JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

43, 1, pp. 111-118, Warsaw 2005

ELASTIC ELECTROCONDUCTING SURFACE IN

MAGNETOSTATIC FIELD

Marek Rudnicki

Faculty of Civil Engineering, Warsaw University of Technology

e-mail: marr@siwy.il.pw.edu.pl

The dynamical linear theory of amaterial surface placed in vacuum and
subjected to an external strongmagnetostatic field is considered.Motion
of the surface is described by a position function. The material of the
surface is assumed to be an isotropic elastic non-magnetizable electric
conductor.The residual stress is taken into account.Displacement-based
field equations are obtained in a coordinate-free notation.

Key words: magnetoelasticity, material surface, membrane theory, sur-
face current, real electric conductor, perfect electric conductor, residual
stress

1. Introduction

A three-dimensional thin body may be represented by a two-dimensional
continuum as a result of reduction of the thickness dimension or by a direct
approach. A deformable surface with usual kinematics (one deformation func-
tion) serves as a directmodel underlying themembrane theory. In this paper,
we develop the theory of Gurtin and Murdoch (1975) providing an exten-
sion necessary formagnetoelastic interactions. Themechanical part is directly
obtained as two-dimensional, however, the electromagnetic part is subsequ-
ent to three-dimensional considerations. Displacement of the surface, normal
magnetic induction at the surface and scalar potentials of outward magne-
tic induction are unknowns involved in the final field equations. The MKSA
system of units is used.



112 M.Rudnicki

2. Initial state

2.1. Surface

Let s denote a surface in the three-dimensional Euclidean point space Σ
endowedwith an appropriate structure (seeGurtin andMurdoch, 1975), espe-
cially the tangent space Tp and the unimodularvector field a3 : s → V , where
V is the translation space, such that a3(p)∈ T

⊥
p at each point p ∈ s.We use

the following notation: I(p) for the inclusion map from Tp into V , P(p) for
the perpendicular projection from V onto Tp. If c : s → R, where R stands
for the reals, u : s → V , S : s → V ⊗ V , where S(p) ∈ V ⊗ Tp, then
gradsc(p)∈ Tp, gradsu(p)∈ V ⊗Tp, gradsS(p)∈ V ⊗V ⊗Tp. Moreover, we
have

u=Pu+ua3 S=PS+a3⊗S (2.1)

where u(p)∈ R and S(p)∈ Tp are defined by

u =u ·a3 S=S
>
a3 (2.2)

with S> being the transpose of the tensor S. Given surface gradients and
making use of the following notations

skw(a⊗b)=
1

2
(a⊗b−b⊗a) Λ(a⊗b−b⊗a)=a×b

tr(a⊗b)=a ·b tr (1,3)(a⊗b⊗c)= (a ·c)b

δ(2,1,3)(a⊗b⊗c)= b⊗a⊗c

(2.3)

where × and ·mean the cross product and the inner product, respectively, we
define surface divergence and curl operations as

divsu= tr(Pgradsu)

curlsu=−Λ[2skw(Pgradsu)] (2.4)

divsS= tr(1,3)Pδ(2,1,3)gradsS

Thus, divsu(p)∈ R, curlsu(p)∈ T
⊥
p , and divsS(p)∈ V .

2.2. Static bias magnetic field

The bias magnetic induction B is governed in a certain neighbourhood of
the surface s by equations

curlB=0 divB=0 (2.5)



Elastic electroconducting surface... 113

Introduce surface vector fields: L,G : s → V by

L=B
∣

∣

∣

s
G=

∂

∂x3
B
∣

∣

∣

s
(2.6)

where x3 is the metric coordinate in the normal direction. Then, when calcu-
lating on the surface s, Eqs (2.5), take the form

PgradsL−K(PL)−PG=0
(2.7)

curlsL=0 divsL+G =0

where K denotes theWeingarten map.

3. Present state

3.1. Kinematics

Deformation of the surface s during the time interval T is a mapping
χ : s×T → Σ.Thedisplacement corresponding to χ is thefield u : s×T → V
defined by

u(p,t)= χ(p,t)−p (3.1)

where t is time. Thus

gradsχ = I+ gradsu (3.2)

where gradsχ(p,t) ∈ V ⊗ Tp. The rotation field corresponding to u is a
mapping r : s×T → V defined by

Pr=(gradsu)
>a3 r =

1

2
a3 · curlsu (3.3)

The infinitesimal strain reads

E=sym(Pgradsu)= sym[Pgrads(Pu)]+uK (3.4)

where ”sym” means the symmetrical part of a tensor.



114 M.Rudnicki

3.2. Magnetic field outside the surface

Let Ω+ and Ω− denote certain outwardmaterial-free regions touching the
surface s fromtheupperand lower side, respectively, and b represents induced
magnetic induction governed in the regions Ω+ and Ω− by the equations

curlb=0 divb=0 (3.5)

accompanied by the continuity condition at the surface s in the form

[b] = 0 (3.6)

where [·] denotes the jump across the surface. Introducing scalar potentials
ψ+ : Ω+×T → R and ψ− : Ω−×T → R with the use of the space gradient

b= gradψ (3.7)

Eqs (3.5) lead to the Laplace equations in the regions Ω+ and Ω−

∆ψ+ =0 ∆ψ− =0 (3.8)

with the Neumann boundary conditions on the surface s

∂

∂x3
ψ+ = b

∂

∂x3
ψ− = b (3.9)

3.3. Electromagnetic field within the surface

The surface current density on the surface s is determined by the relation

j
sur =

1

µ
a3× [b] (3.10)

where µmeans themagnetic permeability of vacuum.Moreover, thequantities
b and Pe, where e denotes the electric field, are identical at both sides of the
surface s. The corresponding differential equation reads

curls(Pe)−
∂

∂t
b =0 (3.11)

Making use of the inverted Ohm law

Pe=
1

λ
Pj
sur+P(L×v)=

1

µλ
P(a3× Igrads[ψ])+P(L×v) (3.12)



Elastic electroconducting surface... 115

where v denotes the velocity vector and λ is the electric surface conductivity,
Eq (3.11) becomes

∆s(ψ
+−ψ−)−µλ

∂

∂t
b+µλ

∂

∂t
[Ldivsu+G ·u− (PL) · (Pr)] = 0 (3.13)

where ∆s stands for the surface Lagrangian. In the case of perfect conduction,
Eq (3.13) simplifies to the relation

b = Ldivsu+G ·u− (PL) · (Pr) (3.14)

3.4. Electromagnetic momentum and energy

The following linearized identity is derivable from three-dimensional Ma-
xwell equations when simplified by neglecting the displacement current

fL = divTM (3.15)

where fL and TM are the electromagnetic force andmagnetic stress, respec-
tively, defined by (see Costen and Adamson, 1965)

fL = j×B TM =
1

µ
(b⊗B+B⊗b)−wM1 (3.16)

where, in turn, j is the conduction current density, 1 denotes the identity
on V , and wM means the electromagnetic energy density in the form

wM =
1

µ
B ·b (3.17)

Similarly, the power per unit volume lost by the fields equals

PM =−divSM −
∂

∂t
wM =0 (3.18)

where

S
M =

1

µ
e×B (3.19)

denotes the Poynting vector. In an integral form, the electromagnetic momen-
tum and energy laws are

∫

V

fL =

∫

∂V

T
Mn

∫

∂V

SMn+

∫

V

∂

∂t
wM =0 (3.20)



116 M.Rudnicki

where n represents the outward unit vector normal to the surface ∂V . In the
limit for the surface s, setting n = a3, the electromagnetic momentum law
reduces to

fsur = [TM]a3 (3.21)

Using Eqs (3.16), (3.10) and (3.17), we find

[TM] = (jsur×a3)⊗B+B⊗ (j
sur×a3)− [(j

sur×a3) ·B]1 (3.22)

Hence

[TM]a3 =(j
sur×a3)(B ·a3)− [(j

sur×a3) ·B]a3 = j
sur×B (3.23)

Similarly,

[SM] =
1

µ
[e]×B=

1

µ
[e]a3×B (3.24)

Thus, the electromagnetic energy law for the surface s takes the form

[SM] ·a3 =0 (3.25)

3.5. Stress-based equations of motion

The stress equation of motion of a material surface has the local form

divsS+f
mech+fsur = ρ

∂2

∂t2
u (3.26)

where S denotes the surface stress tensor, ρ is themass density per unit area,
and fmech stands for the mechanical force. Using Eqs (2.1), (2.6), (3.10) and
(3.7), Eq (3.26) may be put in a more detailed form

Pdivs(PS)+KS+Pf
mech+

1

µ
Lgrads[ψ] = ρ

∂2

∂t2
(Pu)

(3.27)

divsS−K · (PS)+f
mech−

1

µ
(PL)grads[ψ] = ρ

∂2

∂t2
u

where ”·” denotes the inner product of two tensors.

3.6. Stress-strain relation

The constitutive relation for the stress S reads

S=(gradsχ){S
res+C[E]} (3.28)



Elastic electroconducting surface... 117

where Sres is the residual stress and C denotes the elasticity tensor. If the
material is isotropic relative to the reference configuration, then

S
res = σ1s C[E] = λL(trE)1s+2µLE (3.29)

where λL and µL areLame constants, and 1s(p) is the identity on Tp.Making
use of Eqs (2.1) and (3.2), we arrive at

PS= σ1s+σPgradsu+λL(trE)1s+2µLE S = σPr (3.30)

3.7. Displacement-based equations of motion

Nowassumethat σ,λL andµL are constant onthe surface s.Then,making
use of Eqs (3.30) and (3.4), Eqs (3.27) are transformed to the displacement-
based form

(σ+2µL)Pdivs[Pgrads(Pu)]+λLgradsdivs(Pu)−

−2µLa3× [Igrads(a3curlsu)]−σK[K(Pu)]+

+2(σ+µL)Kgradsu+2HλLgradsu+2(σ+2µL+λL)(gradsH)u+

+Pfmech+
1

µ
Lgrads(ψ

+−ψ−)= ρ
∂2

∂t2
(Pu)

(3.31)

σ∆su− (σ+2µL)(K ·K)u−λL(2H)
2u−2Hσ−2σ(gradsH) · (Pu)−

−2λLHdivs(Pu)−2(σ+µL)K · [Pgrads(Pu)]+

+fmech−
1

µ
(PL) · grads(ψ

+−ψ−)= ρ
∂2

∂t2
u

where H is the mean curvature.

4. Conclusions

• In order to incorporatemagnetoelastic effects in the theory of amaterial
surface, the concept of surface electric current is required, even in the
case of real conduction.

• The obtained model is not entirely two-dimensional because Eqs (3.8)
are needed for completeness.



118 M.Rudnicki

• The lack of a term including normal bias magnetic induction in the
second equation of motion seems to be the most significant difference
occurring within the electromagnetic part between the presentedmodel
and shell-like models based on the electromagnetic thickness hypotheses
(cf. Rudnicki, 1995).

References

1. Gurtin M.E., Murdoch A.I., 1975, A continuum theory of elastic material
surfaces,Arch. Rational Mech. Anal., 57, 1, 291-323

2. Costen R.C., Adamson D., 1965, Three-dimensional derivation of the elec-
trodynamic jumpconditionsandmomentum-energy lawsatamovingboundary,
Proc. IEEE, 53, 9, 1181-1196

3. Rudnicki M., 1995, On 2D approximations for magnetoelastic non-
magnetizable plates, Int. J. Applied Electromagnetics and Mechanics, 6, 2,
131-138

Elektroprzewodząca powierzchnia sprężysta w polu magnetostatycznym

Streszczenie

Przedmiotem rozważań jest teoria liniowa powierzchni materialnej umieszczonej
w próżni i poddanej działaniu silnego zewnętrznego pola magnetostatycznego. Ruch
powierzchni opisuje funkcja położenia. Założono, że materiał powierzchni jest izotro-
powy, sprężysty, niemagnetyzowalny i przewodzący prąd elektryczny. Uwzględniono
naprężenia rezydualne. Otrzymano równania rozwiązujące z użyciem przemieszczeń.

Manuscript received March 16, 2004; accepted for print May 10, 2004