A Boundary Integral Formulation of the Plane Problem of Magneto-Elasticity for an Infinite Cylinder in a Transverse Magnetic Field

The objective of this work is to present a boundary integral formulation for the static, linear plane strain problem of uncoupled magneto-elasticity for an infinite magnetizable cylinder in a transverse magnetic field. This formulation allows to obtain analytical solutions in closed form for problems with relatively simple geometries, in addition to being particularly well-adapted to numerical approaches for more complicated cases. As an application, the first fundamental problem of Elasticity for the circular cylinder is investigated.


Introduction
An early version of the present boundary integral formulation was suggested by one of the authors (M.S. Abou-Dina) for the study of certain problems in the electrodynamics of current sheets [1].It was later on applied for the solution of a general problem of nonlinear gravity wave propagation in water [2].Due to its efficiency, the method was used by the authors of the present work to study the static, plane strain problem of the linear Theory of Elasticity in stresses for bounded, simply connected regions [3].The thermoelastic problem was later on treated along the same guidelines [4].Recently, the authors presented a boundary integral formulation for the static, linear plane strain problem of uncoupled Thermomagnetoelasticity for an infinite cylinder carrying a uniform, axial electric current [5].A new representation of the mechanical displacement vector allowed to obtain the complete solution of the problem.
The proposed method relies exclusively on the use of boundary integral representations of harmonic functions and is suitable for both the analytical and the numerical treatments of the problem.The numerical aspect of the proposed formulation was carried out by the authors for pure Elasticity [6].An implementation of the method for boundaries with mixed geometries was investigated in [7].
In the present paper, the formulation presented in [5] is modified and adapted to fit the case of an infinite cylinder of a magnetizable material, subject to an external, transverse uniform magnetic field.The first and the second fundamental problems of Elasticity are treated.An application is given for the first fundamental problem only for a circular region.This application is meant to stress the capability of the method to handle cases where analytical solutions are possible and to provide these solutions explicitely.The second fundamental problem may be treated in a similar way.

Problem Formulation and Basic Equations
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   x s and   y s are assumed continuously differentiable twice on C.
Here,   , , x y z denote orthogonal Cartesian coordinates in space with origin O in D and unit vectors respectively.Let , , i j k s 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 where the dot over a symbol denotes differentiation w.r.t.s .Also, .  k n τ (2b) The unknown functions of the problem are assumed to depend solely on the two coordinates   , x y .

Equations of Magnetoelasticity
The general equations of static, linear Magnetoelasticity may be found in [5].In what follows, we shall quote these equations for non-conducting media, to be used throughout the text.The condition for the external magnetic field is incorporated appropriately.

Equations of Magnetostatics
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.
, , 1, 2,3, where the indices 1, 2 and 3 refer to the , x y andcoordinates respectively and a repeated index denotes summation.Here, ij z  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 0 1 ,   and 2  are constants with obvious physical meaning, 1 I is the first invariant of the strain tensor with components ij  and ij  denote the Kronecker delta symbols.Constant   refers to the magnetic permeability of vacuum with value 7 1 10 H m .
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, 0 .
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 A 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  ,  x y -plane and is independent in magnitude of the third coordinate .A vector potential producing such a field must be of the form (7) 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 0, 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 0 1 and one uses the following decomposition: .
Function r A  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     x y Function A  accounts for the unperturbed, original constant magnetic field.If the intensity of this initial field is 0 H and its direction is inclined at an angle  to the x -axis, then The separation of the expression for A  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 2 0 1 , say on .
These conditions, together with the vanishing condition at infinity of r A  , are sufficient for the complete determination of the two harmonic functions A and r A  .The magnetic field components are expressed as where A c 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, r A  , subject to the boundary conditions ( 14) and (15).

Equations of Elasticity
1) Equations of equilibrium.
In the absence of body forces of non-electromagnetic origin, the equations of mechanical equilibrium in the plane read 0, , 1, 2, where ij  are the components of the "total" stress ten- sor and j  denotes covariant differentiation.It is worth noting here that the total stress tensor is sometimes decomposed into two parts: mechanical and electromagnetic [10], in which case the boundary conditions may take different forms. It 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 , E  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 ij  and the displacement vector components where and stand for and respectively.u v The condition of solvability of Equation (21b) for and for given R.H.S. is u v 2 .
These equations are complemented with the proper boundary conditions, to be discussed in detail in subsequent sections.

Equation for the stress function
An equation for the stress function may be obtained from the general field equations written in covariant form [11].For the present purposes, however, we prefer to derive this equation for the special, two-dimensional problem under consideration.Solving (20) for the strain components and using (18), one obtains 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 C D C D Ι , D denotes the closure of D and superscript "c" denotes the harmonic conjugate.Function p U is any particular solution of the equation and may be expressed in the form of Newton's potential after the function 2 H 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 where and A 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 D .
Introducing two new functions it can be easily verified using the equations of Magnetostatics that . .
with these notations, equations (28a,b) take the form It is to be noted that the addition of a constant to the function c A amounts to adding linear terms in y to H u and linear terms in x to H v , which do not alter (33a,b).

A representation for the mechanical displacement vector components
Differentiating (33a) w.r.t.y and integrating the resulting equation w.r.t.x after using (31a) and (32a), one gets where   f y is an arbitrary function of y.A similar procedure with (33b), using (31b) and (32b), yields where   g x 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 c A .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 D .
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 0 M (which may be arbitrarily chosen in D ) to a general field point M, one obtains and where the integration constants being absorbed into functions  and c  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).

Boundary Integral Representation of the Solution
The problem now reduces to the determination of seven harmonic functions: and , , , , , 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).
or, in the equivalent form where is the distance between the field point R   , x y in and the current integration point D where The representation of the conjugate function is given by or, in the equivalent form when point  , x y tends to a boundary point, relations (36a) and (36b) are respectively replaced by

Solution for the Magnetic Vector Potential
As noted above, each of the two functions c A and c r A  by is defined up to an arbitrary constant, to fixed assigning a given value to the function at an arbitrarily chosen point in its domain of definition.
In order to obtain the boundary values of the two ha where Equation ( 40 the solution of which allows to determine rywhere in space, while Eq derivatives, may be determined. Finally, Equations (36b) d (39a) yield the values of the magnetic vector potential eve uation (36c) gives the harmonic conjugate c A in the body.

Solution for the Stress and Displacement
Components ield ever and in the region D occupied by the Having obtained the solution for the magnetic f ywhere in space material, we now turn to solve the mechanical problem for the stress and the displacement components in D .The stresses are given through the stress function U from relations (20a,b,c), and these may be rewrit using expression (25) in terms of the harmonic functio , , from which, using (21), one obtains Thus, once the magnetic field has been uniquely m deterined, the derivative The mechanical displacement components are given (35a,b), which may rewritten using (25) in te ued function.from relations rms of the harmonic functions , c   and  as In view of the integral representations (37a,b) and expressions (43) and (45), it is sufficient for th of the mechanical problem to determine the values of the harmonic functions and e solution boundary , , un This requires four independent these kn relations in owns, 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 elim e poss e rigid body motion.Following [5], we formulate the conditions for the two following fundamental problems: The first fu amenta problem, where the stresses are specified on the boundary, and the second fundamental problem, where the displacements are specified on the boundary.

Conditions for Eliminating the Rigid Body Trans
inate th ibl nd l lation In terms of the boundary values of the unkn monic functions, condition (46) becomes own har-

Conditions for Eliminating the Rigid B Rotation
This condition, like the first two, is applied only for the first fundamental problem.We shall require that which may be written in terms of the boundary v 4 1 0,0 0,0 alues of the unknown harmonic functions as

Additional Simplifying Conditions
 of the boundary, in order to determine th f the arbitrary integration constants appearing throughout the solution process.These additional conditions have no phy implications on the solution of the problem.For de concerning these additional conditions, the reader is kindly referred to [3].
first order We shall require the following supplementary conditions to be satisfied at the point 1) The vanishing of the function U and its partial derivatives at which, in terms of the boundary values of the unknown harmonic functions, give and 2) The vanishing of the combination is last additional condition amounts to determining the value of Th c  at and is chosen for the uniformity of presentation as in [5 Let us finally turn to the boundary condit to the equations of Elasticity.For this, we consider separately two fundamental boundary-value problems.

4.
In this problem, we are given the force distribu boundary C of the domain D. Let The force f is divided into two parts: , where ex f is the force of non electromagnetic origin and H f is the force due to the action of the magnetic field, per unit length of the boundary.The second force may b pressed in terms of the Maxwellian stress e ex tensor  σ as   and yy  in terms of the stress function U and for x n and y n and taking con-ditions (51) into account, the last two relations yield Using expressions ne may easily obtain the tangential and norm l derivative of the ress function U at the boundary point Q .
or, in terms of the unknown harmonic functions Equations ( 58), together with relation (38) written for

The Second Fundamental Problem
In this problem, we are given the displacement vector on the boundary C the do ain D .Let this vector be denoted x s s y s s e m ssion by Similarly, if on ultiplies the restriction of expre-(45a) to the boundary These last two relations may be conveniently rewritten as x s s y s s x s s y s s where  

Practical Use of the Method
In practice, if the form of the boundary is simple enough (e.g.th le or th e), one may attempt to find alytical forms for the solution as shown below in the application.However, for more complicated boundaries, one has to recur to numerical approaches.In this case, the differential and integral operators appearing in the equations are to be discretized as usual and the problem of determination of the boundary values of the unknown functions reduces to finding the solution of a linear ll solution inside system of algebraic equations.The fu D is then obtained by numerical integration of boundary integrals of the type (36) [cf. 6,7].
In a later stage, if it is required to determine boundary values of some unknown functions (for example, the boundary displacement for the first fundamental problem or the boundary stresses for the second fundamental problem), this may be achieved at once if the solution is obtained analytically as in the worked examples.Otherwise, if a numerical approach is adopted, the calculation may proceed by calculating the first and the second derivatives w.r.t.x and y 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).

The Circular Cylinder
As an illustration of the proposed scheme, we present here below the solution of a problem which can be handled analytically, namely the infinite, non-conducting, circular elastic cylinder pla ed in a transverse constant external magnetic field.
Let the normal cross-section of the cylinder be bounded by a circle of radius a centered at the origin of coordinates, with parametric equations Let a circular cylinder of a weak electric conducting, magnetizable material be placed in an external, transversal constant magnetic field 0 H , which we take along the x -axis.

Solution for the Equations of Magnetostatics
The solution for the magnetic vector potential component ned foll ceding section in the form: is obtai owing steps similar to those of the pre- where 0 H is the intensity of the applied magnetic field.

*
The linear part in r in the expression for A is just the function A  in the general formulation of the problem.

Choosing c
A to vanish at the origin, one gets The corresponding magnetic field components are or, related to the system of polar coordinates The boundary values of the magnetic field outside the body are used to calculate the Maxwellian st components for the formulation of the boundary condi elasticity.One obtains ress tensor )

The Elastic Solution
Turning now to the determination of the stresses and displacements, one has where we have introduced the dimensionless parameter The stress function inside the domain is then The suppression of the rigid body rotation is idensatisfied tically .There remains now the boundary conditions to be satisfied, which may be simply written as the conditions of continuity of the two stress components rr The first of the elastic boundary conditions then gives Finally, the mechanical displacement components are It is worth noting here that the material constants 1

c
We use the well-known integral representation of a harmonic function f at a general field point   , x y inside the region D in terms of the boundary values of the function and its harmonic conjugate (after integrating by parts and rearranging) as be rmonic functions A and r A  , write down equation (38b) ) is the canonical form of the well-known linear Fredholm integral equation of the second kind for the determination of the boundary values of   A s .Having solved this integral equation, the boundary of values   r A s  may then be obtained from (25a).Also , using Copyright © 2013 SciRes.ENG M. S. ABOU-DINA, A. F. GHALEB 400 (40) and its solution, Equation (38b) written for   A s is r to the following Fredholm integral equation of educed n: the first kind for the normal derivative of this functio boundary values of A and r A  , as well as o al f their norm an force per unit length of the boundary.Then, at a general boundary point Q the stress vector is taken to satisfy the condition of continuity

.
), together with (59a,b) form the required set of simultaneous integro-differential equations for the determination of the boundar the unknown harmonic functions Φ and Ψ and harmonic conjugates.The full solution of the problem proceeds as for the first fundamental problem.
angle in the associated polar system of coordinates   , r  .
68) g yieldThe four simplifyin conditions taken at the point Of the two conditions expressing the suppression of the rigid body translation, one is identically satisfied, while the other gives

,
and r  related to the system of polar coordinates  

 and 2 
appear only in the expression for the ra ent.A measurement of this displaceme surface of the cylinder provides the numerical value