Tabakalı Bir Hiperelastik Yarım Uzayda Nonlineer Yüzey SH Dalgalarının Yayılması

Abstract

Bu çalışmada farklı elastik özelliklere sahip uniform kalınlıklı iki tabaka ile kaplı bir yarım uzayda nonlineer yüzey SH dalgalarının yayılmasını modelleyen bir sınır değer probleminin, bu ortamdaki çeşitli yayılma olaylarını karakterize eden çözümleri, değişik ölçekler yöntemi olarak adlandırılan bir asimptotik yöntem kullanılarak elde edilmişlerdir. Çalışma dört bölümden oluşmaktadır. İlk bölümde elastik dalgaların ve elastik yüzey dalgalarının yayılması ile ilgili problemlerin incelenmesinin tarihi gelişimi özetlenmiştir, ikinci bölümde önce, iki tabakalı bir elastik yarım uzayda nonlineer yüzey SH dalgalarının yayılmasını yöneten hareket denklemleri ve onlara eşlik eden sınır koşulları verilmiştir. Bu bölümün sonraki kısımlarında cx, c2 sıra ile en üst ve ara tabakayı meydana getiren ortamlardaki, c3 de yarım uzayı meydana ge tiren ortamdaki lineer kayma dalgalarının hızlarım göstermek üzere, bu hızlar arasında c\ < c2 < c3 ve c2 < c% < c3 eşitsizliklerinin sağlanması hallerinde lineer dalgaların yayılması incelenmiş ve dispersiyon bağıntıları türetilmiştir. Böylece daha önce çeşitli araştırıcılar tarafından elde edilen sonuçlar özetlenmiş tir. Daha sonra c\ < c2 < c3 eşitsizliğinin sağlandığı yarım uzaylarda, seçilen farklı tabaka kalınlık oranları, cj., c2, c3 hızları ve diğer lineer malzeme parametreleri için dispersiyon bağıntıları hesaplanmıştır. Üçüncü bölümde c\ < c2 < c3 eşitsizliğini sağlayan bir yapılanmaya sahip iki tabakalı bir nonlineer yarım-uzayda yüzey SH dalgalarının modülasyonu problemi değişik ölçekler yöntemi ile incelenmiş ve bu dalgaların self modülasyo- nunun asimptotik olarak bir nonlineer Schrödinger (NLS) denklemi ile karek- terize edilebileceği gösterilmiştir. Çeşitli limit durumlarda bu denklemin kat sayılarının daha önce tek tabakalı bir yarım uzay için verilen NLS denkleminin katsayılarına dönüştüğü gözlemlenmiştir. Çözümlerin nonlineerliğe bağlılığını incelemek için, tabakaları ve yarım uzayı meydana getiren malzemelerin lineer özellikleri sabit tutulmuş, nonlineer sabitler ve tabakaların kalınlık oranları değiştirilerek bu katsayıların dalga sayısına göre değişimleri nümerik olarak hesaplanmışlardır. Dördüncü bölümde aynı yönde ilerleyen iki yarı-monokromatik yüzey dal gasının etkileşimleri gene değişik ölçekler yöntemi kullanılarak incelenmiştir. Etkileşim olayını asimptotik olarak karakterize eden ve iki küple denklem den oluşan bir nonlineer denklem takımı türetilmiştir. Bu denklem sistemi iki yüzey dalgasının etkileşimini karakterize ettiği gibi, eşit grup hızlarına sahip iki yüzey dalgasının ve üçüncü harmonik rezonans hariç, beşinci ve daha yüksek harmonik rezonans koşulların sağlandığında, böyle iki yüzey dalgasının rezo nans etkileşimlerini de karakterize eder. Bu durumda, denklemlerin katsayıları çalışmada türetilen katsayı ifadelerinden kolayca hesaplanabilirler. Üçüncü harmonik rezonans etkileşimi hali ayrı bir inceleme gerektirmektedir. Ek.A nonlineer hiperelastik ortamlarda genelleştirilmiş kayma hareketi ile ilgilidir.

In this work, we have considered the propagation of small but finite am plitude shear horizontal (SH) surface waves in an elastic half-space covered by two different elastic layers each of uniform thickness. It is assumed that, in a rectangular cartesian reference frame (X, Y, Z), the top layer occupies the region between the planes Y = h\ and Y = 0, the intermediate layer occupies the region between the planes Y - 0 and Y = -h2 and the half- space occupies the region Y < -h2 where h\ > 0 and h2 > 0. It is also assumed that the free boundary Y - hi is free of tractions and stresses and displacements are continuous at the interfaces Y = 0 and Y = - h2. Then, in this layered half-space, an SH surface wave (a Love-type wave) is supposed to propagate along the X-axis. The constituent materials of the layered half- space are assumed to be homogeneous, isotropic and hyperelastic. The aim of the present work is to deal with the small but finite amplitude waves; therefore, it is assumed that the wave motion under consideration takes place in media for which the components tn, tX2, t22 of the Cauchy stress ten sor are identically zero as in the corresponding linear problem. To investigate the small but finite amplitude wave motions, proceeding with the approximate field equations instead of the exact ones is more convenient. Therefore, the following approximate expressions for the governing equations and boundary conditions of the problem, involving terms not higher than the third degree in the deformation gradients, are produced; 0 0 as Y - > - oo (7) where u,v,w are the displacements of a particle in the Z direction in the top layer, intermediate layer and half-space, respectively. The constants c", v = 1,2,3 are the linear shear wave velocities and nu are nonlinear material constants. The operator )C(ip) and the constants 71,72 and /3" are defined as 71 = pilpu 72 = ^3/^2, Şu = nvpvlıı". (9) In this work two different wave propagation problems governed by the equations (1-7) are considered. First the nonlinear self modulation of a group of surface SH-waves centered around a wave number k and a frequency u> is investigated. Thus the harmonic-resonance phenomena is excluded in this examination. The amplitude of waves is assumed to be small but finite, and therefore the problem is investigated by employing the method of multiple scales. Following the usual procedure of the method the functions u, v, and w 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 = Ş2 tnun, v = J2 ^v* ' w = 5Z ^n (10) n=l n=l n=l where un, vn and wn are assumed to be the functions of the variables x0, xi, X2, y, to, *i, ti defined as xt = iX, U = eH i = 0, 1, 2 y = Y (11) in which x,- and £,. are the multiple scales introduced to specify the slow vari ations of the amplitude compared with the phase of the carrier waves. Hence, writing first the equations and boundary conditions (1-7) in terms of the new independent variables {xo, x\, X2, to, t\,t2, y}, then employing the expansions (10) 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, vn and wn successively. Up to third order in e these are given as follows IX 0(t) : 4V = O, 42)^ı = 0, 4V = 0 (12a, b, c) -7^ = 0 on y =hi (I2d) oy ui = v1 and - ^~q~ = ° on !/ = ° (12e,/) uı = wx and - ^~q~ = ° on 2/ = ~^2 (12^, /i) iui - > 0 as y - > - oo (12i) c(e2): 41)«2 = 41)«i, 42)«2 = 42)«i, 43)^ = 4!V (13a, 6,c) -z- = 0 on y =h\ (13d) oy u2 = u2 and - 7l- - = 0 on y = 0 (13e,/) u2 = u;2 and - ^"â- = ° on V = ""^2 (13$, A) tu2 - * 0 as y - > - oo (13?) 0(e3) ': 41)«3 = 4V + 4X)«i + n1H0(u1) (14a) 4%3 = 42)U2 + 4V + n2Wo(vi) (146) 43)^3 = 43) ^2 + >C23) ioi + n3?io(tOi ) ( 14c) ^ = 0 on y = h, (Ud) du3 dv3 dvx.. dux -dy--ll-dy-= ^A-^-JCo(ui) - ft-gj-JCoM and U3 = u3 on y = 0 (14e, /) dv3 dw3 öt»! 5uı and U3 = iü3 on y = -h2 (14g,h) w3 - > 0 as y ->? - oo (14i) where 4 > 4 > 4 are linear, "Ho and /Co are nonlinear differential operators, and their definitions are given in Section 3 in (3.11a - e). Note that, as is usual in this type of asymptotic analysis, the perturbation problems (12-14) are linear. Moreover the first order problem is simply the clasic linear wave problem which was first investigated by Stoneley. We will assume that between the linear shear velocities of the layers and the half space the following inequalities are valid c\ < c2 < c3. Then, it is well known that for the existence of a surface SH wave, the phase velocity c of this wave must satisfy either Ci < c2 < c < c3 or Ci < c< c2 < c3. (15) We proceed first by assuming that the first inequality is satisfied by the phase velocity of the surface SH wave. Then by using the separation of variables method and also by employing the condition (I2i), the solutions of the equa tions (12a, b,c) are found to be oo «a =^[A?{x1,x2Jut2ykpiV + B[l){xllx2it1,t2)e-ilkpiV}eil* + c.c. (16a) 1=1 oo vi =Y1 {CP(xi> x*MMylkT + D{l\xu x2, tu t2)e-ilkT}eil* + c.c. (166) oo Wl =J2E[l\xux2,t1,t2)elku3yeil'l> + c.c. (16c) where 2 detW, ^0. (20) Hence, considering (20) the solutions of the homogeneous algebraic equations (17) are found to be V^] = Aı{x1,x2,tı,t2)R, U? } = 0 for />2 (21) where A\ is a complex function, representing the first order slowly varying amplitude of the wave modulation, to be determined in higher order perturba tion problems, and R is a column vector satisfying WiR = 0. By using (21) in (16a - c) the first order solutions ui, vi and w\ are written explicitly as «i = Aı{Rıeikpıy + R2e-ikpiy)e^ + c.c. (22a) vi = At (R3eikT + R4e-ikp'y)ei4, + c.c. (226) wx = A1R6ekvty+i* + c.c. (22c) where Rm, m = 1, 2,..., 5 are the components of R. We nuw proceed to solve the second order problem. The solutions u2, v2 and w2 of this problem are sought as «2 = «2 + «2, v2 = v2 + v2, w2 = w2 + w2 (23) where «2, v2 and w2 are the particular solutions of the nonhomogeneous differential equations (13a, 6, c) while u2, v2 and w2 are solutions of the homogeneous equations £q 'u2 - 0, O0 v2 = 0, Cq w2 = 0 satisfying the nonhomogeneous boundary conditions derived from (13a* - i) by considering the decompositions (23). The solutions u2, v2 and w2 are found by the method of undetermined coefficients. For u2, v2 and w2 the solutions are written as in the first order problem. The second order slowly varying amplitudes A2, B\\ C2\ D2\ and E2 of the surface waves will be determined by using the nonhomogenous boundary conditions. Hence the use of u2, v2 and w2 together with the solutions ü2, v2 and w2 in these boundary conditions yields W, U? = h[l) (24) Xll where Since detWi = 0 and b^ ^ 0, in order that the equation (24) is alge braically solvable for U2 the compatibility condition L.b^ = 0 (26) must be satisfied, where L is a row vector defined by LWi = 0. The con dition (26) leads to the result M,+V.^ = 0, V.= £ (27) &tx a dxx ' 9 dk r(i) and then U2 ; is found to be where A2 = A2(xi,X2,tx,t2) is a complex function representing the sec ond order slowly varying amplitude of the wave modulation, and it can be determined from higher-order perturbation problems. But, since this work is centered around the small but finite amplitude waves, the aim is here to obtain just the uniformly valid first-order solution. Therefore it is sufficient to obtain A\ only, and this will be done at the third order. Note that, the equation (27) only implies that the amplitude A\ remains constant in a frame of reference moving with the group velocity Vg of waves. That is, A\ = Ai(x\ - Vgti, x2,t2). For I >2, since it is assumed that det W/ ^ 0 for / 7^ 1 and since b2 ' = 0 for I ^ 1, the solutions of (24) are U2° = 0, />2. (29) Thus, the solutions of the second order perturbation problem are completed. The solution of the third order problem can be sought as in the second order. That is we decompose the solutions as, U3 = U3 + U3, V3 = V3 + v3, w3 = w3 + w3. (30) The particular solutions u3, v3 and w3 are found by the method of undeter mined coefficients. For «3, v3 and w3 the solutions are written as in the previ ous cases. For the third order slowly varying amplitudes A3, B^', C3, D3, xiii and.Eg ' of the surface waves from the boundary conditions of the third order perturbation problem we get w, uw = bw (31) In (31) hf = 0 for all I ^ 1,3, hf] ^ 0 and b^x) is defined as u(D_ D3 -. (dWxdA2 8W1dA2\.fdW1dAl 8W1dA1 + \ duj dti 1 /a2Wx d2Ax - I -2 + dk dxi a2Wx d2Ax 2 V du2 dt\ ~ dkdu dxxdtx /&Wi (PA _ dWr d2Ax \ (dR \ dk dx\ du dxidtij \dk + \ du dt2 d2W1 d2Ax dk2 dx\ "dR dk dx2 R + F\Ai\'A1 (3) (32) Here F is a constant vector. The explicit form of the vector h^ ' is not given, since it is not required in the construction of the first order solution. For I = 1, in order that (31) is algebraically solvable for U3 ' the com patibility condition L.b« = 0 (33) must be satisfied. If we assume that A2 depends on xx and tx through the combination x\ - Vgt\ as A\, then the compatibility relation (33) yields the following nonlinear Schrödinger (NLS) equation MA ^d2A A., l2 t n with the following definitions t =ut2, £ = ke~1(x2 - Vgt2) = k(x! - Vgt\), A = kA\ k2 0 or TA < 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 as | £ |- » oo tends to become a series of envelope solitary waves if TA > 0, while it evolves into the decaying oscillations if TA < 0. As the properties of solutions of the NLS equation strongly depend on the sign of the product TA, the variation of it with the nondimensional wave number k[h\ + h2) is evaluated for the lowest branch of the dispersion relation giving appropriate values to the materials constants and the ratio hi/hi. As a re sult of the numerical evaluation of TA for fixed linear material properties, it is observed that the existence of the envelope solitary waves depends on the nonlinear constitution of the layered half-space as well as the ratio h2/hi. In the later part of this work the interaction of two co-directional nonlinear surface SH waves having the phase factors a ~ kax-uat and <^ = k^x-uibt, respectively; is considered. That is, the propagation of two co- directional non linear quasi-monocromatic waves centered around the respective wave number ka and k\, is examined. In the analysis, as in the previous parts, the method of multiple scales is employed. Hence by assuming the asymptotic expansions (10), the same perturbation problems given in (12-14) are written. It is first supposed that the harmonic resonance does not exist between the waves. That is, between (ka,uja = u)(ka)) and (&6,a>& = uj(kb)) the following conditions are satisfied; h^Mka, ub^Mua for M ?{3,5,7,...} (36) Hence wo write detWİa) = 0, detw!6)=0 (37) and detW,(tt)^0, detWf^O for l^l (38) where W, is defined as xv wfm) = İlPml cİlPrn\ UPml c-HPml 0 0 0 hi hi UPml UPmi ?v, UPml., UPm2 f| hi hi I1 h2 'l h2 u 1 1 -1-1 0 0 0 İlPm2 Ç~İ1Pm2 İlPm2 CilPm _~ fym3,e-tVm3 h2 h2 '" h2 Q Q e-UPm2 eilPm2 _g-Wrr 'm3 (39) in which Pml = kmpmihi, Pm2 = kmpm2h2, Vm3 = kmVmzh2 (40) and Pmi = (c2Jcl - l)1'2, pm2 = (cl/ct - l)1/2, um3 = (1 - 4/c2)1/2. (41) In the first instant it is assumed that c\ < c2 < cm < C3. Then by taking (37) and (38) into account the solutions of the first order perturbation problem are obtained as +A?{R?eikbPuy + 4V^P6iy)e^ + c.c. (42a) Vl = A[a){Rİa)eikaPa2y + R{a)e-ik«Pa2y)ei,l>a +A{i\RfeikbPh2V + Rfe-ik*pT)ei't'b + c.c. (426) wt = ^la)4a) e*d""sl'+*- + 46)46) e*"*8tf+i*6 + c.c. (42c) where.Aj and.4^ are complex functions of (£1, £25^1,^2) representing the first order slowly varying amplitude of the interacting waves, and ift and Rİ,, n = l,...,5; are the components of the column vertors R(°) and RW defined as W«£-=0 m =.,6 (43) where Vg and Vg ' axe the group velocities of the interacting waves xvi The equation (43) implies that the amplitudes A* and A\ remain con stants, respectively, in the reference frames moving with the group velocities Vg and Vg '. That is A^' depends on X\ and t\ through the combination x\ - Vg ti, whereas A[ depends on the combination x\ - Vgt\. From the compatibility conditions similar to (33) at the third order prob lem, assuming that the second order slowly varying amplitudes of the waves depend on X\ and t\ as A[ and A\', we deduce the following coupled nonlinear equations for A* and A\ fl/j(a) f)2 /1(a) z^- + r(a>^£- + A I AT |2 AT + A^ | AV>) p AN = 0 (45a) ar o£2 i^ğl + iAW^ + r<«^f + aw I >i« |2.*« + a(°6> | >tw |2 a® = o ar d£ <9£2 (456) where r = uat2, e = *.(*ı - V^h), ^ = kaA^, r(m) = ^«0 ^ A(«) = Â(nm)/u;^a2, A« = (WajaXVM - VT) A(nm) = L^m).F{nm) /L(m) (dWl ) R(m) (46) 2dkl in which F is a constant vector defined when solving the third order problem, and l/m> is a row vector satisfying l/m>W{m) = 0. For the case in which c\ < cm < c2 < c3, the coupled nonlinear equations (45) are also valid with different coefficients. The equations (45a, b) are also govered by the nonlinear interactions of two co-directional waves having equal group velocities, i.e. Vga' = Vg are also governed by the equations (43a, b). In this case A(6) = 0. Beside this, from these equations, the coupled equations describing the fifth and higher order harmonic resonances may be obtained by putting K = k, kb=Mk, and w"=w, ub = Mu, M e {5, 7, 9,...} (47) in the expressions defining the coefficients of (45a, b). The third harmonic resonance interaction needs a special treatment.