Solitary waves in micropolar elastic crystals

Inl. J. Engng Sci. Vol. 24, No. 9, pp. 1477-1499, Printed in Great Britain

OOZO-7225/86 $3.00 + .30 Pergamon Journals Ltd

1986

SOLITARY

WAVES IN MICROPOLAR ELASTIC CRYSTALS

G. A. MAUGIN and A. MILED Laboratoire de Micanique Theorique Associe au C.N.R.S., Universite Pierre-et-Marie Curie, Tour 66, 4 Place Jussieu, 75230 Paris CCdex 05, France Abstract-By analogy with the domain structure in ferromagnetics and ferroelectrics the notion of Bloch and Neel walls is introduced in the description of micropolar elastic crystals. By establishing a one-dimensional version of the nonlinear theory of micropolar continua set forth by Kafadar and Eringen, it is shown that the dynamics of such walls can be represented by solitary waves, jointly in the microrotation angle and one or two elastic displacements depending as whether a Bloch or Neel wall is considered. In the first case, it is the longitudinal displacement which couples with the angle of microrotation about the axis of propagation; in the second case, both the longitudinal elastic displacement and the transverse displacement component polarized in the plane of the microrotation are coupled to the latter. Exact closed-form solutions are obtained along with the pseudo dispersion relations that they must satisfy. The stability of these solutions is briefly discussed. This approach should apply to crystals with a molecular group such as potassium nitrate KNOJ.

1. INTRODUCTION

between the equations governing elastic ferromagnets and Cosserat continua on the one hand (e.g., [I]) and those governing elastic ferroelectrics and micropolar continua (e.g., [2]) on the other hand, has been pointed out by several authors. This is especially clear when one studies the propagation of linear (harmonic) waves where, in all cases, classical elastic wave modes couple through resonance [3] with a wave mode exhibiting a cutoff, which is related to the internal structure of the elastic crystal. This internal structure, or microstructure, may have a magnetic origin (case of elastic ferromagnets), an electric origin (case of ferroelectrics) or a pure mechanical origin (case of micropolar continua). In the latter case, it is simply assumed that the deformable body is endowed with additional rotational degrees of freedom that characterize the rotation of a rigid microstructure. The allied kinematical and dynamical description for elastic bodies originated in the early works of the Cosserat brothers at the beginning of this century. A modern comprehensive formulation was later developed in the sixties to the same degree of rigor as classical elasticity (see the definitive presentation of linear micropolar elasticity by Eringen [4]). A fully nonlinear framework was further developed culminating in a paper by C. B. Kafadar and A. C. Eringen [5] and an encyclopaedic synthesis [6]. The obvious complexity of the nonlinear theory of micropolar elastic solids has probably hindered the study of exact solutions of which only a very few can be found in the technical literature (e.g.,.[7]). The present work aims at giving some exact dynamical solutions of the solitary-wave type for such media. More precisely, having noticed that the above-mentioned analogy can be carried to the nonlinear framework in which only nonlinearities related to the additional internal degrees of freedom are retained, classical elastic deformations remaining within the framework of small-perturbation theory [8], we introduce the notions of Bloch and N&e1walls in micropolar elasticity and we study the representation of such moving walls by solitary waves, the existence and the stability of such waves and whether these are true solitons or simply solitary waves. The latter point is clarified not through analysis, but only by analogy with what has already been achieved in elastic ferroelectrics [9] and elastic ferromagnets [lo]. A reminder concerning the equations of nonlinear micropolar materials as set forth by Kafadar and Eringen [5-61, is given in Section 2, the only restriction being eventually the choice of a simple expression for the internal energy for elastic solids (crystals). The notion of one-dimensional (nonlinear) motion of micropolar elastic crystals with a microstructure made of elongated rigid “particles” is introduced in Section 3. For the sake of comparison with the upriori linear theory of micropolar elasticity (e.g., in Ref. [4]) the linearized version THE FORMAL ANALOGY

1477

1478

G. A. MAUGIN

and A. MILED

of the nonlinear theory is presented in Section 4. This has allowed us to identify some parameters of the nonlinear theory and, simultaneously, this provides some clues for the discussion of dispersive effects which, when compensated by nonlinearities in the nonlinear theory, will yield the possibility of solitary-wave motions. The notion of Bloch walls separating domains in micropolar elastic solids is introduced in Section 5 by analogy with the equivalent, but already well known, notion in ferromagnetic and ferroelectric crystals. The corresponding nonlinear system of hyperbolic dispersive equations is obtained and appears to be built of a double sine-Gordon equation for the internal rotation and a linear wave equation for the longitudinal elastic displacement, the two being nonlinearly coupled. Exact solitary-wave solutions are obtained for such a system, some being unstable and some others stable. In the last case, however, the stable solitary-wave solutions are not true solitons. The parallel or complementary notion of Niel walls is introduced in Section 6 where the problem becomes somewhat more involved and, in general, would require a numerical approach. Through some approximations it is nonetheless possible to exhibit solitary-wave solutions, which are unfortunately unstable. Concluding comments concerning further problems and the applicability of the present solutions to real elastic crystals are given in Section 7. 2. EQUATIONS

OF

NONLINEAR

MICROPOLAR

MATERIALS

We use the Eringen-Kafadar formulation [4-61 of nonlinear micropolar materials. A direct intrinsic notation and the Cartesian tensor notation in rectangular coordinate frames are indifferently used. Lower case Latin indices refer to the current configuration Kt at time t of the material while upper case Latin indices refer to the reference, natural, field-free configuration KR at time t = to. A. Elements of deformation theory The macro- and micro-motions of the micropolar described by

continuum

between

KR and K, are

x = %(X, t)

(2.1)

x = x(X, t)

(2.2)

where (T = transpose) xxT = xrx The gradients of the generalized motion the finite-strain tensors are defined by

Xk,K

=

dxk ZK

(2.1)-(2.2)

x,;

>

det x = +I.

= I,

=

the Jacobian

(2.3) of the macromotion

z

and

(2.4) I

J = detlXk,KIo

(2.5)

aXkM =

ax,

f.EKL

=

xk,KxkL

(Cosserat

r

=

hMN%M,LXkN

(wryness

KL

where tm,v is the permutation

strain compatibility

=

bkl>

xK,kxk,L

conditions

%f,d@ML,N

tKMdrLN,M

strain tensor)

(2.7)

tensor)

(2.8)

6KL

(2.9)

in KR. We have

symbol xk,KxK,l

and the generalized

(2.6)

XkM$L

+

+

=

read [5]

tLPQEMQrNP)

+LPQrPMrQN)

=

0

=

0.

(2.10)

1479

Solitary waves in micropolar elastic crystals

The classical velocity field v, its gradient rotation velocity v are introduced by

uk,/, the microgyration

v

Vk[and the micro-

=

agatl,

(2.11)

;

(Xk,K)

=

vk,lxl,K

(2.12)

;

(XkK)

=

Vk/X/k

(2.13)

=

fcjk[Vk/

(2.14)

vj

where t,k/ is the permutation It is readily checked that

tensor

symbol in Kt and D/Dt is the so-called material

$

@KL

=

(u/,k

&

rKL

=

vk,lxI,LxkK.

In the linear theory of micropolar

vkl)xk,Kx/L

(2.15)

elastic solids [4] one has the following @KL

FKL

+

=

=

&KL +

@KL

+K

*K,L,

where aK is the vectorial “rotation angle” introduce the Eulerian quantities [4]

time derivative.

=

+

approximations:

uL,K

(2.16)

-$tKMN@MN,

and UL is the displacement

field. One can also

(2.17)

(2.18) with uk = &&UK and $& = &&@K. Within

the same approximation, vk

=

(bk.

(2.19)

B. Balance equations at regular points of a body The local balance laws of nonlinear micropolar [4-61: (i) conservation of mass and inertia:

DP -= Dt

Dj/ci -Dt =

-P%?,*

2v(klml.&n

or

media

for adiabatic

PJ = PO(X)

or

JKL

=uf&(x)

processes

read

(2.20)

(2.21)

(ii) balance of linear momentum:

Dvl tk1.k+ @fi = P E

(2.22)

(2.23) (iv) batarz~~ofenergy (here equivalent to Gibbs’ equation): [k:l(Vl,k

-

vkl) +

rnkIV1.k = P -

Dt

(2.24)

.

In these equations valid at any regular point in the material body j, = j& and JK~ = JLK; refated by .ikl =

(2.25)

XkKXu..JKL

are the current and material tensorial densities of inertia, and tk/ is the (nonsymmetric) stress tensor, .I; is the body force per unit mass, mkf is the (nonsymmetric) couple-stress tensor, II is the body couple per unit mass and e is the internal energy per unit mass. The tensor JKL describes the geometrical distribution of mass within the “particles” which. in the usual nonmicropolar mechanics, would carry only a mass density pa(X) and no inertia tensor. The axial vector (Tis the intrinsic spin of these micropolar “particles”. Thus ck =

(2.26)

Am.

For further use it proves to be more convenient to use the “material” form of eqns (2.22)(2.23). This is achieved by introducing the “Piola-Kirchhoff” tensors TK, and MKf by (2.27) It follows from eqn (2.20) and the identity (J-‘.hfk,K).k

(2.28)

0

=

that eqns (2.22) and (2.23) yield the “material” expressions TKI.K +

(2.29)

~‘0.6 = POx

(2.30) where all quantities are now functions of the independent

variables X and t.

C. Constitutive equations For (nonnecessarily linear) elastic micropolar materials, ignoring the dependence on entropy because of the working hypothesis of adiabaticity, one takes (2.31)

e = q(Q, l-y

so that the material-frame-indifference (or objectivity) requirement is automatically fulfilled and we have the following constitutive equations on account of eqns (2.31), (2.15) and (2.24): dP fkf = Pxk,KxlL

g2-Q

ar mkl

= PXk,KxfL

%K

(2.32)

or

or

n/r,,

de

= POxlL ~ arLK

.

(2.33)

Solitary waves in micropolar

elastic crystals

1481

If, in addition, the medium is macroscopically isotropic, then I? must be invariant under the full orthogonal group of transformations of the material frame, from which there classically follows that e can only be a function of joint isotropic invariants of the tensors Q and I’; i.e., (2.34)

cy = 1,2,. . ) 15

e = NJ,

where, according to Kafadar and Eringen [5] (tr = trace): II = tr 9,

Jz = f tr ($2,

Z3 = 4 tr g3,

Z, = f tr@QT)

Z5 = trV%T),

Z6 = f tr[92(Q7)2],

I, = tr(Qr),

Z8 = tr(@Y2)

Z9 = tr(@),

Ii0 = tr r.

I,, = 1 tr I‘*,

Zr2 = f tr r3

Zi4 = tr(PP),

Z15= f trlr2(Pj2&

113

=

5

tr(lV),

(2.35)

In the sequel, to fix ideas, we shall consider the following simple expression for e(Z,): I; = pee =

fAZ: +

pZ2 + (p + K)& +

$cvZ,~+ @Z,, - yZ13

(2.36)

where X, H, K, a, @ and y are constant coefficients which can be identified to those introduced in linear micropolar elasticity [4], although the expression (2.36) still involves the nonlinear strain measures. In addition, to simplify the forthcoming formulation to some extent, we shall eventually consider the following constraint relating A, p and K (this is some kind of “Stokes” hypothesis for micropolar materials [4]) (2.37)

3h + 2/J + I( = 0.

In the linear theory the positive-definite character of the energy density imposes the following inequalities: 3x + 2/J -t

K

2

2@ + K 2 0,

0,

3a + 2p L 0,

rI-+$Z--_r,

Kzo

r&O

(2.38)

so that the more stringent constraint (2.37) is admissible as a limiting case. 3.

ONE-DIMENSIONAL WITH

MOTION

ELONGATED

OF

MATERIALS

PARTICLES

We consider so-called one-dimensional motions; i.e., generalized motions (2.1)-(2.2) that depend only on one material coordinate, say X, and time. Thus,

u = fUf0 =

wy = U(X,

(U, Y,

t),

x = x(X 0

(3.1)

and eqn (2.4), provides the following matrix

F=

The microrotation

(3.2)

x is taken as the multiplicative composition of two rotations x

=

x2

a xi

(3.3)

1482

G. A. MAUGIN

and

A. MILED

where X, is a rotation of angle H about the e3 axis in Fig. 1 and x2 is a rotation of angle cp about e’, , where the latter is the image of el by X, (Fig. 1 gives the Euler angles of the microrotation (3.3)). Then eqn (3.3) provides the rotation matrix cos 0 sin 0 sin cp sin 0 C

-sin H cos cp sin 0 sin cp cos 0

cos p

X=

It follows from this that tJ and I’ can be represented frame:

1 +g

cos(I+gsinDcos(aI /

+

by the following

-

1

0 -sin cp . cos cp :1 matrices

’

I --

ItgsinH

:(

1

I

dW

sin 0 sin cp

z

’

dX

\ sin cp

I I

sin cp

ax

I

dV.

in the material

’ dW ; +zcosq

+~~cosHcosip

I l aw I + -costI

--------------_---

(3.4)

1

(3.5)

I

l___________l___----

cos cp sin

I

0

I

I

cos $0 cos 0 I l___________l___--_I sin cp cos 8 I

-----------------sin cp sin 0

-sin

1

p

cos p

\

/

and /

dP rxcos

0

0

-zxsinO

0

0

do dX

0

0

acp

r=

i

On account

of eqns (2.32) TKI

MKI

0

.

r

(2.33) and (2.36) we have the following =

aT,,

z=POp

K,-Kp) Fig.

equations:

= dtr r)x/~ + bx[LrK~ - ?'xd~K.

a2u

a~,,

constitutive

A(tr @)XNC+ PXL&LK + b + K)XIL~KL

In the absence of body forces and body couples and for motions and (2.30) admit the following components:

-=Pop ax

(3.6)

I. Microrotation

a2v

aT,, ax=

K, for one-dimensional

motions.

(3.7) (3. I), the equations

a2w POat2 -

(2.29)

(3.8)

Solitary waves in micropolar elastic crystals

1483

and dMxx

-

ax

dW + dv TX2 -,,r,,+(Gax

po dt

aMx, + dW TX1 au -zTxz+(TzxdX ax

52, J3)

and thus the intrinsic

spin (J (2.26)

aa,

tensor

J, = const,

on account

(3.9)2

po at

Tyx)=PO --g

In agreement with Fig. 1 we assume that the micro-inertia reference configuration KR of coordinates (X, Y, 2); i.e.,

(3.9)1

aa,

TX=)=

ahf, au av -ax + ax Txy--gi+x+Kx,-

(JKL) = diag (A,

aax

Tz,J=

a=

(3.9)3

.

JKL is diagonal

1,2,3

in the

(3.10)

of eqn (3.4) is represented

by the column

(3.11)

In writing down the explicit form of the equations of motion (3.6) and (3.9) we shall neglect terms involving the products of two space or time derivatives, e.g.,

au a0 -axax'

au2 ax' (>

ap ad

at at

apav

-

-

at at

’

, etc.

(3.12)

so that, on account of eqns (3.7) (3.4), (3.5) and (3.1 l), we finally have the following system of field equations for the present, simplljied, theory of one-dimensional motion of nonlinear elastic, macroscopically isotropic, micropolar materials made of elongated particles:

balance of linear momentum.

p0

a2u

7gy

=

[p

+

K +

(A

+

p)C0s2B]

a2U

z

+

i

2

(X

+

a2v

p)

jgj

cos

$

sin (a sin 28

19+ (X + y)( 1 + cos cp)sin 201

+ g

X

&y

>

(

-$[A cos p sin

2 cp +

cos 2~ sin 20 + &y

[p sin cp sin28 + A( 1 + cos @sin cp cos f?] (3.13),

sin 2+0 sin20 >

X COSq COS20 - p sin2q sin 01 - g -

p

+ 5:

[A cos2~ cos 0 + (X + P)( 1 + cos p)

[(A + y)(sin 2~ sin B + 4 sin p sin 28) sin 0 cos 0 cos cp + a(~ + 2X)sin 2+0 sin 281

(3.13)2

1484

G.

a2w

p. dt2

A.

MAUGIN

>

and

2

2

$

sin cp sin 20 + $$

MILED

I

= [(w + K) + (A + p)sin’O sin3p] $

X

A.

+ i (A + I*)

7

sin 2~ sin’0

+ $j (X + P)( 1 + cos cp)sin cp cos 28 >

+ g

(X + ~)[cos cp cos B + (1 + cos @sin 0 cos 2971 (3.1 3)3

balance of angular momentum:

a

p. Z

+/3 +-f)$ cos*8 +y2 1=(a

aq

(J1cos% + J2sin20) z

[

+ g

sin*0

[A sin cp cos 0 + (A cos*O - CLsin’f3)sin cp] + i 5; (A + CL)

X(sin28sinrp+sin28sin2rp+2sinHsin2~)-~$~(h+p) X (sin 20 cos cp + sin 28 cos 2~ + 2 sin I3cos 2~) + 2X cos 19sin 2~ + 2X sin cp cos*8 (3.14),

1

do

PO &

[; (Jl - 52) $

cos cpsin 20 + J3 z

sm cp

2 a*9

sin cp + ;
= y g

+ 5:

cos cp sin 20 + f(A + p) g

(sin 8 + 1 sin 28)sin cp

[(A coS28 - p sin’8)cos cp + {(A + p)(cos 0 + cos 19cos y? - sin28 sin2+9)

av

sin cp cos 0 + 4(X + p)sin 2cp sin281 - ;(A + p)(sin 0 + sin 20)

- ,,[P

X sin 2~ + (CL- X)sin cp sin 0

PO &

[;

X2

+ g

+ 5:

(JI

-

J2)

$

sin cp sin 28 +

sin 0 sin 20 + g

do

J3 z

cos

cp

(3.14)2

+$(a! +p) 1=yf$coscp

[(A cos20 - P sin*@sin 0 + f(h + j~)( 1 + cos cp)*sin 201

[(A + P)(COS~+O sin28 - cos*O - cos 20 cos cp) - X cos (0 cos fI]

[ +(A + p)sin 2~ sin20 - (A sin*0 - I* cos*O)sin ‘p] + f(A + fi)(l + cos cp)*sin 20 + (A ~0s’~ - p sin’p)sin

8.

(3.14)3

The search of exact propagative solutions for the above system in which no precise symmetry is specified for the micro-inertia tensor and the angles 19and (o may have arbitrarily large excursions in amplitude, represents a formidable task and we shah have to envisage special cases where only either one of the two angles 6’and cp intervenes in an active manner (see the notions of Bloch and NCel walls below). Before doing so, we need to examine if the equations (3.13) and (3.14) of the nonlinear theory reduce to those of the classical, a priori linear, theory of micropolar elasticity (as stated, for instance, in Ref. [4]) when they are linearized about certain reference states.

1485

Solitary waves in micropolar elastic crystals 4. LINEARIZED

THEORY

We assume that the departure in displacement components erence configuration remains small for all X’s and t’s. Thus,

IUI, IVI, IWIare O(4

and angles from some ref-

101and IpI are O(tr) = O(t)

(4.1)

where t is an infinitesimally small and L is a macroscopic characteristic length. As initial configuration K0 coinciding with KR, we have (cf. Fig. 1) 80 =

$00

=

0,

u, = vo = w, = 0.

(4.2)

The inertia parameters J, and p. are taken as pure constants. Noting by a superimposed tilde the linearized fields, the linearization of eqns (3.13) and (3.14) about the spatially uniform state (4.2) yields the following set of small-amplitude field equations:

(4.3) and

(4.4) In the same conditions, but using Eringen’s notation [4], the equations of the a priori linear theory of micropolar elasticity (Eringen [4]) yields (A + 2p +

K) 2

a2uy a~, (P+K)=-Kx-Po-C$=O

(P+K)&$+K$+$=O

-

PO 3 a22+

= 0

(4.91

(4.5)~

(4.93

and

(4.6)~

y~-tK(~-2pz)-pojz~=o.

(4.6)3

The two systems (4.3)-(4.4) and (4.5)-(4.6) fully coincide with x = X in an a priori linear theory with (0, F, I@,+, 0, 5) identified to the set (u,, uY, uZr cpx, ‘pY,pa,) and with the Stokes’ hypothesis (2.37) holding true.

G. A. MAUGIN

1486

and

A. MILED

Dispersion relation gfthe linearized theor):!’ We consider

the system (4.3)-(4.4)

and harmonic

plane wave solutions

of the type

( 0. P, 0, +. f?) = ( L!. 1,; 0, cp, B)exp[i(qX - wt)]

(4.7)

where q is the wave number, w is the circular frequency and W has been taken equal to zero [see eqns (4.3h and (4.4)3] without loss in generality. One will obtain nontrivial solutions for the amplitudes of the solution (4.7) if and only if the following dispersion relation holds:

DL(w, q) = (w* - w;)(w2 - w;) (w* - w;.)(w* - w;) - (y);,4q2]

= 0,

(4.8)

where we have set

w:.= c2q2, w$-= &* w;= v2 + c$q2, w; = (J,/Jh2

(4.9)

+ dq2

(4.10) (4.11)

d = (~/PoJI)(Q + P + ~1,

c% =

(lI~oJ3h,

v2= @/PoJI)K.

(4.12)

A graphical representation of the dispersion relation (4.8) is given in Fig. 2 in the first positive quadrant of the (w, q) plane. The modes w = wL and w = w. travel independently of others and they correspond, respectively, to a longitudinal elastic mode and a dispersive optical mode (as J, tends to zero, the latter escapes by the top of the figure going to the infrared region and up). The remaining two modes, of which the frequency spectra are solutions of the equations obtained by making the last factor in eqn (4.8) to vanish, are coupled by resonance [3] at q = q* and w* defined by wT(q*) = wR(q*). Their repulsion is O(q). They are mixed transverse displacement-microrotation modes. As J3 goes to zero, the skeleton branch wR of this diagram coalesces to zero, leaving alone the uncoupled transverse elastic branch wr. As the typical micropolar parameter K goes to zero, both rotational and optical modes, wR and w. coalesce to zero and we recover the one-dimensional model of linear isotropic elasticity. Therefore, the dynamical linearized theory here exhibits two typical features encountered in other circumstances in condensed-matter physics: (i) the existence of a high mode with cutoff of an optical nature, resulting from the existence of a high-

Fig. 2. Dispersion

relation

for the linearized

case.

Solitary

waves in micropolar

1487

elastic crystals

frequency internal motion and (ii) a resonance coupling between a transverse acoustic mode and a mode describing “rather” low-frequency oscillations in an internal degree of freedom. This situation occurs as well in elastic ferromagnets (so-called magnon-phonon couplings [ 1 l]-[ 121) and elastic ferroelectrics (coupling between a polariton and an acoustic wave [ 131) and was already exhibited in micropolar crystals on the basis of a lattice-dynamics approach (Askar [ 141). The present analysis will shed some light on the much more involved nonlinear case. 5.

THE

BLOCH

WALL

IN

MICROPOLAR

CRYSTALS

A. The notion of wall In a ferromagnetic Bloch wall [ 15- 161 the magnetic spin transits, smoothly but throughout a relatively narrow layer, from a uniform state, pointing up, at X = -cc, to another uniform state, pointing down, at X = +a~, having effected a 180” rotation by rotating about the X-axis. In a ferromagnetic Nkel wall [ 17- 181, the same limit situations are considered at X = ?CG, but the 180” rotation takes place in a plane [say the (X, Y) plane] containing the X-axis. In both cases the bulk of the rotation takes place within a characteristic distance, called the thickness of the wall, which results from a competition between exchange forces of quantum mechanical origin and the magnetic anisotropy energy which is minimal for spatially uniform, perfectly ordered, ferromagnetic states. A rather similar picture occurs in elastic ferroelectrics of the molecular-group type such as sodium nitrite where the notion of NCel wall can be introduced [9]. These pictures hint at a direct analogy in the case of micropolar crystals. To that purpose, it is sufficient to consider a micropolar structure in the form of elongated ellipsoids (Fig. 1) with the longest principal axis directed parallel to the Y-axis both at plus and minus infinity. Obviously, unless a marker (say a point of color) is used on these ellipsoids to indicate the upward and downward directions, we have to consider that the two limit conditions are indeed physically the same, modulo ?r (for the magnetic spins of the ferromagnetic case there obviously exists an intrinsically defined direction). Other limit conditions can be envisaged but only those paralleling the ferromagnetic and ferroelectric cases are considered in the present work. Then if we allow for finiteamplitude excursions of cp (with 13kept equal to zero) or 0 (with (o kept equal to zero), in a rather small X interval, we shall have the picture of Bloch and NCel walls, respectively (Fig. 3). This schematization will obviously greatly simplify the coupled nonlinear equations (3.13)-(3.14) in both cases. B. The governing .ry.stem of equations We consider the scheme of a Bloch wall, thus 8 = 0 at all times and places, Fig. 3a. The reference configuration I& is taken as the physical situation at X = -co. This defines the “zero” position of the angle cp. Then eqns (3.1 l)-(3.14) immediately simplify to 2

(5.1)

p,$=(h+2p+x)$$--22hgsinp

a2v PO jp

Fig. 3. Notation

2 =

c/J

+

K)

(5.2)1

$

for walls in micropolar

elasticity.

(a) Bloch wall, (b) Niel wall.

G. A. MAUGIN

1488

and

A. MILED

(5.2)2 and poJ,$

= ((.y + p + y)$

+ 2X g

sin cp + 2[Xsin cp + 2(X+ p)sin 2cp] (5.3)

~&Jsin~+[xcos~+(X+p)(l

+cos~)cos+0]$~=0

r~~,,,~+Ixcos0+(X+~)(2+cosp)l~=0.

(5.4)l

(5.4)~

As eqns (5.2) that govern the transverse elastic displacement are not coupled to other degrees of freedom, we shall set I/= w=o

(5.5)

without loss in generality. Then eqns (5.1) and (5.3) reduce to the following nondimensional system of two nonlinearly coupled equations:

a2U ~-

a72

a20

a

ax2

ax

vi--=(Y-_

4 2

( 1 cos-

a24 a24 sin $I = e sin g + (Yau sin ?L! 2 ax 2 aT2 aX2

(5.6)

(5.7)

where we have set

Izl = 2P 7 = w,&ft,

(5.8) x = X/6

(5.9) (5.10)

a= J72u,

v2 = JI(~ + 2~ + K) L (a +

P +

(5.11)

7)

and IS = [2(cr + p + y)(X + /.#‘2,

e = X/2(X + /L).

(5.12)

The nonlinear hyperbolic system (5.6)-(5.7) consists in a (nonlinear dispersive) double sineGordon equation [ 191 for I$ (twice the microrotation angle) and a (linear, nondispersive) d’Alembert wave equation for the nondimensional longitudinal elastic displacement, the two being coupled in a nonlinear manner through the parameter G. None of the parameters is a priori infinitesimally small, but Vi ti1 as wave speeds are normalized with respect to the characteristic speed (here one) of eqn (5.7) and rotational modes are usually very slow in the long-wavelength limit. C. Solitary-wave solution Equation (5.6) for Cr = 0 possesses propagative solutions at the speed +VL. Equation (5.7), with 6 = 0, is a double sine-Gordon equation which, under proper limit conditions at _8?= +cc is known to possess solitary-wave solutions which are indeed solitons (see Refs. [ 191 and [20]). The question therefore arises of the possible existence of solitary-wave and soliton solutions for the coupled system (5.6)-(5.7). To answer this question we look for solutions of this system that depend on x and 7 only through the phase variable t = q;u - w7 + to,

to = const.

(5.13)

I489

Solitary waves in micropolar elastic crystals

where q and w are now pseudo wavenumber and frequency. The system (5.6)-(5.7) therefore implies that fl and 4 satisfy the following coupled system of nonlinear ordinary differential equations: d2u (02 - Cl&)-_? = 4 - sin 4 - e sin $ = iiq g

(w2 - 8) $

This must be supplemented

by limit conditions

a0 - - 0,

dJ-

a.2

where 4. is a constant

determined

implies

VLS

sin : .

(5.14)

(5.15)

at *cc. We shall assume that

+40

1x1 -

as

CL

(5.16)

by

e sin 5 This condition

wdq) -

+ sin $0 = 0.

(5.17)

that c$~ = 2kr,

either

cos -40 zz - e 2 2’

or

Being interested in a situation consider only the first class of longitudinal stress vanishes at and after one integration, eqn

kEZ ;
(5.18)

which closely parallels the one met in ferromagnetism, we conditions (5.18) while the first of eqns (5.16) means that the fee. Suppose that o f wL, on account of(5.18)2 and (5.16), (5.14) then provides dU/d[ as

(5.19)

Substituting

from this into eqn (5.15) and setting

(5.21) G = w/G, the resulting

equation

4 = ql&

that governs

(5.22)

4 is found as 2

(A2

-

i2)

C-9

-

sin C#I- < sin $ = 0.

(5.23)

dE2 If o and q are fixed, then eqn (5.23) may be considered as the equation of motion (where the role of time is played by 6) of a “particle” of “mass” (&’ - G2) in a periodic potential. and eqn (5.23) is the nonlinear ordinary differential equation associated with a double sineGordon equation for a functional dependence of the type (5.13). The first integral of energy associated with eqn (5.23) is

(&I2-

=2[1

-cosC#J]+4j-

(-I)‘-cos$

1

(5.24)

1490

G. A. MAUGIN and A. MILED

with

d4

as IEI-

z-O

(5.25)

a.

Solutions of eqns (5.23)-(5.24) are sought in the form (5.26)

@G) = 4 tan-‘JIG). It is found that #([) satisfies the following ordinary differential equation:

(2 -

B2@=[l + ;{(-l)k]J,

+ lr[l

+ (-1)~]~3.

(5.27)

We clearly have to distinguish for two cases depending on whether k is even or odd. (i) Case where k is even. We take k = 0. We thus have the limit condition

4-O

(modulo 4n)

as

IEI-

03.

(5.28)

After a short calculation the explicit solution (5.26) is found as 4(E) = 4 tan-r[(-&)“2cosh

I]’

(5.29)

with the accompanying pseudo dispersion relation:

IjCBVen(W, q) = (i2 - i2) - 1P + r(@,411= 0 or D7”(w, q) = cd2- [q2 + i(e + 2)] = 0. Furthermore,

(5.30)

using eqns (5.21)-(5.22), it is found that

2 + rt(4 4) < rtw> 4)

o

if

e-c -2.

(5.31)

The graphical representation of the “dispersion relation” (5.30) is given in Fig. 4a. Branch (a) in this figure corresponds to the case e < -2 and thus 2 + { < 0, but according to eqn (5.3 l), we then have r//(2 + {) < 0, so that we have an incompatibility. Branch (a) does not, therefore, correspond to a real solution. Branch (b) does not yield an incompatibility but, unfortunately, upon using the stability criterion of the linearized theory [2 l] we see that it corresponds to an unstable solution. Therefore, for the limit condition (5.28) with an even k, there does not exist any stable solitary-wave solution.

c.

! I

/ tiizl

9

(b)

Fig. 4. Pseudo dispersion relation for Bloch walls in micropolar elasticity.

1491

Solitary waves in micropolar elastic crystals

(ii) Case where k is odd. We take k = f 1. We thus have the limit conditions (a)

4 -

+27r

(b)

C#J - ~27r

as

(5.32)

~-+Cco.

Then, on account of eqns (5.26), (5.27) and (5.32), the solution 4(t) is shown to read 4(t) = 4 tan~‘[(&)“zsinh

t]

+(D = 4 tan-‘[ (--&)lilsinh

(1,

(5.33)

or e^=e+

s

(5.34)

with a pseudo dispersion relation

dpyq q) = (3 - G2) - f[2 - l(w, 411= 0 or DFd(w, q) = w2 - [q2 + $(2 - e)] = 0. The graphical representation that

(5.35)

of this “dispersion relation” is given in Fig. 4b. It is checked !: 2-{
for

e> 2.

(5.36)

In Fig. 4b, only the branch (b) (o < q, e > 2) corresponds to a stable branch in agreement with the linear stability criterion [21]. Thus a stable solitary-wave solution (5.34) in the microrotation angle Q,is possible for limit conditions of the type (5.32). The accompanying elastic displacement and stress fields are obtained by integrating eqn (5.19) (for odd k’s) and directly from this equation, respectively. Thus

and 2cFw2 sinh2[ zxx = - (02 - w$.)[i/(2 - e)] + sinh’< ’

(5.38)

The thickness of the corresponding Bloch wall is computed from eqn. (5.34) to be 6 = ~ (e - 2)(e - 2 - .c?)~“2 B t?(e- 2 - 263 t I ’

(5.39)

This completes the solitary-wave representation of a “Bloch wall” in a micropolar elastic crystal. We know from a similar analysis (which, however, involves only a sine-Gordon equation to start with and not a double sine-Gordon equation) in elastic ferroelectrics 1221 that the obtained solitary waves are not true solitons since, because of the coupling with the wave equation (5.6) radiations always accompany the interaction of two such waves. 6. THE

Nl?EL

WALL

IN MICROPOLAR

CRYSTALS

A. The governing system of equations We consider the scheme of a pure “Niel wall”, thus (o = 0 at all times and places (Fig. 3b). The reference configuration is still taken as the physical situation at X = --co. Then eqns (3. I3)-(3.14) immediately simplify to

1492

p.

G. A. MAUGIN

a2u

7gp

and

2

=

[(p

+

K) +

4(X

+

p)]

A. MILED 2

$$

+

i

(A

p)

+

2

213+ $$sin

f$cos

(

28 >

- [A sin 0 + 2(X + r)sin 201 g

a2v

PO 7g

2

=

[(p

+

K) +

+
+

p)]

2

$-$

+

i

(A

+

p)

(6.1),

2

sin 28 - $

s

cos 20 >

+ [A cos 13+ 2(X + /I)COS201 g

(6.1)2

and poJ3 $

= y $

+ [A sin 0 + 2(X + p)sin 201 g - [A cos e + 2(X +

~)COS

281 g

+ 2[h sin 0 + 2(X + p)sin 281. (6.2)

Without loss in generality we have considered that the transverse elastic displacement polarized orthogonally to the plane of the microrotation vanishes since it independently satisfies a homogeneous wave equation. The system (6.1)-(6.2) couples, in an intricate manner the microrotation 0 and the elastic displacement vector components in the plane of this rotation in a way rather similar to, but more involved than, what is observed in magnetoelastic NCel walls [lo]. To the same degree of approximation as the one underlying the neglect of contributions (3.12), the system (6.1)-(6.2) can be cast in a more convenient form. The following trick has to be used. Introduce the circular components of the elastic displacement vector in the (X, I’) plane by U’=

U&iv,

i=G.

(6.3)

Then eqns (6.1) and (6.2) yield

a2u* POT

=

a2u’ exp(f2i0)

[(j.l + K) + $(A + /.L)]g

+ ; (A + /L) 5

+ I” -$ exp(W)

(6.4)

and

au+

aZe

I- exp(-i0) x

poJ3 s

-

r+

exp(if3) g]

+ 2[h sin

e + 2(X +

p)sin 281

(6.5) where we have set I* = X + 2(h + p)exp(+ifI).

(6.6)

U’ = F*exp(+iB)

(6.7)

Now set and neglect terms of the type

au’ de --

au* ae --

ax ax’

..a2e ax2 ’

at at ’

(6.8)

which are of the same type as those in eqn (3.12). Equations (6.4)-(6.5) then reduce to aZF’

PO -z&y

@F’

= (P + K) z

1

d2p.k

+ 2 (A + wCL)e

a2p

+ -$-

>

*

ir*

g

(6.9)

1493

Solitary waves in micropolar elastic crystals

and

2 poJ3 g

= y $$

+ f

l--

I

g

-

r+ dx aF-

F,

Finally, we return to Cartesian components a2F 2 po at2

-

(A +

2p + K) 2

2

+

Fy

and

ax

(6.10)

such that

= 2(h + p) -& (cos 0)

d2F

Po$+(P+K)-$$=a

2[X + sin 0 + 2(X + p)sin 281.

[Ae +

2(X + p)sin e]

(6.1 I)I (6.11)2

and p0J3 g

=

2(X + p)sin 0 $$ - [X + 2(X + p)cos B] 2

+ 2[X sin ~9+ 2(X + p)sin 201. (6.12)

This reduced problem is now made nondimensional 7

=

by setting [compare eqns (5.9)-(5.1 l)]:

x = X/6

cd&,

(6.13)

6 = (r/2X)“2

wM = (~X/P~J~)“~

(6.14) (6.15)

g = F,lG

f = &I%

a! = (X + p)(2J3/yX)“2,

e = 2(X + jL)/X

6 = (XJ,/2#‘,

(6.16)

and v:. =

X+2p++

I/-‘, = ~P+K

PO(YIJ~)

PO(Y/J~)

’

(6.17) .

The system (6.1 l)-(6.12), after removal of all superimposed bars to simplify the notation, reads

A??_ V2aZS=aa(COS~) ar2 a%

-aT2

Lax2

ax

gg a VZ,~=~(@3+asin@

(6.18)1

(6.18)2

and

a2e

---aT2

a2e

ax2

sine - esin 28 = a j$sin

e - (p+ (Ycose)$

(6.19)

This nonlinear hyperbolic dispersive system is interesting and its response can be studied numerically. Analytically, however, the nontrigonometric function appearing in the righthand side of eqn (6.18)2 is rather annoying, so that we shall focus the attention on the following special case: p = 0, which is equivalent to the original modulus X being zero. In this case, eqns (6.18)-(6.19) are replaced by the perfectly symmetric system:

a

a’f_v2#f_

a72

L ax2

e'gq2a2g_ Tax2 a72

+

(y ax ( cosY )

a . 4 "ax (Yi 1

(6.2O)i

(6.20)~

1494

G. A. MAUGIN

and A. MILED

and

a%$ a! . 4 ag 4 sin C$= ff - sin - - - cos ax2 ax 2 dX 21

a24 ----

(

d? where the nondimensionalization

has been performed

I#)= 20

a = /.4Jd2y)“=, of eqns (6.14) and (6.16). Limit conditions

4 -

2k?r

(modulo

--

(6.22)

accompanying

kE Z,

4x),

af ag_

by setting

d = j(y/2p)“2

WM= 2(2d~oJd”*>

instead are

(6.21)

o

as

ax'ax

as

[XI-

the system (6.20)-(6.2

IXI -

0;

1)

(6.23)

co.

This first of these means that, as noticed before, the two limit conditions for the microrotation angle are not distinguishable (modulo 47r for twice the microrotation angle), while eqns (6.24) mean that there are no (usual) strains at +cc. The system (6.20)-(6.21) is quite remarkable in that it couples nonlinearly in a perfect symmetric manner a sine-Gordon equation for twice the microrotation angle of the Neel wall and both longitudinal and transverse elastic displacements, hence the elastic displacement components in the plane of this rotation. It is thus similar to the system obtained for a magnetoelastic NCel wall [lo] except that (i) it is more symmetric (the same coupling coefficient (Yintervenes in both eqns (6.20)), and (ii) the asymptotic conditions (6.13) are quite different. This last difference comes from the “nondirectional” nature of elongated particles if they are not equipped with electric or magnetic property (e.g., dipole) which allows one to define up and down, or right and left, pointing positions. However, we know from this partial analogy and the analytical and numerical results of Ref. 22 that if stable solitary-wave solutions of the system (6.20)-(6.2 1) exist at all, they will not be true solitons. The only question which needs an answer is whether such solitary-wave solutions exist and what is their stability if they do exist. This is answered in the following paragraph. B. Solitary-wave solutions We consider solutions ($,f; g) which depend [ = QXThen eqns (6.20)-(6.21)

517+

yield the following

(Q2- 0;)

on X and 7 only through

to,

if0 = const.

set of nonlinear

(6.25)

ordinary

-$=CYQ$ (co, ;) ,

(n2 - n$) 3

= CYQi

(sin z) ,

On account

of eqns (6.23)-(6.24)

dC; 2

first integrals

(Q2- a;)

df =CYQ d4‘ ds

differential

equations:

Q; = V:Q’

(6.26)1

Q$ = V$Q2

(6.26)~

d2d df 4 dg (fi2 - Q2) 7 - sin 4 = CYQ- sin - - - cos g

dt

the phase variable

dt

.

(6.27)

of eqns (6.26) are given by cos ;

-

(-l)k 1

4

(fi2 - n’,) - = LYQsin - . 2 dt

(6.28)2

1495

Solitary waves in micropolar elastic crystals

Suppose that if a wave propagates at all, its phase velocity is different both from VL and Vr. Then dfld[ and dg/d[ can be substituted from eqns (6.28) into eqn (6.27) which yields a unique nonlinear ordinary differential equation for 4. On setting [compare eqns (5.20) through (5.22)]

Vi(l -

t = Ly/2v;

4Q2,

(6.29)

522- ti1, n’, - 6:. Q2 - a2 _ @.

(6.30)

a/J>

(6.3 1)

Q = Q/A

and

1

-2(-l)k

(6.32)

we find that this equation reads

(a2 - Q”) 3

= sin # + {(a, Q)sin

$.

(6.33)

With eqns (6.23) and the additional limit conditions

dQ_0

z

a first energy integral of eqn (6.33)-at (62 -

Q’,(!!T

=

as

El - m

fixed ($2, Q)-is

(6.34)

given by

2(1 - cos 4) + 4s[(-l)k

- cos g1.

I

(6.35)

LJ

Again a solution can be sought in the form # = 4 tan-‘+([).

(6.36)

Then J/(E) satisfies the following differential equation

(~2-d2)~=[1+lr(-f)k]~+ 5i-11+ wYllt,3. dt2 2

We shall distinguish between the two cases whether k is even or odd. (i) Case where k is even. We take k = 0. Then eqn (6.37) and the corresponding energy integral (with $J - 0 as ][I - co) integrate to give $ as 4(E) = 4 tan-1{[-(2:

‘)]“2sech [}

(6.37)

first

(6.38)

along with the pseudo dispersion relation ~~*(~,

Q) = (G2 - Q”) - f[2 + {(a, Q)] = 0

or, explicitly, o”T”“(fi, Q) = (@ - @)(n2 - S$.) + &‘Q2 = 0

(6.39)

where [compare eqn (3.23)2] flR = 1 -t Q2.

(6.40)

1396

G. A. MAUGIN

and

A. MILED

A graphical representation of the solution (62, Q) of eqn (6.39) is given in Fig. 5a. Only that part of the bottom branch (a) situated below the straight line 0 = nL corresponds to red solutions. Unfortunately, according to the linear-stability theory criterion [21], these are unstable for they correspond to points such that Ii/Q > 1 with a change in sign of d@/d[. (ii) Chse where k i.s odd. We choose k = 1. Then with *-*EL

after some calculations

we obtain

the solution

$([) = 4 tan-1{[&]“2sinh

together

with the following

(6.41)

.$-Tee

as

psrudo

dispersion

@jrd(fi, Q) = (fig-

I}

(6.42)

relation:

Q’) - f(2 - <) = 0

or. explicitly. Dgd(s,

Q) = ( fi2 - @)(fi’

- L!;) + ;a2Q2

= 0,

(6.43)

which is formally the same as (6.39) although the solutions are different. On the graphical representation in Fig. 5b, only that part of the bottom branch (a) situated above the straight line 12 = nL corresponds to real solutions, which are unfortunately unstable according to the linearized-stability criterion [21]. The longitudinal and transverse “elastic displacements” f and g and the corresponding compressional and shear stresses Z XXand Zxr can be calculated from eqns (6.28). It is found that

VLQ J-=.1; ~

g = go +

(Q2

_

p,

@?

\JZClQ (92

_

Fig. 5. Pseudo dispersion

Q’,)

I@?

relation

tan’

[ (-&)‘12tanh

tan-’ [( - [/2)‘/2cosh

for N&e1 walls in micropolar

t]

[]

elasticity

(6.44)

(6.45)

1497

Solitary waves in micropolar elastic crystals

a2 2CYn2 xx = ($I2 - Ri) a2 + sinh2[ ’

a2_2

z z

(6.46) r

a sinh t; 2a!f12 xy = (n2 - Q2’T> a2 + sinh2[ f

(6.47)

The thickness of the wall can be estimated as (6.48)

djv = 7r[2/(1 - 2{)]“2.

The true displacements and stresses are computed by returning to U, V, etc. For instance, one obtains

W3 =

G{h+

VLYQ

(a2

_

ni)

(C- W2tan-’[(&)“2tanh VZYQ

-

G{go

+

(02

_

fp)

(11 z:L ?$:

({ - 2)“2tan-‘[(-{/2)1’2cosh

(6.49)

{]

and ECYQ ni) (c - 2)“2tan-‘[(2/(r

“6 + (n2 _ vu = \/-JJ

- 2)‘j2tanh E]

V&Q (Q2 _ n2) (!: - 2)1’2tan-‘[(-U2)“2cosh T

This completes the solitary-wave representation (5.18),] in a micropolar elastic crystal. 7. CONCLUDING

f” “nh t a + smh [ []

a2 - sinh2t a2 + sinh2[ .

(6.50)

of a “N&e1 wall” [with limit conditions

REMARKS

The possible existence of solitary waves in nonlinear micropolar elasticity has been established in the foregoing sections. In particular, such exact dynamical solutions which relate two asymptotic states and correspond to an exact compensation between nonlinear and dispersive effects, provide the notions of Bloch and Neel walls in micropolar bodies by analogy with what is already known in ferromagnets and ferroelectrics. A stable solution has been obtained in the case of a Bloch wall in which the axis of propagation is also the axis of the large-amplitude microrotation. Within the present scheme of approximations, no stable solitary-wave solution was exhibited for the case of a NCel wall where the axis of the large-amplitude microrotation is orthogonal to the direction of propagation. This kind of instability also exists in ferromagnets with an easy plane of magnetization and this was proved both analytically [23-241 and numerically [25], leading to a warping of the wall and its transformation from one type in the other one. This means that, often, it is not sufficient to consider the case ofpure Bloch and NCel walls; the general case where both microrotations of the composition (3.3) must be regarded as nonzero and possibly of finite amplitude, has to be considered. The expressions of eqns (3.13) and (3.14) give an idea of the complexity of the analytical problem that one then has to face. Other difficult points or points which need further clarification in the present analysis are worth mentioning. First, in the description (2.3 1) and (2.36) of nonlinear micropolar elasticity we have used absolute measures of finite strains and not relative ones as is usual in classical nonlinear elasticity. This has, in turn, obliged us to impose constraints concerning the coefficients of eqn (2.36) to be able to define a natural reference state. These constraints appear to be somewhat artificial [for instance, the kind of “Stokes’s hypothesis” (2.37) without which we would not recover the linearized theory]. There is therefore need to introduce good relative finite-strain measures in micropolar elasticity. Also, two working

1498

G. A. MAUGIN

and

A. MILED

hypotheses may not be fully justified, in particular, the neglect of terms such as (6.8) once circular components have been introduced in the Neel case, and the very strong condition (X = 0) used to set forth the simplified system (6.20)-(6.2 I). In the latter case only pragmatism and the will to arrive at a manageable system that closely resembles the one already obtained by us in the case of elastic ferromagnets justify such a strong hypothesis which one doubts to be realized for many materials. Finally, isotropy has been considered as material symmetry while we often speak of the applicability of the present work to the case of ctystals, which are much less symmetric. Here also, isotropy has been considered for the sake of simplicity in the nonlinear theory (because representation theorems are at hand and easily applied), but it is understood that the present scheme essentially applies to elastic crystals that exhibit a truly independent rotational microstructure as is the case for crystals with molecular groups whose prototype may be chosen as potassium nitrate KN03. The good correspondence between this physical example of material and micropolar elasticity is already known for the linearized theory [ 141; also [26]. Possible extensions of the present work obviously include the study of the influence of external stimuli on the solitary-wave motion. Of particular interest here are those stimuli that directly excite the additional rotational degrees of freedom, hence those which can act through volume couples. Such external stimuli can be operative, for instance, if the rigidly rotating microstructure carries rigidly with it an electric dipole (this is the case in molecular crystals such as sodium nitrite NaNOJ or a magnetic dipole. Then applying either an electric field or a magnetic field-this held being spatially uniform but possibly time dependent-will cause a volume couple in the region of the wall where strong spatial disuniformities in the dipole exist. The “sum” of these elementary couples will act as a “motive power” for the wall. For instance, if one supposes that the microstructure at KR = K, (X = -CL) in Fig. 1 carries an electric dipole PO along the Y-axis, and one applies an electric field E0 in this direction, then within a pure Neel wall there will act a volume couple given by &P,,sin B or E0P0sin($/2), which must be added to the right-hand side of the relevant equations, e.g. eqn (6.19) or (6.21). If ]E,$ is constant in time, then we see that in the first case the applied couple has for effect to modify the coefficient of sin 19in the sine-Gordon operator, hence to change the definition of 6 and to “stiffen” the wall. In the second case (eqn (6.2 1)) this new term has for effect to transform the sine-Gordon operator into a double sine-Gordon operator. Whenever IEo] is a function of time, then one must have recourse to perturbation methods to study the forcing of the wall motion by the applied couple. Several methods are known [27] for perturbations of the sine-Gordon soliton dynamics. For the perturbation of systems such as (6.19) or (6.2 l), one can apply the various methods of perturbation (modulation of free parameters and use of the energy conservation at the first order, Whitham’s averaged Lagrangian method, numerical study) already considered by Maugin and Pouget for another case [28-301. The same scheme applies if the rotating microstructure carries a magnetic moment M0 and a magnetic field Ho is applied. Then the couple I in eqn (2.30) is M0 X Ho/p instead of PO X Eo/p. Another problem amenable by perturbation theory would be the influence of dissipative effects (classical viscosity and rotational viscosity) on the properties of the obtained solitary waves. .4cknowledgements-The Materials” of A.T.P. France).

present work was supported by Theme “Nonlinear “Mathtmatiques Appliqutes et Methodes Numeriques

Waves in Electromagnetic Elastic Performantes” (M.P.B.. C.N.R.S..

REFERENCES

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and A. C. ERINGEN, Deformable Magnetically Saturated Media-i-Field Equations. J. Math. Phys. 13, 143-155 (1972). [2] A. ASKAR, J. POUGET and G. A. MAUGIN. Lattice Model for Elastic Ferroelectrics and Related Continuum Theories. In The Mechanical Behavior qfElectromagnetic Solid Continua (ed. G. A. Maugin), North-Holland, Amsterdam, pp. 151-156 (1984). [3] G. A. MAUCIIN, Elastic-Electromagnetic Resonance Couplings in Electromagnetically Ordered Media. In Theoretical I yd Applied Mechanics (eds. F. P. J. Rimrott and B. Tabarrok), North-Holland, Amsterdam. pp.

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(Received 3 1 December 1985)