Progress In Electromagnetics Research Symposium Proceedings, Xi’an, China, March 22–26, 2010 1269
Huijuan Bai and Xiaojing Zheng
Department of Mechanics and Engineering Science, Lanzhou University
Lanzhou, Gansu 730000, China
Abstract — A mathematic model was established for electrified cantilever conductive thin plates under external transverse multi-pulse magnetic field. The distribution of eddy current induced by the transverse multi-pulse magnetic field would be influenced due to a subsistent inner uniform electric field in the conductive thin plates by adopting finite element numerical method, and also obtained the distribution of temperature field in thin plate from the law of heat conduction.
Subsequently the influences of multi-pulse magnetic field and uniform electric field on the inplane magnetic volume forces and the maximum deformation of thin plate were quantitatively simulated on the basis of electro-magneto-thermo-elastic theory. The simulation results indicate that the dynamic buckling phenomenon is caused by the in-plane magnetic volume compression force arising from the interaction between the eddy current induced by multi-pulse magnetic field and inner uniform electric field. The dynamic buckling phenomenon of cantilever thin plate is determined by the value of maximum magnetic field and impulse parameter, and also the inner uniform electric field.
1. ITRODUCTION
Abundant conductive shells and structures are broadly applied to the complicated electromagnetic circumstance in modern industry and electronic devices, so developing a relevant electro-magnetothermo-elastic theory is important to make the guidance on the metallic thermo forming, optimum design of configuration for the electronic mechanical devices run under the electromagnetic field.
Much attention had been paid to the mechanical behaviors of a conductive thin plate under the magnetic field changing with time. Hua et al. [1, 2] took the limiter blade of a Tokamak fusion reactor as a cantilever beam in applied magnetic fields delayed exponentially with time. Zhou et al. [3] developed a coupled magneto-mechanical model, which can describe various mechanical behaviors for the ferromagnetic structures. Takagi et al. [4] conducted an experimental of thin plate deflection in the magnetic field and gave some computational analyses by using the T -method [5]. Recently,
Wang et al. [6, 7] presented a theory model to analyze the magneto-thermo-elastic instability of ferromagnetic plates subjected to thermal and magnetic loading. Zhang et al. [8] and Zhu et al. [9] revealed some rules for the dynamic stability of a cantilever conductive plate under transverse impulsive magnetic field and strong magnetic field. Higuchi et al. [10] studied the magneto-thermo-elastic stressed of a conductive solid circular cylinder in a transient magnetic field. Laissaoui et al. [11] estimated the thermal impact on induction machines, for which a coupled electromagnetic thermal analysis was carried out.
Most researchers are focus on the dynamic behaviors of rectangular conductive beams or plate structures which are in transverse monopulse magnetic field, and fewer considered the existence of inner electric field. The present study considers the electrified cantilever conductive thin plate subjected to external transverse multi-pulse magnetic field. By using finite element numerical method, we obtain the distribution of eddy current induced by the transverse multi-pulse magnetic field and inner uniform electric field. Subsequently, the distribution of temperature, the in-plane magnetic volume forces and the maximum deformation of thin plate were quantitatively gained on the basis of electro-magneto-thermo-elastic theory. We also discuss the main influencing factors on the dynamic stability of the conductive plate.
2. BASIC EQUATION
Considering that a cantilever conductive thin plate is applied by the transverse multi-pulse magnetic field B
0
= B
0
( t ) k and the inner electric field E = E
0 i as shown in Fig. 1, where i and k are the unit vectors in the direction of x and z respectively. Ignore the displacement current, and Maxwell

1270 PIERS Proceedings, Xi’an, China, March 22–26, 2010 z
( ) = B
0
( − ( − t τ z h
E
0 y x b a
Figure 1: Schematic diagram of the electrifiedcantilever conductive thin plate in transverse multi-pulse magnetic field.
electromagnetic equations related to the distribution of eddy current can be described as
∇ × H = J e
∇ × E e
= −
∂ B e
∂t
+
∂B
0
∂t
¶
.
(1a)
(1b) where H , J e and E e are the vector of magnetic field, eddy current density and electric field intensity induced by time-varying magnetic field respectively.
B
0 the induced magnetic field duo to the eddy current.
and B e are the applied magnetic field and
The constitution relations for the magnetic field and the eddy current can be respectively expressed as
B = B
0
+ B e
= µ
0
H , J e
= σ ( E e
+ V × B ) .
(2) where V is the velocity of vibration of the plate, σ , µ
0 are respectively the electrical conductivity of the conductive plate and the magnetic permeability of vacuum.
Applying divergence operator to both sides of Eq. (1a), we can get ∇ · J e current vector potential T [5] can be defined as J e
= 0, and the eddy
= ∇ × T . Introducing Coulombian gauge condition ∇ · T = 0, and using the Helmholtz’s formula and Biot-Savart’s law for the thin plate, the magnetic flux density arise from the eddy current in the plate can be written as
B e
= µ
0
T −
µ
4
0
π
Z s
T 0 n
∂
∂z
µ
1
R
¶ ds 0 .
(3) here T = T k , B e
= B e k .
T 0 n is the component of T along the thin plate surface s , and R is a distance from an arbitrary point in the plate to a point occupied by the eddy current. Substituting
Eq. (3) into Eq. (1b), and considering Eq. (2), we can obtain the basic equation of the vector potential of the eddy current T as
∇ 2 T − µ
0
σ
∂T
∂t
−
σµ
0
4 π
Z s
∂T 0 n
∂t
∂
∂z
µ
1
R
¶ ds 0 = σ
∂B
0
( t )
∂t
.
(4) with the boundary condition ∂T / ∂s = 0 and the initial condition T ( x, y, 0) = 0.
In the Eqs. (3) and (4), the eddy current vector potential T for the thin plate is simplified as the only component T in the direction of z , which makes the magnetic flux density B e the eddy current has the component in the direction of z .
induced by
Since an inner uniform electric field E is existence in the thin plate, the current J can be expressed as,
J = J e
+ σ E = σ ( E e
+ E + V × B ) .
(5)
Assumed that the current is uniform along the thickness of the thin plate, the electromagnetic force density vector F is given by Lorentz force formula,
F =
Z h /2
− h /2
( J × B ) dz = − h ( B
0
+ B e
)
∂T
∂x i − h ( B
0
+ B e
)
µ
∂T
∂y
+ σE
0
¶ j (6) the Eq. (6) reveals that electromagnetic force exerted on the plate only has two in-plane component
F x and F y with applied transverse time-varying magnetic field. If these in-plane components

Progress In Electromagnetics Research Symposium Proceedings, Xi’an, China, March 22–26, 2010 1271 were ignored, the thin plate would be not subjected to electromagnetic forces so that it would be impossible to analyze the mechanical behavior.
The plane temperature field induced by the heating effect of eddy current for the conductive plate can be expressed as
λ
µ
∂ 2
∂x
T
2
+
∂ 2
∂y
T
2
¶
= −
µ
1
σ
J 2 − cρ
∂T
∂t
¶
(7) where λ , c and ρ are respectively the thermal conductivity, specific heat ratio and mass density, T is the temperature of one point in the plate. The convective boundary condition is λ and β is the convection coefficient.
∂T
∂n
= − β ( T sc
− T e
)
Based on the analyses of the electromagnetic force mentioned above, the in-plane displacement of the plate under the in-plane electromagnetic force F x and F y and thermal stress is considered.
The movement equations for the displaces u and v can be expressed as
1
Eh
− µ 2
Eh
1 − µ 2
µ
∂ 2 u
µ
∂x 2
∂ 2 v
∂y 2
+
+
1 − µ
2
∂ 2 u
∂y 2
1 − µ ∂ 2 v
2 ∂x 2
+
+
1 +
2
µ ∂ 2 v
∂x∂y
¶
2
1 + µ
∂x∂y
∂ 2 u
¶
+ F x
( x, y, t ) −
+ F y
( x, y, t ) −
Ehα ∂θ
1 − µ ∂x
1
Ehα
− µ
∂θ
∂y
=
=
ρh
ρh
∂ 2
∂t
∂ 2
∂t u
2 v
2
.
, (8)
(9)
The boundary conditions and the initial conditions are respectively written as x = 0 : v = 0; x = a, y = 0 , b : N |
Γ
= 0 , u ( x, y, 0) = v ( x, y, 0) = 0 , ∂u ( x, y, 0)/ ∂t = ∂v ( x, y, 0)/ ∂t = 0 .
(10)
(11) where E , µ , α and θ are the Young’s modulus, the Poisson’s ratio, bending strength and the temperature difference between the instantaneous temperature of one time and the next time. Γ represents the boundary of the plate except for the edge x = 0; N |
Γ is the component of internal membrane force N at the boundary Γ. Since the internal membrane force N is related to the displaces u and v of the plate, its all components can be written
N x
=
N xy
=
1
Eh
− µ
Eh
2(1 +
2
µ
µ
)
∂u
∂x
µ
+
∂v
µ
+
∂x
∂v
∂y
∂u
∂y
¶
¶
−
.
Eα
1 − µ
Z h /2
− h /2
θdz, N y
=
Eh
1 − µ 2
µ
∂v
∂y
+ µ
∂u
¶
∂x
−
Eα
1 − µ
Z h /2
− h /2
θdz,
(12)
The vibration equation of the plate under the internal membrane forces N x written as
, N y
− D ∇
+
µ
2
∂N
∇ 2
∂y y w
+
−
1 − µ
¶
∂N xy
∂x
Eα
Z
∂y h /2
− h /2
∂w
=
∇
ρh
2
∂
θzdz
2 w
∂t 2
+
µ
N x
∂ 2
∂x w
2
+ 2 N xy
∂ 2 w
∂x∂y
+ N y
∂ 2
∂y w
2
¶
+
µ and
∂N
∂x x
+
N xy
∂N
∂y xy can be
¶
∂w
∂x
(13)
Considering the cantilever plate as shown in Fig. 1, the boundary conditions and the initial conditions are given x = 0 : w ( x, y, t ) = 0 , ∂w ( x, y, t )/ ∂t = 0; x = a, y = 0 , b : M n
| Γ = 0 , V n
| Γ = 0 , (14) t = 0 : ∂w ( x, y, t )/ ∂t = 0 , w ( x, y, 0) = 0 .
(15) where M n
|
Γ of the plate.
and V n
|
Γ are a bending moment and an equivalent shearing force at the boundary Γ
In the process of simulation calculation for the dynamic response of the electrified conductive thin plate under the multi-pulse magnetic field, we need to solve Eq. (4) first to obtain the eddy current vector potential T , and then get the total current density J and the distribution of temperature.
After having got the membrane forces N x
F y solution of Eqs. (13)–(15).
, N y and N xy arising from in-plane forces F x
( x, y, t ) and
( x, y, t ) by solving Eqs. (8)–(11), we will predict dynamic response of the plate according to the

1272 PIERS Proceedings, Xi’an, China, March 22–26, 2010
3. NUMERICAL RESULT
In order to simulate the electro-magneto- thermo-mechanical multi-field coupling behaviors of the cantilever conductive thin plate under the inner uniform electric field and external transverse multipulse magnetic field, as shown in Fig. 1, the theoretical model developed in preceding section is preformed. The parameters of geometry and material of the plate are taken as a = 0 .
2 m, b = 0 .
1 m, h = 4 mm, E = 6 .
89 × 10 10 Pa, µ = 0 .
3, ρ = 2 .
713 × 10 3 kg/m 3 , σ = 3 .
65 × 10 7 S/m,
µ
0
= 4 π × 10 − 7 H/m, λ = 263 W/mK, varying magnetic field is B
0
= B
0
β = 5 W/ m 2 K, cρ = 2 .
44 × 10 6 J/m 3 K. The applied time-
[1 − exp( − t / τ )] k , and adopting the critical value B
τ = 0 .
06 s. The maximum number of impulse adopted here is N max
0
= 2 T esla
= 5000. We propose a numerical technique of the finite element method using the Crank-Nicolson method and Newmark method
, in time for simulation, taking the element number of the plate as 20 × 10 and the time step as
∆ t = 0 .
01 s.
Choose one point of ( a/ 2 , b/ 2) in the thin plate and its deflection response are plotted respectively for E
0
= 0 V/m and E
0
= 0 .
3 V/m in Fig. 2. It is found from the Fig. 2(a) that the vibration of the plate with a small amplitude tends to stability when no inner added electric field. However, with the uniform inner electric field in the Fig. 2(b), the amplitude increases greatly and the vibration will last long. Also, the center of the eddy current is moved and the distribution of the eddy current density in the conductive plate is changed owing to the added uniform electric field, as shown in
Fig. 3.
(a) (b)
Figure 2: Response of the vibration of the conductive plate ( (a) no electric field; (b) inner electric field
E
0
= 0 .
3 V/m).
) (b)
(a) (b)
Figure 3: The distribution of the eddy current density ((a) no inner electric field; (b) inner electric field
E
0
= 0 .
1 V/m).

Progress In Electromagnetics Research Symposium Proceedings, Xi’an, China, March 22–26, 2010 1273
4. CONCLUSIONS
A numerical analysis for the dynamic stability of the conductive thin plate subjected to the inner uniform electric field and transverse multi-pulse magnetic field is displayed. When the inner electric field is applied, the vibration of the plate becomes instable and the amplitude increases greatly.
The changes of the distribution of eddy current density certainly alter the in-plane components of the electromagnetic force and the thermal stress, which can bring the influence on the distortion of the conductive thin plate.
ACKNOWLEDGMENT
This research was supported by the Fund of HSFC’s program (No. 90405005), the Fund of “973 program” (2007CB607506), the Ph.D. Fund (No. 20050730016), and the Fund of No. WUT2005Z04.
REFERENCES
1. Hua, T. Q., M. J. Knott, and L. R. Turner, “Experimental modeling of eddy currents and deflection for Tokamak limiters,” Fusion Tech.
, Vol. 10, 1047–1052, 1986.
2. Turner, L. R. and T. Q. Hua, “Experimental study of coupling between eddy currents and deflections in cantilevered beams as models of Tokamak limiters,” Electromagnetomechanical
Internationals in Deformable Solids and Structures , 81–86, North Holland, Amsterdam, 1987.
3. Zhou, Y. H. and X. J. Zheng, “A general expression of magnetic force for soft ferromagnetic plates in complex magnetic field,” Int. J. Eng. Sic.
, Vol. 35, No. 15, 1405–1417, 1997.
4. Takagi, T., J. Tani, S. Matsuda, and S. Kawamura, “Analysis and experiment of dynamic deflection of a thin plate with a coupling effect,” IEEE Trans. Magn.
, Vol. 28, No. 2, 1259–
1262, 1992.
5. Miya, K., J. E. Akin, and S. Hanai, “Finite element analysis of an eddy current induced in thin structure of magnetic fusion reactor,” Int. J. Numer. Mech. Eng.
, Vol. 17, 1613–1629, 1981.
6. Wang, X. Z., Y. H. Zhou, and X. J. Zheng, “A generalized variational model of magnetothermoelasticity for nonlinearly magnetized ferroelastic bodies,” Int. J. Eng. Sci.
, Vol. 40, No. 17,
1957–1973, 2002.
7. Wang, X. Z., Y. H. Zhou, and X. J. Zheng, “Magneto-thermo-elastic instability of ferromagnetic plates in thermal and magnetic fields,” Int. J. Solids Struct.
, Vol. 40, 6125–6142, 2003.
8. Zheng, X. J., J. P. Zhang, and Y. H. Zhou, “Dynamic stability of a cantilever conductive plate in transverse impulsive magnetic field,” Int. J. Solids Struct.
, Vol. 42, 2417–2430, 2005.
9. Zhu, L. L., J. P. Zhang, and X. J. Zheng, “Multi-field coupling behavior of simply-supported conductive plate under condition of a transverse strong impulsive magnetic field,” Acta Mech.
Solida Sin.
, Vol. 19, No. 3, 203–211, 2006.
10. Higuchi, M., M. Kawamura, and Y. Tanigawa, “Magneto-thermo-elastic stresses induced by a transient magnetic field in a conducting solid circular cylinder,” Int. J. Solids Struct.
, Vol. 44,
5316–5335, 2007.
11. Laissaoui, S., M. R. Mekideche, D. Sedira, and A. Ladjimi, “Dynamic modeling of induction motor taking into account thermal stresses,” COMPEL , Vol. 26, No. 1, 36–47, 2007.