Mikropolar ortamlarda nonlineer dalga yayılması
Nonlinear wave propagation in micropolar media
Yazar: SAADET ERBAY
Danışman: PROF.DR. ERDOĞAN S. ŞUHUBİ
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü
Derece: Doktora, 1988
Konu: Fizik ve Fizik Mühendisliği = Physics and Physics Engineering
Dizin: Asimptotik analiz = Asymptotic analysis ; Dalga yayılması = Wave propagation ; Mikropolar ortam = Micropolar media

Özet:
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-

Abstract:
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. 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= |w| exp(i6) (6= constant). Assuming w and its deriv atives tend to zero for Ç-00, |w| is 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,