Copyright 7 Tech Science Press CMC vol.53 no.4 pp.37-34 7 Generalized Rayleigh Wave Dispersion in a Covered Half-space Made of Viscoelastic Materials S.D. Akbarov and M. Negin 3 Abstract: Dispersion of the generalized Rayleigh waves propagating in a covered halfspace ade of viscoelastic aterials is investigated by utilizing the exact equations of the theory of linear viscoelasticity. The dispersion equation is obtained for an arbitrary type of hereditary operator of the aterials of the constituents and a solution algorith is developed for obtaining nuerical results on the dispersion of the waves under consideration. Dispersion curves are presented for certain attenuation cases and the influence of the viscosity of the aterials is studied through three rheological paraeters of the viscoelastic aterials which characterize the characteristic creep tie long-ter values and the echanical behaviour of the viscoelastic aterial around the initial state of the deforation. Nuerical results are presented and discussed for the case where the viscoelasticity of the aterials is described through fractional-exponential operators by Rabotnov. As the result of this discussion in particular how the rheological paraeters influence the dispersion of the generalized Rayleigh waves propagating in the covered half-space under consideration is established. Keywords: Generalized Rayleigh wave viscoelastic aterial rheological paraeters dispersion fractional-exponential operator. Introduction Surface waves propagating in viscoelastic layered edia are of a particular iportance for nuerous scientific and engineering applications fro aterial science to biological science and fro vibration reduction of different structural or echanical eleents to earthquake engineering and geophysical explorations. Several atheatical odels have been used by any authors to study the dispersion and the attenuation behaviour of guided waves in such viscoelastic edia. However in ost cases either they have described the viscoelasticity of the aterials through soe siple odels such as the classical Kelvin-Voigt spring-dashpot odels [Chiriţă Ciarletta and Tibullo (4); Quintanilla FanLowe et al. (5); Mazzotti Marzani Bartoli et al. (); Manconi Yildiz Technical University Faculty of Mechanical Engineering Departent of Mechanical Engineering Yildiz Capus 34349 Besiktas Istanbul Turkey. e-ail: akbarov@yildiz.edu.tr. Institute of Matheatics and Mechanics of National Acadey of Sciences of Azerbaijan 374 Baku Azerbaijan. 3 Bahcesehir University Departent of Civil Engineering 34353 Besiktas Istanbul Turkey. e-ail: asoud.negin@eng.bau.edu.tr.
38 Copyright 7 Tech Science Press CMC vol.53 no.4 pp.37-34 7 and Sorokin (3)] or they have used coplex elasticity odulus instead of the real one in the stress-strain relations of the viscoelastic aterials [Vishwakara and Gupta (); Barshinger and Rose (4); Addy and Chakraborty (5); Garg (7); Jiangong ()]. Consequently in general such a siple viscoelastic odels and the nuerical results obtained within these odels cannot illustrate the real character of the influence of the rheological paraeters of the viscoelastic aterials on the corresponding wave dispersion and attenuation. Recent efforts to use ore realistic odels for the wave propagation and attenuation probles in viscoelastic edia were ade by Meral Royston and Magin (9 ) by utilizing fractional order Voigt odel to investigate Lab wave propagation. In this way a new rheological paraeter which is the order of the fractional derivatives is introduced into the odel and through this paraeter the results are agreed ore accurately with experients as copared with conventional odels. Here we review in detail investigations related to the propagation of the Rayleigh waves in viscoelastic edia which is close to the topic of the current paper and begin this review with the paper by Carcione (99) in which the inelastic characteristics of the Rayleigh waves fro the standpoint of balance energy is investigated. He calculated the quality factors as a function of the frequency and depth and showed that the viscoelastic properties calculated fro energy considerations are consistent with those obtained fro the Rayleigh secular equation. Based on the Cauchy residue theore of coplex analysis Lai and Rix () presented a technique which perits siultaneous deterination of the Rayleigh dispersion and attenuation curves for linear viscoelastic edia with arbitrary values of aterial daping ratio. Fan (4) obtained the analytical solution of the Rayleigh wave propagation phenoena considering the nonlinear daping echanis of seisic waves by applying the perturbation ethod. Pasternak (8) analysed the Rayleigh wave propagation proble in the elastic half-space and viscoelastic layer interface using in the Fourier-Laplace space using the Biot viscoelastic solid odel. Shara Shara and Shara (9) derived the coplex secular equations for Rayleigh wave propagation in closed and isolated atheatical conditions and studied the theros-elastic interaction in an infinite Kelvin-Voigt type viscoelastic therally conducting solid bordered with viscous liquid half-spaces/layers of varying teperature. Zhang Luo and Xia et al. () studied the dispersion of Rayleigh waves in viscoelastic edia by applying pseudo spectral odelling ethod to obtain high accuracy for Rayleigh wave odelling in viscoelastic edia. In pseudo spectral ethod the spatial derivatives in the vertical and horizontal directions are calculated using Chebyshev and Fourier difference operators respectively. Chiriţă Ciarletta and Tibullo (4) studied the propagation of surface waves over an exponentially graded half-space of isotropic Kelvin-Voigt viscoelastic aterial by eans of wave solutions with spatial and teporal finite energy. They showed that when there is just one wave solution it is found to be retrograde at the free surface while when there is ore than one viscoelastic surface wave one is retrograde and the others are direct at the free surface. We also note investigations carried out in the papers by Shara (5) Shara and Othan (7) Kuar and Parter (9) Shara and Kuar (9) Abd-Alla Aftab Khan and Abo-Dahab (7) and others listed therein in which the Rayleigh-Lab
Generalized Rayleigh Wave Dispersion in a Covered Half-space 39 waves dispersion in the plate ade of viscoelastic or thero-viscoelastic aterials are investigated. However it should be noted that in these works the viscoelasticity of the plate aterial is described by the Voigt odel. Moreover it should be noted that the investigations carried out in these works are also have iportant significance in the ethodological sense i.e. in these works the functional iteration nuerical technique is developed for deterination of the coplex roots of the secular equation. Detailed consideration of this ethod and its advantage and disadvantages are discussed in the paper by Shara (). We will again turn below to this ethod in the text of the paper during the discussions the solution algorith of the secular equation. As follows fro the foregoing discussion and works reviewed above the investigations on the dispersion of the guided waves in the half-space or covering half-spaces ade fro viscoelastic aterials ainly were carried out by eploying siple Kelvin-Voigt classical odels or by using frequency dependent coplex odulus of viscoelastic aterials which is obtain fro the experients. These siple ethods were not actually connected with the ore coplicated and real behaviour of viscoelastic aterials and they do not illustrate the influence of the rheological paraeters of the viscoelastic aterials on this dispersion. These considerations led the authors to study the generalized Rayleigh waves dispersion and attenuation for a syste consisting of a viscoelastic covering layer and a viscoelastic half-space utilizing ore realistic atheatical viscoelastic odel using Rabotnov (98) fractional exponential operator which are already used in the papers by Akbarov and Kepceler (5) Akbarov Kocal and Kepceler (6a 6b) and Kocal and Akbarov (7) under investigations of the axisyetric torsional and longitudinal waves respectively in the layered hollow cylinders ade of viscoelastic aterials. Moreover in the paper by Akbarov (4) this odel is eployed to study of the axisyetric tie-haronic Lab s proble for a syste consisting of s viscoelastic covering layer and viscoelastic half-space. Note that these results in these papers are also detailed in the onograph by Akbarov (5). Moreover this study actually extends the authors previous works Negin Akbarov and Erguven (4) Negin (5) and Akbarov and Negin (7) on propagation of the generalized Rayleigh waves in an initially stressed elastic covered half-space to viscoelastic cases where the constitutive relations for the covering layer and the halfspace aterials are described through the fractional exponential operator by Rabotnov (98). The investigations are carried out within the fraework of the piecewise hoogeneous body odel by utilizing exact equations of otion of the linear theory of viscoelasticity and it is assued that perfect contact conditions take place on the interface surface between the layer and the half-space. The theoretical results obtained in this paper can be utilized in any relevant practical probles of wave propagation in viscoelastic layered edia which play roles in areas like engineering earthquake and geophysical sciences etc. Soe nuerical calculations discussions and conclusions will be discussed in their proper places. Forulation of the proble
3 Copyright 7 Tech Science Press CMC vol.53 no.4 pp.37-34 7 The syste consists of a layer with thickness h which covers a half-space as shown in Figure. The layer and the half-space occupy the regions x x h x 3 and x x x 3 respectively. We assue that the aterials of the constituents are isotropic hoogeneous and hereditaryviscoelastic. Positions of the points are deterined in the Cartesian syste of coordinates Ox xx 3 and a plane-strain state in Oxx plane is considered. Below the values related to the layer and half-space are denoted by upper indices and () respectively. The governing equations of otion and echanical relations for the case under consideration are as follows: The equations of otion: u x x t u x x t. Constitutive and geoetrical relations: * * * * * ( ) ( ) ( ) ( ) u u i j ii ij i j () i xj x i * where * * are the following type viscoelastic operators: t ( t) ( t) ( t ) ( ) d t * ( t) ( t) ( t ) ( ) d. (3) In Eq. (3) are the instantaneous values of Lae s constants and are the corresponding kernel functions for describing the hereditary properties of the aterials of the constituents. We assue that the following boundary conditions on the free face plane and contact conditions on the interface of the covering layer and half-space satisfy: Boundary conditions: xh xh. (4) Contact conditions:
Generalized Rayleigh Wave Dispersion in a Covered Half-space 3 u u u u () () x x x x. (5) () () x x x x We also assue that the following decay conditions are satisfied: () () i j ui x x. (6) Figure : Geoetry of the covered half-space. This copletes the forulation of the proble under consideration the novelty of which is the atheatical odelling of the near-surface (or generalized Rayleigh) wave propagation in the syste consisting of the covering layer and half-space ade of viscoelastic aterials with arbitrary hereditary properties. 3 Method of Solution According to the proble nature we can represent the sought values in the following for: u ikx t vi x e ( ) i ( ) x e i kx t ij ij x e ij v x i kx t i j j v x i i j. where k is the wavenuber and is the circular frequency. Note that the representation (7) is siilar forally with the corresponding one related to the purely elastic case. However in the present case as it will be detailed below in section 4 the wavenuber k ( k ik) is selected as a coplex one the iaginary part k of which characterizes the attenuation of the aplitudes of the stresses strains and displaceents in the constituents. Consequently under investigations of the dispersion of the guided waves in the eleents of constructions ade of viscoelastic aterials it is assued that there is not any attenuation of the aplitudes with respect to tie and the attenuation of the aplitudes takes place only with respect to the coordinate x on the coordinate axis which is directed along the wave propagation direction. This eans that the agnitude of the aplitudes of the sought values at the (7)
3 Copyright 7 Tech Science Press CMC vol.53 no.4 pp.37-34 7 point x is less than that at the point x if x x. Note that the sae approach is also used in the papers Mazzotti Marzani Bartoli and Viola () Barshinger and Rose (4) Jiangong () Shara (5) Shara and Othan (7) Kuar and Parter (9) Shara and Kuar (9) and in any other investigations related to the study of the wave propagation in the viscoelastic aterials. It is evident that the presentation of the sought values with the ultiplying factor exp( i( kx t)) has unrealistic eaning only in the case where x (as I k ) nevertheless all researchers have to put up with this contradiction. Consequently the case where x has a eaning without any doubt. If free vibration of the eleents of constructions ade of viscoelastic aterials is investigated then all the sought quantities are presented with ultiplying factor exp( i t) where the circular frequency in this factor is assued to be coplex one (i.e. the natural frequencies are deterined as coplex frequencies). Under this free vibration the attenuation of the aplitudes of the sought values with respect to tie takes place. Thus after the foregoing discussions we turn to consideration of the solution ethod and using the relation t t f ( t ) f ( ) d f ( t ) f ( ) d (8) in () and (3) and taking Eq. (7) and (8) into account we can write the following relation: x e x e i kx t ( ) i kxt ii ii ikx i x e ( t ) e d x e ( t ) e d. t t ( ) ikx ii Eploying the transforation t s we can ake the following anipulations of the integrals which enter into (9) t i it is ( ) ( ) where t e d s e e ds it is it c s i e ( s) e ds e i () c ( s)cos sds s ( s)sin sds. () In a siilar anner we obtain t where ( ) i i t ( ) ( ) t e d e c i s ( ) c ( s)cos sds s ( s)sin sds. (3) (9)
Generalized Rayleigh Wave Dispersion in a Covered Half-space 33 Taking the relations (-3) into account we can write the following expressions for the stresses: ikx t x M x e ( ) ikx t x M x e ( ) ikxt M x e (4) ( ) where i M c s i (5) c s. ( Thus we obtain the coplex odulus ) M instead of Lae constants in the relations () and (3) the real and iaginary parts of which are deterined through the expressions () (3) and (5). This eans that the coplete syste of field Eqs. () () and (5) for the viscoelastic syste can also be obtained fro those written for the purely elastic syste by replacing the elastic Lae constants and ( with the coplex constants ) and M respectively. In other words the foregoing atheatical calculations confir the dynaic correspondence principle (see Fung (965)) for the proble under consideration and the solution ethod used here agrees with this principle. Thus according to the foregoing procedures the syste of Eqs. (-3) with boundary conditions (4) and (6) and contact condition (5) can be solved analytically by eploying the so-called ethod of separation of variables. Also it follows fro the foregoing procedures that the presentation of the sought values through the variables x and t is ade through the known functions exp( ikx ) and exp( i t) respectively. However the presentation of the sought values through the variable x is ade through the unknown functions v ( x ) and v ( x ) ( ) and substituting the expressions (7) and (4) into the equation of otion we obtain the following equations for these unknown functions: d v d kx dv b v c d kx d v d kx where dv b v c (6) d kx
34 Copyright 7 Tech Science Press CMC vol.53 no.4 pp.37-34 7 b b M ( ) ( ) M k M M M i k c. M M ( ) ( ) c M i (7) It can be written fro the second equation in (6) the following expressions dv d v b v d kx c d kx c 3 4 d v d v b d v (7 3 4 ) d kx c d kx c d kx and it can be written the following expression fro the first equation in (6) 3 d v dv d v b 3 c. (7 ) d kx d kx d kx Thus substituting the expressions in (7 ) into the expression (7 ) we obtain: d v d v B C v d kx 4 4 d kx B b b c c C b b (8). The general solution of the Eq. (8) for the -th layer can be written as follows: 3 4 v x Z exp R kx Z exp R kx Z exp R kx Z exp R kx v x Z exp R kx Z exp R kx. 3 Z R Z R Z3 R exp exp exp c c c v x R kx R kx R kx 4 exp R kx exp R kx exp R kx c R c R c Z R Z b Z b 3 4 exp R kx exp R kx R c Z b Z b R c Z R Z b Z3 R Z3 b v x exp R kx exp R kx (9) R c R c where
Generalized Rayleigh Wave Dispersion in a Covered Half-space 35 B B R C 4 B B R C () 4 Finally substituting the expressions in () into the Eq. (7) and () and eploying wellknown usual procedure we obtain the following dispersion equation fro the boundary (4) and contact (5) conditions (4-6): det i; j... 6. ij The explicit expressions of the coponents of the atrix ij are given in Appendix A through the expressions (A). Thus in the present section the analytical expression for the dispersion equation related to the near-surface (or generalized Rayleigh) wave propagation in the syste consisting of the covering layer and half-space ade of viscoelastic aterials with arbitrary hereditary properties are obtained. 4 Nuerical results and discussions As we consider the tie haronic wave propagation in a viscoelastic aterial it is necessary to assue that the wave nuber k is a coplex one and can be presented as follows: k k ik k ( i) k k () where k (or paraeter in ()) i.e. the iaginary part of the wave nuber k defines the attenuation of the wave aplitude under consideration and is called the coefficient of the attenuation. It should be noted that this attenuation takes place in the wave propagation direction (i.e. in the Ox axis direction) and can be called as horizontal attenuation. We deterine the phase velocity of the studied waves through the expression: c (3) k We assue that the viscoelasticity of the aterials of the constituents is described through the fractional exponential operator by Rabotnov (98) i.e. we assue that ( ) ( ) 3 ( ) ( t ) ( t ) ( ) ( ) ( ) R ( ) t ( ) 3 ( ) 3 ( ) ( t ) ( t ) ( ) ( ) ( ) R ( ) t where t ( ) R x ( t) R x t ( ) d ( ) (4)
36 Copyright 7 Tech Science Press CMC vol.53 no.4 pp.37-34 7 R x t ( t) t. n n( ) ( x ) t (5) n (( n)( )) In (5) ( x) is the gaa function. Moreover the constants and in (4) and (5) are the rheological paraeters of the -th aterial. The echanical eanings of these rheological paraeters are ore explained in the papers by Akbarov (4) and Akbarov and Kepceler (5). Introducing the following diensionless rheological paraeters d Q ( ) ( ) R the following expressions are obtained for the long-ter values of the echanical ( ) constants and for ( ) ( ) ( ) and which enter into the Eq. (5): c s c ( ) li( ) t () ( ) (3 / (( )) d ( ) ( ) ( ) 3 li( ) t () ( ) (3 / (( )) d ( ) ( ) ( ) 3 c d R ( ) ( ) ( ) c ( ) ( ) ( ) ( ) ( ) 3 s d R s ( ) ( ) ( ) ( ) ( ) ( ) 3 3 c d R ( ) ( ) ( ) c ( ) ( ) ( ) ( ) ( ) 3 3 s d R s ( ) ( ) ( ) where c R s ( ) sin ( ) ( ) sin cos ( ) s R ( ) sin (6) (7) (8) (9)
Generalized Rayleigh Wave Dispersion in a Covered Half-space 37 c ( ) ( ) Q k h. (3) c The diensionless rheological paraeter d in (6) characterizes the long-ter values of the viscoelastic aterials and the rheological paraeter Q characterizes the creep tie of the viscoelastic aterials and finally the rheological paraeter characterizes the for of the creep (or relaxation) function for the -th aterial and the case where corresponds to the standard viscoelastic body odel (or the odel by Kelvin). With respect to solution of the dispersion Eq. as the values of the deterinant obtained in are coplex the dispersion equation can be reduced to the following one det ij (3) where det ij eans the odulus of the coplex nuber det ij. Consequently for construction of the attenuation or dispersion curves it is necessary to solve nuerically the Eq. (3) for the selected proble paraeters. Therefore we use the algorith which is based on direct calculation of the values of the oduli of the dispersion deterinant 9 det and deterination of the sought roots fro the criterion det and the ij values of the wave dispersion velocity are deterined under fixed values of the proble paraeters. According to the physico-echanical consideration in finding the velocity c which is the root of the Eq. (3) for the selected kh and it is assued that this velocity is greater (less) than that obtained in the corresponding purely elastic case with the elastic constants calculated at t (at t ). This ensures the existence of the roots of the Eq. (3). Thus we consider the nuerical results obtained fro the solution of the dispersion Eq. (3) by eploying the algorith discussed above. First we analyse the case where the attenuation of the aterials is low and according to Ewing Jazdetzky and Press (957) and Kolsky (963) we assue that () s s () () c c ( ) ( ) or. ( ) ( ) ij It should be noted that the attenuation deterined by the relation (3) relates to the dispersive attenuation case. At the sae tie the non-dispersive attenuation case under which the selected values for kh (or ) in () do not depend on the wave frequency also is considered in the present investigations. Note that the solution technique of the secular equation described above has also been used in the paper by Barshinger and Rose (4). In general as noted in the paper by Shara () there is no general ethod for finding the coplex roots of the transcendental secular equations. Theoretically it is known that the functional iteration (3)
38 Copyright 7 Tech Science Press CMC vol.53 no.4 pp.37-34 7 ethod detailed by Shara () can be applied for deterination of the coplex roots of an analytical function. In this ethod the analytical function is represented in the finite power series for and the obtained algebraic equation is solved through the iteration ethod. Naely this ethod is developed for solving the secular equations with coplex roots and eployed in the papers Shara (5) Shara and Othan (7) Kuar and Parter (9) Shara and Kuar (9) and others listed therein. For instance in the paper Shara (5) the Rayleigh-Lab wave s dispersion in the viscoelastic plate is studied and the corresponding secular equation is solved by the use of the aforeentioned functional iteration ethod. The key step in the application of this ethod is the successful selection of the initial guess. However we have not found in the papers Shara (5) and Shara and Othan (7) what initial guess is taken under eploying the iteration procedure. Nevertheless it can be predicted that the sought coplex root ust be near to the certain coplex root of the secular equation obtained in a special liit cases in which it is possible to obtain an analytical expression for the coplex root. Note that such liit cases and deterination of the exact coplex roots in these cases take place in the investigations carried out in Shara (5) Shara and Othan (7) Kuar and Parter (9) and Shara and Kuar (9). We think that naely this and siilar type coplex roots can be taken as initial guess for eploying the functional iteration ethod. In the cases where there is no the aforeentioned situation the selection of the initial guess for the coplex root is difficult and there is no any rule for selection the entioned initial guesses. This stateent is the disadvantage of the functional iteration ethod. At the sae tie this ethod allows to find the real and iaginary parts of the coplex roots siultaneously. This is the advantage of the functional iteration ethod. However the solution ethod described above and applied in the present and earlier works Akbarov and Kepceler (5) Akbarov Kocal and Kepceler (6) the values of the attenuation coefficient are given a priori for finding the wave propagation velocity or as in the paper Barshinger and Rose (4) the values of the wave propagation velocity are selected a priori for finding the attenuation coefficient. Of course this is disadvantages of the used ethod. However this ethod does not require the selection of any initial guesses which is advantage of ethod. Note that as noted above in the case under consideration we prefer to use the solution algorith described above and used also in the papers [Barshinger and Rose (4) Akbarov and Kepceler (5) and Akbarov Kocal and Kepceler (6) Kocal and Akbarov (7)]. This is because eploying the functional iteration ethod for the solution of the secular equation obtained in the present paper requires special consideration and developent of this ethod which has not been done up to now. Thus we turn to the analysis of nuerical results which are obtained for such cases () () where the conditions Re{ R k} and Re{ R k} are satisfied siultaneously where the R and R are deterined through the expression in () and k is a coplex wavenuber presented as in (). Naely satisfaction of these conditions provides the existence of the near-surface (or generalized Rayleigh) waves in the biaterial viscoelastic syste under consideration. According to the entioned conditions
Generalized Rayleigh Wave Dispersion in a Covered Half-space 39 () nuerical investigations are ade within the scope of the assuptions.3 () () and c / c / in the cases where () / and () for which it can be provided the satisfaction of the conditions () Re{ R k} throughout all the calculation procedures. / 9 () () Re{ R k} and First we analyse the results obtained in the case where the viscoelasticity properties of the covering layer are the half-space are the sae i.e. the case where Q Q () ( Q) d d () ( d) () ( ) ; and denote it as the V.V. case. Moreover unless otherwise specified the results discussed below are obtained within the scope of the attenuation relation (3). Consider the graphs given in Figure and 3 which are constructed in the cases where () () / and / 9 respectively under.5. The graphs in Figure (a) and 3(a) illustrate the influence of the paraeter Q on the dispersion curves under a fixed value of the paraeter d (i.e. under d ) and the graphs in Figure (b) and 3(b) illustrate the influence of the paraeter d on the dispersion curves under a fixed value of the paraeter Q (i.e. under Q 5 ). According to the discussions ade in the paper by Akbarov Kocal and Kepceler (6) it can be predicted that the wave propagation velocity obtained for the all selected values of the paraeter Q under a fixed value of the paraeter d ust have the sae liit velocity as kh and this liit velocity coincides with that obtained for the corresponding purely elastic case with long-ter values of the elastic constants deterined with expressions in (7). Consequently according to the expressions in (7) it can be concluded that these liit values of the wave propagation velocity ust depend on the rheological paraeter d and ust not depend on the rheological paraeters Q and. Note that this conclusion is confired with the results illustrated in Figure and 3 and with corresponding ones which will be discussed below. Moreover these results show that the dispersion curves obtained under fixed values of the paraeter d are liited with the corresponding dispersion curves obtained for the purely elastic cases under instantaneous values of the elastic constants (upper liits) i.e. under t and under long-ter values of the elastic constants (lower liits) i.e. under t. Now we note the following stateent. According to the definition of the phase velocity c / k of the wave propagation the finite liit value of this velocity as kh can be obtained only in the cases where. Moreover according to the expressions ( ) ( ) ( ) ( ) ; ; ; as. This eans (8) (9) and (3) it is obtained that c s c s that the liit values of the phase velocity of the wave propagation as kh for fixed finite value of the layer thickness h ust approach to the corresponding ones obtained in the purely elastic case with the instantaneous values of elastic constants at t. This is because the agnitude of the influence of the aterial viscosity on its vibration decreases with the vibration frequency and this influence disappears copletely as
3 Copyright 7 Tech Science Press CMC vol.53 no.4 pp.37-34 7. This is well-known physico-echanical stateent which is confired again with the results given in Figure and 3 and other corresponding ones given below. (a) (b) () Figure : Dispersion curves in V.V. case / obtained for (a) various values under a fixed value of the paraeter d (b) for () of the paraeter Q Q Q () various values of the paraeter d d d Q 5. under a fixed value of the paraeter Note that the foregoing conforations of the obtained nuerical with the related predictions and agreeent of those with the known physico-echanical considerations can be taken as validation and trustiness of the used calculation algorith and PC progras which are coposed by the authors and are realized in MATLAB. Unfortunately we have not found any related nuerical results in literature in order to copare these results with those. We turn again to the consideration of the liit values of the wave propagation velocity as kh. Thus in general according to the foregoing discussions and conclusions we can write the following relation c in( cr ; cs ) as kh (33) where c R is the Rayleigh wave propagation velocity in the covering layer aterial in the corresponding purely elastic case and c S is the Stoneley wave propagation velocity for the selected pairs of aterials for covering layer and half-space also in the purely elastic case. As in the cases under considerations the Stoneley wave does not exist; therefore the relation (33) can be replaced with the relation
Generalized Rayleigh Wave Dispersion in a Covered Half-space 3 (a) (b) () Figure 3: Dispersion curves in V.V. case / 9 obtained for (a) various values under a fixed value of the paraeter d (b) for () of the paraeter Q Q Q () various values of the paraeter d d d Q 5. under a fixed value of the paraeter c c R as kh. (34) Now we analyse in detail the results given in Figure and 3 fro which first of all it follows that the viscoelasticity of the aterials of the constituents causes a decrease in the wave propagation velocity. Moreover these results show that the dispersion curves obtained for the viscoelastic case approach to the corresponding one obtained for the purely elastic case with instantaneous (long-ter) values of the elastic constants at t (at t ) with increasing (decreasing) of the rheological paraeters d and Q. It should be noted that the entioned increase (decrease) has onotonic character and considerable effect in this increasing (decreasing) are observed in the cases where kh.. According to the character of the dispersion curves obtained for the viscoelastic case and given in Figure 3 and in other ones which will be illustrated below it can be concluded that for each value of the rheological paraeter Q and for each value of the rheological paraeter d there exist the case where () d( c / c ) d( kh). (35) The wave propagation velocity and diensionless wavenuber related to this case we denote by c cr and ( kh ) cr respectively. Note that for the dispersion curves related to the purely elastic waves there is not the case where the relation (35) takes place. Consequently the appearing of the cases where the relation (35) takes place is caused
3 Copyright 7 Tech Science Press CMC vol.53 no.4 pp.37-34 7 naely with the viscoelasticity of the aterials of the constituents of the syste under consideration. We analyse the physico-echanical eaning of the relation (35). In the wave propagation sense the relation (35) eans that in the case where this relation satisfies the phase velocity of the wave propagation becoes equal to the corresponding group velocity. According to this stateent in the cases where kh ( kh ) cr (in the cases where ( k h) cr k h ) the dispersion of the waves for viscoelastic case is anoalous (noral) dispersion. At the sae tie the relation (35) eans that the velocity c ccr is root of the second order of the dispersion Eq. (or (3)). According to the nuerous investigations on the dynaics of the oving load acting on the covering layer + halfspace systes for instance in the onograph by Akbarov (5) and any others listed therein the velocity which is second order root of the dispersion equation i.e. under which the relation (35) takes place is the critical velocity of the corresponding oving load. We recall that under this velocity of the oving load the resonance type accidents appear. Table : The values of the c/ c obtained in the case indicated in Figure a under kh.5 Table : The values of the c/ c obtained in the case indicated in Figure b under kh.5 Q c / d c/ c Q ( t ).6 d ( t ).6 Q =.588 d =.59 Q =.584 d = 5.58 Q = 5.576 d = 5.574 Q =.55 d = 5.558 Q = 5.57 d =.534 Q =.444 d = 5.456 Q ( t ).6 d =.96 Thus it follows fro the foregoing discussions that the viscoelasticity of the aterials of the constituents influences on the dispersion curves of the generalized Rayleigh waves not only in the quantitative sense but also in the qualitative sense. Moreover it follows fro the foregoing results that under investigations of the dynaics of the oving load acting on the systes which can be odelled as the covering layer+half-space the appearing of the critical velocities as a result of viscoelasticity of aterials of the constituents ust be taken into consideration. For a clear illustration of the aount of the influence of the aterials' viscosity on the wave propagation velocity in the cases considered in Figure and 3 the values of these velocity are presented in Tables 3 and 4 in the cases where kh.5 (Tables and
Generalized Rayleigh Wave Dispersion in a Covered Half-space 33 ) and kh.(tables 3 and 4). It follows fro these data that in the quantitative sense the influence of the rheological paraeter d on the wave propagation velocity is ore significant than that of the rheological paraeter Q. Table 3: The values of the c/ c obtained in the case indicated in Figure 3a under kh. Table 4: The values of the c/ c obtained in the case indicated in Figure 3b under kh. Q c / d c/ c Q ( t ).799 d ( t ).799 Q =.77 d =.77 Q =.754 d = 5.745 Q = 5.76 d = 5.795 Q =.73 d = 5.744 Q = 5.758 d =.769 Q =.68 d = 5.687 Q ( t ).697 d =.539 We recall that the all foregoing results on the dispersion curves and the results which will be discussed below are obtained within the scope of the attenuation deterined by expression (3) and in the case where () Q Q ( Q) () d d ( d) () ( ) this expression can presented as () s s () () c c ( ) ( ). (36) ( ) ( ) Consider graphs of the dependence of the attenuation coefficient and diensionless frequency h/ c which are given in Figure 4 for various values of the rheological paraeter d under a fixed 5 paraeter Q under a fixed Q (Figure 4a) and for various values of the rheological d (Figure 4b) when.5. Using these results and the results obtained for the dispersion curves for instance for the dispersion curves given in Figure and 3 one can easily deterine the value of the attenuation coefficient for each selected value of the wave propagation velocity. Note that this stateent reains valid also for all the results which will be considered below.
34 Copyright 7 Tech Science Press CMC vol.53 no.4 pp.37-34 7 (a) Figure 4: (a) The influence of paraeter d d () d () Q Q Q 5 and (b) the influence of paraeter Q Q () Q () d d d. (b) on under on under Now we attept to answer a question what contribution is ade the viscoelasticity of each constituent of the syste under consideration on the dispersion curves and for this purpose we consider the graphs given in Figure 5-8 which are obtained in the case where ().5. Note that in these figures the graphs grouped by letter a (letter b) relate to the case where the rheological paraeters related to the half-space (to the covering layer) aterial are changed and the rheological paraeters related to the covering layer (to the half-space) are fixed. Moreover note that the results given in Figure 5 and 7 () () (Figure 6 and 8) relate to the case where / (to the case where / 9). The results given in Figure 5 and 6 (in Figure 7 and 8) illustrate the influence of the () () rheological paraeters Q and Q (rheological paraeters d and d ) on the dispersion curves. Thus it follows fro the graphs given in Figure 5-8 that the ain contribution on the dispersion curves is ade by the viscoelasticity of the half-space aterial.
Generalized Rayleigh Wave Dispersion in a Covered Half-space 35 (a) Figure 5: Dispersion curves in V.V. case paraeter paraeter () Q when Q when () / Q 5 and d d d () () Q 5 and d d d (). (b) (a) for different values of (b) for different values of (a) Figure 6: Dispersion curves in V.V. case paraeter paraeter () Q when Q when () / 9 Q 5 and d d d () () Q 5 and d d d (). (b) (a) for different values of (b) for different values of
36 Copyright 7 Tech Science Press CMC vol.53 no.4 pp.37-34 7 (a) Figure 7: Dispersion curves in V.V. case paraeter paraeter () d when d when () / d and Q Q Q () 5 () d and Q Q Q () 5. (b) (a) for different values of (b) for different values of (a) Figure 8: Dispersion curves in V.V. case paraeter paraeter () d when d when () / 9 d and Q Q Q () 5 () d and Q Q Q () 5. (b) (a) for different values of (b) for different values of We recall that the foregoing results are obtained in the cases where both the aterials of the covering layer and the half-space are viscoelastic i.e. the V.V. case. Now we consider the results obtained in the case where the aterial of the covering layer is purely elastic but the aterial of the half-space is viscoelastic and denote this case as the
Generalized Rayleigh Wave Dispersion in a Covered Half-space 37 E.V. case. Note that the all foregoing results for this case are obtained within the scope of the attenuation deterined by second expression in (3). Figure 9 shows the graphs obtained for the case E.V. when () / for different values of paraeter () (Figure 9a) under fixed values of paraeter d and for different values of () () paraeter d (Figure 9b) under fixed values of paraeter Q 5. () Q Coparison of the graphs given in Figure 9a with the corresponding V.V. ones given in Figure 5a shows that the influence of the rheological paraeters of the half-space on the dispersion curves obtained in the E.V. case under () / () Q is alost the sae as in the V.V. case. However coparison of the graphs given Figure 9b with the corresponding V.V. ones given in Figure 7a shows that the influence of the rheological () paraeters of the half-space d on the dispersion curves obtained in the E.V. case is ore significant than that obtained in the V.V. case. We do not consider here the () nuerical results obtained for / 9 in the E.V. case because these results in the qualitative sense are the sae. Now we consider the results related to the effect of rheological paraeters () () () on the wave dispersion curves in the case where d d d and Q Q Q. Note that the influence of this rheological paraeter on the wave dispersion is considered for the first tie in the present paper and this effect has not been exained in the papers by Akbarov and Kepceler (5) and in the paper by Akbarov Kocal and Kepceler (6). Thus we consider graphs given in Figure -3 which illustrate the entioned influence. () Note that these graphs are constructed in the cases where / (Figure and () ) and / 9 (Figure and 3). Moreover note that the graphs given in Figure and (Figure and 3) show the effect of the rheological paraeter on the dispersion curves under various values of the rheological paraeter Q (of the rheological paraeter d ) for a fixed value of the rheological paraeter d (for a fixed value of the rheological paraeter Q ). The graphs given in Figure and (in Figure and 3) and grouped by letters a b c and d correspond the cases where Q 5 and 5 (the case where d 5 5 and 5) respectively.
38 Copyright 7 Tech Science Press CMC vol.53 no.4 pp.37-34 7 (a) Figure 9: Dispersion curves in E.V. case paraeter () Q when () / () d (b) for different values of paraeter (b) (a) for different values of () d when () Q 5. Thus it follows fro the results given in Figure -3 that in the all considered cases there exists such value of the diensionless wavenuber kh (denote it by ( kh )* at which the change in the values of the rheological paraeter does not influence on the values of the wave propagation velocity. However in the cases where kh ( kh )* (in the cases where k h ( k h)* ) an increase in the values of the paraeter causes a decrease (an increase) in the wave propagation velocities. According to the aforeentioned nuerical results it can be concluded that the ( kh )* depends on the values of the rheological paraeters Q and d and an increase in the values of these paraeters causes to decrease of the ( kh )*. Moreover it can be concluded the values of the ( kh )* () depends also on the ratio / and an increase of this ratio causes a decrease in the values of the ( kh )*.
Generalized Rayleigh Wave Dispersion in a Covered Half-space 39 (a) (b) Figure : Dispersion curves in V.V. case for different values of paraeter and Q () and /. when d d d ()
33 Copyright 7 Tech Science Press CMC vol.53 no.4 pp.37-34 7 (a) (b) (c) (d) Figure : Dispersion curves in V.V. case for different values of paraeter and d when () () Q Q Q and /.
Generalized Rayleigh Wave Dispersion in a Covered Half-space 33 (a) (b) (c) (d) Figure : Dispersion curves in V.V. case for different values of paraeter and Q () and / 9. when d d d ()
33 Copyright 7 Tech Science Press CMC vol.53 no.4 pp.37-34 7 (a) (b) (a) (b) Figure 3: Dispersion curves in V.V. case for different values of paraeter and d when () () Q Q Q and / 9. Note that besides of all these it can be concluded that the change in the values of the rheological paraeter does not influence on the liit values of the wave propagation velocity as kh. However in the near vicinity of this liit case if to say ore precisely in the region kh ( kh )* the influence of the rheological paraeter on the dispersion curves is significant not only in the quantitative sense but also in the qualitative sense. So that under sall values of the for instance under. the dispersion curves have well-defined iniu in the region kh ( kh )* and at this iniu the relation (35) takes place. Moreover in the near vicinity of this iniu
Generalized Rayleigh Wave Dispersion in a Covered Half-space 333 the wave propagation velocity obtained for the viscoelastic cases becoe less than that obtained for the purely elastic case with long-ter values of elastic constants at t. Consequently in the region kh ( kh )* a decrease in the values of the causes to change the character of the dispersion curves. However with increasing of the aforeentioned iniu disappears in the dispersion curves and wave propagation velocities are liited with the wave propagation velocities obtained for the purely elastic cases with instantaneous values of elastic constants at t (upper liit) and with longter values of elastic constants at t (lower liit). Thus the all foregoing results and discussions allow us to conclude that in investigations of the wave propagation in eleents of constructions ade of viscoelastic aterial the viscoelastic relations of which are described through the fractional exponential operators it is necessary to take into consideration the influence of the rheological paraeter on this propagation. We recall that all the foregoing results are obtained for the dispersion attenuation case with the attenuation coefficient deterined through the expression (3) according to which as kh. This stateent can be proven as follows: According to the expressions in (3) we can see that and ( n) as kh. This is because ( ) in the first expression in (3). Taking this liit value of the paraeter into account we obtain fro the expressions given in (9) that R c and R ( ) as kh. According to these liit values it follows fro s the expressions in (8) that s and also it follows fro the expressions in (3) (or fro the expressions in (36)) that as kh. Thus for estiation of the influence of this character of the dispersion attenuation on the wave dispersion curves we also consider a few results obtained for a non-dispersive attenuation case. Note that under this non-dispersive attenuation the value of the k (or kh ) in () given a priori as a constant and then the dispersion curves are deterined fro the dispersion Eq. (3). Let us call the kh as an attenuation order. Thus analyse the graphs given in Figure 4 which illustrate the dispersion curves () obtained for the non-dispersive attenuation case under kh.5 d d d and () Q Q Q for the case where () /. Note that these graphs are constructed for the various values of the paraeter Q under fixed value of the d (Figure 4a) and for the various values of the paraeter d under fixed Q 5 (Figure 4b). It follows fro these results that as a result of the non-dispersity of the attenuation of the waves propagated in the viscoelastic aterials the cut off values of the wavenuber kh appear. We denote this cut off value through ( kh ) cf.. and note that if we ultiply
334 Copyright 7 Tech Science Press CMC vol.53 no.4 pp.37-34 7 ( kh ) cf.. with the corresponding wave propagation velocity c then we obtain the corresponding cut off frequency c. f. c( k h) c. f.. (a) (b) Figure 4: Dispersion curves obtained in the non-dispersive attenuation case under kh.5 in the V.V. case (a) for various values of the paraeter Q under a fixed value of the paraeter d( ) and (b) for various values of the paraeter d under a () fixed value of the paraeter Q( 5) in the V.V. case under /. Figure 5: The influence of the attenuation order kh on the cut off values of kh i.e. on the values of ( kh ) cf.. in the case considered in Figure 4.
Generalized Rayleigh Wave Dispersion in a Covered Half-space 335 It follows fro Figure 4 that the change in the rheological paraeter Q does not influence on the values of the ( kh ) cf.. however an increase in the values of the rheological paraeter d causes to increase the values of the ( kh ) cf... Figure 5 shows the dispersion curves constructed for various values of the attenuation order kh in the case where Q 5 and d fro which follows that the values of the ( kh ) cf.. increase onotonically with the kh. Thus the results given in Figure 4 and 5 allow us to conclude that as a result of the dispersity of the attenuation of the waves under consideration the cut off wavenubers or frequencies disappear. The experiental evaluation application of the nuerical results obtained in the present paper can be ade in the following anner. First it is necessary to odel the viscoelasticity of the aterials of the constituents of the syste through the fractional exponential operators by Rabotnov and to deterine the values of the corresponding rheological paraeters which enter to these operators. After it can be used the traditional ethods on the easureent of the near surface wave propagation velocities for deterination of the influence of the rheological paraeters of the aterials to this velocity. The obtained nuerical results allow to use those not only in the aforeentioned for easureents procedures in the concrete selected cases but also these results allows to have theoretical knowledge on the character of this influence in the principal sense. This copletes the discussions of the nuerical results. 5 Conclusions Thus in the present paper dispersion of the generalized Rayleigh waves in the covering layer+half-space syste ade of viscoelastic aterials is investigated. The investigations carried out within the scope of the piecewise hoogeneous body odel by utilizing the exact equations of otion of the theory of linear viscoelasticity in the plane strain state. The ain processing flow of investigations carried out in the present paper is as follows: ) the analytical expressions of the sought values are deterined for arbitrary kernel functions in the operators described the viscoelasticity of the aterials by eploying the ethod of separation of variables; ) the corresponding dispersion equation is also obtained for arbitrary hereditary type viscoelastic operators; 3) for nuerical investigations the viscoelasticity operators (the kernel functions in these operators) of the aterials are specialized through the fractional exponential operators by Rabotnov (98) according to which diensionless rheological paraeters characterizing the characteristic creep tie (denoted by Q ) the long-ter values of the elastic constants (denoted by d ) and the for of the creep (or relaxation) function of the aterials in the beginning region of deforations (denoted by ) are introduced and through these
336 Copyright 7 Tech Science Press CMC vol.53 no.4 pp.37-34 7 paraeters the viscoelasticity of the aterials of the covered layer and the half-space on the dispersion curves is studied; 4) nuerical investigations are ade for the cases where the wave attenuation coefficients are deterined through the expressions in (3); 5) These results are obtained for the cases under which the generalized Rayleigh waves () () exists i.e. under which the conditions Re{ R k} and Re{ R k} satisfy siultaneously where the R and and k is a coplex wavenuber deterined as in (); R are deterined through the expression in () and finally 6) The nuerical results are presented and discussed ainly for the attenuation dispersion case and at the sae tie a few nuerical exaples are also presented and discussed for the non-dispersive attenuation case. Moreover these results are obtained for the case where the viscoelasticity properties of the covering layer and the half-space aterials are the sae (denoted as V.V. case) and for the case where the aterial of the covering layer is purely elastic but the aterial of the half-space is viscoelastic (denoted as E.V. case). According to these nuerical results the following ain conclusions can be drawn: In both V.V. and E.V. cases in the considered attenuation dispersion case the viscoelasticity of the aterials causes the generalized Rayleigh wave propagation velocity to decrease. The agnitude of this decrease increases with a decrease in the aforeentioned diensionless rheological paraeters d and Q ; The character of the influence of the paraeter on the wave velocities and on the dispersion corves depends on the values of the diensionless wave nuber kh and on the values of the rheological paraeters d and Q. There exist such value of the kh after (before) which an increase in the values of the causes to decrease (to increase) the wave propagation velocity; The lower wavenuber liit values of the wave propagation velocity depends only on the rheological paraeter d and coincide with that obtained for the corresponding purely elastic case with long-ter values of elastic constants at t and as a result of this stateent the viscoelasticity of the aterials of the constituents causes to change the character of the dispersion of the waves under consideration and to appear the cases where the relation (35) satisfies; The satisfaction of the relation (35) confors the appearing of the critical velocities of the oving load acting on the syste under consideration as a result of the viscoelasticity of the aterials of the constituents of the syste; In general the dispersion curves obtained to viscoelastic cases are liited by the dispersion curves corresponding to the purely elastic case with instantaneous values of the elastic constants (upper liit) and by those obtained for the purely elastic case with long-ter values of the elastic constants (lower liit) however this rule is violated in the relatively sall values of the rheological paraeter under which the wave propagation velocity can becoe less than the entioned lower liit;