Plane waves in electrically conducting and magnetizable viscoelastic isotropic solids subjected to a uniform magnetic field

hr. I. Engng Sci. Vol. 17. pp. 193-214 I@ Pergamon Press Ltd.. 1979. Printed in Great Brilain

PLANE WAVES IN ELECTRICALLY CONDUCTING AND MAGNETIZABLE VISCOELASTIC ISOTROPIC SOLIDS SUBJECTED TO A UNIFORM MAGNETIC FIELDt YASAR ERSOYS Department of Engineering Sciences, Middle East Technical University, Ankara, Turkey Abstract-The propagation of plane electromagneto-mechanical (EM M) waves in homogeneous, initially unstressed and isotropic, electrically conductive magnetoviscoelastic solids in a uniform primary magnetic field is investigated. Depending upon the direction of the applied magnetic field, several modes of the waves arise such as coupled mechanical (M) and electromagnetic (EM) waves. The phase velocities and the attenuations of the waves are obtained both analytically and numerically. Some interesting behavior of the phase velocities and the attenuations are, in particular, detected for certain frequencies, intensity and direction of the primary magnetic field, viscoelastic parameters and conductivity. For example, there are anomalous dispersions of the coupled modes of EM- and M-waves depending upon the intensity of the primary magnetic field, viscoelastic parameters and the conductivity of the material.

I. INTRODUCTION THE PROPAGATIONof

electromagneto-thermo-mecanical (EMT M) waves in solids either in externally primary electromagnetic fields or not have been the subject of many theoretical and experimental investigations during the last two decades. These investigations have been fastly developed in recent years because of the possibilities of their extensive practical applications in diverse fields such as optics, acoustics, geophysics and so on. The purpose of the present article is to describe the propagation of plane electromagneto-mechanical (EM M) waves in a homogeneous, initially unstressed and isotropic, electrically conductive magnetizable viscoelastic solid subjected to a uniform primary magnetic field in an arbitrary direction, both analytically and numerically. The governing equations of this investigation are based on the theory and linearization procedures presented in [ 11. There are many research articles dealing with the propagation of electromagnetic (EM) and EMT M-waves in rigid or deformable media. In certain studies, the propagation of EM- or mechanical (M) waves in a deformable continua without an external electrical or magnetic field is considered, and in the others the effects of the primary electric or especially magnetic field on the propagation of EM M-waves are taken into account. To mention a few, Mckenzie[2] and Potekin[3] present the propagation of EM-waves in moving isotropic and stationary anisotropic rigid materials, respectively. Smith and Rivlin[4], and Ersoy and Kiral[Sl discuss the propagation of plane EM-waves in solids being anisotropic due to the special deformations. In view of external electromagnetic fields, early attempts to describe the effects of the earth magnetic field on elastic waves in the conducting core of the earth are investigated by Knopoff [6] and Chadwich[7]. Dunkin and Eringen[8] discuss the coupling of plane EM- and elastic waves in a moving, initially isotropic medium and they cite a large number of previous researches. While initiating the study of plane EMT M-waves the interactions between electromagnetic and thermal fields are investigated in [9-l 11. Later, Wilson[l2] and Parushothoma [ 131reinvestigate the problem of plane waves in the presence of uniform thermal and magnetic fields in different orientations. More recently, the propagations of EM M-waves in tThis paper is based on a portion of the Ph.D. Thesis by the author referred to in [I], and the summary of this paper was accepted as a contributed paper to the 6th International Conference on Internal Friction and Ultrasonic Attenuation in Solids (ICIFUAS-6). 4-7 July 1977.Tokyo, Japan. *Present Address: Department of Mathematics, Eindhoven University of Technology, Eindhoven, The Netherlands. 193

I!94

YASAR ERSOY

isotropic and anisotropic elastic dielectric crystals in a uniform magnetic field are presented in [14,151. Further, the propagation of plane EM M-waves in soft ferromagnetic isotropic solids is numerically presented by Hutter[l6,17] considering the Amperian formulation of the Maxwell equations in terms of the potentials.? The above articles neglect spin wave effects, magnetic saturation and the anisotropy due to the deformation as does the present investigation. On the other hand, none of these previous articles contain the propagation of EM M-waves in mechanically dissipative media under an externally primary electric or magnetic field. Furthermore, all the aforementioned authors neglect a true dynamic behavior or use special forms of the body force and/or constitutive equations of elastic material, and do not present the results numerically with the exception of the works of Hutter[l6,17]. Together with electrical conductivity, however, internal friction of the material also effects the propagation of waves in a rather significant manner. Thus, the difference between present investigation and the others can be emphasized as the following: (i) The equations presented here are the ones in linear theory of viscoelasticity (Kelvin-Voigt type) combined with a linearized electromagnetic theory. However, the ponderomotive Lorentz force and the actual stress tensor are taken into account. (ii) The Chu formulation of the Maxwell equations in terms of the electric and magnetic fields is used. More specifically, Section 2 is devoted to the governing equations of magnetoviscoelastic solids and their linearization. The observation of the 9 linear partial differential equations shows that the electromagnetic fields influence the mechanical ones through the ponderomotive Lorentz force and the actual stress tensor while the mechanical fields in turn affect the electromagnetic ones by modifying the Ohm law of electrical conduction and the Maxwell equations. In the linearization, the existence of a strain field, in fact, changes the initial isotropy of the constitutive equations. However, this effect is neglected in the present study. The change in initial isotropy is, in particular, considered in [19,20]. The solution of the coupled system of equations is sought in a form of steady harmonic waves in the x-direction in Section 3, since the simultaneous solutions of these equations, in general, being a tedious work. For a possible wave motion, the propagation condition leads to a polynomial equation with complex coefficients. In Section 4, the influence of the primary uniform magnetic field, when the field is parallel or transverse to the direction of the propagation, on the phase velocities and the attenuations are discussed both analytically and numerically.$ The plots reveal some interesting behavior of the phase velocities and the attenuations for certain frequencies, intensity and direction of strong magnetic field, electrical conductivity and certain values of the viscoelastic parameters. Section 5 is devoted to the discussion and the conclusions of the present investigation. 2. GOVERNING

EQUATIONS

OF MAGNETOVISCOELASTICITY

AND THEIR LINEARIZATIONS

The theory of magnetoviscoelasticity is concerned with the interacting effects of magnetic field on the deformation of viscoelastic solids with the inverse effects. In this section, the governing equations of magnetoviscoelasticity and the linearized equations are presented. If an electrically conductive deformable continuum is subjected to a mechanical load while immersed in a time varying electromagnetic fields, the equations of Maxwell and Cauchy still determine the electromagnetic and mechanical fields. (a) Governing equations of magnetoviscoelasticity As discussed in [18], the Maxwell equations (using SI units), in the Chu formulation for moving and deformable media, are expressed in the form§

tAs reviewed in a recent monograph[lE], there are at least four different formulations of the Maxwell equations for polarizable and magnetizable moving media, and each of them has several pros and cons. SA computer program written in FORTRAN IV language (IBM 370/145)with double precision complex algebra is developed for the determination of the roots of the secular equation. §In what follows, the Cartesian tensor notation is used and the Einstein summation convention being understood. Furthermore, a comma indicates differentiation in the sense that ( ).k= a( )/dxk and ( ),K = J( )/ax, where xk and X, are, respectively, the spatial and material coordinates of the same matertal particle.

Plane waves subjected to a uniform magnetic field

195

where p(m)= -pJ&,

p(p) = -&;

J”’ = a*Pi

+

Cjfikmn

tpm%hj;

(2.2)

JI”’ = &/JON;: +

6jk~knm(&WKU~)gj-

In (2.1)andW), E, H, P, M, P U,, p @), p (mf,J, J@‘,trn) and v are, respectively, the electric field, magnetic field, polarization per unit volume, magnetization per unit volume, free charge density, polarization charge density, magnetization charge density, conduction current vector, polarization current vector, magneti~tion current vector and the velocity of material particles defined by the material time derivative of the position vector x: v = i. Furthermore, and 8, are, respectively, the symbols for permutation and partial derivative with respect to time, and e. and fiOare two universal constants with eopo = c-*, c being the speed of light in vacuum. Two more conservation laws which are analogous to that of p@’and J are satisfied by the q~tities in (2.2). The sources (2.2) in the Maxwell eqns (2.1) contain the velocity field which is governed by the equation of motion Tfk,I + P(f?’ + f(km)) = Pak (2.3) f!ijk

where 7, p, fern’, em) and a are the actual stress tensor, density, electromagnetic body force, mechanical body force (to be neglected in this study) and the acceleration defined by (2.4) With regard to I181 and [21], the electromagnetic body force on the element of unit volume ptccm)is expressed in the form

On the other hand, the local deformation in the neighborhood of a particle is characterized by the deformation gradient Fk =

Xk,K

(2.Q

or the more familiar Lagrangian strain tensor EKL defined by

in the large deformation theory of elasticity where 8KLbeing the Kronecker delta. Equations (2.1) and (2.3) are to be supplemented by constitutive equations since the balance equations are, in number, inadequate to determine the unknowns. The general constitutive equations for a material either polarizable or magnetizable, and both are considered in several researches, e.g. [1,22-291. On the other hand, the governing equations, the balance and constitutive equations, are not used in the applications of electrodynamics of deformable continua in their general forms. Usually, the quasistatic electric or magnetic field system and isotropic materials are taken into account. Therefore, we now consider here electrically conducting and magnetizable viscoelastic isotropic solids the constitutive equations of which are obtained by substituting the polarization in the rest frame 8 = 0 and neglecting all the thermal effects in the associated equations in [ 11.Thus, the constitutive equations are

1%

YASAR ERSOY

where the script letters denote the electromagnetic quantities in the rest frame, and given by

(2.11)

In (2.10), i, 6, po, & & and 5, 6 are, respectively, the magnetic susceptibility, electrical conductivity, density in the undeformed state, elastic and viscous Lame constants. In (2.1 I), the terms containing the factor (v/c)* are neglected since high velocities are assumed to be not attained by bulk materials. It should be noted that T in (2.10)j is not the symmetric Cauchy stress tensor in the usual theory of viscoelasticity for Kelvin-Voigt type anisotropic materials because of the motion of the magnetizable material. However, it is possible to define a symmetric tensor for anisotropic material through (2.12) uki = Tk/ - /.b&.&. Furthermore, since the material is not polarizable (8 = 0) and does not have the free charge (p”’ = 0) in this investigation, the first and third terms in (2.5) vanish. In view of anisotropy, the anisotropy due to the second order terms for the electric and magnetic fields in (2.11) is considered only. (b) Linearized governing equations Even if the linear behavior of the isotropic material is considered, the sets of the governing equations is nonlinear due to finite deformations and the motion. In the remainder of this section, the governing equations are linearized. Various linearizations concerning the infinitesimal deformations are possible and are applied in [l, 10, 11,22,23,29,30]. Since we are interested in the propagation of plane EM M-waves through the moving material subjected to uniform magnetic field and the material deformed infinitesimally the governing equations can be linearized under the following assumptions [ 11. (i) Displacement gradients are small, i.e. (2.13) where u is the relative displacement vector given by ,‘k = xk -

x,&

(2.14)

and the norm introduced may conveniently be defined by

[normutK(t)]* = lim sup !&,K(T)&,K(r). rsl

(2.15)

In (2.14), X is the material coordinates of the particle. The assumption (i) does not suffice to furnish a theory linear in the displacement, therefore, the assumption which guarantees the smallness of the material time derivative of u is considered. (ii) The velocities of the particles are assumed to be small compared to a characteristic wave speed co in the strained body, i.e.

d norm (a&k) + 1.

(2.16)

Next, all the field variables in the present configuration are decomposed into two groups (2.17)

Plane waves subjected to a

uniformmagneticfield

197

where the quantities indicated by capital letters with over-head bars are assumed to be uniform and the quantities represented by either prime or lower case letters stand for the effect of the infinitesimal deformations. Thus (iii) The electromagnetic variables, their space and time derivatives associated with the infinitesimal deformation are small in magnitude when compared with the uniform electromagnetic fields, i.e. (2.18) It will be noted that (

hk

=

XK.k(

hK

=

(6kK

-

UK,k)(

=

($K

-

~kLuK,d(

hK

(2.19)

)rK

and p=

pO(l

-

UK,K)

due to the assumption (2.13). It is commonly believed that ( ),k= ( ).K&K and p =:po in the linear theory of elasticity, but the first order terms due to u in (2.19) and its time rate are not neglected in the linearization because of the electromagnetic interactions. By means of the assumptions above, the governing equations in the previous section can be linearized for initially isotropic, electrically conducting and magnetizable viscoelastic solids. Thus, the linearized equations are ek,k

=

0;

Eijkek,j

+

Poh,k

/drhi

= Lfcrn),

= -j$“‘;

Eijkhtj - $&f$ = ji,

(2.20)

pa fui = t&k + pfyrn’

where

(2.21) and

In this case, the constitutive eqns (2.10) become

where & = 0 since the uniform primary electric field Ek is assumed to vanish in the present study. By inserting (2.22) into (2.21) and the resulting into (2.20), one obtains 6jkek.j + Po(l

+i)&hi

+,f(&ruk.k

- Hkath,k)

=

0,

Eijkhk,j- (& + Eo&)ei + /&$kijk~jdfl(k = 0,

(2.23)

-/.&~ijktijek + L?/LoiCijkhj,k + (i + gd,)uk,ki +(/L + ,&a,)(&$ + ui,kk)- ~&ijk&#j#$&

-

@:Ui

=

0.

I

This coupled system of nine linear partial differential equations forms the basis of the theory of electromagnetic and viscoelastic disturbances when the material is subjected to a primary

198

YASAR ERSOY

magnetic field. This system is to be solved with the initial and boundary conditions which are not written here expticitly since the wave propagation in an unbounded medium is considered in this study. 3. PLANE WAVES IN AN UNBOUNDED

MEDIUM

Solutions of (2.23) are being sought in the form of steady harmonic waves moving in the x-direction. Therefore, e, h and u must be functions of x and t only. Thus, if {e, h, u} = (e*, h*, u*} exp [I”(k1x - wt)]

(3.1)

are substituted into (2.23), one obtains nine equations of the form A&k,

w, ii, i, 6, i, j.Z,i, ,&,O*, = 0, (a, /3 = 1,. . . ,9)

(3.2)

where e*, h* and u* are constant complex amplitudes, and k and o are, respectively, wave vector and frequency.? In (3.2), A,, is the coefficient matrix of order 9 x 9 and *$ is the column vector in the form

For a wave to exist, the amplitude lug must not be a null vector. Therefore (3.4) is the condition for the wave propagation. Now, relatively general eqn (3.4) may be simpli~ed in the case when the planes of constant phase are also planes of constant amplitude, i.e. k. x = const.,

(3.5)

and a specific direction for E is assigned. Since k=k,+I”ki

(3.6)

the planes of constant phase and those of constant amplitudes are, respectively k,

+ x =;

const.;

ki

+x =

const.,

(3.7)

and k=kn

(3.8)

when these two planes coincide where n is a unit vector in the direction of the normal to the wave front. On the other hand, one has a free choice of coordinate axes in an infinite medium, no loss of generality is involved in taking the xl-direction to coincide with the direction of the normal vector n, i.e. nk = 4t.S Then a primary uniform magnetic field is assumed in the form Hk = H(&, cos cp+ Sk2sin cp)

(3.9)

where Q is an angle between the magnetic field and n. The choice of a zero x3-component of the magnetic field is no restriction on the problem since the selected fi has components both parallel and transverse to the direction of wave motion. tk and o are, in general, complex, however, it is usual practice to regard either k or o to be a real constant so that waves of given wavelength or a specified frequency may be studied. Here the waves of a given frequency are considered. SWe can then associate the displacement component u, with logitudinal (primary, P) wave, and the transverse components ti2 and u3 with shear (secondary, S) waves.

Plane waves subjected to a uniform magnetic field

As a result of (3.8) and (3.9), A,, in (3.2) becomes

(3.10) A%= 2&,fkG

Aw=

cos

A78= Ag7= --~&Lu#~ sin 2Q

Q;

An = po2 - [h^+ 2b - 1%(6+ 2b)]k2 + &&K+&~ Asa= po2 - (c; - b&k2

+ &$oc+fi2

cos’

Q,

AW= po2 - (6 - hb)k2

A9, = i,& sin Q;

sin’ Q

+ .$&c#~.

The resulting equation of (3.2) can be rearranged so that the determinant of the coefficient matrix will be deduced easily. After some lengthy elementary row operations, (3.2) is put into a diagonal matrix except the last row

&I

0

0

0

0

0

0

0

0

0

a22

a23

a24

0

0

0

0

0

0

a32

d-la

034

0

0

0

0

0

0

a42

R43

a+4

0

0

0

0

0

0

0

0

0

a,,

a6

a7

0

0

0

0 0

0 0

0

%5

a6

f&7 %a

0

0

0 0

0 0

n,

n,

0%

0

0 0 0

=0

(3.11)

0

f&6 a277 J-b8 0 f&7 i&7* 0 i-&76 0 R99 0 0

where the elements Q,, are given in terms of the original ones Aea as 01, =

A114;

a24

b&9

=

a22 -

=

-hd9,;

A,9A%;

a23 a32

=

=

h9bb

A4,b

-

bdv

-

A49A91,

-

Awb,

-A49A92; &=-A49A%,

&=fk,,=

-

A.59Ag2;

G,

=

As&9

A2Aia

-

A2ah

=

Aa

a43

=

As2b

a,,

=

A2~h;

fb

=

A2Aa

f&i

=

A63; &,, = A6,; f-l@= Aa;

f&h = A,,

f-I,, = A,;

f&8= A,a; 0% = A,,;

OS7= Aa,

&=A,;

%2=A9,;

fix -

=

A2dm;

flu

(3.12)

f193=A92; &=A%;

fl,=A,.

For a possible wave motion, the propagation condition (3.4) is expected to yield a nontrivial solution for the amplitudes e*, h* and u* with a real nonvanishing speed of propagation. Thus this leads to a polynomial equation, the so called dispersion relation, with complex coefficients.

4. DISPERSION

RELATIONS,

PHASE VELOCITIES

AND ATTENUATIONS

From (3.4) and (3.1 l), the propagation condition is firr&lQijl*IQ~~~l= 0, (i,j = 1,2,3; a’, p’ = 1,2,3,4)

(4.1)

200

YASARERSOY

which is the secular equation for the waves in the considered material subjected to the uniform magnetic field. In (4.1), ]Qii]and ]&a,] are in the form

and LPI

= ~*,D-L(&7& - &x7%8)+ f&7(&%6 - &&8) + QfLx(%7~76 - i-MM1

+ fk&-Mhfll37 - &7&s) + fw-h&% - fh&6)1

(4.3)

respectively. It can be seen immediately from (3.11) and (4.1) that the fluctuating magnetic field h has no component along the x,-direction, i.e. hi = 0, however this is not true for the electric field e. It will be also noted that e and h are no longer perpendicular to each other.

On the other hand, the number of wave speeds has been increased from 3 to 6 in the considered material as a result of the effects of the terms containing displacement gradient and uniform magnetic fields and the motion of the material. There are, in fact, 3 different mechanical and electromagnetic wave speeds in isotropic materials when the material is not subjected to a uniform magnetic field. Nevertheless the EM- and M-wave speeds are different in each direction when anisotropic materials are taken into account. The coupled EM M-waves are naturally due to the coupling of u with both e and h. However, it will be seen that some components of u, e, and h do not affect the coupling depending upon the direction of fi. Furthermore, Rw = 0, in (4.1), implies that there is one mode of mechanical S-waves coupled with the primary magnetic field whether the direction of which is parallel or transverse. For this degree of generality maintained up to now, the order of the polynomial equations in k is rather high, and in consequence all of its roots may not possibly be determined. In the remaining part of this investigation, we shall confine our attention to the study of the waves for the special direction of the primary magnetic field. 4a.PROPAGATIONOFWAVESUNDERTHEPRIMARYLONGITUDINALMAGNETICFIELD

Wave propagation along the primary magnetic field is obtained by substituting cp= 0, i.e. sin cp= 0 into (3.10) and the resulting equations into (3.12). Thus the determinant (4.1) assumes the form (4.4) with A,, = (1- i&k2-to’, A,2= (1 - i%)V:k2 - (1 + i’i+r+,)02,

- 1”(1+/+,V”

‘k2+ [1+2&,

+ 1”(1- y~)yr](&~)~

and

where some dimensionless quantities are defined by

(4.6)

In (4.5) and (4.6), the speed of light V,, in the magnetizable medium with the magnetic

Plane waves subjected to a uniform magnetic field

201

susceptibility R, the speeds of elastic S- and P-waves V,, V,, and the viscoelastic parameters W,and w, are defined by

vo=(] +;),,*;

v, =

(f)“‘;

VP =

(+q

(4.7)

and h”t2b - __. w, - Fa, 0” = @2/j’ LL

(4.8)

respectively. In (4.6),,*, 0 and 0 are called the inverse of acoustic quality factors. Two obvious solutions of (4.4) are A,, = 0 and Al2= 0. The first one represents the dispersive and dissipative uncoupled mechanical P-wave and its behavior is well known, i.e. this mode of M-wave propagates without being influenced by the primary magnetic field, the direction of which coincides with the direction of wave propagation. The second one is a dispersive and dissipative coupled mechanical wave which is now affected by the primary longitudinal magnetic field (PLMF). The dimensionless phase velocity V (= V/V,) and the attenuation per wavelength & follow from (4.5)* (4.9) where 6, = [(I + &(I -I-&rf)]“2;

62= c;iV& - 1.

(4.10)

As seen explicitly from the solution (4.9), PLMF influences the phase velocity and the attenuation in a significant manner. If YH = 0 (no magnetic field) or v, = 0 (nonconductor) is substituted into (4.10), (4.9) reduces to the uncoupled mechanical S-wave. The numerical solution of (4.5), is carried out for a hypothetical test material either keeping the electrical conductivity constant, i.e. & = 5MS or changing the conductivity. The numerical calculations are performed for the following domains of the variables I? = (I-IO? kA.m-‘; & = (1-1O’5)mHz;

o = (l-104 kHz, C?= ( 1-1O’2)mS.

Then, V and rp are plotted for several selected values of the other variables, and they are shown in Figs. 1-5. In Fig. 1, it is shown that log V and log 4 are linear functions of log o for small values of ii and PLMF in the considered domain of frequency. As long as the applied magnetic field is small enough (1 kA.m-‘), V and C?do not change with respect to PLMF. However, when the order of PLMF is 1 MA.m-’ or greater, both log V and log a! are nonlinear functions of log w in a certain domain of o, but beyond certain values of w, they are again linear functions of log o. On the other hand, there is an anomalous dispersion of the propagated mechanical S-wave depending upon the intensity of PLMF and the value of the parameter 6, Figures 2 and 3 give the magnetic field dependence of V and & in a certain interval of PLMF at several values of o and (3,. Figure 2 displays, however, that V and a of the ultrasonic wave in a very high viscous solid (i.e. 0 < 1) are not affected by PLMF of the order 1 MA-m-*. If the solid becomes more elastic, V and i are now affected by the same magnetic field. A detailed numerical investigation indicates that when PLMF+m, V becomes very small, and 6 very large. Physically, this means that when PLMF +w, mechanical S-wave can not propagate. In ‘Fig. 3, it is shown that V and & for very high viscous solids are greater than those of elastic solids. Furthermore, the figure exhibits that there is no difference in the attenuations of the waves &, = 1 mHz and r3, = 1 kHz when the order of PLMF is ld kA.m-’ or less. Figure 4 displays the dependence of V and G to the viscoelastic parameter 6”. It is performed that for a certain interval of & there is a linear relationship between log V and log (5, for constant values of PLMF and w. As seen in Figs. 1-4, a special material is considered with respect to the electrical conductivity. Now, the numerical solution is carried out for the altering ci. Figure 5 discloses

202

YASAR ERSOY

that v and 5 change smoothly as functions of &, and depending upon the values of PLMF, o and (3, v has maximums at certain values of 6. For the solutions of Ai3= 0 and Ai4= 0 in (4.9, it is appropriate to introduce some new dimensionless quantities defined by

kd;

(4.11)

0

where k. and w. are, respectively, the reference wave number and frequency. Substituting (4.11) into (4.5bd and equating the quadratic expresssions to zero, one obtains

ad *4 - b,3k ** + cl3 = 0; a,4k*4 - b14k*’ + cl4 = 0

(4.12)

where al3 = al4 = 1 - fti,

(4.13)

These two eqns (4.12) lead to coupled modes of EM- and M-waves, and in the absence of PLMF (vH = z& = 0) they reduce to the well known uncoupled EM- and mechanical S-waves. If the material is assumed to be electrically nonconductive (v, = 0), (4.12), becomes a13k*4-

b;3k*2+ cij = 0

(4.14)

which is associated with the coupled modes of nondispersive EM- and dispersive mechanical S-waves affected by PLMF. The constants bh and c;3 are, respectively, obtained by substituting u, = 0 into b13 and cl3 in (4.13). However, (4.12)2reduces to the uncoupled EM-wave which propagates at a constant speed without an attenuation and to the uncoupled mechanical S-wave. Since (4.12) is a quadratic equation in k *2, its solution is easily obtained by using the formula for the roots of the quadratic equation. However, the phase velocities and the attenuations can be obtained from approximate or numerical solution of (4.12). The material constants for the test material are assumed to be p = 7.8 x IO3kg.mm3; @= 81 GN.me2, i= 112GN.mm2; i=

104: &=5MS,

and the numerical solution of (4.12) is considered in the intervals 1Hz~o~lTHz;

10 mHz s 0, w, < 10 GHz

and 1 kA.m-’ d fi c 10GA.m-‘. Then, the order of the quantity &.I used for the roots of k* in the expression (1 + _)“2 is (lOeM- 10e4) and obviously small compared to 1. Therefore approximate solutions for the phase velocities and the attenuations are obtained by using the Binomial

203

Plane waves subjected to a uniform magnetic field

-61

-\

1 (

a).

I 3

v5

J

1 7

-1

9

3

5

4

7

9 logW(

logW(Hz)

11 Hz)

*

I

Id

I>

E

$

3

2

3

2

1

1

0

C

-1

-1

-2

-2

-3

-1

3

(b)

logW( Hz)

(a)

Fig. I(a). Phase velocity v; (b) attenuaticn 6 of coupled mechanical S-wave as function of frequency o for different values of PLMF H,parameter 6. = I kHz, and conductivity 6 = 5 MS.

expansion retaining the first order terms. From (4.12),, one obtains “T = R”fK&+$]112;

(y, = 2+cA?)“*,

(4.15)

vt = R”:[2r&+;2)l”7;

(y2= 2*(&z&*

(4.16)

where 6

=

&=

1 +

KIKz;

1 -

iiK,;

&

=

[(I

+

2 l/2 K2)1 ,

+ hi2)(1

+

K;)]“*.

K~,

K4are defined by

+

K:)(l

(4.17) 56 =

[(I

In (4.15)-(4.17), the new dimensionless quantities

. . . ,

(1 - Vrf)& 2~Wf + (vc - G)( VY W2 . K’ = 1+ 2iVH + (1+ Ol.Q)(v:/ Vi;)“’ K*= 1+2&f

(4.18) K3= 1 + 2i2VH + (1+ W&)(v:/ v$)*. 1+2ufv* ’ 17.No.Z-C

IJES Vol.

K4= ;f;(l + 2&).

204

YASAR ERSOY

t

1

3

5 7 log ii ( kA.6’)

fa1

1

3

s logFi(

fb)

7 kAdf

Fig. 2(a). Phase velocity ‘ii-;rb) attenuation 6 of coupled mechanical S-wave as function of PLMF fi for certain values of frequency U, parameter c3,= 1 mHz and conductivity & = 5 MS.

5

6-

Id m ,o

IW ‘)

$

4

5-

3

4”

)W,

3”

1

z__I l_

3

1

faf

5 Log 3tkA. cii’j

7

1 tb)

3

5 log5

7 I kA.6’)

Fig. 3(a). Phase velocity c; (b) attenuation 5 of coupled mechanical S-wave as function of PLMF g for certain values of parameter 0, frequency o and conductivity 3 = 5 MS.

Observation of (4.15) and (4.16) shows that the phase velocities and the attenuations are, respectively, associated with the predominantly EM- and mechanical S-waves. Similarly, from the approximate solution of (4.12)*the phase velocities and the attenuations are obtained. They are (4”19) (4.20)

Plane waves subjected to a uniformmagneticfield L-

I

I

I

I

___---_

-3

-1

I,g+j

Hz)

3

-1

-3

1

3

log Q

lb6

Hz)

1

E

I

b.

1 GA.ni’

4-

2-.-. O-

.-*-*-*-*-*w. f Hd

‘\.

-

lMA.6' [J

-2-W (a)

I

_2 1

3

(b)

logiiv,(Hz)

5

I

log$iv(Hz)

-

Fig. 4(a). Phase velocity v; (b) attenuationI of coupled mechanicalS-wave as functionof parameter8, for several values of PLMF i?, frequency o = I Hz and o = I MHz, and conductivity15= 5 MS.

where 2

I/2

&=l+K,&;

&=[(l+l?:)(l+KZ)]

& = 1 -OK,;

2 l/2 .1 56= [(l + 62)(1 + K,)]

(4.21)

In (4.19)-(4.21), the new dimensionless quantities Cr,. . . , Kq are, respectively, defined by (1+ &r’H)Y, - (I, E1= 1+ ov, + (VS/ v:P

K2= (1 + VH)U,

k3=

&=A&

(4.22) 1+(1+wv,)(Vf/Vfj)2:

1

Thus (4.19) and (4.20) are, respectively, the phase velocities and the attenuations of the predominantly EM- and mechanical S-waves. Further insight concerning the behavior of the phase velocities and the attenuations with respect to the frequency and the primary magnetic field is gained for the coupled modes (4.12), when numerical solutions are obtained. For the test material, the wave speeds V,, and V, of

206

YASAR ERSOY

(b)

21 -1

log (5

1

I 1

3

5

(a)

7 log?

9 (S 1

1

-1

( b)

3

5

1og7F(5 P

Fig. 5(a). Phase velocity P: (b) attemmtio_nd of coupled mechanical S-wave as function of conductivity & for different values of PLMF If, frequency o, parameter 6, = 1mHz and (3, = I kHz.

uncoupled elastic waves, and that of EM-wave V. in the nonconductor are V, = 5.9269 km.s-‘;

V, = 3.2225 km-s-‘;

V. = 2.9998 Mm-s-‘.

Thus, Figs. 6-8 represent the behavior of VT, Vly, a1 and a2 with respect to I?, o and (3, at &=5MS. Observation of Fig. 6 indicates that the influence of the existing PLMF of the order lOOMA.m-’ and less on the phase velocities and the attenuations of the coupled modes is negligible. However, if the existing PLMF is sufficiently strong, the effects on the waves are si~ificant. If the order of PLMF is beyond 1 GA-m-‘, while VT and aI increase, Vq and a2 decrease very rapidly with respect to the variation of PLMF. Physically, this means that energy might be transfered from mechanical S-wave to EM-wave beyond a certain value of PLMF. Therefore, in a large interval of the magnetic field, it is sufficient to consider the predominantly EM- and mechanical S-waves propagating without the influence of PLMF. The attenuation of the predominantly EM-wave in a strong PLMF is seen in Fig. 7. The attenuations are seen to be sufficiently large so that they may be detected experimentally if a strong enough magnetic field could be produced. Figure 8 displays V? and CY~ of the predominantly mechanical S-wave as the function of w.

207

Plane waves subjected to a uniform magnetic field i.295

x2

Ml 6.2 33

K293

&66

6.291 I5.291

t.565

1.7850-

6.269 ?.7866-

/ I.7842 //

_._._.-+

/*

6.265 6.285

4.562

2

I 4

3 (a)

_:;

/6.207

4.563

i

1.7838<

6.289

c.564

logi?(

MA.2

2

’

)

fb)

4 lo$i(MA.n?)

“;[~WblL:TJ vio,1k8

i

15 -

12 -

9-

_*_._-.-* 6-

L._._.--.-

5.5 4

I

2 (a)

3

i

2

log ii( b&n-‘)

3 (b)

L logFi( MA.ni’)

Fig. 6(a). Phase velocities VT, V$; (b) attenuations (I~, a, of coupled modes of predominantly EM- and mechanical S-waves as functions of PLMF H for frequency o = 1kHz and parameter 13,= 1mHz, and o = I MHz and (3, = 1kHz. and conductivity ci = 5 MS.

Observation shows that the effect of PLMF disappears if the wave is ultrasonic or very high ultrasonic. Similar to Fig. 5, the variations of VT, Vt, aI and (Y~as functions of & can be shown graphically by carrying out the numerical solutions of (4.12),. Similarly, the phase velocities and the attenuations of the predominantly EM- and mechanical S-waves associated with (4.12), can be plotted. 4b. PROPAGATION

OF WAVES UNDER THE PRIMARY TRANSVERSE

MAGNETIC

FIELD

Substituting cp= 7rl2 i.e. cos cp= 0 into (3.10) and the resulting equations into (3.12), one obtains the secular equation for the propagation of waves in the considered material subjected to the primary transverse magnetic tield (PTMF). The determinant (4.1) now.assumes the form &&&A&

=0

(4.23)

208

YASAR ERSOY

with

AZ,= V;k* - (1-t bc)02, AZ2= (I - Ip;i)V:k* - co’, A23=(1-~~)V5k2-(l+tO~,~H)~2, A~~=(l-~~)(l+iv~)V~k2-[l+I’(l+v,)u,]02

(4.24)

where I*OB2

?H

=x

(4.25)

is another dimensionless number. Three obvious solutions of (4.23) are At, = 0, A22= 0 and A2J= 0. Now, AZ1= 0 and A22= 0 give the phase velocities and the attenuations of the purely EM-wave propagating in the electrically conductive magnetizable solid, and the purely mechanical S-wave, respectively. Thus, these two modes of the plane EM M-waves propagate with being influenced by the magnetic field if its direction is transverse to the direction of the wave propagation. Furthermore, AZ3= Ai2 hence no extra discussion is needed. The analytical solution of A24 = 0 is possible and it leads to 2( 1 + t32) v,=[ 1+(l+&)u2

if*_ (1 + #‘21 ’ Gl=2n [ (I+h)‘“+*l

1 iI2

(4.26)

where (4.27) This is a coupled mechanical S-wave, and either PTMF or G is zero yields the same result as that of AZ = 0. For example (4.26) is plotted and the observation of the figures shows that PTMF in the interval 1 kA.m-’ 6 B < 1 GA.m-’ has no effect on vii,and Gi. In particular, the numerical side of the viscous versus electrical dissipation is carried out for the same equation. It is shown that ii, and Ei are constants when Ei varies in the interval (1-10’2) mS, and PTMF, o and U, alter in the considered domains. However, when the material is very viscous and PTMF is very strong, vi becomes minimum at a specific value of ci. For example, when wI,= 1 mHz, o = 1 GHz and @ = 1 GA.m-‘, ii, = 6.465 X 10’ at ci = 20 mS. Using the dimensionless quantities in (4.11) and introducing the dimensionless velocity V*,= Vpjc into (4.24)5,the quadratic equation a2,k”’ - b2,k*2 + c21= 0

(4.28)

is obtained where a21= 1 - &,

(4.29) and

The roots of (4.28) are determined in the usual way, and in the absence of PTMF (i.e. yH = 0, vr, = 0) they are associated with the modes of uncoupled EM- and mechanical P-waves. If the

209

Plane waves subjected to a uniform magnetic field

L

6 logW( k Hz

25

Fig. 7, Attenuation (I~of predominantly EM-wave as function of frequency for certain values of PLMF I?, parameter t3, and conductivity i = S MS.

is electrically nonconductive, i.e. V,= 0, the coupling between these two different modes of waves and the effect of PTMF disappear, which was not observed in the case of PLMF. Similar to Case (a), the phase velocities and the attenuations af the coupled modes of waves in (4.28) cannot be deduced expiicitiy unless an approximate analytical treatnient or a numericaI solution is employed. For the same test material, the order of the quantity tcH in the expression [ 1 T (1 - 4&H}“231’2 used for the roots of k* is (10-20- 1W4)and smah compared to 1. In a similar material

210

YASAR ERSOY

2.5

,

I 2.2 (0)

I

1

L

2.6

3.0 log

3.

2.2

W(kHi

3.0 3. c togW(kHl !)

2.6 (b)

Fig. 8(a). Phase velocity Vy; (b) attenuation q of predominantly mechanical S-wave as function frequency o for certain values of PLMF H. parameter & and conductivity 6 = 5 MS.

of

manner, one obtains the phase velocities and the attenuations as follows (4.30) and a2 = ~,fd)“*

VT = RV;(&)“‘;

(4.31)

where

&l= 1 +

59 =

KIK2i

[(I

+

Ki)(l + &)1”2, (4.32)

510= 1 -MCI;

51, = [(l + 02)(1 + &I”*. I

In (4.30)-(4.32), the new dimensionless quantities K,, . . . , g4 are defined by (l+!Qfbc--Ir

K’=l+@W,+(v$/v*,)2; K3 =

I+

(1+

wvc)(V*,/

vQ)*;

K2 =

(I+

VH)Vc, 1

K4 =

l/&.

(4.33)

J

Observation of (4.30) and (4.31) shows that VT and aI, and V5 and a2 are, respectively, associated with the predominantly EM- and mechanical P-waves. Numerical solution of (4.28) is carried out for the same test material in certain ranges of the variables and the behaviour of the phase velocities and the attenuations is shown in Fig. 9. As seen in Figs. 6 and 9, whether the primary magnetic field is longitudinal or transverse there are similarities between the behavior of the phase velocities and the attenuations of the predominantly EM- and M-waves. For example, as long as PTMF is less than 1 GA.m-‘, E/* and e of the coupled modes are not affected. However, for PTMFs 1 GA.m-’ they change very rapidly, such that YT and (yl of the predominantly EM-wave increase while JJt and CY* of the predominantly mechanical P-wave decrease. Observation of Fig. 10 leads one to conclude that both waves are dispersive and dissipative, and the phase velocity of the predominantly mechanical P-wave is affected by the existing

211

Plane waves subjected to a uniform magnetic field 1

xlli’ .4005

5.284

17044

.ux)o 1.1843

‘.3995

6.282

‘.3990

6.281

I.3985

6.280

1.7t942

1.7841

1.78rO

6.282d 4 logii(MA.~')

3

2

2

3

b!

1.7845 L6‘

a.3820

6.

1.7844

8.3015

283

1.784: 0.3810

-6

6.2 85 -

.282

1.784; 8.3605

j.Bl

6.284 -

I

1.784

i -.-._.-.I

.0’

i 03aoo

_._._._.-.d.

X80

6.203 -

1.7w

I

OCJ

ii(t.lA.n?)

Fig. 9(a). Phase velocities yt, vt; (b) attenuations g,, (12of coupled modes of predominantly EM- and mechanical P-waves as functions of PTMF H for several values of frequency o, parameter 0. and conductivity d = 5 MS.

magnetic field. However, the influence of PTMF on very high ultrasonic waves is not significant. As seen in Fig. 10, while there is no anomalous dispersion of EM-wave in the case of strong PTMF. Another interesting result is that the interval of the frequency for the anomalous dispersion depends upon the viscoelastic parameter of the medium. Of course, these results may provide some information about the electromagnetic and mechanical properties of the solid. 5. CONCLUSIONS

This investigation has enabled us to arrive at the following conclusions: (i) The plane EM M-waves in the material considered are dispersive and dissipative due to the conduction and the viscosity of the material.

YASAR

ERSOY

’ 10-6 16

f

1

(a) Fig. IO(a). mechanical

logW(

kHz)

2

0 (b)

4

6 logW(kHz)

Phase velocities _Vf, vj; (b) attenuations CI,, e2 of coupled modes of predominantly EMP-waves as functions of frequency o for several values of PTMF I?, parameter 0, conductivity & = 5 MS.

and and

(ii) Whether the direction of the primary magnetic field is longitudinal or transverse h, = 0 and el # 0, and a coupled mode of mechanical S-wave arises. The electric field e and the magnetic field h are not perpendicular to each other. (iii) Depending upon the direction of the primary magnetic field several modes of the plane EM M-waves arise such as uncoupled mechanical P-(or S)waves, coupled mechanical P-(or S)waves, coupled EM-waves, and coupled predominantly mechanical P-(or S) and EM-waves. (iv) Either the direction of the primary magnetic field is longitudinal or transverse the number of EM M-wave speeds is 6. (v) When the primary magnetic field is not present, the modes of the waves are uncoupled and the 3 different phase velocities and the attenuations of EM M-waves vary with frequency smoothly. (vi) There is only one mode of mechanical P-waves which is not affected by the primary magnetic field if the direction of the magnetic field coincides with the direction of wave propagation. (vii) There are two modes of uncoupled waves, one is the mechanical S-wave and the other is EM-wave, when the direction of the primary magnetic field is transverse. (viii) The dependence of the speeds and the attenuations of the coupled modes varies quadratically on the primary magnetic field obtained is in agreement with the previous results. (ix) For an electrically nonconductive material, the coupling exists if the primary magnetic field is longitudinal and it is not the case for the transverse magnetic field. Little physical insight is gained in the analytical solutions of the dispersion relation, therefore, the numerical solutions are presented for the hypothetical test material. Thus (x) The effect of the primary magnetic field on the coupled M-wave is negligible up to the order 1 MA.m-‘, but when it is stronger, its effects become significant depending upon the frequency of the propagating waves and the viscoelastic parameter of the material. (xi) The phase velocity and the attenuation of the ultrasonic M-wave in a very high viscous solid are not affected by PLMF of the order 1 MA.m-‘, however, this is not the case for the elastic material. (xii) The primary magnetic field has little influence on the coupled modes of the predominantly EM- and M-waves compared with the effect on the coupled M-wave. The magnetic

Plane waves

subjectedto a uniformmagneticfield

213

field has its effect equivalent to increasing the speed of the predominantly EM-wave, and decreasing the speed of the predominantly M-wave. (xiii) Depending upon the modes of EM M-waves and the viscoelastic parameters of the material, the electrical conductivity may have significant effects on the phase velocities and the attenuations of the waves. (xiv) There is an anomalous dispersion at a certain small interval of the frequency of the propagating M- and EM-waves depending upon the intensity of the primary magnetic field and the viscoelastic parameters of the material. We were not able to find experimental investigations for comparison. The experimental verification would be very desirable. We are planning to work on the propagation of EM T M-waves in electrically and thermally conductive isotropic and anisotropic solids subjected to uniform primary electromagnetic and thermal fields and report these in subsequent papers. Acknowledgements-This work partially supported by a grant from the Scientific and Technical Research Council of Turkey. The author is grateful to Prof. Erhan Kiral for many helpful discussions and suggestions and the Referees for their valuable comments.

NOMENCLATURE EM EMM EMTM M P PLMF PTMF s

electromagnetical electromagneto-mechanical electromagneto-thermo-mechanical mechanical primary primary longitudinal magnetic field primary transverse magnetic field secondary

Units

A ampere HZ hertz (rad.s-‘) kg kilogram metre : newton (kg.m.s-*) s simens (Mho.m-‘) second S Prefix of units G

k M m k T

giga (109) kilo (IO’) mega (106) milli (lo-‘) micro (IO+) tera (IO”)

REFERENCES [I] Y. ERSOY, A Dynamic Theory for Polarizable and Magnetizable Magneto-Electra Thermoviscoelastic Anisotropic Solids with Thermal and Electrical Conductions. Ph.D. Thesis. METU. Ankara (1976). I [2] J. F. MCKENZIE, Proc. Phys. Sot. 91,532 (1967). [3] A. I. POTEKIN, Radiation and Propagation of Electromagnetic Waves in an Anistropic Medium, NASA TT F-743. Washington D.C. (1973). [4] G. F. SMITH and R. S. RIVLIN, Zeitschrifr fiir an Angewandte Math. und Phys. 21, 101(1970). [S] Y. ERSOY and E. KIRAL, METU 1. Pure and Appl. Sci. 7, 11 (1974). [6] L. KNOPOFF, J. Geophys. 60,441 (1955). [7] P. CHADWICK, 9th Inf. Congr. Appl. Mech. 7, 143 (1957). [8] J. N. DUNKIN and A. C. ERINGEN, Int. 1. Engng Sci. 1,461 (1%3). [9] S. KALISKI, Proc. Vibration Prob. 3, 231 (1%5). [lo] G. PARIA, In Adv. in Appl. Mech. Academic Press, New York (1967). 1111H. PARKUS. Moaneto-Thermo-Elasticitv. PISM. Udine 11972). [I21 A. J. WILSON, Pyoc. Camb. Phil. Sot. 5;: 483 (l&3). ’ [13] C. M. PARUSHOTHOMA, Proc. Camb. Phil. Sot. 61,939 (1%5). (141 T. TOKUOKA and M. KOBAYSTI, Lett. Appl. Engng Sci. 2,203 (1974). [15] M. SAITO and T. TOKUOKA, Inf. J. Engng Sci. 14, 537 (1976). 1161K. HUTTER, ht. .I. Engng Sci. 13, 1067(1975). [17] K. HUTTER, ht. I. Engng Sci. 14, 883 (1976). [IS] P. PENFIELD and H. A. HAUS, Electrodynamics of Moving Media. M.I.T. Press, Cambridge (1%7).

214

YASAR ERSOY

[I91 D. S. CHANDRASEKHARAIAH, Tensor N.S. 22,285 (1971). [20] D. S. CHANDRASEKHARAIAH, Tensor N.S. 23,293 (1972). [Zl] Y. H. PA0 and K. HUTTER, Proc. IEEE 63, 1011(1975). [22] R. A. TOUPIN, hf. J. Engng Sci. 1, 101(1%3). [23] H. F. TIERSTEN, /. Math. Phys. 6, 779 (l%S). [24] N. F. JORDAN and A. C. ERINGEN, ht. J. Engng Sci. 2.59 (1964). 1251 R. A. GROT and A. C. ERINGEN, ht. /. Engng Sci. 4,639 (1966). [26] G. A. MAUGIN, Ann. Inst. Henri-Poincari IS, 275 (1971). [27] G. A. MAUGIN, IA. Appl. Engng Sci. 2, 293 (1974). [28] Y. ERSOY, Pm. GAMM/DcAMM Congr. Lyngberg, Denmark (1977). [29] Y. H. PA0 and C. YEH, Int. J. Engng Sci. 11,415 (1973). [30] E. SUHUBI, Int. L Engng Sci. 2,441 (1%5). (Receiued

II May

1978)