İki tabakalı elastik ortamlarda nonlinear dalga modülasyonu

Abstract

Bu çalışmada iki tabakalı bir elastik ortamda nonlineer dalgaların yayılmasını modelleyen bir sınır değer probleminin asimptotik çözümü inşaa edilmiştir. Çalışma beş bolümden oluşmaktadır. İlk bölümde önce elastik dalga yayılması problemlerinin tarihi gelişimi kısaca özetlenmiştir, ikinci bölümde. nco-IIookean malzemelerden meydana gelen iki tabakalı bir ortamda genelleştirilmiş kayma dalgalarını yöneten hareket denklemleri ve onlara eşlik eden sınır koşullan verilmiştir. Üçüncü bölümde böyle bir ortamda nonlineer kayma dalgalarının modülasyonu bir singüler pertürbasyon yöntemi yardımı ile incelenmiş ve bu dalgaların self modülasyonunun asimptotik olarak bir nonlineer Schrödinger (NLS) denklemi ile karakterize edilebileceği gösterilmiştir. Bilindiği gibi NLS denkleminin çözümlerinin davranışları, denklemin katsayılarının işaretine kuvvetli olarak bağlıdır. Çözümlerin nonlineerliğe bağlılığını incelemek için, tabakaları meydana getiren malzemelerin lineer özellikleri sabit tutulmuş, nonlineer sabitler değiştirilerek katsayıların dalga sayısına göre değişimleri elde edilmiştir. Bu sonuçlardan zarf soliton tipi dalgaların varlığının tabakalı yarım uzayın nonlineer yapısına bağlı olduğu görülmüştür. Ayrıca, c\ ve c-ı tabakalardaki lineer yayılma hızlarını, c 'de sistemdeki kayma dalgalarının faz hızını göstermek üzere c\ < c < c-ı için bulunan sonuçların, ikinci tabakanın kalınlığı sonsuza götürüldüğünde, daha önce tabakalı bir yarım uzayda Love dalgalarının modülasyonu için bulunan sonuçlara dönüştüğü gösterilmiştir.

In tli is work, we have considered Uie propagation of small but finite am plitude waves in a plate of uniform thickness which is composed of two layers occupying the regions A = {(XuX2,X3)\0 < X2 < hx,-oo < (XUX3) < 00} (la) and P2 = {{XUX2,X3)\ - h2 < X2 < 0,-oo < (XUX3) < 00} (16) in the reference frame Xk, k = 1,2,3; where Xk denotes the material co ordinates of a point referred to the rectangular Cartesian system of axes. It is assumed that the layers have different material characteristics and stress and displacements are continuous at the interface X2 - 0 and the bound aries X2 - h\ and X2 = - h2 are free of tractions. Then, a shear horizantal (SH) wave described by the equations xk = XK8kK + u.:;(XA, t)6k-, v = 1, 2 (2) is supposed to propagate along the Xi-axis in the media. In (2) the super- cipts v refers to the layer P\ and P2 respectively; u^ is the displacement of a particle in the ^-direction in a layer, Xk are the spatial rectangular coordinates, t is the time and 8ki< is the Kronecker symbol. The summa tion convention on repeated indiced is implied in (2) and in the sequel of this section, and Latin and Greek indices have respective ranges (1,2,3) and (1,2) Let tki be the Cauchy stress tensor field accompanying the deformation field (2); in the absence of body forces, the equations of motion in the reference state take the following forms iv v iv ri ıv u ıu n lll,l ~~ u3,llll,3 - u.> l22,2 u3,2l22,3 ~ u ^13,1 + ^23,2 "I" ''33,3 = PoU3 (o.C, 0, C) where subscripts preceded by a comma indicate partial differention with re spect to coordinates Xk and an over-dot represents the partial differentiation with respect to t. When the constituent materials of the media are hyperelastic then there exit strain energy functions £" characterizing the mechanical properties of the materials. If the material are homogeneous and isotropic then E" are functions of the principle invariants of the Finger deformation tensor c^1 = Xk,K%i,K ? VI Here it is assumed that the constituent materials are homogeneous, isotropic and incompressible and their strain energy functions are of the form E" = E"(7"). (4) Where r = ire"1 = 3 + «3V.3.A- (5) This class of materials is called generalised neo-llookean and for such a material the stress constitutive equation can be expressed as tki = -phi + W (6) where p(Xx,t) is an arbitrary pressure function, and $ = 2Ş>0. (7) For the deformation field (2) the component of the Cauchy stress tensor arc found to be tap = 0, t a3 = *«3l«, *33 = («3,1 + Uh) $ > P = $ (8) Hence it is seen that the first two equations of (3) are satisfied identically, and the third gives A{$( Dibi) + _L(*< A = p^ 8XK dX> ^ dYK dY' Podi2' (9) dX{ dX)+dY{ dY' PodV { ' where X = Xi, Y = X2, Z = X3, u = ul, v = u2. (11) The conditions of vanishing tractions on the free surfaces and continuity of stresses and displacements at the interface yield the following boundary con ditions accompanying the equations (9-10); du dY=° °n u = v and *<»!£ = *W*L dv w = 0 °n The problem under consideration is the nonlinear self modulation of a group of waves centered around the wave number k and the frequency u),i.e., a quasi- monochromatic wave train. The harmonic-resonance phonemane is excluded. The amplitude of the wave is assumed to be small but finite. Therefore the vn problem is investigated by employing the method of multiple scales. Following the usual procedure of the method u and v are expanded in the following asymptotic power series in a small parameter e > 0 which measures the degree of nonlinearity and, at the same time, the narrowness of the side-band width of the carrier wave number centered around a specific wave number; u = ^2 cn«n(*o, si, «a, y, *o, tuU) n=l oo v = JŞ2envn(x0,xux2,y,t0,tı,t2) (15) n=î where Xi = iX, U = cH (16) are the multiple scales introduced to specify the slow variations of the ampli tude compared with the phase of the carrier waves, and y = Y. Hence, writing first the equations and boundary conditions (9-14), in terms of the new independent variables (3.9) and collecting the terms of like powers in e yield a hierarchy of equations and boundary conditions from which it is possible to determine un and vn successively. Up to third order in e these are given as follows 0(e): 41)«i = 0, 4V = 0 (17a, b) -5- = 0 on y = hi (17c) ay ıtı = v\ and - 7- - = 0 on y - 0 (17d, e) dy dy (17/) (18a, b) (18c) on y = 0, (18a*, e) (18/) uy 0(e3) : 4X)«3 = C^u2 + 41)«i + »iJvo(ut) (19a) viii C{2)vz = Ö2)v2 + £22)Vl + n2JV0(ı>ı) (196) -5- = 0 on y = /ii (19c) u3 = u3 and ^ = 0 on y = -h2 (19/) where £0 ', £0 ',... C2 are linear and Q0 ve jV"0 are nonlinear differential operators defined respectively as C^^-^±-c2(^t + ^t) *-12 L° ^~ dig H&rg + oyW ' M ^U of ^ c2 ^ 1 **-[Ş+«â-î(3+ı&]' (20) *w-(£)'+(S)'. Here c", 1/ = 1,2 are the velocities of shear waves, and nv are nonlinear material parameters of the layers and they are defined as; cl = HuİPvQ, n" = 2EW"(3)/rt (21) and /z" = *"(3), 7 = ^1, Pv = n"/4 (22) Note that, as usual in these types of asymptotic analysis, the problems at each step are linear. Moreover the first order problem is simply the clasic linear wave problem. Accordingly we take the solutions of (17a,b), for ci < c2 < c, as; 00 Ul = Y^{Ai)(x^x^tut2)eilkny + B{t\xl,x2,tuh)e-ilkrnv}J1'1, + c.c. (23a) /=i 00 t;1 = ^{C{'î(xı,xa,f1,i3)c,7^ + i?S,?(sı,x3,<,1<0e-i'*ww}e,"'* + c.c. (236) 1=1 where = kx0-u;t0, pu = (c2/cl-l)1/2 (24) IX and A\, B\, C\' and D[' are functions of slow scales (xı,x2,tı,i2), k is the wave number, u> is the angular frequency, c = uj/k is the phase velocity and c.c. denotes the complex conjugate to the preceding terms. Then the substitution of (23a,b) into the boundary conditions (17c-f) yields where and W/ = (25) (26) (27) Note that detWj = 0 gives the dispersion relation of the linear waves, i.e. pi tan(fe/iipi) + 7p2 tan(fe/i2p2) = 0. (28) Since the harmonic resonance phonemona is excluded in the analysis then for I > 2 detW,^0 (29) Hence the solutions of the homogenous algebraic equations (30) are found to be V^) = A1{xt,x2,tut2)R U?) = 0forZ>2 (30) where Ai is a complex function repesenting the first order slowly varying am plitude of the wave modulation, to be determined in higher order perturbation problems, and R is a column vector satisfying W!R = 0 (31) The solutions u2 and v2 of the second order problem can be sought as u2 = u2 + u2, v2 = v2 + v2 (32) where v,2 and v2 are the particular solutions of nonhomogeneous differential equations (18a,b) while u2 and v2 are the solutions of the homogeneous equa tions 42)«2 = 0, 42)?2 = 0 (33) satisfying the following nonhomogeneous boundary conditions defined from (18c-f) by considering the decompositions (32); du2 _ dv,2 dy ~ dy on y = ht, (34a) u2 - v2 = -(u2 - v2) and (346) du2 dv2 (dü2 dv (Ou2 ov2\ on y = -h2. (34d) dy dy \ dy dy dv2 dv2 dy dy The solutions u2 and v2 are found by using the method of undetermined coef ficients. For u2 and u2 as in the first order problem we let oo «2 = X)[4',(^,^,ii,Me^,* + ^(x1)x3,«I,*0e-,7fcp,v]e"* + c.c. (35a) (=1 oo «a = ]T)[Cf (*i, *2, U ' t2Wlkp2V + D{l\xx, x2> U, t2y-iXkTYl* + c.c. (356) Then the use of (35) together with the solutions Jl2 and v2 in (34a-d) yields WiU2l) = b2l) (36) where Since det Wi = 0 and b^ ' ^ 0 in order that the equation (36) is algebraically solvable for U^ ' the compatibility condition L.bt = 0 (38) must be satisfied where L is a row vector defined by LW1=0. (39) The condition (38) leads to the result ^+^ = 0, V. = % (40) dh 9 dxx ' " dk r(i) and then XJ2 is found to be where A2 = A2(xl,x2,ti,t2) is a complex function representing the second order slowly varying amplitude of the wave modulation, and it can be de termined from higher-order perturbation problems. But, since this work is centered around the weakly nonlinear waves the aim is here to obtain just the uniformly valid first-order solution. Therefore it is sufficient to obtain A\, and this will be done at the third order. (40) only implies that the amplitude XI remains constant in a frame of reference moving with the group velocity Vg of waves. That is, A{=Ax{xx-Vgti,x2,t2) (42) Note that, for / > 2 since it is assumed that detWj ^ 0 for / ^ 1 and since b2 = 0 for I ^ 1, then l4'J = 0 for / > 2 (43) Thus the solution of the second order perturbation problem is completed. The solutions of the third order problem can be decomposed as in the second order, i.e. «3 = «3 + «3 ve v3 = v3 + v3 (44) The particular solutions of the nonhomogeneous differential equations (19a,b) are found by the method of undetermined coefficients. For «3 and v3 as in the previous case we let 00 «3 = ^[40(^,^,<ı,^e,^ıw + 4')(^ı.^,«ı,<3)e"''*pıl']e*'* + c.c. 1=1 00 v3 = J2 [Cf(*i, *2, h, t2)eilkT+Dİl)(xux2, h, t2)e-ilkT] eil*+c.c. (45a, 6) 1=1 Then the boundary conditions (19c-d) yield: W,Uj° = b3° (46) where b3° = 0 for all/ ^ 1,3, b33) ^ 0 and b3n is defined as b(D =./dW1dA2 _ dWldA2\R./9WiMt _ 8W1dA1\Jl 3 \ du dti dk dxi J \ dw dt2 dk dx2 ) l/d2Wld2Al d2Wy d2At d2W1d2Al\ + 2\ du* dt2 dudkdxidh+ dk2 dx\) fdW, d2At mx d2Ax \(9R ÖRn 2 H dk dx\ ~ &, dx1lH1)Vdk+V'fo)+mi]Al (4?) Here F is a constant vector. The explicit form of the vector b3 ' is not given, since in the sequel it will not be required. For / = 1, in order that (3.38) is algebraically solvable for U3 ' the com patibility condition L.b^ = 0 (48) must be satisfied. If we assume that A2 depends on x\ and t\ through the combination x\- Vgt\ as A\ then the compatibility condition yields the following equation for A\\ C^H^*^-0 (49) xn where By defining the following non-dimensional variables and constants T-uiii, ( = ke~1(x2-Vgl2) = k{xi -Vgh) A=kAt, r = ifc2r/u;, A = A/uk2. (51) (49) can be written in the standard NLS equation form as BA B2A i^ + r^ + A\A\*A = 0. (52) Thus our task is completed, since a solution for A is derived from (52) for a given initial value of the form A(t,0) = AQ(O (53) then the first-order solutions u\ and v\ can be constructed by (23). Note that the initial value for A is related to the initial values of displacements in the layers via (23). The analysis is also carried out for the case in which c\ < c < c2. The nonlinear modulation of the waves is also governed by a NLS equation whose coefficients F and A are different from the ones given in (50). It is also observed that in the limit hi/h\ - ? oo the NLS equation obtained here recovers the result for the nonlinear modulation of Love waves. It is well known that the criterion whether FA > 0 or FA < 0 is impor tant in determining how a given initial data will evolve for long times for the asymptotic wave field governed by the NLS equation. An initial disturbance vanishing a.s |£| -* oo tends to become a series of envelope solitary waves if TA > 0, while it evolves into the decaying oscillations if FA < 0. As the properties of solutions strongly depend on the sign of the product TA, the variation of it with the nondimensional wave number kh\ is evaluated for the lowest branch of the dispersion relation giving appropriate values to the material constants. As a result of the numerical evaluation of TA for fixed linear material properties, it is observed that the envelope solitary waves may exist depending on the nonlinear constitution of the layered media.