
( 1)
( 1) ( 1) ( 1) ( 1) ( 1)cos sin cos sink
k k k k ks
s s T
dQ
Q vN Eh Eh u w
ds r r r r
2 ( 1)
( 1)
2
( ) ;
k
k w
F f h
t
(30)
( 1)d
d
k
sM
s
3
( 1) ( 1) ( 1) ( 1) ( ) ( ) ( 1) ( ) ( )cos cos
( 1)
12
k k k k k k k k k
s s T s s s s s s
Eh
v M N N N
r r
3
( 1) ( ) ( ) ( 1) ( ) ( ) ( 1) ( ) ( ) 2
2
sin sin cos
2 ( )
12
k k k k k k k k k
s s s s s s s s s
Eh
M M M
r r
2 ( 1)3 3
( 1) ( ) ( )( ) ( 1) ( ) ( 1)
2
sin
12 12
k
k k kk k k k s
s s s sT T T
Eh h
Q
r t
; (31)
( 1) ( 1) ( 1) ( ) ( )
( 1) ( ) ( 1) ( )1
( )
2
k k k k k
k k k k
s s
dB w u u u
E B B B B B
ds t t t t
;s sB B
rh
(32)
( 1)( 1)
( 1)cos
.
kk
kBdE
E
ds t r
(33)
The Lorentz forces in Eqs. (26) – (33) can be written as
( 1) ( 1) ( 1) ( ) ( ) ( 1) ( ) ( )( )k k k k k k k k
s clf h J B h E B E B E B
( 1) ( ) ( )
( ) ( 1) ( )1
( )
2
k k k
k k k
s s
w w w
h B B B B B
t t t
(34)
( 1) ( ) ( )
( ) 2 ( ) ( 1) ( ) 2( ) 2 2( ) ;
k k k
k k k ku u u
h B B B B
t t t
( 1)
( 1) 2 21 1 1
( ) ( ) ( )
2 12 4
k
k
cl s s s s s s
w
f h J B B h B B B B
t
133
( 1) ( ) ( )
( 1) ( ) ( 1) ( )1 1
( ) ( )
2 2
k k k
k k k k
s s s s
u u u
B B E B B B B B
t t t
(35)
( 1) ( ) ( )
( ) ( 1) ( )( )
12
k k k
k k ks s s
s s
h
B B B B B
t t t
.
4. Electromagnetic Temperature Effect.
By using the electrodynamics equations and generalized Ohm’s law, and considering
the side electric current sclJ and clJ , we obtain [7]:
;s scl s
v w
J J E B B
t t
.cl s
w u
J J E B B
t t
(36a, b)
Because of Joule’s heat effect in the electromagnetic field, the heat source is certainly
produced in the current-carrying conical frustum. Considering the thin shell discussed and
the electromagnetic field of low frequency, so the skin effect is not obvious. We can consid-
er approximately that the distribution of the electric current density is symmetrical in the
thin shell. Thus, the heat source power can be expressed as [23]
2 22
0,86 0,86 .sJ JJ
Q
(37)
Without considering external heat source and the local temperature effect of the conical
frustum ends, considering only the heat exchange between the outer, inner surfaces of the
conical frustum and the exterior. We consider that the distribution of the heat source power
density along the thickness of the thin shell is symmetrical. At the neutral surface 0 , the
heat flux 0q ; at the outer surface 2h , the heat flux 2q Q h , and wT T .
Thus, the equation of the temperature distribution in the -direction can be written as
22 4
1 4
8 2
T
f w
T F F
Q h hc
T T T
h h
, (38)
where T is the heat exchange coefficient of the material; c is the specific heat capacity of
the material; F is the surface heat exchange coefficient of the material; wT is the surface
temperature of the thin shell; fT is the temperature of the medium bordering upon the thin
shell; wT is the change rate of the surface temperature.
Using Eqs. (10), (36), (37), and (38), we obtain:
( 1)k
T
22 ( 1) ( ) ( )
2 2 ( 1) ( ) ( ) 2 2 243
2 ( ) 2
400
k k k
k k kT
cl s
T
h w w w
J E E E B
t t t
2 2( 1) ( ) ( ) ( )
( ) 2 ( 1) ( ) ( ) 22 ( ) 2 3 ( )
k k k k
k k k ku u u u
B B B B
t t t t
( 1) ( ) ( ) ( 1) ( ) ( ) ( ) ( )
( ) ( ) ( 1) ( )2 2
k k k k k k k k
k k k k
s
w u w u w u w u
B B B B B
t t t t t t t t
134
( 1) ( 1) ( ) ( )
( 1) ( ) ( 1) ( )2 2
k k k k
k k k k
cl cl s
w u u u
J E J B B B B
t t t t
( ) ( 1) ( )
2 ( 1) ( ) ( )2
k k k
k k k
s
w w w
B E E E
t t t
(39)
( ) ( ) ( 1) ( )
( 1) ( ) ( ) ( 1) ( ) ( ) ( ) ( )2
k k k k
k k k k k k k ku u u u
E B E B E B E B
t t t t
( 1)4 2
;
3 2
kT T
w
F F
hc
T
h
( 1) 0k
T
. (40)
The ( 1)k
T
and ( 1)k
T
are substituted in Eqs. (26) – (33), the thermal-magnetic-elastic
coupling equations for the thin conical frustum shell are obtained. All unknown variables
can be found by the discrete-orthogonalization method.
5. Numerical Results and Discussion.
Fig. 1 shows a thin cantilever conical frustum shell made of aluminum in a magnetic
field , 0, 0 .sBB It bears the mechanical load 0, 0, ,FF alternating electric current
that the density is 0, , 0clJclJ is exteriorly imported the shell. Let 71GPa;E 0,34;
32670kg m ; 7 13,63 10 (Ω m) ; 61,256 10 H m; 934J ( k C);c g
237 W ( m C);T 2235W ( m C );F 5 12,35 10 C ;T
2sin A m ;clJ J t
2 110 sec ; 2
0 0( ) tan N m ;F F s s 0 100;F 12; 32 10 m;h
0 0,25 m;r 4; 0 0,35 m;s 0,85 m.Ns
The boundary conditions are
0s s : 0u , 0w , 0s , 0,1 sin T;B t (41a-d)
Ns s : 0sN , 0sQ , 0sM , 0.B (42a-d)
Fig. 1. A thin cantilever conical frustum shell in a magnetic field.
135
The initial conditions are
0
( , ) 0
t
s t N ;
0 0 0 0
( , ) ( , ) ( , ) ( , ) 0;st t t t
u s t w s t s t B s t (43a-e)
0 0 0
( , ) ( , ) ( , ) 0.st t t
u s t w s t s t (44a-c)
In order to acquire stabilizing computational process, mass coordinate m s is taken.
Programming Eqs. (26) – (33) and conducting computations for the known data and the
boundary and initial conditions yield the eight basic unknown variables u , w , s , sN ,
sQ , sM , B , and E . Then the relations and variation laws between the mechanical and
electromagnetic variables can be determined by changing the relevant parameters.
Fig. 2 shows the deflection distribution in the thin conical frustum shell for
22 MA/mJ , 0,1T,sB and 11msec,t the mechanical load being the same as above.
According to Fig. 2, the coupling action effect of different loads can be seen. Fig. 3 and fig. 4
Fig. 2. Effects of the coupling action on the deflection in the thin conical frustum shell.
Fig. 3. Curves of the deflection distribution for 0,2T,sB
10 msec,t and different values of .J
136
Fig. 4. The deflection at 0,6 ms versus t for
0,2TsB and different values of J .
Fig. 5. The deflection at 0,6 ms versus t for
2MA/m3J and different values of .sB
Fig. 6. Effects of the temperature on the stress
in the thin conical frustum shell.
137
show the deflection distribution in the thin conical frustum shell ( 10 msec)t and the varia-
tion of the deflection at 0,6 ms with time for 0,2TsB and different electric current
density, respectively. The deflection in the thin shell increases with increase in the electric
current density. Fig. 5 shows the variation of the deflection at 0,6 ms with time for
23MA/mJ and different magnetic induction intensity. Initially, the deflection varies a
little with increase in the magnetic induction intensity. As time goes on, the deflection rapid-
ly increases with the magnetic induction intensity. Fig. 6 shows the distribution of the nor-
mal stress s
and s
in the s -direction on the inner and outer surfaces of the shell for
25MA/m ,J 0,3 T,sB and 9 msec.t According to Fig. 6, the effect of the temperature
on the stress of the thin shell is relatively remarkable. Fig. 7 shows the distribution of the nor-
mal stresses s
and s
in the s -direction on the inner and outer surfaces of the shell for
0,2T,sB 10 msec,t and different electric current density. Fig 8 shows the distribution
Fig. 7. Curves of the stresses s
and s
distribution for
0,2T,sB 10 msec,t and different values of .J
Fig. 8. Curves of the stresses
s and
s distribution for
2MA/m3J , msec10t , and different values of sB .
138
of the normal stresses s
and s
in the s -direction on the inner and outer surfaces of the
shell for 23MA/m ,J 10 msect , and different magnetic induction intensity. Fig. 9
shows the variation of the normal stress s
in the s -direction on the outer surface at the
fixed end ( 0,35m)s with time for 23MA/mJ and different magnetic induction inten-
sity. Initially, the stress varies a little with increase in the magnetic induction intensity. As
time goes on, the stress rapidly increases with the magnetic induction intensity, peaking general-
ly at 29 msec.t
Fig. 10 shows the distribution of the normal stress s
and s
in the s -direction on the
inner and outer surfaces of the shell for 23MA/m ,J 0,2T,sB 10 msec,t and differ-
ent thickness of the shell. According to Fig. 10, the effect of the thickness on the stress of
the thin shell is very remarkable.
Fig. 9. The normal stress
s at the fixed end ( 0,35m)s
versus t for 2MA/m3J and different values of .sB
Fig. 10. Curves of the stresses s
and s
distribution for
2MA/m3J , T2.0sB , msec10t , and different values of h .
139
Fig. 11. The electric field intensity at the free end ( 0,85m)s
versus t for 0,2TsB and different values of .J
Fig. 12. Curves of the electric field intensity distribution for
23MA/m ,J 18msect , and different values of .sB
Fig. 13. The Lorentz force f at 0,6 ms versus t
for 0,2TsB and different values of .J
140
Fig. 11 shows the variation of the electric field intensity E at the free end ( 0,85m)s
with time for 0,2TsB and different electric current density. Initially, the value of the E
is small, and the E varies a little with increase in the electric current density. As time goes
on, the E rapidly increases with the electric current density and changing tempestuous. It
predicates that coupling effect is increasingly stronger. Fig. 12 shows the distribution of the
electric field intensity in the thin conical frustum shell for 23MA/mJ , 18msect , and
different magnetic induction intensity. According to Fig. 12, the distribution of the electric
field intensity in the thin shell is asymmetric. The electric field intensity varies tempestuous-
ly with increase in the magnetic induction intensity. Fig. 13 shows the variation of the Lo-
rentz force f at 0,6 ms with time for 0,2TsB and different electric current densi-
ty. According to Fig. 13, as time goes on, the Lorentz force rapidly increases, peaking gen-
erally at 25msect . The Lorentz force varies tempestuously with increase in the electric
current density. Fig. 14 shows the temperature distribution in the thin conical frustum
Fig. 14. Curves of the temperature distribution for 0,2T,sB
19 msec,t and different values of .J
Fig. 15. The temperature at the free end ( 0,85m)s
versus t for 0,2TsB and different values of .J
141
shell for 0,2T,sB 19 msect , and different electric current density. According to Fig. 14,
the value of the temperature at the fixed end ( 0,35m)s is the highest. Fig. 15 shows the
variation of the temperature at the free end ( 0,85m)s with time for 0,2TsB and dif-
ferent electric current density. Initially, the temperature varies a little with increase in the
electric current density. As time goes on, the temperature rapidly increases with the electric
current density. The results show that the temperature in the thin conical frustum shell can
be controlled by changing the electric current density. Fig. 16 shows the temperature distri-
bution in the thin conical frustum shell for 25MA/m ,J 0,2T,sB and different mo-
ment. Fig. 17 shows the temperature distribution in the thin conical frustum shell for
23MA/mJ , 0,2T,sB 7 msect , and different thickness of the shell. According to
Fig. 17, the effect of the thickness on the temperature of the thin shell is very remarkable.
Fig. 16. Curves of the temperature distribution for
25MA/m ,J 0,2T,sB , and different moment.
Fig. 17. Curves of the temperature distribution for 23MA/m ,J
0,2T,sB 7 msec,t and different values of .h
142
6. Conclusions.
A thermal-magnetic-elastic problem for a thin current-carrying conical frustum shell in
a magnetic field is studied. Using Newmark’s stable finite equidifferent formulas and the
quasi-linearization method, we have reduced nonlinear partial differential equations with
eight basic unknown variables to normal Cauchy form linear ordinary differential equations,
which can be solved by the discrete-orthogonalization method. Numerical solutions for
magnetoelastic stresses and deformations in a thin conical frustum shell under the interac-
tion of an electromagnetic field, a temperature field, and a mechanical field have been ob-
tained. This lays the analytical foundation for practical applications of the thermal-
magnetic-elastic problems on plates and shells. By the results, we now know that:
(1) the effect of an electromagnetic field of low intensity on the temperatures, defor-
mations, and stresses of structures and components is weak, this effect becoming stronger
with increase in the electromagnetic field intensity; it is shown that the thermal-magnetic-
elastic analysis on the structures and components in an electromagnetic field is necessary
and very important;
(2) the deformations and stresses in a thin conical frustum shell nonlinearly increase
with the electric current density or magnetic induction intensity;
(3) the stress, strain, and temperature in plates and shells can be controlled by changing
the electromagnetic and mechanical parameters;
(4) when the electric current density is large, the electromagnetic heat effect can not be
ignored; when the electric current density is enough, the thermal stress will dominate.
Acknowledgements.
This research was partially supported by a grant from the National Natural Science
Foundation of China, the Foundation of Key Laboratory of Nonlinear Continuum Mechan-
ics, Institute of Mechanics of Chinese Academy of Sciences. The authors gratefully
acknowledge these supports.
РЕЗЮМЕ. Вивчено термомагнітопружну задачу для тонкої конічної зрізаної оболонки, що
проводить струм і перебуває у магнітному полі. Отримано методом заміни змінних нелінійні дифе-
ренціальні рівняння нормального за Коші типу, які включають всього вісім невідомих змінних. За
допомогою стійких скінченних формул Ньюмарка і методу квазілінеаризації нелінійні диференціаль-
ні рівняння з частинними похідними зведені до послідовності квазілінійних диференціальних рів-
нянь, які далі можуть розв’язуватися методом дискретної ортогоналізації. З розгляду теплового ефек-
ту Джоуля в електромагнітному полі і рівняння теплової рівноваги визначено температурне поле в
тонкій конічній зрізаній оболонці і інтегральні власні значення. На спеціальному прикладі проаналі-
зовано зміну напружень, зміщень та температури зі зміною електромагнітних параметрів. Отримані
результати матимуть теоретичне продовження в дослідженні поставленої задачі.
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From the Editorial Board: The article corresponds completely to submitted manuscript.
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