Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

41, 3, pp. 521-536, Warsaw 2003

STUDY OF THE THERMOSTRESSED STATE OF

ELECTRICALLY CONDUCTIVE NONFERROMAGNETIC

SHELLS

Oleksandr Hachkevych1,2

Zygmunt Kasperski2

Borys Chornyj1

Oleh Dzyubachyk1

1 Pidstryhach Institute for Applied Problems ofMechanics and Mathematics, NASU, Lviv, Ukraine;

e-mail: dept13@iapmm.lviv.ua

2 Technical University of Opole, Poland; e-mail: kaspersk@po.opole.pl

A method of determination of parameters describing electromagnetic,
temperature and mechanical fields in electrically conductive shells cau-
sed by external quasisteady electromagnetic fields under presence of the
strong skin effect is considered in this paper.

Key words: electrically conductive shell, temperature fields, stresses,
quasisteady electromagnetic field, resonance phenomena

1. Introduction

In recent years, treatment and production of structural elements as well
as traditional and new materials widely incorporates the use of the electro-
magnetic filed (EMF). For determination of the rational parameters of such
a treatment, development of the corresponding methodology of the mathe-
matical modelling and effective ways of research of mechanical, thermal and
electromagnetic processes that happen in a deformable material medium un-
der conditions of various-typed EMF influence is necessary. Also it concerns
the exploitation of elements and products that work under electromagnetic
loading.

The method of determination of parameters describing electromagnetic,
thermal and mechanical processes in electrically conductive shells, caused by
theaction of external quasisteadyEMFunderpresenceof the strong skin effect



522 O.Hachkevych et al.

is discussed in the paper. Themethod is a development of the known one for
electrically conductive bodies in the case of electrically conductive shells.

2. Problem formulation

Consider a thin electrically conductive shell with the thickness 2h with li-
near electric andmagnetic material properties. The shell space D is free from
strange charges and currents. The shell is placed in a dielectric medium D0,
close to vacuumwith respect to electric andmagnetic properties. It is subjec-
ted to the quasisteadyEMF.The field is created by the system of given in D0
solenoidal currents (inductor) of AM diapason (radiofrequency with amplitu-
de modulation) with the density (Tamm, 1976; Gaczkiewicz and Kasperski,
1999)

j
(0)(r, t)= j∗0(r, t)cos(ωt+ψ0)≡Rej

(0)
∗ (r, t)

(2.1)

divj∗0(r, t)= 0

where
r – radius vector
ω – circular frequency
t – time
ψ0 – initial phase

j
(0)
∗ (r, t) – complex vector of current density;

j
(0)
∗ (r, t)= j

∗

0(r, t)e
i(ωt+ψ0) ≡ j(0)(r, t)eiωt

j∗0(r, t) – modulated amplitude; j
∗

0(r, t)= j(t)j0(r)
j0(r) – amplitude of carrying signal

j
(0)(r, t) – complex modulated amplitude of vector of current den-

sity; j(0)(r, t)= j∗0(r, t)e
iψ0

j(t) – function, describing the lawof changing the signal in time,
modulating the amplitude of electromagnetic oscillations
(amplitude of carrying regime).

This function is slowly changing in the period f∗ = 2π/ω (close to a
constant), so that the condition

∣∣∣
dj(t)

dt

∣∣∣≪ ω|j(t)| t > 0 (2.2)



Study of the thermostressed state... 523

Fig. 1.

is satisfied (condition of the regime quasisteadiness (Tamm, 1976;Gaczkiewicz
and Kasperski, 1999)).

We accept that the parameters of electromagnetic action (j0(r),j(t),ω)
are such that the action concerns ”non-shocking” EMF under the value of
magnetic field strength less than 106A/m (H0 < 10

6A/m, where H0 – the
maximal value of themagnetic field strength in the body (Tamm, 1976;Wain-
berg, 1967). Under such parameters of the electromagnetic action we assume
that displacements, deformations and their velocities are so small that the
assumptions of the linear elasticity theory are satisfied, and the influence of
movement on the characteristics of quasisteady EMF in the shell is negligible
(Moon, 1978; Nowacki, 1986). We consider materials for which electromecha-
nical and thermoelectric effects are small, and can be neglected. Thus, we
treat the EMF as an external action to the shell. Its influence on the heat-
conduction and deformation processes reduces to taking into account heats
and ponderomotive factors (electromagnetic (ponderomotive) forces and tur-
ningmoments). Formaterials, being linearwith respect to electric andmagne-
tic properties, vectors of strengths and inductions (displacements) of electric
and magnetic fields are parallel (Rawa, 1994; Tamm, 1976), and electric and
magnetic turning moments per unit volume are equal to zero. As a result,
the connections between electromagnetic, temperature and mechanical fields
in the considered case are taken into account through Joule’s heat, pondero-
motive forces, and also through the dependence between the deformation and
temperature fields (thermoelastic dissipation of energy). In such an approach,
accordingly to the knownmathematical model of thermomechanics of electri-
cally conductive shells under an action of a quasisteady EMF (Gaczkiewicz
et al., 1997), the initial problem of determination of parameters describing
thermomechanical behavior of the shell, will be resolved through two stages.



524 O.Hachkevych et al.

At the first stage, from the equation of electrodynamics of shells (neglecting
the influence of displacement currents in the shell region), we determine EMF
parameters in it, and then, on their base – Joule’s heat and ponderomotive
forces (as functions of electrodynamic parameters). At the second stage, we
find the temperature andmechanical fields.With it we set forth the equations
of the coupled problem of thermomechanics under the known sources of heat
and volume forces, which are correspondingly the Joule heats andponderomo-
tive forces, under given conditions of thermal exchange of the shell with the
external medium and conditions of mechanical fixation.

Consider frequencies, for which the parameter δ = (2µωσh2)−1/2 (cha-
racterizing relative depth of the inductive current penetration into the shell
(Podstrigach et al., 1977; Tamm, 1976); µ –magnetic penetrance; σ – coeffi-
cient of electrical conductivity) is small in comparison with unit

δ =
1

√
2µωσh2

≪ 1 (2.3)

(condition of presence of the strong skin effect (Rawa, 1994; Tamm, 1976)).

For determination of the parameters of examined fields we will use:

• method of solving singularly perturbed equations of electrodynamics in
the form of asymptotic expansions on powers of the small parameter δ

• method of spectral expansions on the shell thickness for determination
of the temperature

• knownmethods of solving the equations of thermoelasticity of thin shells
for determination of mechanical fields.

In the region D of the shell and in the certain round area D0 we introduce
a mixed curvilinear coordinate system (α1,α2,γ), in which αj (j =1,2) are
the lines of the main curvatures kj of the medium surface of the shell, and
where the coordinate γ determines the place of the point on the line normal
to this surface. In the further part of the paper, all metric characteristics are
given with respect to the half of the shell thickness h.

3. Mathematical model

At the first stage, i.e. in the determination of the electromagnetic field pa-
rameters under the given distribution of external currents, we assume that the
equations of the electrodynamics of shells are simplifieddue to negligibly small



Study of the thermostressed state... 525

values kjγ in comparison with the unit.We also assume that the parameters
of EMF in the system ”shell – external medium” approximately look like (qu-
asisteady approximation) (Gaczkiewicz and Kasperski, 1999; Podstrigach et
al., 1977):

— vector of the current density

j
∗
= j(t)Re

{
j(r)eiωt

}

—electric field strength

E∗ = j(t)Re
{
E(r)eiωt

}
E
(0)
∗ = j(t)Re

{
E
(0)(r)eiωt

}

—magnetic field strength

H∗ = j(t)Re
{
H(r)eiωt

}
H
(0)
∗ = j(t)Re

{
H
(0)(r)eiωt

}

in the region of the shell D and the region of the external medium D0. Then,
the determination of this parameters reduces to solving of the boundary pro-
blem, governed by the equations (Gaczkiewicz et al., 1997; Podstrigach et al.,
1977)

[
δ2
( ∂2

∂γ2
+∇2−k2

)
− i
2

]
E=0 x ∈ D (3.1)

– for the shell region and

(∆+k20∗)E
(0)
±
= iµ0ωh

2
j
(0)
±

x ∈ D±0 (3.2)

– for the subregions D±0 of the region D0 of the external medium, external to
the surface γ =±1 of the shell. On the surfaces γ =±1, dividing D and D0,
conditions of the ideal electromagnetic contact (Podstrigach et al., 1977) are
satisfied

E±j = E
(0)±
j

( ∂
∂γ
+kj −k

)
E±j −

1

Aj

∂E±γ
∂αj
= µ∗

[( ∂
∂γ
+kj
)
E
(0)±
j −

1

Aj

∂E
(0)±
γ

∂αj

]

(3.3)

E±γ =2iµ∗k
2
0∗δ
2E(0)±γ

( ∂
∂γ
+k
)
E±γ =

( ∂
∂γ
+2k
)
E(0)±γ



526 O.Hachkevych et al.

In infinity, the radiation conditions (Gaczkiewicz and Kasperski, 1999;
Gaczkiewicz et al., 1997; Podstrigach et al., 1977; Tamm, 1976) is

lim
γ→±∞

γ
( ∂
∂γ
± ik0∗

)
E
(0)
±
=0 (3.4)

In (3.1)-(3.3) the repeated indices are not the summation ones; they, i.e. j,
(j =1,2) correspond to αj coordinates. In the formulas above, the following
denote:

E
(0)
±
,j
(0)
±

– complex amplitudes of the electric field strength
anddensity of current in the subregions D±0 , exter-
nal to the surface γ =±1 of the shell

ε0,µ0 – dielectric andmagnetic penetrances of vacuum
Aj, (j =1,2) – coefficients of the first quadratic formof themedial

surface (Korn and Korn, 1968)

and

E
(0)+ =E

(0)
+

∣∣∣
γ=1

E
(0)− =E

(0)
−

∣∣∣
γ=−1

k20∗ = ε0µ0ω
2h2 µ∗ = µµ

−1
0 2k = k1+k2

The components of the vector ∇2E can be written down as (Gaczkiewicz
et al., 1997)

(∇2E)j = L2jEj +(−1)lLEl +L−1j Eγ l,j =1,2 (l 6= j)
(3.5)

(∇2E)γ = L2γEγ −L+1 E1−L
+
2 E2

where

L2j =∇2−B2−k2j L2γ =∇2−k21 −k22

L =
B1
A1

(
2
∂

∂α1
− 1
A1

∂A1
∂α1

)
− B2
A2

(
2
∂

∂α2
− 1
A2

∂A2
∂α2

)
+

+
1

A1A2

( 1
A1

∂2A1
∂α1∂α2

−
1

A2

∂2A2
∂α1∂α2

)

L±j =
2kj
Aj

∂

∂αj
+(kj ±kl)Bl +

1

Aj

∂kj
∂αj

l,j =1,2 (l 6= j)

∇2 = 1
A1A2

[ ∂
∂α1

(A2
A1

∂

∂α1

)
+

∂

∂α2

(A1
A2

∂

∂α2

)]

B2 = B21 +B
2
2 Bj =

1

AjAl

∂Aj
∂αl



Study of the thermostressed state... 527

and the Laplace operator in equation (3.2) is

∆=
∂2

∂γ2
+2k

∂

∂γ
+∇2

∗

∇2
∗
=
1

A1A2

[ ∂
∂α1

(A2
A1

∂

∂α1

)
+

∂

∂α2

(A1
A2

∂

∂α2

)]

In the case when one of the external subregions D±0 (for example D
−

0 ) is
constrained, it is necessary to replace radiation condition (3.4) in this subre-

gionwith the condition of the finiteness of the function E
(0)
−
to be determined.

While considering nonclosed shells, constrained by surfaces of the coordi-
nate system αj = α

±

j , we set conditions similar to (3.3) on these surfaces.
On the base of condition (2.3) equations (3.1), (3.2) for complex amplitu-

des of the electric field strength (of an elliptic type) are singularly perturbed.
Therefore, to construct their solution we use the known method of asympto-
tic expansions (Gaczkiewicz et al., 1997; Vishyk and Liusternik, 1960). In the
environment of bases γ = ±1 of the shell (subregions with the thickness
β± = β±(δ), 0 < β± < 1) we introduce the regularizing substitution
ξ± = ρ±/δ, where ρ± =1±γ. Further, the solution of the initial problem in
the subregions D±0 and D

± is given in terms of serieswith respect to the small
parameter δ. Each next approximation is searched in the form of a function
of the boundary-layer type that corresponds to quick attenuation of EMF in
the direction normal to the surface γ =±1. As a result, the solutions in the
chosen shell subregions (Gaczkiewicz andKasperski, 1999) will be

E±j = δ
∞∑

n=0

δn

[
n

2

]
∑

k=0

B
±(n)
j,k (α1,α2)(ξ

±)ke−ξ
±

(3.6)

E±γ = δ
2
∞∑

n=0

δn

[
n+1

2

]
∑

k=0

B
±(n)
γ,k (α1,α2)(ξ

±)ke−ξ
±

The coefficients B
±(n)
j,k , B

±(n)
γ,k are determined consecutively with the help of

recurrent correlations through their previous values and solutions of corre-
sponding boundary problems for the external medium region

(∆+k20)E
±(n)
0 =(iωµ0h

2
j
±

0 )δn0 n =0,1,2, . . .

E
±(n)
0j = B

±(n−1)
j,0 (3.7)

( ∂
∂γ
+2k
)
E
±(n)
0γ

∣∣∣
γ=±1

=±
i+1

2
B
±(n−1)
γ,0 ∓B

±(n−1)
γ,1 +kB

±(n−2)
γ,0



528 O.Hachkevych et al.

under radiation conditions in infinity. There δn0 denotes Kronecker’s symbol,
values with negative indices (n−p) are identically equal to zero.
While solving the sequence of boundary problems (3.7) we use a represen-

tation in the formof a serieswith respect to the small parameter k0∗. Thefirst
approximation (that corresponds to the zero power) is the solution to problem
(3.7) under quasisteady conditions, i.e. under neglecting the displacement cur-
rents in the region D±0 of the external medium.

The solution to the formulated problem for the shell region D is construc-
ted by themethod of prolongating solutions to (3.6) from the subregions D±

on thewhole region D with the help of the smoothingmultipliers ψ± (Vishyk
and Liusternik, 1960) in the appearance

E= ψ+E++ψ−E−

For such a solution it is possible tomake the evaluation of the reminder in the
metric L2 (Gaczkiewicz et al., 1997).

Notice, that the problem of determination of EMF parameters can be for-
mulated in a similar way with respect to the magnetic field strength H as
well.

At the second stage of solving, i.e. during determination of the thermo-
stressed state parameters, the initial ones will be Joule’s heats Q∗ and pon-
deromotive forces F∗, which in the considered case can be written down as

Q∗ = j∗∗ ·E∗∗ = σE2∗∗ ≡
1

σ
rotH2

∗∗

(3.8)

F∗ =F∗A = j∗∗×B∗∗ = µσE∗∗×H∗∗ = µrotH∗∗×H∗∗

where j∗∗, E∗∗, H∗∗ are real current density and sterengths of the electric
and magnetic field, respectively. For considered quasisteady EMF that corre-
sponds to real parts of complex vectors taking into account the dependence
Rea = (a+a)/2 (where a is a complex conjugate to a), we obtain (Gacz-
kiewicz andKasperski, 1999)

Q∗ = Q(1)+Q(2) F∗ =F (1)+F (2) (3.9)

In dependences (3.9)



Study of the thermostressed state... 529

Q(1) =
σ

2
E∗ ·E∗ =

σ

2
ϕ(t)E(r) ·E(r)

Q(2) =
σ

4
(E2
∗
+E

2
∗
)=
1

4
ϕ(t)
(
E
2(r)e2iωt +E

2
(r)e−2iωt

)
≡

≡
1

2
ϕ(t)Re

(
E
2e2iωt

)

(3.10)

F (1) =
1

2
σµϕ(t)Re

(
E(r)×H(r)

)

F (2) =
1

2
σµϕ(t)Re

(
E(r)×H(r)e2iωt

)

ϕ(t) = j2(t)

Taking into account (3.6) and writing down theMaxwell equation

H =− 1
iµω
rotE

for the shell case

Hj =
(−1)j+1
iµω

[( ∂
∂γ
+kl
)
El −

1

Al

∂Eγ
∂αl

]
l,j =1,2, l 6= j

(3.11)

Hγ =−
1

iµωA1A2

[ ∂
∂α1
(A2E2)−

∂

∂α2
(A1E1)

]

we express the functions Q(j) and F (j) (j = 1,2) in the form of asymptotic
expansions on the powers of the parameter δ.
Correspondingly, due to the formal structure of E and H for quasisteady

EMF representation (3.9) of heats and ponderomotive forces in function of
Q(1), F (1) and Q(2), F (2) we search the temperature and components of the
tensor of stresses as

w = w(1)+w(2) w ≡{T,σ̂} (3.12)

Slow-changing components T(1), σ̂(1) are searched in a quasistatic formu-
lation neglecting the coupling between temperature and deformation fields
(Kovalenko, 1975; Nowacki, 1986), i.e. on the base of correlations of the quasi-
static thermomechanics of shells (Kovalenko, 1975;Nowacki, 1986;Podstrigach
and Shvets, 1978). The components T(2), σ̂(2) are searched in a quasisteady
representation

w(2) =2
∞∑

m=1

Re
(
w
(2m)
∗ (r, t)e

2imωt
)
≡ 2ϕ(t)

∞∑

m=1

Re
(
w(2m)(r)e2imωt

)
(3.13)



530 O.Hachkevych et al.

(there w
(2m)
∗ (r, t) = {T(2m)(r, t), σ̂(2m)(r, t)}) – slow-changing in the pe-

riod f∗ functions) from the correspondingdynamic task of coupled thermoela-
sticity.Taking into account theknowndata (Gaczkiewicz andKasperski, 1999)
indicating negligible influence of quite periodical components of Joule’s heat
on the thermostressed state of electrically conductive bodies under the condi-
tions of strong skin effect in comparison with the influence of ponderomotive
forces, we assume that the components T(2), σ̂(2) are caused by the quasiste-
ady component F (2) of the ponderomotive force. Since thermal perturbation
in this case is caused by the deformation from a quick-changing dynamic ac-
tion, the process of deformation is considered as adiabatic (Kovalenko, 1975;
Nowacki, 1986), and the components T(2), σ̂(2) in (3.12), (3.13) are determi-
ned from the correlations of the dynamical problem of thermoelasticity in an
adiabatic quasisteady approximation. Thus, from the correlations for shells,
we obtain an increment in the temperature T(2) equal to (Kovalenko, 1975;
Nowacki, 1986)

T(2) =−(3λ∗+2µ∗)αtaT0εkk
λ

(3.14)

where λ∗ and µ∗ are isothermal Lamé moduli; εkk – the first invariant of the
deformation tensor. Then

T(2) =−(3λ∗+2µ∗)αtT0
cε(3λs +2µ∗)

σ
(2)
kk ≡−

αtaT0(
1+3ε∗

1−ν
1+ν

)
λ
σ
(2)
kk (3.15)

In (3.15), (Kovalenko, 1975; Nowacki, 1986)

λs = λ∗+
(3λ∗+2µ∗)

2α2taT0
λ

≡ νE
(1+ν)(1−2ν)

(
1+ε∗

1−ν
ν

)

is the adiabatic Lamé module (Kovalenko, 1975), and

ε∗ =
(3λ∗+2µ∗)

2α2tT0
(λ∗+2µ∗)cε

≡
(1+ν)α2taET0
(1−ν)(1−2ν)λ

denotes the parameter of conjugation of the deformation and temperature
fields (Kovalenko, 1975; Nowacki, 1986), where

σ
(2)
kk = σ

(2)
11 +σ

(2)
22 +σ

(2)
33 cε =

λ

a

and
αt – coefficient of linear thermal expansion
ν – Poisson’s coefficient
T0 – initial temperature, [K]



Study of the thermostressed state... 531

λ – coefficient of heat-conduction
a – coefficient of heat diffusion.

Notice, that the slow-changing component T(1) on the assumption that
the shell stays in conditions of convective heat transfer with the external me-
dium, can be effectively determined from the simplified (due to the fact that
the shell is thin-walled) equation of heat-conduction (Podstrigach and Shvets,
1978) and corresponding initial and boundary conditions making use of the
spectral expansions on the thickness coordinate (the method of finite integral
transformations) (Galitsin and Zhukovskii, 1976). The solution is searched in
the form

T(α1,α2,γ,τ)=
∞∑

m=1

Km(γ)T̃m(α1,α2,τ) (3.16)

The coefficients T̃m are determined from the solution to the equation

(
∇2− ∂

∂τ
−β20m

)
T̃m =−Q̃∗m T̃m

∣∣∣
τ=0
=0 (3.17)

under given conditions on the shell edges. In equation (3.17)

Q̃∗m =
h2

λ

1∫

−1

Q(1)Km(γ) dγ (3.18)

where
Km(γ) – kernel of the transformation; Km(γ)= Φm(γ)/‖Φm‖
β0m – characteristic values
Φm(γ) – characteristic functions of the corresponding Sturm-

Liouville task;
Φm(γ)= (Bi

−+k)sin[βm(1+γ)]+βmcos[βm(1+γ)]

‖Φm‖ – normalizing multiplier, ‖Φm‖= [
∫1
−1Φ

2
m(γ)dγ]

1/2

βm – positive roots of the equation

[β2m − (Bi+−k)(Bi−+k)
]
tan2βm =(Bi

++Bi−)βm

βm =
√
β20m −k2

τ = at/h2 – Fourier criterium
Bi± = hH∗± – Biot criterium
H∗± – coefficients of convective heat exchange from the sur-

faces γ =±1.



532 O.Hachkevych et al.

Under condition that EMF is harmonic (j(t) = 1 and the density of heats
Q∗ = Q(1)h

2/λ is a function of coordinates only), we can obtain an asymptotic
solution to the three-dimensional problem of heat-conduction for the initial
period of heating (τ ≪ 1). As a small parameter we take the value 1/s, where
s is theparameter of Laplace’s transformation. Specifically,wewill obtain such
an expression for the temperature field (in the second approximation)

T = τQ∗−ψ+
√
(4τ)3I3erfc

(1+γ
2
√
τ

)[ ∂
∂τ
Q∗+

(
Bi+−k

)]

γ=1
+

(3.19)

+ ψ−
√
(4τ)3I3erfc

(1+γ
2
√
τ

)[ ∂
∂τ
Q∗−

(
Bi−+k

)]

γ=−1

where (Lykov, 1967)

Iperfc(x)=

∞∫

x

Ip−1erfc(ξ) dξ I0erfc(x)= erfc(x)

In expression (3.19) components corresponding to theboundary layer take into
account the influence of the heat exchange process on the temperature field
during short heating times.

4. Calculations results

In Figures 2 and 3 the distributions of the temperature T and compo-
nents of the stress tensor σ̂ for a cylindric shell of radius R = 0.40m sub-
ject to induction heating by the external currents of a constant amplitude j0
(j(t) = 1) applied coaxially to the shell surface of radius R =0.42m in the
direction tangential to the line of the cross-section are shown.The shell, which
is made of the rustless steel (X18H9T) (Gaczkiewicz and Kasperski, 1999) is
heat-insulated on the bases γ =±1.
In Fig.2 and Fig.3 the solid lines depict functions of temperature and

stress on the surface nearest to the inductor (γ = 1), the dash-dotted ones
on the medium surface (γ = 0), and the dashed – on the interior surface
(γ =−1) of the shell. Lines 1 correspond to δ =0.1, and lines 2 – to δ =0.2.
The value of the current j0 was determined from the condition that at the

instant τ∗ the shell is heated up to the given temperature T∗.
In Fig.2 the dependance of the temperature on the parameter τ/τ∗ for

τ∗ = 1 is given. One can see that the temperature level essentially decreases
with a drop in γ. On the inductor side of the surface the temperature level is



Study of the thermostressed state... 533

Fig. 2.

higher for lower δ, and on the medium and interior ones on the contrary —
higher for greater values of the parameter δ.

In Fig.3 the dependance of stresses σ∗0 = σφφ = σxx in time is given. The
stresses on the surface γ =1 are compressive, and on γ =−1 – tensile. The
stresses for γ = 1 are growing quicker, and their absolute value exceeds the
stresses on the interior surface.With decreasing δ the level of stresses on the
exterior surface is growing, and on the interior – decreasing.

Fig. 3.

Notice that for the considered parameters δ the components of the tem-
perature and stresses T(2), σ∗0

(2) are negligible in comparison with T(1) and
σ∗0
(1)

σ∗0
(1) =

αtE

1−ν
(T1−T) T1 =

1

2h

1∫

−1

T dγ



534 O.Hachkevych et al.

Thequasistatic stresses causedby theponderomotive forces are also negligible.

Carried out investigations on the thermomechanical behavior of the shells
in functionof thepenetrationdepth (frequencies of the externalEMF) showed,
that as well as for the bodies of a simple geometrical structure (Gaczkiewicz
andKasperski, 1999) in theneigbourhoodof theEMFfrequencies ωn =0.5ω

∗
n,

n =1,2, ... (where ω∗n – eigenfrequency of the thermoelastic shell oscillations)
the levels of quite periodic components of the temperature and stresses in a
non-polarized ferromagnetic significantly grow and become proportional (re-
sonance phenomena take place). High levels of the quite periodic components
of temperature are caused by the coupling between deformation and tempe-
rature fields. With the growth in the number n of resonance frequency the
amplitudes of quite periodic components T(2), σ̂(2) decrease.

With respect to theweakness of the coupling between the deformation and
temperature field parameter ε∗ (ε∗ ≪ 1) for steel shells (for which the pheno-
mena of the strong skin effect take place), we notice that each eigenfrequency
of the thermoelastic oscillations ω∗ practically equals to the corresponding
eigenfrequency of the elastic oscillations of the considered shell. Fig.4 the so-
lid lines illustrate the dependance of the first two resonance frequencies of
EMF ωn (curves 1 and 2 correspondingly) on the shell thickness (for a shell of
radius R = 0.40m made of the steel X18H9T). The dashed lines correspond
to the dependance of the ”resonance” parameter of the relative depth of the
currents penetration on the thickness. For the given shell thickness lower δ
corresponds to higher resonance frequencies.With the growth of the frequency
the resonance frequencies decrease.

Fig. 4.



Study of the thermostressed state... 535

5. Conclusions

The area of the resonance frequency (value of the deviation ∆ω1 of the
EMF frequency ω from the first resonance ω1, for which themaximal value of
the stress σ̂(2) constitutes not less than 10%of themaximal value of σ̂(1) in a
steady regime) does not depend in linearmaterials on theEMFcharacteristics
and is narrow (∆ω1 ¬ 10−4÷10−5ω1 – for non-magneticmaterials and ∆ω1 ¬
10−4÷10−5µ2

∗
ω1 – formagnetic ones). In this area themaximal values of T

(2)

and σ̂(2) are caused by the ponderomotive forces (the influence of the Joule
heats is negligible) and not significantly depend on Biot’s criterium. Outside
of the resonance area the thermostressed state of the shell is determined by
the slow-changing components Q(1) and F (1) of the heats and ponderomotive
forces (which coincide with the averaged ones over the period f∗, i.e.

M∗ =
2π

ω

t+2π
ω∫

t

M dt

where M ≡ {Q∗,F∗}). For µ∗ < 30 it is possible to neglect outside of the
area of resonance frequencies the effect of the ponderomotive force as well, i.e.
to use the approach, which is usually used for solving problems of induction
heating in shells with the skin effect condition (Gaczkiewicz and Kasperski,
1999; Podstrigach and Shvets, 1978).
The given research method can be used for study of thermomechanical

behavior of shells undergoing an action of EMF due to quasisteady currents
that are used in practice for inductive thermotreatment of longitudinal and
cross welding seams in welded shells, especially cylindrical ones.

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Badania termicznego stanu naprężenia nieferromagnetycznych powłok

przewodzących prąd elektryczny

Streszczenie

W pracy zaprezentowano metodę wyznaczania parametrów elektromagnetycz-
nych, temperaturowych oraz mechanicznych w powłokach przewodzących prąd elek-
tryczny, znajdujących się pod wpływempola elektromagnetycznego przy uwzględnie-
niu silnego efektu naskórkowości.

Manuscript received December 3, 2002; accepted for print March 5, 2003