Mikropolar ortamlarda nonlineer dalga yayılması

Abstract

ÖZET Bu çalışmada, nonlineer mikropolar ortamlarda bir boyutlu dalga yayılması problemi asimptotik olarak incelenmiş, asimptotik analiz sonucunda elde edilen Kompleks Modifiye Kortewegde Vries denkleminin integre edilebilirliği tartışılmıştır. Çalışma beş bölümden oluşmaktadır. Birinci bölümde nonlineer dalga denklemleri kısaca gözden geçirilmiş, nonlineer dalga hareketleri sınıflandırılmıştır. Hiper bolik ve dispersif olarak sınıflanan denklemlerin çözüm yöntemleri anlatılmıştır. İkinci bölüm, nonlineer mikropolar katıya ait alan ve bünye denklemlerinin özetlenmesine ayrılmıştır. Üçüncü bölümde nonlineer alan denklemlerini incelemek için seçilen İndirgeyici (Redüktif) Pertürbasyon Yöntemi tanıtılmıştır. Dördüncü bölümde bir boyutlu alan denklemleri İndirgeyici Pertürbasyon Yöntemi ile incelenmiş, denklemlerin uzak alan davranı şını karakterize eden küple nonlineer evrim denklemleri elde edil miş, katının izotrop olması durumunda evrim denklemlerinin Kuple Modifiye Korteweg-de Vries denklemlerine indirgendiği gözlenmiştir. Bu duruma ait soliter dalga çözümleri aranmış ve özel durumlar tartışılmıştır. Beşinci bölümde KMKdV denkleminin integre edilebilirliği tartışılmış, bu anlamda Painleve analizi yapılmıştır. -IV-

NONLINEAR WAVE PROPAGATION IN MICROPOLAR MEDIA SUMMARY In this work, the plane wave propagation in nonlinear micropo- lar solids is considered. The reductive perturbation method is used to examine the plane wave propagation in weakly nonlinear dispersive media. The theory of microelastic solids is developed by Eringen and Şuhubi [2], [3]. Such solids are affected by the local deformations, not encountered in the classical elasticity, of the material parti cles in a volume element. The aim of the theory is to enlarge the limits of classical continuum theory so that it can explain the mechanical properties in a volume element AV smaller than some critical volume element AV*. It is assumed that a material point in a microelastic solid has six degrees of freedom, three of them related to micro rotations and the other three related to an affine deformation. Micropolar solids which constitute a subclass of microelastic solids are named as couple stress theory in the works mentioned above. Later Eringen recapitulated and renamed it micro- polar theory [5]. A material particle in a micropolar elastic medium has only three degrees of freedom associated with micro rotations. General nonlinear theory of micropolar media is given by Kafadar and Eringen [6]. Throughout this work field equations given in [6] are used. In material coordinates, equations of motion in the absence of body forces and body couples may be written as TKk,K = PoVk `Kk.K + E, x., TT, = p o, kmn m,K Kn *o k where TK is the first Piola-Kirchhoff stress tensor, p is the mass density associated with undeformed state, v is the velocity, Mj. is the material couple stress tensor, p o is the spin density. Stress and couple stress tensors in material coordinates are defined as 82 az TKk = ~ ' MKk = Amk ~ 8xk,K 9Vk -1 where A is given as -1 e e e e «pm9k 1 1/2 A. = - cot - 6, + ( 1 - cot - ) 5- + - e. <p, 6 = (cp <p ) mk 2 mk _ 2 8 2 n arm and £ is free energy density per unit mass in reference configuration. -V-Finally, field equations for a general nonlinear anisotropic polar medium may be written as 8 az 9XK d/,K P x, *o k a az az ax a<p a<p. K 9k,K vk where X and *y.. are defined as *ki = Po JKL Ark ( XrK XmL Am£ + XrK XmL Am* + XrK XmL /l ) /SL ~- po JKL Ark XrK XmL Am£ for microanisotropic materials, and X,,:p JA. A,, y, » = P J A, A. ki o mk mZ ' 'k£ o mk m£ for microisotropic materials. Here A is given as sine sinG `'Pj.'Pn 1-cos6 Ak£= - 6k£ + (1- - )7-`(~7`)e^v In the present study, the plane wave motion in a homogeneous, macro- and microisotropic micropolar solid is investigated. This motion may be described as xk(X,t) = XK6Kk + uk(X,t) where X is a coordinate along the direction of propagation. In the reference state equations of motion may be written as = 0 ax at -VI-a2 2 ap£ a2 2 a2 z a<p£ 82 aw£ a<j,kaPje ax e*^ * e^e^ ax a<pk at aw a* - - - = o ax at a<pk w, - = 0 k at where v., p., w. and <» are.defined by relations 8uk 811 8<pk 8<Pk at K ax at ax In the region where the long acoustical waves propagate the dispersion relation has the form tu= ak + bk +..., k-0 and the phasor of waves is kX - cot = k (X - at) - bk5 t + 0(k5 ) where a and b are constants, the relevant Gardner-Morikawa coordi nate stretching is taken as £ = e1/2(X-at), T = e3^2t where e is & small parameter measuring the weakness of dispersion, i.e. E=0(k ). In order to balance the dispersive effects with nonlinear effects, we expand the dependent variables about the undeformed uniform state as a power series of the same parameter e: ` (n) (n) (n) (n) (n) _ n (pk' V V V V = n?1(pk' V V V V (E ) After using the suitable coordinate stretching and expanding dependent variables in half powers of e, it is arrived at sets of equation for each power in e-*' - Eliminating the higher order terms in these equations, it is finally found the following third order coupled nonlinear dispersive equations -VII-ap2 (poJa2-T) 83p2 i a ap3 aP + - - î- + (a.p, + a p ) 8t 8p a &V Ipa 8Ç 3Ç 8Ç o o 1 8 5 » 2 ` 2 ` D3 4p a aç o (B1P2 + B2P2P3 + B3 P2P3 + B4B3) = 0 2 3 8p, (P0Ja -Y) 8 P* 1 9 8P2 9P2 + r- + (a p + a p- ) ax 8p a ar ^p a aç aç ' aç + - (~^?l * B2p3p2 + B3P3P2 + ^p3} = ° ho a 9Ç where a..,a?,&,, 0, are material constants containing derivatives of free energy function with respect to p, <p., <p. Because of the isotropy o'f the medium there exist some restrictions on the admissi ble forms of 2, direct approach in employing isotropy of the medium is to assume that I is a function of all invariants of its arguments under the orthogonal group and calculate all the_relevant derivatives thereafter to obtain coefficients a a, B.,..., 3,. However since this way is quite tedious and provides in `the meantime several informa tion which we do not need in the analysis we have adopted instead an indirect approach exploiting the effect of isotropy on the field equations proper. If the solid is isotropic, the equations satisfied by the pair (P2, P* ) must be the same equations satisfied by the pair (p2,pi) which can be obtained by rotating plane coordinates with an arbitrary angle a. In this way we obtain that for an iso tropic solid we should have a1 = <x2 = 0, B1 = B3 = g1 = İ3, e^e^ = -($2 = -^. Nonlinear evolution equations may then be written as ap2 (poJa2-ir> a3p2 1 a 3 2 23 - + - - ¦ -3-+ - - tVp2 + P2P3^> + B2(p2p3 + P3)] = ° 8t op a 8Ç 4p a 8Ç o o 8p (p - JaZ-Tf>- 93P, 18, ? ?, 77*-irr-i?*^Tirhi^*p'Pi)-,i(p'PitP')1-0- o o These equations are of appropriate invariant form under the proper orthogonal group. Therefore they are valid for hemitropic solids. For isotropic solids the invariance is under the full orthogonal group, that is, reflections are allowed. Hence if the pairs (p,.-p, ) -VTT-and (-p.,p, ) both satisfy the evolution equations, it is easily seen that the coefficient B« should vanish. Thus for a general iso tropic solid, evolution equations take the following form 8p2 a, a3p2 a 2 + a (p? ) + B =- + a (P?Px ) = ° ax aç açp aç ? aP a, a3P, a 2 - - + a (p,) + B - - =-+a (p,p2) = 0. at aç 3 &v aç These equations are called as Coupled Modified Korteweg de-Vries equations. If a complex function w is defined as w = p_ + ipc where p2 and p, are real, the equations may be written as a single one aw a 2 aw + o ( w w) + B 5- = 0. 8t aç dV It can be named as Complex Modified Korteweg-de Vries (CMKdV) Equation. In order to find a solitary wave solution to CMKdV equation we try the form w= wexp(i6) (6= constant). Assuming w and its deriv atives tend to zero for Ç-00, wis found as <* 1/2 1 2 w= asech[( ) a(Ç aa t)+6] 2B 2 where a and 6 are defined as 2 2C *, * 0/2 a =, 6 = ( ) ax. a 2B For p_ and p,, expressions written below are valid a 1/2 1 2 p? = a sech [ ( ) a(Ç aa x) + 6] cos 6 2& 2 a 1/2 1 2 p, = asech[( ) a(Ç aa x) + 6]sin6. 2B 2 They are reminicent of plane polarized waves in finite elasticity [2], If w, amplitude of wave is taken as a constant value, then the phase function 6(Ç,t) will become -IX-e(Ç,T)sB [Ç-(«A^- BB2)x] o o o where A = l w I and B are constants. In this case p` and p_ are o ' ' o 2 3 p, = A cos [B (Ç- (aA2- 3B2) t)] l O O o o p, = B sin [B (Ç- (aA2 -& B2)t)] 3 o o o o Similar to the case mentioned above, they are reminicent of the circularly polarized waves in finite elasticity [2]. For quadratic solids, _second order longitudinal displacement gradient p'^is e- qual to Y (Po + P? 2)and nas a solitary wave solution: 1 (Z) -2,2r, `,1/2,_ 1 2 x Al p =ifa sech L( ) a(Ç aa t) + 6J. 1 20 2 At the end of these calculations, it is observed that first order components pjj y and my' are equal to zero, and first order shear components (pi1', py') and (ay J, <pi^) satisfy coupled Nonline ar Evolution equations (Coupled MKaV Equations). These couples have both solitary wave solutions which are similar to plane polarized waves of classical finite elasticity. If a solid is taken as a quadratic one, the second order longitudinal displacement gradient pir' has also solitary wave solution. In the last section integrability of CMKdV equation is dis cussed. Related to the integrability of nonlinear partial differen tial equations (NPDE) Painleve Property is defined. Painleve proper ty is discussed for both NPDEs and Nonlinear Ordinary Differential Equations (NODD). In order to verify whether CMKdV equation is integrable, Painleve analyses are carried out for both situations. In both analyses, CMKdV fails to pass the Painleve test. This result conforms with Carney, Sen and Chu's observations [29]. They obtained CMKdV while they were studying electrostatic waves in a magnetized plasma. They observed that there were only four constants of motion. This situation is in contrast to equations soluble by the inverse scattering method which have an infinite number of constants of motion. -X-