DISPERSION OF THE SEISMIC RAYLEIGH WAVES IN ELASTIC CRUSTAL LAYER ON VISCOELASTIC MANTLE S.D. Akbarov, ad M. Negi 3 Prof. Dr., Mechaical Eg. Departmet, Yildiz Techical Uiversity, Istabul Istitute of Mathematics ad Mechaics of the Natioal Academy of Scieces of Azerbaija, Baku ABSTRACT: 3 Assist. Prof. Dr., Civil Eg. Departmet, Bahcesehir Uiversity, Istabul Email: masoud.egi@eg.bau.edu.tr Dispersio of the seismic Rayleigh waves propagatig i a elastic crustal layer of the Earth layig o liear viscoelastic matle, is ivestigated. The ivestigatios are made by applyig the exact field equatios of viscoelasto-dyamics utilizig the piecewise homogeeous body model. The dispersio equatio is obtaied for a arbitrary type of hereditary operator of the materials of the matle ad a solutio algorithm is developed for obtaiig umerical results o the dispersio of the waves uder cosideratio. Dispersio curves are preseted for certai atteuatio cases ad the ifluece of the viscosity of the matle is studied through the rheological parameters which characterize the characteristic creep time, log-term values of the elastic costats ad through the parameter which characterizes the form of the iitial part of the creep curves. Numerical results are preseted ad discussed for the case where the viscoelasticity of the matle s material is described through the fractioalexpoetial operators by Rabotov. I particular, how the rheological parameters ifluece the dispersio of the geeralized Rayleigh waves propagatig i the covered half-space uder cosideratio is established. We coclude that i the cosidered dispersive atteuatio cases the viscoelasticity of the materials causes a decrease i the geeralized Rayleigh wave propagatio velocity. It is also established that the dispersio curves obtaied for the viscoelastic matle case are limited by those obtaied for the purely elastic matle with istataeous ad with the log-term values of the elastic costats. KEYWORDS: Geeralized Rayleigh Waves, Wave Dispersio, Viscoelastic Material, Rheological Parameters, Fractioal-Expoetial Operator, Dispersive Atteuatio. INTRODUCTION The Rayleigh waves propagatig i the Earth s surface ca be the most destructive type of seismic waves produced by earthquakes. Therefore they play a importat role i seismology, earthquake egieerig ad structural egieerig i studyig of the damages caused by those earthquakes. O the other had, as seismic waves propagate i Earth s crustal layers, they are iflueced by itrisic viscosity of deep layers of the Earth. Rayleigh waves propagatig i liear viscoelastic materials have bee studied itesively i the literature ad several mathematical models have bee used by may authors to study the dispersio ad the atteuatio behavior of guided waves i such viscoelastic media. However, i most cases either they have described the viscoelasticity of the materials through some simple models such as the classical Kelvi-Voigt sprig-dashpot models (see, for istace, Chiriţă et al. (4, Quitailla et al. (5, Mazzotti et al. (, Macoi ad Soroki (3 or they have used complex elasticity modulus istead of the real oe i the stress strai relatios of the viscoelastic materials (see, for istace, Vishwakarma ad Gupta (, Barshiger ad Rose (4, Addy ad Chakraborty (5, Garg (7, Jiagog (. Cosequetly, i geeral, such a simple viscoelastic models ad the umerical results obtaied withi these models caot illustrate the real character of the ifluece of the rheological parameters of the viscoelastic materials o the correspodig wave dispersio ad atteuatio.
These cosideratios led the authors to study the seismic Rayleigh waves dispersio ad atteuatio for a system cosistig of a elastic coverig layer ad a viscoelastic half-space utilizig more realistic mathematical viscoelastic model usig Rabotov (98 fractioal expoetial operator which are already used i the papers by Akbarov ad Kepceler (5, Akbarov et al. (6. Moreover, this actually exteds the authors previous works Negi et al. (4, Negi (5 ad Akbarov ad Negi (7 o propagatio of the geeralized Rayleigh waves i a iitially stressed elastic covered half-space to viscoelastic cases. The ivestigatios carried out withi the scope of the piecewise homogeeous body model by utilizig the exact equatios of motio of the theory of liear viscoelasticity. The correspodig dispersio equatio is obtaied for arbitrary hereditary type viscoelastic operators, however cocrete umerical results are preseted for the case where the viscoelasticity of the materials is described by the fractioal expoetial operator by Rabotov (98. Accordig to this operator, dimesioless rheological parameters characterizig the characteristic creep time (deoted by Q, the log-term values of the elastic costats (deoted by d ad the form of the creep (or relaxatio fuctio of the materials i the begiig regio of deformatios (deoted by α are itroduced ad through these parameters the viscoelasticity of the materials of the half-space o the dispersio curves is studied. A icrease i the values of these parameters Q ad d meas a decrease i the viscosity properties of the related material.. FORMULATION OF THE PROBLEM The system cosists of a elastic layer with thickess which covers a hereditary-viscoelastic half-space as show i Figure. Positios of the poits are determied i the Cartesia system of coordiates Ox xx 3 ad a plae-strai state i Oxx plae is cosidered. Below the values related to the layer ad half-space are deoted by upper idices ( ad (, respectively. For simplicity of the writig of the field equatios we assume that both the coverig layer ad half-space materials are viscoelastic. The goverig equatios of motio ad mechaical relatios for the case uder cosideratio are as follows. The equatios of motio: u, x x t u,,. x x t Costitutive ad geometrical relatios: * * * * *,,, where, *, u, x x h u u u, x x * are the followig type viscoelastic operators: t * ( ( t ( t ( t ( d, * t ( t ( t ( t ( d. (3 Other otatio used i ( ad ( are covetioal. ( ( ( Represetig the displacemets ad strais as ( i kx t ( ( ( pq pq x e ad ( i kx u t p vp x e, ad doig mathematical maipulatios as described i the papers by Akbarov ad Kepceler (5 ad Akbarov et al. (6 ad itroducig otatios c s ( scos s ds, ( ssi s ds. c s (4 ( scos s ds, ( ssi s ds, ( (
we obtai the followig expressios for the stresses where Figure. Geometry of the covered half-space. ik x t i kx x M x e, ikx t x M x e, M x e, (5 ( ( ( c i s, M c i s ( Thus, we obtai the complex modulus (,. t (6 M, istead of Lame costats i relatio (. We use the fractioal expoetial operators by Rabotov (98 for describig the viscoelasticity of the costituets, accordig to which, we get the followig relatios: ( 3 ( 3 ( ( t ( t ( (, ( R ( t where, ( ( 3 ( ( t ( t ( (, ( R ( t t R x ( t R x, t ( d, ( m m m( ( x t R x, t( t t, (( (7 (8 ( x is the gamma fuctio ad, ad are the rheological parameters of the viscoelastic -th materials. Accordig to Akbarov ad Kepceler (5, we itroduce the followig dimesioless rheological parameters: d c,, R ( Q (9
d The dimesioless rheological parameter i (9 characterizes the log-term value of the viscoelastic materials, the rheological parameter characterizes the creep time of the viscoelastic materails, ad fially Q the rheological parameter characterizes the form of the creep (or relaxatio fuctio for the -th material. Cosequetly, accordig to the foregoig expressios, the ifluece of the viscoelasticity of the material o the dispersio curves will be estimated through these three dimesioless rheological parameters. Cosiderig the boudary ad cotact coditios, we obtai the system of the liear algebraic equatios where from the existece of the o-trivial solutio we get the dispersio equatio: det, i, j,,3,...,6. ( ij For solutio to the dispersio equatio ( we employ the algorithm developed i Akbarov ad Kepceler (5, ad uder this solutio procedure we use the relatio k k ik k ( i for the complex waveumber. We aalyze the case where the atteuatio parameter preseted as follows: ( (. ( ( ( s ( ( c, accordig to Ewig et al. (957, is 3. NUMERICAL RESULTS AND DISCUSSIONS ( ( ( 3 ( 3 ( We assume that.3,.8 g/cm, 3. g/cm, c 3 km/s ad 5 km/s ( ad c are the Poisso ratio, mass desity ad shear wave velocity of the elastic layer ad ( c ( c ; where are the Poisso ratio, mass desity ad the shear wave velocity of the viscoelastic half-space, respectively. Note that we cosider the results obtaied i the case where the material of the coverig layer is purely elastic, but the material of the half-space is viscoelastic. Thus, for elastic layer the operators with the istataeous values of the Lame costats ad ( ( (* ad (* (, ( ( (, ad are coicide at t. Cosequetly, the ifluece of the viscoelasticity of the material o the dispersio curves is estimated through three dimesioless rheological parameters of the viscoelastic half-sapce material, i.e. the dimesioless rheological parameter d (=d ( which characterizes the log-term value ad the rheological parameter Q (=Q ( which characterizes the creep time of the viscoelastic half-sapce material, ad fially the rheological parameter α (=α ( which characterizes the form of the creep (or relaxatio fuctio of this materail. Moreover, the results discussed below are obtaied withi the scope of the atteuatio relatio (. The graphs i Figure illustrate the ifluece of the parameter Q o the dispersio curves uder a fixed value of the parameter d (i.e. uder d ad the graphs i Figure 3 illustrate the ifluece of the parameter d o the dispersio curves uder a fixed value of the parameter Q (i.e. uder Q 5. Note that these graphs are costructed i the case where.5. Accordig to the discussios made i the paper by Akbarov et al. (6, it ca be predicted that the wave propagatio velocity obtaied for the all selected values of the parameter Q uder a fixed value of the parameter d must have the same limit velocity as kh ad this limit velocity coicides with that obtaied for the correspodig purely elastic case with log-term values of the elastic costats. Cosequetly, it ca be cocluded that these limit values of the wave propagatio velocity must deped o the rheological parameter d, ot o the rheological parameters Q ad. These results show that the dispersio curves obtaied uder fixed values of the parameter d are limited with the correspodig dispersio curves obtaied for the purely elastic cases uder istataeous values of the elastic costats (upper limits, i.e. uder t, ad uder log-term values of the elastic costats (lower limits, i.e. uder t. It follows from Figure ad 3 that, first of all, the viscoelasticity of the half-space material causes a decrease i the wave propagatio velocity. Moreover, these results show that the dispersio curves obtaied for the viscoelastic
Figure. Dispersio curves obtaied for various values of the parameter Q uder a fixed value of the parameter d=. Figure 3. Dispersio curves obtaied for various values of the parameter d uder a fixed value of the parameter Q=5. case approach to the correspodig oe obtaied for the purely elastic case with istataeous (log-term values of the elastic costats at t (at t with icreasig (decreasig of the rheological parameters d ad Q. It should be oted that the metioed icrease (decrease has mootoic character ad cosiderable effect i this icreasig (decreasig are observed i the cases where kh..
Accordig to the character of the dispersio curves obtaied for the viscoelastic case ad give i Figures ad 3 it ca be cocluded that for each value of the rheological parameter Q ad for each value of the rheological parameter d there exist the case where ( d( c / c. ( d( kh The wave propagatio velocity ad dimesioless waveumber related to this case we deote by ad ( kh, respectively. Note that for the dispersio curves related to the purely elastic waves there is ot the case where the relatio ( takes place. Cosequetly, the appearig of the cases where the relatio ( takes place is caused amely with the viscoelasticity of the materials of the costituets of the system uder cosideratio. Thus, it follows from the foregoig discussios that, the viscoelasticity of the materials of the costituets iflueces o the dispersio curves of the geeralized Rayleigh waves ot oly i the quatitative sese but also i the qualitative sese. cr Now we cosider the results related to the effect of rheological parameters o the wave dispersio curves. Thus, we cosider graphs give i Figure 4 ad 5 which illustrate the metioed ifluece. Note that the graphs give i Figure 4 (Figure 5 show the effect of the rheological parameter o the dispersio curves uder various values of the rheological parameter Q (of the rheological parameter d for a fixed value of the rheological parameter d (= (for a fixed value of the rheological parameter Q (=. The graphs give i Figures 4 ad 5 are grouped by letters a ad b correspod the cases where Q= ad 5 (the case where d=5 ad 5, respectively. Note that the case where coicides with the case where viscoelasticity of the halfspace material is described through the "stadard solid body" model. It follows from the results give i Figures 4 ad 5 that i all the cosidered cases there exists such value of the dimesioless waveumber kh (deote it by ( kh * at which the chage i the values of the rheological parameter does ot ifluece the values of the wave propagatio velocity. However, i the cases where kh ( kh* ( k h ( k h* a icrease i the values of the parameter causes a decrease (a icrease i the wave propagatio velocities. Accordig to the aforemetioed umerical results, it ca be cocluded that the ( kh * depeds o the values of the rheological parameters Q ad d ad a icrease i the values of these parameters decrease the ( kh *. It ca also be cocluded that the chage i the values of the rheological parameter does ot ifluece the limit values of the wave propagatio velocity as kh. However, i the ear viciity of this limit case, if to say more precisely i the regio kh ( kh*, the ifluece of the rheological parameter o the dispersio curves is sigificat ot oly i the quatitative sese but also i the qualitative sese. So that uder small values of, for istace uder., the dispersio curves have well-defied miimum i the regio kh ( kh* ad at this miimum the relatio ( takes place. Moreover, i the ear viciity of this miimum the wave propagatio velocity obtaied for the viscoelastic cases become less tha that obtaied for the purely elastic case with log-term values of elastic costats at t. Cosequetly, i the regio kh ( kh* a decrease i the values of causes to chage the character of the dispersio curves. However, with icreasig of the aforemetioed miimum disappears i the dispersio curves ad wave propagatio velocities are limited with the wave propagatio velocities obtaied for the purely elastic cases with istataeous values of elastic costats at t (upper limit ad with log-term values of elastic costats at t (lower limit. c cr 4. CONCLUSIONS Accordig to the foregoig umerical results, the followig mai coclusios ca be draw: - The viscoelasticity of the half-space material causes the geeralized Rayleigh wave propagatio velocity to decrease ad the magitude of this decrease icreases with a decrease i the values of the dimesioless rheological parameters d ad Q ;
(a (b Figure 4. Dispersio curves for differet values of parameter α whe d is fixed. (a (b Figure 5. Dispersio curves for differet values of parameter α whe Q is fixed. - The character of the ifluece of the parameter o the wave velocities ad o the dispersio corves depeds o the values of the dimesioless wave umber kh ad o the values of the rheological parameters d ad Q. There exist such value of the kh after (before which a icrease i the values of the causes to decrease (icrease the wave propagatio velocity;
- The lower waveumber limit values of the wave propagatio velocity depeds oly o the rheological parameter d ad coicide with that obtaied for the correspodig purely elastic case with log-term values of elastic costats at t ; - The high waveumber limit values of the wave propagatio velocity do ot deped o the rheological parameters of the materials. - I geeral, the dispersio curves obtaied to viscoelastic cases are limited by the dispersio curves correspodig to the purely elastic case with istataeous values of the elastic costats (upper limit ad by those obtaied for the purely elastic case with log-term values of the elastic costats (lower limit, however this rule is violated i the relatively small values of the rheological parameter α. REFERENCES Addy, SK. ad Chakraborty, NR. (5 Rayleigh waves i a viscoelastic half-space uder iitial hydrostatic stress i presece of the temperature field. Iteratioal Joural of Mathematics ad Mathematical Scieces 5:4, 3883-3894. Akbarov, SD. ad Kepceler, T. (5 O the torsioal wave dispersio i a hollow sadwich circular cylider made from viscoelastic materials. Applied Mathematical Modellig 39:3, 3569-3587. Akbarov, SD., Kocal, T. ad Kepceler, T. (6 O the dispersio of the axisymmetric logitudial wave propagatig i a bi-layered hollow cylider made of viscoelastic materials. Iteratioal Joural of Solids ad Structures, 95-. Akbarov, SD. ad Negi, M. (7 Near-surface waves i a system cosistig of a coverig layer ad a halfspace with imperfect iterface uder two-axial iitial stresses. Joural of Vibratio ad Cotrol 3:, 55-68. Barshiger, JN. ad Rose, JL. (4 Guided wave propagatio i a elastic hollow cylider coated with a viscoelastic material. ieee trasactios o ultrasoics, ferroelectrics, ad frequecy cotrol 5:,547-556. Chiriţă, S., Ciarletta, M. ad Tibullo, V. (4 Rayleigh surface waves o a Kelvi-Voigt viscoelastic halfspace. Joural of Elasticity 5:, 6-76. Ewig, WM., Jazdetzky, WS. ad Press, F. (957 Elastic Waves i Layered Media. McGraw Hill, New York. Garg, N. (7 Effect of iitial stress o harmoic plae homogeeous waves i viscoelastic aisotropic media. Joural of soud ad vibratio 33:3, 55-55. Jiagog, Y. ( Viscoelastic shear horizotal wave i graded ad layered plates. Iteratioal Joural of Solids ad Structures 48:6, 36-37. Macoi, E. ad Soroki, S. (3 O the effect of dampig o dispersio curves i plates. Iteratioal Joural of Solids ad Structures 5:, 966-973. Mazzotti, M., Marzai, A., Bartoli, I. ad Viola, E. ( Guided waves dispersio aalysis for prestressed viscoelastic waveguides by meas of the SAFE method. Iteratioal Joural of Solids ad Structures 49:8, 359-37. Negi, M., Akbarov, SD. ad Erguve, ME. (4 Geeralized Rayleigh wave dispersio aalysis i a prestressed elastic stratified half-space with imperfectly boded iterfaces. CMC Comput Mater Cotiua 4:, 5-6. Negi, M. (5 Geeralized Rayleigh wave propagatio i a covered half-space with liquid upper layer. Structural Egieerig ad Mechaics 56:3, 49-56. Quitailla, FH., Fa, Z., Lowe, MJ. ad Craster, RV. (5 Guided waves' dispersio curves i aisotropic viscoelastic sigle-ad multi-layered media. IProc. R. Soc. A 47, 568. Rabotov, YuN. (98 Elemets of hereditary solid mechaics. Mir, Moscow. Vishwakarma, SK. ad Gupta, S. ( Torsioal surface wave i a homogeeous crustal layer over a viscoelastic matle. Iteratioal Joural of Applied Mathematics ad Mechaics 8:6, 38-5.