Deformation of an elastic magnetizable square rod ...

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Deformation of an elastic magnetizable square rod due to a uniform electric current inside the rod and an external transverse magnetic field

Mathematics and Mechanics of Solids 1–20 Ó The Author(s) 2015 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1081286515582872 mms.sagepub.com

AR El Dhaba Departement of Mathematics, Faculty of Science, Damanhour University, Damanhour, Egypt; Faculty of Engineering, International Telematic University, Rome, Italy AF Ghaleb Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt L Placidi Faculty of Engineering, International Telematic University, Rome, Italy Received 07 November 2014; accepted 21 November 2014 Abstract We find the deformation and stresses in an infinite rod of an electric conducting material with square normal crosssection, carrying uniform electric current and subjected to an external, initially uniform magnetic field. The complete solution of the uncoupled problem is obtained using a boundary integral method. The results are discussed in detail.

Keywords Boundary integrals, electric current, long rod, magnetic vector potential, Maxwell stress, plane problem, thermo-magneto-elasticity

1. Introduction The rapid development of the different branches of technology and applied physical sciences in the last decade has revealed the importance of the subject matter of electromagnetic interactions in deformable solids [1–9]. There are many books englobing the subject of electromagnetic interactions in deformable media [10–13], others are confined to magneto- and thermo-magneto-elasticity and applications [14–17], General field equations, boundary conditions and jump conditions for the electrodynamics of deformable bodies may be found in [18]. In [19] one can find contributions to magneto-elasticity both in theory and applications, in particular magnetostriction and related subjects. An extensive literature covering a wide range of topics in dynamic cases may be found in [20–30], and in quasi-static cases in [31–35]. Contributions to the continuum theory of electro- and magneto-elasticity, including magnetostatics and continuous media with complex constitutive relations such as elastomers and polymers, including Corresponding author: Departement of Mathematics, Faculty of Science, Damanhour University, Damanhour 22511, Egypt, and Faculty of Engineering, International Telematic University, Corso Vittorio Emanuele II, 39 00186 Rome, Italy. Email: [email protected]

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mathematical and computational aspects of the modeling of these materials are exposed in [36]. Nonlinear constitutive relations for magnetostriction were investigated in [37] and particular problems of magneto-elasticity for the half-space were considered in [38, 39], see also [6, 40] for the relaxed micromorphic case. It is believed that elastic magnetizable square rod could be used as an external instrument for a better analysis the dynamics of cables [41, 42] and in the aerodynamic identification [43, 44]. Damping uncertainties reduction [45] and complex methods for the dynamic energy reduction [46, 47] could be considered by the use of an elastic magnetizable square rod in the presence of an external magnetic field. Thermo-magneto-elasticity for long electrical conducting rods carrying a uniform current was investigated analytically in [35] for the circular boundary and in [33] for the elliptical boundary. An electrical version with electrostriction was treated in [48].The calculation of Lorentz force-induced stresses in long straight conductors carrying a uniform current density was treated in [49], with application to rods of elliptic or narrow rectangular cross-sections having a stress-free lateral surface. The presence of an external magnetic field was also considered [32]. Particular solutions of the problem of the interaction with the external field could be obtained by considering the so-called ‘‘total stresses’’ which allow to reduce the non-homogeneous equilibrium equations to the homogeneous case. In contrast to [49], where attention was concentrated on Lorentz-force induced stresses, the present study relies on a rigorous set of field equations, boundary conditions and other additional relations previously introduced in [31]. All quantities of practical interest, such as the stresses and the displacements, are calculated at the boundary. Quantities in the bulk and the magnetic field in the surrounding space may then be obtained at a later stage. In the same framework, the case of the elliptic contour was treated analytically in [50]. The approach may be easily generalized to the case of transverse isotropy considered in [49] which correspond to two particular cases of practical interest concerning composite conductors. The application of an additional external magnetic field to the body does not require significant changes. The presently used model relies in principle on the use of a free energy of the medium which has a quadratic dependence only on strain, magnetic field and temperature [51, 52], and thus leads to a linear Hooke’s law of thermo-magneto-elasticity involving magnetostriction. The model allows for a dependence of the magnetic permeability on strain. However, under some conditions one may use an uncoupled scheme, in which the magnetic field can be determined independently of the mechanical problem [53]. Details may be found in [33, 35, 48]. Other, more complex, formulations are also possible, which take in consideration a generalization of the celebrated Cauchy postulate concerning the notion of contact force to cases where the element of surface is not plane. It is shown that a generalized argument of the tetrahedron is needed in that case, together with a generalized contact interaction and the introduction of line densities of force on the edges. This in turn necessitates the use of second gradient order theories or more. Details of this may be found in work by dell’Isola and others [54–57]. Here we deal exclusively with a normal metallic conductor which does not belong to the class of the so-called generalized continua, or continua with microstructure, and therefore lie beyond the scope of the present research. A long elastic cylindrical electrical conductor with square normal cross-section carrying a steady axial current and subjected to uniform transverse magnetic field deforms under the combined action of Joule heat, the magnetic forces and an external distribution of pressures on its boundary. The cross-section boundary of the cylinder is subjected to two contact forces: an external given pressure and the Maxwellian force due to the magnetic field distribution in space. It is required to find the distribution of magnetic field in space and the deformed state of the conductor. This difficult mathematical problem will be solved numerically by a boundary integral method, following the guidelines introduced earlier in [31]. The method was applied to two cases: The currentcarrying rod and the non-conducting magnetic rod placed in an external transverse magnetic field. The present work may be considered as the numerical realization of the field equations, boundary conditions and other relations presented in [31] to a case in which the boundary has corner points, see also [53]. This fact requires particular attention, since the proposed method cannot efficiently treat the corners, unless a smoothing of the boundary is undertaken first. For more details for analytical boundary integral method one can return to the references [58, 59].

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Figure 1. Description of the problem.

2. Problem formulation Consider a long elastic, electric conducting rod with square normal cross-section made of a homogeneous, isotropic material. The rod carries a steady, axial electric current of constant density J and, in addition, is subjected to an external transverse magnetic field. The boundary of the rod is subjected to a given distribution of pressure. The parametric representation for the first quadrant of the square in a set of cylindrical polar coordinates (r, u, z) with origin at the center of the square and z-axis along the axis of the rod is (see [60, 61]) ( a, 0  u  p4 ð1Þ xðuÞ = a cot u, p4  u  34 p ( yðuÞ =

a tan u, a,

p 4 p 3p 4 u 4

0u

ð2Þ

The unit vectors along the normal and the tangent to the square contour with the usual orientation are     y_ x_ x_ y_ , , n= , t= ð3Þ v v v v pffiffiffiffiffiffiffiffiffiffiffiffiffiffi respectively, with v = x_ 2 + y_ 2 . The ‘‘dot’’ over a symbol refers to differentiation with respect to the parameter u. Figure 1 shows a normal cross-section of the rod with the direction of the applied magnetic field. Following [60] and [61], a smoothing of the square boundary (1), (2) is carried out. This is a necessary requirement for the proposed method to be efficient. Sections of the square boundary around the corners are replaced with smooth curves expressed as Fourier expansions in terms of trigonometric functions as follows: 8 0  u  p4  d > < a, P2M2+ 2 ð1Þ P +2 p 1 ð1Þ xðuÞ = ð4Þ fm cos mu + 2M m = 1 m = 2M2+ 2 + 1 gm sin mu, 4  du 4p+d > : p 3p a cot u, 4 +du 4  d

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yðuÞ =

8 u, > < aPtan 2M + 2 2

> :

m=1

fmð2Þ cos mu +

P2M + 2

ð2Þ m = 2M2+ 2 + 1 gm

0u p 4

sin mu, p 4

a,

p 4

d

 du

+du

3p 4

p 4

+d

ð5Þ

d

where M is a positive integer depending on the required degree of smoothness and d is half the angle facing the removed part of the boundary at the center of the square. The coefficients fmð jÞ , gmð jÞ , j = 1,.,2, are determined so as to guarantee the continuity of the functions x(u), y(u) and their derivatives up to and including the Mth derivatives. Other methods of smoothing are also possible.

2.1 Equation of heat conduction The temperature T, as measured from a reference temperature T0, satisfies Poisson’s equation and the heat flow vector Q is given by Fourier law for heat conduction. r2 T = 

J2 , Ks

Q =  K rT,

ðx; yÞ 2 B

ð6Þ

with the boundary condition Q  n= 

Bi ðT  Te Þ, K

ðx; yÞ 2 ∂B

ð7Þ

K, s, B and ∂B are the coefficient of heat conduction, the coefficient of electrical conductivity, the domain occupied by the body and the boundary of the domain. By the superposition principle, the general solution of equation (6) is ð8Þ

T = Th + Tp

where Th is the harmonic part of T and Tp is any particular solution of Poisson’s equation (6). It is easily verified that Tp = 

 J2  2 x + y2 4Ks

ð9Þ

A thermal radiation condition (7) is taken at the boundary:   ∂  Bi  Th + Tp =  Th + Tp  Te ∂n K

ð10Þ

where Bi is Biot constant. Also, it is required that Thc ð0Þ = 0

ð11Þ

superscript ‘c’ stands for the harmonic conjugate. The ‘‘thermal displacements’’, which will go into the expressions for the mechanical displacement components, are defined as [31] Z M Z M    c  uT ðM Þ = aT ð1 + nÞ Th dx  Thc dy , vT ðM Þ = aT ð1 + nÞ Th dx + Th dy ð12Þ M0

M0

where n and aT are Poisson’s ratio and the coefficient of linear thermal expansion respectively. Here M0 is an arbitrarily chosen point in the domain of the cross-section, usually taken to coincide with the origin O, M is the general boundary point at which the function is calculated. Both integrations are path independent [31]. Note that the ‘‘thermal displacements’’ do not involve Joule heat, but only the external heat sources.

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2.2 Equations of magnetostatics Following [62, pp. 450–452], one solves for the magnetic vector potential, which is parallel to the axis of the rod (the z-axis), with component denoted by A. In the quasi-static approximation one has r2 A =  m  m 0 J

ð13Þ

Here m0 is the magnetic permeability of the rod and m* is the magnetic permeability of vacuum, with value m* = 4p × 1027H × m21. In the surrounding vacuum labeled ‘‘*’’m0 = 1 and r2 A = 0

ð14Þ

The solutions of equations (5), (14) are looked for in the form   1 A = Ah  m m0 J x2 + y2 , 4

A = A‘ + Ar

ð15Þ

Ah and Ar being the harmonic parts and AN is the expression for the vector potential far away from the axis of the cylinder. The functions Ah and Ar have a regular behavior and tend to zero at infinity, and AN is a known function which satisfies Laplace’s equation but does not vanish at infinity. Function Ar represents the modification occurring in the magnetic vector potential outside the body, due to the presence of the body. The continuity requirements for the magnetic field reduce to the condition of continuity of the vector potential, and the continuity of the normal component of the magnetic field (in the absence of surface electric currents). These imply [63] A = A ,

1 ∂A ∂A = m0 ∂n ∂n

on C

ð16Þ

If an external transverse magnetic field H0 = ðH0 cos aÞi + ðH0 sin aÞj

ð17Þ

is applied to the rod, the corresponding expression for the magnetic vector potential far away from the body is A‘ = 

m J SD r ln + m H0 ðy cos a  x sin aÞ a 2p

ð18Þ

SD is the normal cross-sectional area. The boundary conditions for the magnetic problem, in the absence of surface electric currents, are Ah  Ar = A‘

ð19Þ

1 ∂Ah ∂Ar ∂  = ðA‘ Þ m0 ∂n ∂n ∂n

ð20Þ

The magnetic field may be obtained from the magnetic vector potential through the well-known relation H=

1 m m0

ð21Þ

curl A

The ‘‘magnetic displacements’’, which are part of the expressions for the mechanical displacement components, are given by [31] in the form Z M Z M H H u ðM Þ = ðMH dx + NH dyÞ, v ðM Þ = ðRH dx + SH dyÞ ð22Þ M0

M0

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where    J2  2 2 2 2 x +y MH = W Hy  Hx + , 2    J2  2 2 2 2 x +y , SH = W Hx  Hy + 2



 1 c NH =  2W Hx Hy +  JAh m m0   1 c RH =  2W Hx Hy   JAh m m0

ð23Þ ð24Þ

and   1 m 1 W = ð1 + n Þ m0  m1 2 2 E

ð25Þ

The integrations in (22) are path independent [31].

2.3 Equations of elasticity 2.3.1 Field equations. 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 [33, 35, 48]   nE E aT E 1 sij = I1 dij + eij  T dij + m m0  m1 Hi Hj (1 + n)(1  2n) 1+n 1  2n 2 ð26Þ 1  2  m ðm0 + m2 ÞH dij , 2

where E, n and aT are Young’s modulus, Poisson’s ratio and the coefficient of linear thermal expansion respectively for the considered elastic medium. Also, H2 = HiHi is the squared magnitude of the magnetic field. The strain tensor components eij in terms of the displacement vector components ui is eij =

 1 ri uj + rj ui , 2

i, j = 1, 2

ð27Þ

Solving (26) for the strain components and using (27) one obtains 2E 2E ∂u ∂2 U ∂2 U exx [ = ð1  2nÞr2 U + 2  2 + 2aT ET 1+n 1 + n ∂x ∂y ∂x      1  1 2  m1 + m2 H + m m0  m1 Hy2  Hx2 + ð1  2nÞm 2 2

ð28Þ

2E 2E ∂v ∂2 U ∂2 U eyy [ = ð1  2nÞr2 U + 2  2 + 2aT ET 1+n 1 + n ∂y ∂x ∂y      1  1 2  + ð1  2nÞm m1 + m2 H + m m0  m1 Hx2  Hy2 2 2

ð29Þ

 

E 1 E ∂u ∂v ∂2 U 1 exy [ +  m m0  m1 Hx Hy = 1+n 2 1 + n ∂y ∂x ∂x∂y 2

ð30Þ

and

The compatibility condition is ∂2 exy ∂2 exx ∂2 eyy + = 2 : ∂y2 ∂x2 ∂x∂y

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ð31Þ

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Substituting from equations (28–30) into (31), performing some transformations using the equations of magnetostatics and (6), one finally arrives at the following inhomogeneous biharmonic equation for the stress function U [33]:     aT EJ 2 1  2n  1 m 1 4 2 2 m r U= ð32Þ m + m2 r H +  m0  m1 J 2 ð1  nÞsK 2ð1  nÞ 2 1 2 1n of which the general solution is expressed in terms of two basic harmonic functions F,C as U = xF + yFc + C + Up

ð33Þ

and Up is any particular solution of the equation:   aT E 1  2n  1 r Up =  Tp  m m + m2 H 2 1n 2ð1  nÞ 2 1     m 1 m0  m1 J 2 x2 + y2 + 2 4ð1  nÞ 2

ð34Þ

The constants m1 and m2 appearing in the above formulae express the linear dependence of the magnetic permeability of the rod’s material on strain. The stress tensor components sxx, syy and sxy are defined through the stress function U by the relations sxx = U, yy ,

syy = U, xx ,

sxy =  U, xy

ð35Þ

where ‘‘comma’’ means differentiation with respect to the shown variables. The mechanical displacement vector components are expressed as [69]  E E  T u =  U, x + 4(1  n)F + u + uH 1+n 1+n

ð36Þ

 E E  T v =  U, y + 4(1  n)Fc + v + vH 1+n 1+n

ð37Þ

where uH and vH are defined in (22). 2.3.2 The boundary conditions. Then, at a general boundary point Q the stress vector is taken to satisfy the

condition of continuity ð38Þ

sn = f

from equations (3)1 and (35), equation (38) can be written as     d ∂U d ∂U  = fx , = fy ds ∂y ds ∂x by integration, then ∂U ðsÞ = ∂y

Z

s 0

fx ds0 = X ðsÞ,

∂U ðsÞ =  ∂x

Z

s

fy ds0 =  Y ðsÞ

ð39Þ

0

with ∂U ∂U ð0Þ = ð0Þ = 0 ∂x ∂y

as an additional simplifying condition. Then we have the following.

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ð40Þ

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1.

By using equations (3) and (39) into the normal and tangential derivatives of the function U, give the boundary conditions,   y_   x_ + xF, xy + yFc, xy + C, xy = Fx xF, yy + 2Fc, y + yFc, yy + C, yy ð41Þ v v





xF, xy + yFc, xy + C, xy

 y

  x_  xF, xx + 2F, x + yFc, xx + C, xx = Fy v v

ð42Þ

where Fx and Fy are given by y_ x_ y_  Up, xy  Up, yy v v v

ð43Þ

x_ x_ y_ + Up, xx + Up, xy v v v

ð44Þ

Fx = fxH  Pðx, yÞ

Fy = fyH + Pðx, yÞ

where P(x, y) is the external pressure applied at the rod and fxH and fyH are the force components per unit length due to magnetic field outside the rod in x, y directions, defined through the Maxwellian stress tensor in vacuum as fH = ðs ÞT n

with sij = m

  1 *2   Hi Hj  H dij 2

ð45Þ

Hi being the magnetic field components outside the domain. Thus, fxH =

 y_ x_ m  *2  m Hx Hy Hx  Hy*2 v v 2

ð46Þ

 x_ m  *2 y_ Hx  Hy*2 + 2 v v

ð47Þ

fyH = m Hx Hy

where H =

2.

1 curl A m

ð48Þ

Conditions for eliminating the rigid-body translation at the origin (see [62, p. 458] and [64]) uð0, 0Þ = vð0, 0Þ = 0

these two conditions are written in terms of the harmonic functions as (3  4n)F  C, x  Up, x +

 1  T u + uH = 0 1+n

ð49Þ

ð3  4nÞFc  C, y  Up, y +

 1  T v + vH = 0 1+n

ð50Þ

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Conditions for eliminating the rigid-body rotation at the origin (see [62, p. 459] and [64]) ∂u ∂v ð0, 0Þ  ð0, 0Þ = 0 ∂y ∂x

these two conditions are written in terms of the harmonic functions as 4(1  n)F, y  aT Thc +

4.

1 1 ðNH  RH Þ = 0 21+n

ð51Þ

The additional simplifying conditions expressing the vanishing of the stress function and its first derivatives at a chosen point of the boundary (these are void of any physical meaning [62, p. 459]): xFc  yF + Cc = 0

ð52Þ

xF + yFc + C + Up = 0

ð53Þ

xF, x + F + yFc, x + C, x + Up, x = 0

ð54Þ

xF, y + Fc + yFc, y + C, y + Up, y = 0

ð55Þ

3. Boundary integral representation of the solution 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 as [32]

I 1 0 ∂ c 0 ∂ f (x, y) = f (s ) 0 ln R + f (s ) 0 ln R ds0 ð56Þ 2p C ∂n ∂s where R is the distance between the field point (x, y) in D and the current integration point (x#, y#) on C. The harmonic conjugate of (56) is

I 1 c c 0 ∂ 0 ∂ f (x, y) = f (s ) 0 ln R  f (s ) 0 ln R ds0 ð57Þ 2p C ∂n ∂s When point (x, y) tends to a boundary point, relations (56) and (57) are respectively replaced by

I 1 ∂ ∂ f (s) = f (s0 ) 0 ln R + f c (s0 ) 0 ln R ds0 p C ∂n ∂s

I 1 c c 0 ∂ 0 ∂ f (s ) 0 ln R  f (s ) 0 ln R ds0 f (s) = p C ∂n ∂s

ð58Þ ð59Þ

 Þ, is harmonic in this region and vanishes at infiIf a function g(x, y) is defined in the outer region C ðD 2 2 2(1 + d) with d . 0, it can be shown that the integral representation (58) is nity at least as (x + y ) replaced by

I 1 ∂ 0 ∂ 0 g(x, y) =  g(s ) 0 ln R  0 g(s ) ln R ds0 ð60Þ 2p C ∂n ∂n

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it being understood that the boundary values g(s#) and ∂n∂ 0 g(s0 ) under the integral sign on the right hand side are calculated at a point with parameter s# on the outer side of C. When the point (x, y) tends to a boundary point with parameter s, then (60) is replaced by the integral relation

I 1 ∂ 0 ∂ 0 g(s) =  g(s ) 0 ln R  0 g(s ) ln R ds0 ð61Þ p C ∂n ∂n

I 1 c c 0 ∂ 0 ∂ g (s ) 0 ln R  g(s ) 0 ln R ds0 ð62Þ g (s) =  p C ∂n ∂s

4. Displacement and stress tensor components The displacement vector components in Cartesian coordinates and intrinsic coordinates are u = ui + vj = un n + ut t

by using equations (3), (36) and (37), one can find the tangential and normal components of the displacement vector in terms of the harmonic functions F and C along with Up, as follows

x_ y_ c E x _ y _c 1 _  C ut = ð3  4nÞ F + F  F  F v v 1+n v v v   ð63Þ   y_  T  E E x_ y_ x_ T H H u +u + v +v  Up, x  Up, y + 1+n1+n v v v v

E y _ x _c 1 _c y_ x_ c  C un = ð3  4nÞ F  F + F  F 1+n v v v v v   ð64Þ   E y_ x_ y_  T x_  T H H u +u  v +v  Up, x + Up, y + 1+n v v v v while the normal and tangential stress components are     snn = sxx nx + sxy ny nx + sxy nx + syy ny ny     snt =  sxx nx + sxy ny ny + sxy nx + syy ny nx

ð65Þ ð66Þ

with sxx, syy and sxy given in terms of the stress function U(x, y).

5. The numerical method In order to obtain a boundary scheme for the numerical solution ofH the above system of equations and conditions, one discretizes the interval of integration of the integral C f ds for a general function f so as to replace this integral, as usual, by the finite sum over M nodes denoted Si (i = 1, 2, ., M) properly chosen on the boundary. For our purposes, these nodes will be scattered on the boundary in such a way so as to keep a fixed difference Du between any two consecutive nodes: I M X f ds = fi Dsi , Dsi = vi Dui , i = 1, 2, . . . , M ð67Þ C

i=1

with ui = ði  1ÞDu,

Du =

2p M

Thus, the first node S1 corresponds to the value u = u1 = 0, and the point (a, 0) introduced earlier is taken to coincide with S1.

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Computational experiments have clearly indicated the necessity of accurate calculation of the first and second tangential derivatives of a general function f(u) along the boundary C, for the considered scheme to perform efficiently. For the present purposes, these derivatives at the ith boundary node have been calculated using the values of the function at N adjacent points from each side of this node by Taylor expansions as fi + j =

2N X ðjhÞn n=0

n!

ðnÞ

fi ,

fij =

2N X ðjhÞn n=0

n!

ðnÞ

fi ,

j = 1, 2, . . . , N

ð68Þ

ðk Þ

where fi denotes the kth derivative of the boundary function f(u) calculated at the ith boundary point. Resolving this set of equations at each node, one obtains the first two derivatives of function f along the boundary. Also, the line integrals introduced above and calculated according to (67) may develop errors during the calculations, especially when the path becomes too long. In such cases, it is recommended to use other integration rules with higher accuracy. If needed, the partial derivatives of the function f(x, y) calculated at the ith boundary point in the discretized scheme are replaced by ðfx Þi =

x_ i _ y_ i fi + 2 f_ic , v2i vi

 c x_ i y_ i fx i = 2 f_ic + 2 f_i , vi vi ðf, xx Þi = 



ji € zi €c .i _ r fi + 4 fi  6 fi  6i f_ic v4i vi vi vi

ð69Þ ð70Þ ð71Þ



 j z . r f, yy i = i4 €fi  i4 €fic + i6 f_i + 6i f_ic vi vi vi vi

ð72Þ



 z j r . f, xy i = i4 €fi + i4 €fic  6i f_i + i6 f_ic vi vi vi vi

ð73Þ

f,cxx

 

  y_ i x_ i fy i = 2 f_i  2 f_ic vi vi   xi y_ i fyc = 2 f_i + 2 f_ic i vi vi

 i

f,cyy

f,cxy

=

 i

 i

=

ji €c zi € .i _ c ri _ f  4 fi  6 fi + 6 fi v4i i vi vi vi

ji €c zi € .i _ c ri _ f + 4 fi + 6 fi  6 fi v4i i vi vi vi

=

ji € zi €c ri _ c .i _ fi + 4 fi  6 fi  6 fi v4i vi vi vi

ð74Þ ð75Þ ð76Þ

where ji = y_ 2i  x_ 2i , di = x_ i€xi + y_ i€yi ,

zi = 2_xi y_ i , si = x_ i€yi  €xi y_ i .i = z i s i  j i di , r i = j i s i + z i di

To complete the ingredients of the boundary scheme, let us note that each of the unknown harmonic functions of the problem, as well as its harmonic conjugate, will have a boundary integral representation according to the well-known formulae of the theory of the potential. If f is a harmonic function in D and fc its harmonic conjugate, then at any boundary point (x0, y0) 2 C one may use the following modified relations which are easily obtained from the previous ones:

I 1 ∂ ln R ∂ ln R c c f ðx0 , y0 Þ = f ðx, yÞ + ðf ðx, yÞ  f ðx0 , y0 ÞÞ ds ð77Þ p C ∂n ∂t

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1 f ðx0 , y0 Þ = p c

I C

∂ ln R ∂ ln R f ðx, yÞ  ðf ðx, yÞ  f ðx0 , y0 ÞÞ ds ∂n ∂t c

ð78Þ

use have been made of the Cauchy–Riemann relations. In discretized form, the terms containing the logarithms will produce singularities which are of the removable type. Following [31] one has qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2 Rij = xj  xi + yj  yi ð79Þ 8 c c < fj fi ðxj xi Þx_2j + ðyj yi Þy2_ j ,   ∂ ln R ij vj c c ðxj xi Þ + ðyj yi Þ = fj  fi : 1 _c ∂tj , f vi i 8 1 ðxj xi Þy_ j ðyj yi Þx_ j ∂ ln Rij < vj ðx x Þ2 + ðy y Þ2 fj , j i j i fj = : x_ i€yi €xi y_ i f , ∂nj i 2v3

i 6¼ j

ð80Þ

i=j

i 6¼ j

ð81Þ

i=j

i

More details for numerical solution can be found in [60, 61, 65–69].

6. Numerical results and discussion 6.1 Dimensionless analysis Dimensionless quantities introduced for convenience: aT Th = T h ,

aT Thc = T ch ,

UT =

uT , að1 + nÞ

VT =

vT að1 + nÞ

Ah Ach Ar A*c c  *c r , A , A , A = = = h r r m a 2 J m a 2 J m a2 J m a2 J uH vH us un , VH = , Us = , Un = UH = að1 + nÞ að1 + nÞ að1 + nÞ að1 + nÞ Hy Hy Hx Hx aBi Hx = , Hy = , Hx = , Hy = , B= K aJ aJ aJ aJ aH0 aT a2 J 2 4m Ks , b= , g= k= aT E J 4Ks Ah =

The following particular values for the geometrical and material parameters of the problem, and for the applied pressures, were used for the calculations: M = 200, B = 1:5,

Du =

2p , M

m1 = m2 = 0:5,

d = 4Du, a = 1,

K = 4, m0 = 1,

n = 0:25 P1 P2 = =0 E E

Accordingly, the boundary parts cut at the corners for smoothing purposes intercept at the center of the square an angle of 4° approximately. The ambient temperature is taken as aT Te = aT T0 + aT T2 cos 2u

which means that the cylinder is cooled by the ambient temperature.

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6.2 Short review of the pervious work Let us take three different values of the thermal parameter b as b = 0:01, 0:02, 0:03,

g = 0:1,

aT T0 =  0:1,

ð82Þ

aT T2 = 0

Applying the numerical scheme proposed in the above section to equations (58) and (59) to the functions T h and T ch , and equations (10) and (11), one obtains a rectangular system of equations in matricial form A1 X1 = B1 ð83Þ  T where A1 is the matrix of coefficients, X1 = T h , T ch is the vector of unknown functions at the boundary nodes and B1 the vector formed by the right-hand sides of equations (58), (59), (10) and (11). Solving the system of equations (83) yields the values of the temperature at the boundary nodes. The next step is to apply the numerical scheme proposed in the above section to equations (58) and (59) to the functions Ah and Ach , equations (61) and (62) to the functions Ar and A*c r and equations (19) and (20) to the two boundary conditions. One obtains a rectangular system of equations in matricial form A2 X2 = B2 ð84Þ  T where X2 = Ah , Ach , Ar , A*c is the vector of unknown functions at the boundary nodes and B2 -the vecr tor formed by the right-hand sides of equations (58), (59), (61) and (62), (19) and (20). Solving the system of equations (84) yields the values of the magnetic  T vector potential at the boundary nodes. Once the boundary values of the functions Ah , Ach , Ar , A*c have been obtained, one can find all of the physical r quantities related to the magnetic potential. The numerical method is next applied to the four functions F, Fc, C, and Cc (equations (58), (59), (61) and (62)), the two boundary conditions (equations (41), (42)) and the seven boundary conditions (equations (49), (50), (51), (52), (53), (54) and (55)). The resulting system of linear algebraic equations is A3 X3 = B3

ð85Þ

where A3 is the coefficient matrix, X3 = ½F, Fc , C, Cc T is the vector of unknown functions at the boundary and B3 -the vector of the right-hand side. Solving the system of equations (85) to obtain the boundary values of the harmonic functions [F, Fc, C, Cc]rmT. aH 

! 0: in J numerical calculation of equations (84) and (85), the results as in [60]. Figure 2 shows deformation occurring due the combination effects of heat Joule and ambient temperature where the thermal parameter is defined by 6.2.1 Case I: Deformed cross-section due to electric current. In this Case we choose the parameter k

b=

aa2 J 2 : 4Ks

aH 

! ‘, this means J that we have only a transverse magnetic field as in [69]. Figure 3 shows the deformation occurring in the cross-section of the rod due to the effect of transverse magnetic field, where the magnetic parameter is defined by

6.2.2 Case II: Deformed cross-section due to external magnetic field. In this case the parameter k

x=

m H02 E

where H0 is the intensity of the external magnetic field.

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Mathematics and Mechanics of Solids

Figure 2. Deformation of the square rod due to the heat Joule and ambient temperature.

Figure 3. Deformation of the square rod due to the transverse magnetic field.

6.3 Analysis of the coupling between electric current and external magnetic field

 Here, we study the effect of the parameter k = contents

aH0 J



for the physical quantities with the following

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Figure 4. Boundary values of the magnetic potential on the boundary for three values of parameter k.

s*

Figure 5. Boundary values of Maxwell’s stress tensor component a2xxE for three values of the parameter k.

b = g = 0:1,

a=

p 4

aT T0 = aT T2 = 0

ð86Þ

Let us take three different values of the parameter k as k = 0:1, 0:2, 0:3

Figures 4, 5, 6 and 7 show the distributions of the magnetic vector potential and the components  of the  Maxwellian stress tensor on the boundary of the square for different values of the parameter k = aHJ 0 . There is no role here to play for the external temperature. These distributions can thus be controlled   by the parameter k. Now, we want to study the effect of different values of the parameter k = aHJ 0 with constants in equation (86), in the displacement of a square rod. Let us take three different values for k as follows k = 0:01, 0:1, 1

  Figures 8, 9 and 10 show the effect of three different values of the parameter k = aHJ 0 ; one can note that by increasing the value of this parameter the deformation occurring in the square rod goes to nonlinear deformation due the term HiHj in equations (26), (32) and (34).

7. Conclusions A boundary integral method in numerical form has been used to investigate the difficult problem of deformation of a long elastic cylindrical rod of square normal cross-section carrying a steady, axial

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Mathematics and Mechanics of Solids

s*

Figure 6. Boundary values of Maxwell’s stress tensor component a2yyE for three values of the parameter k.

s*

Figure 7. Boundary values of Maxwell’s stress tensor component a2xyE for three values of the parameter k.

Figure 8. Total deformation for the square rod due to k = 0.01.

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Figure 9. Total deformation for the square rod due to k = 0.1.

Figure 10. Total deformation for the square rod due to k = 1.

electric current, under: (i) an ambient temperature acting on the boundary; (ii) Joule heat; (iii) an magnetic field to the axis of the cylinder, taken for concreteness to be inclined at 45° to the sides of the square. The scheme allows for additional mechanical forces to be applied to the boundary. All of the unknowns of the problem have been successfully calculated on the boundary of the square through a set of eight basic harmonic functions (two for temperature, three for magnetic potential and three for elasticity), and the effect of the vertices have been put in evidence. Calculations inside the cross-sectional domain may now be carried out easily, either through the well-known integral representation of harmonic functions, or else by expanding the harmonic functions in a proper basis and then calculate the

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coefficients by the boundary collocation method or by any other method. A detailed analysis has been carried out for the different factors contributing to the deformation of the cylinder. Figures have been produced for the boundary distributions of the different functions of practical interest, showing the effect of some physical and geometrical parameters on these distributions. Funding The first author is grateful to the Erasmus Mandus Program 2015 for helping the author in finding a collaboration between Damanhour University in Egypt and International Telematic University in Roma.

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