Let D be a two-dimensional, bounded, simply connected region representing a normal cross-section of the infinite cylinder occupied by the elastic medium and let its boundary C have the parametric representation
Functions 
and 
are assumed continuously differentiable twice on C.
Here, 
denote orthogonal Cartesian coordinates in space with origin O in D and unit vectors 
respectively. Let 
be the arc length as measured on C in the positive sense associated with
, from a fixed point 
to a general boundary point Q and 
is the unit vector tangent to C at Q in the sense of increase of
.
One has
where the dot over a symbol denotes differentiation w.r.t.
. Also,
The unknown functions of the problem are assumed to depend solely on the two coordinates
.
The general equations of static, linear Magnetoelasticity may be found in [
1) The field equations.
Inside the body and in the absence of volume electric charges, the field equations of Magnetostatics in nonconducting media, written in the SI system of units, are:
where H is the magnetic field vector, B—the magnetic induction vector, E—the electric field vector and D—the electric displacement vector.
The magnetic field arises from an external source, in the form of an initially uniform magnetic field.
The equations of Magnetostatics are complemented by:
2) The electric constitutive relation.
where 
is electric permitivity of the body, assumed constant, and 
is the electric permittivity of vacuum, with value
3) The magnetic constitutive relations.
where the indices 1, 2 and 3 refer to the 
and 
- coordinates respectively and a repeated index denotes summation. Here, 
are the components of the tensor of the relative magnetic permeability of the body, assumed to depend linearly on strain according to the law
where 
and 
are constants with obvious physical meaning, 
is the first invariant of the strain tensor with components 
and 
denote the Kronecker delta symbols. Constant 
refers to the magnetic permeability of vacuum with value
Expression (4c) may be deduced from general constitutive assumptions, but this will be omitted here. An electrical analogue for the dielectric tensor components under isothermal conditions may be found elsewhere [8, p. 64 and also 9] .
We shall assume a quadratic dependence of strain on the magnetic field (magnetostriction). Upon substitution of (4c) into (4b) one may neglect, as an approximation, the third and higher degree terms in the magnetic field compared to the linear term. Therefore,
The magnetic vector potential.
In view of the geometry of the problem, the magnetic vector potential has a single non vanishing component along the 
axis:
In view of the property (3b) of the magnetic induction and taking (5) into account, the magnetic field vector may be represented in the form
where 
is the magnetic vector potential. It is usual, for the sake of uniqueness of the solution, to impose the condition
Since we are interested solely in plane problems, the magnetic field lies in the 
-plane and is independent in magnitude of the third coordinate
. A vector potential producing such a field must be of the form
This choice identically satisfies condition (6b), which means that function A still has some indeterminacy. In fact, it is defined up to an arbitrary additive constant.
Equation (3a) reduces to
from which
at each point of the region D.
In the present quasistatic formulation, in view of the fact that the electric and magnetic fields are uncoupled, there are no sources for the electric field. Therefore,
where 
refers to free space surrounding the body.
In the free space, the equations of Magnetostatics hold with 
and
. Hence,
and one uses the following decomposition:
Function 
represents the modification of the magnetic vector potential in free space, due to the presence of the body. This function has a regular behavior at infinity. It is sufficient for the present purpose that this function vanish at infinity at least as 
with
.
Function 
accounts for the unperturbed, original constant magnetic field. If the intensity of this initial field is 
and its direction is inclined at an angle 
to the 
-axis, then
The separation of the expression for 
into two parts as in (12) is of capital importance for the numerical treatment of the problem.
The equations of Magnetostatics are complemented by the following magnetic boundary conditions:
a) The continuity of the normal component of the magnetic induction. This reduces to the condition of continuity of the vector potential, i.e.
b) The continuity of the tangential component of the magnetic field (in the absence of surface electric currents). This implies
These conditions, together with the vanishing condition at infinity of
, are sufficient for the complete determination of the two harmonic functions A and
.
The magnetic field components are expressed as
where Ac denote the harmonic conjugate to A. It follows from (16) that the function 
plays the role of a scalar magnetic potential. Thus, one may invariably proceed with the problem formulation using either the magnetic scalar or the magnetic vector potential. We shall use the latter.
The solution of the electromagnetic problem thus reduces to the determination of two harmonic functions A, 
, subject to the boundary conditions (14) and (15).
1) Equations of equilibrium.
In the absence of body forces of non-electromagnetic origin, the equations of mechanical equilibrium in the plane read
where 
are the components of the “total” stress tensor and 
denotes covariant differentiation. It is worth noting here that the total stress tensor is sometimes decomposed into two parts: mechanical and electromagnetic [
It is well-known that Equation (17) is satisfied if the only identically non-vanishing stress components
, 
and 
are defined through the stress function U by the relations
2) The constitutive relations.
The generalized Hooke’s law may be derived consistently for an appropriate form of the free energy of the medium, using the general principles of Continuum Mechanics. It reads [8, see also 9 for the electric analogue]
where 
is the squared magnitude of the magnetic field. In components, Equation (19) gives
where
are Young’s modulus and Poisson’s ratio respectively for the considered elastic medium.
3) The kinematical relations.
These are the relations between the strain tensor components 
and the displacement vector components
.
or, in Cartesian components
where 
and 
stand for 
and 
respectively.
4) The compatibility condition.
The condition of solvability of Equation (21b) for 
and 
for given R.H.S. is
These equations are complemented with the proper boundary conditions, to be discussed in detail in subsequent sections.
An equation for the stress function may be obtained from the general field equations written in covariant form [
and
Substituting from (23) into (22) and performing some transformations using the equations of Magnetostatics and (3), one finally arrives at the following inhomogeneous biharmonic equation for the stress function
:
The solution of (24) is sought in the form
where 
and 
are harmonic functions belonging to the class of functions
, 
denotes the closure of D and superscript “c” denotes the harmonic conjugate. Function 
is any particular solution of the equation
and may be expressed in the form of Newton’s potential after the function 
on the R.H.S. has been determined.
It follows from (25) that
Using Equations (25) and (27), Equations (23a,b) may be cast in the form
and
where
and
Function 
is defined up to an additive arbitrary constant, which may be determined by fixing the value of the function at an arbitrarily chosen point of
.
Introducing two new functions
and
it can be easily verified using the equations of Magnetostatics that
and
Equations (31), imply the existence of two singlevalued functions 
and 
in D such that
and
with these notations, equations (28a,b) take the form
and
It is to be noted that the addition of a constant to the function 
amounts to adding linear terms in y to 
and linear terms in x to
, which do not alter (33a,b).
Differentiating (33a) w.r.t. y and integrating the resulting equation w.r.t. x after using (31a) and (32a), one gets
where 
is an arbitrary function of y.
A similar procedure with (33b), using (31b) and (32b), yields
where 
is an arbitrary function of x.
Substituting from (34a,b) into (23c), we find that this equation is identically satisfied if and only if
from which it follows that both functions are constants and therefore may be eliminated since their contribution represents a rigid body displacement. A similar argument holds for any constant added to the expression for
. For the following procedure, it will be assumed that each one of these two functions has been completely determined by assigning to it a given value at some arbitrarily chosen point in
.
From (33a) and (34a) once, then from (33b) and (34b), by line integrations along any path inside the region D joining an arbitrary chosen fixed point 
(which may be arbitrarily chosen in
) to a general field point M, one obtains
and
where
and
the integration constants being absorbed into functions 
and 
which are yet to be determined.
The mechanical displacement components u and v given by expressions (35a,b) are single-valued functions in D, since the line integrals in (35c,d) are path independent due to relations (32a,b).
and 
(although the conjugate function 
does not appear in the expressions given above for the stress and displacement functions, it will be required for the subsequent analysis within the proposed boundary integral method).
inside the region D in terms of the boundary values of the function and its harmonic conjugate (after integrating by parts and rearranging) as
is the distance between the field point 
in 
and the current integration point 
on
.
tends to a boundary point, relations (36a) and (36b) are respectively replaced by
is defined in the outer region
, is harmonic in this region and vanishes at infinity at least as 
with
, it can be shown that the integral representation (36b) is replaced by
and
under the integral sign on the R.H.S. are calculated at a point with parameter 
on the outer side of
.
tends to a boundary point with parameter s, then (39a) is replaced by the integral relation
and 
is defined up to an arbitrary constant, to be fixed by assigning a given value to the function at an arbitrarily chosen point in its domain of definition.
, write down equation (38b)
and equation (39b) for 
then use the boundary conditions (14) and (15) to finally get the following integral equation for
:
. Having solved this integral equation, the boundary values of 
may then be obtained from (25a). Also, using
is reduced to the following Fredholm integral equation of the first kind for the normal derivative of this function:
on the boundary using the boundary condition (15). Thus, the boundary values of A and
, as well as of their normal derivatives, may be determined.
in the body.
. The stresses are given through the stress function 
from relations (20a,b,c), and these may be rewritten using expression (25) in terms of the harmonic functions 
and the particular solution 
in the form
must be a uni-valued function.
and 
as
and
. This requires four independent relations in these unknowns, two of which are obtained from relation (38a) written for 
and 
and the remaining two from the boundary conditions. As a matter of fact, other conditions will still be required to eliminate the possible rigid body motion. Following [
of the boundary, in order to determine the totality of the arbitrary integration constants appearing throughout the solution process. These additional conditions have no physical implications on the solution of the problem. For details concerning these additional conditions, the reader is kindly referred to [
at 
and is chosen for the uniformity of presentation as in [
is divided into two parts:
is the force of non electromagnetic origin and 
is the force due to the action of the magnetic field, per unit length of the boundary. The second force may be expressed in terms of the Maxwellian stress tensor 
as
and 
in terms of the stress function 
and for 
and 
and taking conditions (51) into account, the last two relations yield
at the boundary point
.
and
:
and 
and their harmonic conjugates. The full determination of these functions inside the domain D (and hence of the biharmonic part of the stress function U) is then achieved by substitution into the Equation (37) written for 
and
. The stress function U is finally obtained by adding up the particular integral
.
of the domain
. Let this vector be denoted
by 
and that of expression (45b) by 
and adding, one gets
by 
and that of expression (45b) by 
and subtracting, one obtains
and 
are the tangential and normal components respectively, calculated at boundary points, of the vector, the Cartesian components of which are 
and 
given by equations (35c,d).
is then obtained by numerical integration of boundary integrals of the type (36) [cf. 6,7].
and 
of the required functions on the boundary in terms of derivatives taken along the boundary and then substituting these into the proper expressions (for example, expressions (43) for the stresses and (45) for the displacements).
centered at the origin of coordinates, with parametric equations
is the polar angle in the associated polar system of coordinates
.
, which we take along the 
-axis.
is the intensity of the applied magnetic field.
in the general formulation of the problem.
to vanish at the origin, one gets
and 
and of their harmonic conjugates to the boundary are expressed as general Fourier expansions in the polar angle 
of the system of polar coordinates
. For the case under consideration:
inside the domain 
is then
yield
and 
related to the system of polar coordinates
:
and 
may be calculated from 
in the polar system of coordinates as follows:
and 
appear only in the expression for the radial displacement. A measurement of this displacement at the surface of the cylinder provides the numerical value of the combination
. The solution for the elliptical boundary could provide two different relations for the determination of both 
and
.