Copyright 2018 Tech Science Press CMC, vol.54, no.2, pp , 2018

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1 Copyight 08 Tech Science Pess CMC, vol.54, no., pp.03-36, 08 The Influence of the Impefectness of Contact Conditions on the Citical Velocity of the Moving Load Acting in the Inteio of the Cylinde Suounded with Elastic Medium M. Ozisik, *, M. A. Mehdiyev and S. D. Akbaov, 3 Abstact: The dynamics of the moving-with-constant-velocity intenal pessue acting on the inne suface of the hollow cicula cylinde suounded by an infinite elastic medium is studied within the scope of the piecewise homogeneous body model by employing the exact field equations of the linea theoy of elastodynamics. It is assumed that the intenal pessue is point-located with espect to the cylinde axis and is axisymmetic in the cicumfeential diection. Moeove, it is assumed that shea-sping type impefect contact conditions on the inteface between the cylinde and suounding elastic medium ae satisfied. The focus is on the influence of the mentioned impefectness on the citical velocity of the moving load and this is the main contibution and diffeence of the pesent pape the elated othe ones. The othe diffeence of the pesent wok fom the elated othe ones is the study of the esponse of the inteface stesses to the load moving velocity, distibution of these stesses with espect to the axial coodinates and to the time. At the same time, the pesent wok contains detail analyses of the influence of poblem paametes such as the atio of modulus of elasticity, the atio of the cylinde thickness to the cylinde adius, and the shea-sping type paamete which chaacteizes the degee of the contact impefection on the values of the citical velocity and stess distibution. Coesponding numeical esults ae pesented and discussed. In paticula, it is established that the values of the citical velocity of the moving pessue decease with the extenal adius of the cylinde unde constant thickness of that. Keywods: Moving intenal pessue, citical velocity, cicula hollow cylinde suounded by elastic medium, shea-sping type impefection, inteface stesses. Depatment of Mathematical Engineeing, Yildiz Technical Univesity, Davutpasa Campus, 340, Esenle- Istanbul-Tukey; ozisik@yildiz.edu.t Institute of Mathematics and Mechanics of the National Academy of Sciences of Azebaijan, 3704, Baku, Azebaijan; mahimehdiyev@mail.u 3 Depatment of Mechanical Engineeing, Yildiz Technical Univesity, Yildiz Campus, 34349, Besiktas, Istanbul, Tukey; akbaov@yildiz.edu.t * Coesponding autho: M. Ozisik. ozisik@yildiz.edu.t. CMC. doi: /cmc

2 04 Copyight 08 Tech Science Pess CMC, vol.54, no., pp.03-36, 08 Intoduction The development and successful application of moden high-speed undegound tains and othe types of undegound moving wheels equies fundamental study of the coesponding dynamic poblems. As usual, undegound stuctues into which such high-speed wheels move ae modelled as infinite hollow cylindes suounded by an elastic o viscoelastic medium. Consequently, the afoementioned dynamical poblems can be modelled as the poblem of the moving pessue acting on the intenal suface of an infinite hollow cylinde suounded by an elastic o viscoelastic medium. The pesent pape is concened namely with these types of poblems and it studies the dynamics of the moving point-located, with espect to the cylinde axis, nomal foces which ae unifomly distibuted in the cicumfeential diection acting on the inne suface of the cicula cylinde which is suounded by an elastic medium. It should be noted that, in geneal, the main issue in the investigations of the moving load acting on layeed systems is the detemination of both the citical velocity of this moving load unde which esonance type behavio takes place and of the influence of the poblem paametes on the values of this velocity. At the same time, anothe issue which is also impotant is to detemine the ules of attenuation of the petubations of the stesses and displacements caused by the moving load with the distance fom the point at which this load acts. Now we conside a bief eview of elated investigations egading layeed systems and note that the fist attempt in this field was made by Achenbach et al. [Achenbach, Keshava and Hemann (967)] in which the dynamic esponse of the system consisting of the coveing laye and half plane to a moving load was investigated with the use of the Timoshenko theoy fo descibing the motion of the plate. Howeve, the motion of the half-plane was descibed by using the exact equations of the theoy of linea elastodynamics and the plane-stain state was consideed. It was established that citical velocity exists in the cases whee the plate mateial is stiffe than that of the half-plane mateial. Reviews of late investigations, which can be taken as developments of those stated in the pape by Achenbach et al. [Achenbach, Keshava and Hemann (967)], ae descibed in the papes by Dieteman et al. [Dieteman and Metikine (997)] and by Metikine et al. [Metikine and Vouwenvelde (000)]. At the same time, in the pape by Dieteman et al. [Dieteman and Metikine (997)], the citical velocity of a point-located time-hamonic vaying and unidiectional moving load which acts on the fee face plane of the plate esting on the igid foundation, was investigated. The investigations wee made within the scope of the 3D exact equations of the linea theoy of elastodynamics and it was established that as a esult of the time-hamonic vaiation of the moving load, two types of citical velocities appea: the fist (the second) of which is lowe (highe) than the Rayleigh wave velocity in the plate mateial. Late, simila esults wee also obtained by Akbaov et al. [Akbaov and Salmanova (009)], Akbaov et al. [Akbaov and Ilhan (009)], and Akbaov et al. [Akbaov, Ilhan and Temugan (05)], which wee also discussed in the monogaph by Akbaov [Akbaov (05)]. The afoementioned pape by Metikine et al. [Metikine and Vouwenvelde (000)] deals with the study of the suface vibation of the laye esting on the igid foundation and containing a beam on which the moving load acts. In this pape, a two-dimensional

3 The Influence of the Impefectness of Contact Conditions 05 poblem is consideed and the motion of the laye is descibed by the exact equations of elastodynamics, howeve, the motion of the beam is descibed by the Eule-Benoulli theoy. At the same time, it is assumed that the mateial of the laye has a small viscosity and thee types of load ae examined, namely constant, hamonically vaying and a stationay andom load. Note that all the investigations caied out in the papes by Metikine et al. [Metikine and Vouwenvelde (000)], and Dieteman et al. [Dieteman and Metikine (997)] ae based on the concept of the citical velocity which is detemined though the dispesion diagams of the coesponding wave popagation. Up to now, a cetain numbe of investigations have also been made elated to the dynamics of the moving load acting on the pestessed system. Fo instance, in the pape by Ke [Ke (983)], the initial stesses on the values of the citical velocity of the moving load acting on an ice plate esting on wate wee taken into account. In this pape, the motion of the plate is descibed by employing the Kichhoff plate theoy and it is established that the initial stetching (compession) of the plate along the load moving diection causes an incease (a decease) in the values of the citical velocity. Moeove, in the pape by Metikine et al. [Metikine and Dieteman (999)], unde investigation of the lateal vibation of the beam on an elastic half-space due to a moving lateal time-hamonic load acting on the beam, the initial axial compession of this beam is also taken into consideation. In this investigation, the motion of the beam is witten though the Eule-Benoulli beam theoy, howeve, the motion of the half-space is descibed by the 3D exact equations of elastodynamics and it is assumed thee ae no initial stesses in the half-space. The influence of the initial stesses acting in the half-plane on the citical velocity of the moving load which acts on the plate which coves this half-plane was studied in the papes by Babich et al. [Babich, Glukhov and Guz (986, 988, 008a)]. Note that in these studies, the plane stain state was consideed and the motion of the half-plane was witten within the scope of the thee-dimensional lineaized theoy of elastic waves in initially stessed bodies (see Guz [Guz (999, 004)]). Howeve, the motion of the coveing laye, which does not have any initial stesses, as in the pape by Achenbach et al. [Achenbach, Keshava and Hemann (967)], was witten by employing the Timoshenko plate theoy and the plane-stain state is consideed. Fo moe detail, we note that in the pape by Babich et al. [Babich, Glukhov and Guz (986)], the coesponding bounday value poblem fo the highly-elastic compessible half-plane was solved by employing the Fouie tansfomation with espect to the spatial coodinate along the plate-lying diection. Numeical esults on the citical velocity of the moving load wee pesented fo the case whee the elasticity elations of the half-plane mateial ae descibed though the hamonic potential. In the othe pape by Babich et al. [Babich, Glukhov and Guz (988)], the same poblem which was consideed in the pape by Babich et al. [Babich, Glukhov and Guz (986)] was solved by employing the complex potentials of the thee-dimensional lineaized theoy of elastic waves in initially stessed bodies, which ae developed and pesented in the monogaph by Guz [Guz (98)]. Note that in the papes by Babich et al. [Babich, Glukhov and Guz (986, 988)], the subsonic egime was consideed. Howeve, in the papes by Babich et al. [Babich, Glukhov and Guz (008a)], the foegoing investigations of these authos wee developed fo the

4 06 Copyight 08 Tech Science Pess CMC, vol.54, no., pp.03-36, 08 supesonic moving velocity of the extenal load fo the case whee the mateial of the half-plane is incompessible [Babich, Glukhov and Guz (008a)] as well as fo the case whee this mateial is compessible [Babich, Glukhov and Guz (008b)]. Futhemoe, in the woks of the fist autho of the pesent pape and his collaboatos, the investigations of the poblems elated to the dynamics of the moving load acting on the layeed systems wee developed and unde these investigations not only the motion of the half-plane but also the motion of the coveing laye was witten within the scope of the exact equations of the thee-dimensional lineaized theoy of elastic waves in initially stessed bodies. Some of these woks ae as follows. In the pape by Akbaov et al. [Akbaov, Gule and Dincoy (007)], the influence of the initial stesses in the coveing laye and half-plane on the citical velocity of the moving load acting on the plate coveing the half-plane, was studied. The same moving load poblem fo the system consisting of the coveing laye, substate and half-plane was studied in the pape by Dincsoy et al. [Dincsoy, Gule and Akbaov (009)]. The dynamics of the system consisting of the othotopic coveing laye and othotopic half plane unde action of the moving and oscillating moving load wee investigated in the papes by Akbaov et al. [Akbaov and Ilhan (008, 009)] and the influence of the initial stesses in the constituents of this system on the values of the citical velocity was examined in the pape by Ilhan [Ilhan (0)]. Moeove, in the pape by Akbaov et al. [Akbaov and Salmanova (009)], the dynamics of the oscillating moving load acting on the pe-stained bi-layeed slab made of highly elastic mateial and esting on a igid foundation was studied. The 3D poblems of the dynamics of the moving and oscillating moving load acting on the system consisting of a pe-stessed coveing laye and halfspace wee consideed in the pape by Akbaov et al. [Akbaov, Ilhan and Temugan (05)] and, in paticula, it was established that the minimal values of the citical velocities detemined within the scope of the 3D fomulation coincide with the citical velocity detemined within the scope of the coesponding D fomulation. These and othe elated esults wee also detailed in the monogaph by Akbaov [Akbaov (05)]. Finally, we note the pape by Akbaov et al. [Akbaov, Ilhan and Temugan (05)] in which the dynamics of the lineally-located moving load acting on the hydo-elastic system consisting of elastic plate, compessible viscous fluid and igid wall wee studied. It was established that thee exist cases unde which the citical velocities appea. Note that elated non-linea poblems egading the moving and inteaction of two solitay waves was examined in the pape by Demiay [Demiay (04)] and othe ones cited theein. At the same time, non-stationay dynamic poblems fo viscoelastic and elastic mediums wee studied in the papes by Ilyasov [Ilyasov (0)], Akbaov [Akbaov (00)] and othe woks cited in these papes. Moeove, it should be noted that up to now it has been made consideable numbe investigations on fluid flow-moving aound the stagnation-point (see the pape by Aman et al. [Aman and Ishak (04)] and othe woks listed theein). Some poblems elated to the effects of magnetic field and the fee steam flow-moving of incompessible viscous fluid passing though magnetized vetical plate wee studied by Ashaf et al. [Ashaf, Asgha and Hossain (04)].

5 The Influence of the Impefectness of Contact Conditions 07 The investigations elated to the fluid-plate and fluid-moving plate inteaction wee consideed the papes by Akbaov et al. [Akbaov and Panachli (05)] and Akbaov et al. [Akbaov and Ismailov (05, 06)]. It follows fom the foegoing bief eview that almost all investigations on the dynamics of the moving and oscillating moving load elate to flat-layeed systems. Howeve, as noted in the beginning of this section, cases often occu in pactice in which it is necessay to apply a model consisting of cylindical layeed systems, one of which is the hollow cylinde suounded by an infinite o finite defomable medium unde investigation of the dynamics of a moving o oscillating moving load. It should be noted that up to now some investigations have aleady been made in this field. Fo instance, in the pape by Abdulkadiov [Abdulkadiov (98)] and othes listed theein, the low-fequency esonance axisymmetic longitudinal waves in a cylindical laye suounded by an elastic medium wee investigated. Note that unde esonance waves the cases unde which the elation dc dk 0 occus, is undestood, whee c is the wave popagation velocity and k is the wavenumbe. It is evident that the velocity of these esonance waves is the citical velocity of the coesponding moving load. Some numeical examples of esonance waves ae pesented and discussed. It should be noted that in the pape by Abdulkadiov [Abdulkadiov (98)], in obtaining dispesion cuves, the motion of the hollow cylinde and suounding elastic medium is descibed though the exact equations of elastodynamics, howeve, unde investigation of the displacements and stains in the hollow cylinde unde the wave pocess, the motion of the hollow cylinde is descibed by the classical Kichhoff-Love theoy. Anothe example of the investigations elated to the poblems of the moving load acting on the cylindical layeed system is the investigation caied out in the pape by Zhou et al. [Zhou, Deng and Hou (008)] in which the citical velocity of the moving intenal pessue acting in the sandwich shell was studied. Unde this investigation, two types of appoaches wee used, the fist of which is based on fist ode efined sandwich shell theoies, while the second appoach is based on the exact equations of linea elastodynamics fo othotopic bodies with effective mechanical constants, the values of which ae detemined by the well-known expessions though the values of the mechanical constants and volumetic faction of each laye of the sandwich shell. Numeical esults on the citical velocity obtained within these appoaches ae pesented and discussed. Compaison of the coesponding esults obtained by these appoaches shows that they ae sufficiently close to each othe fo the low wavenumbe cases, howeve, the diffeence between these esults inceases with the wavenumbe and becomes so geat that it appeas necessay to detemine which appoach is moe accuate. It is evident that fo this detemination it is necessay to investigate these poblems by employing the exact field equations of elastodynamics within the scope of the piecewise homogeneous body model, which is also used in the pesent pape. Finally, we note the woks by Hung et al. [Hung, Chen and Yang (03); Hussein, Fançois, Schevenels et al. (04); Yuan, Bostom and Cai (07)] and othes listed theein, in which numeical and analytical solution methods have been developed fo studying the dynamical esponse of system consisting of the system consisting of the cicula hollow cylinde and suounding elastic medium to the moving load acting on the

6 08 Copyight 08 Tech Science Pess CMC, vol.54, no., pp.03-36, 08 inteio of the cylinde. Howeve, the main aim of these investigations is the study of a displacement distibution of the suounding elastic medium caused by the moving load and the analyses elated to the citical velocity and to the esponse of the inteface stesses to the moving load ae almost completely absent. Moeove we note the development in ecent yeas numeical methods based on vaious type finite elements and applied fo investigations of the dynamics and statics of the elated bi-mateial elastic and piezoelectic systems (see, fo instance the papes by Fan et al. [Fan, Zhang, Dong et al. (05)], Babuscu [Babuscu (07)], Wei et al. [Wei, Chen, Chen et al. (06)] and othes listed in these papes. With this we estict ouselves to eviewing elated investigations, fom which it follows that up to now thee have not been any systematic investigations on the dynamics of the moving intenal pessue acting on the hollow cylinde suounded by an elastic medium caied out within the scope of the piecewise homogeneous body model by employing the exact field equations of linea elastodynamics. Thee have not been any systematic investigations not only fo the cases whee between the constituents of the hollow cylinde+suounding infinite elastic medium system the impefect contact conditions exist but also in the cases whee these conditions ae pefect. Thee ae also no esults on the inteface stess and displacement distibutions caused by this moving load. Namely these and othe elated questions ae studied in the pesent pape. It should be noted that the elated study of the axisymmetic foced vibation of the hollow cylinde+suounding infinite elastic medium system unde pefect and impefect contact between the constituents is caied out in the pape by Akbaov et al. [Akbaov and Mehdiyev (07)]. Mathematical Fomulation of the Poblem Conside a hollow cicula cylinde with thickness h which is suounded by an infinite elastic medium. We associate the cylindical and catesian systems of coodinates O z and Ox xx 3 (Fig. ) with the cental axis of this cylinde. Assume that the extenal adius of the coss section of the cylinde is R and on its inne suface axisymmetic unifomly distibuted nomal foces act in the cicumfeential diection moving with constant velocity V which is point-located with espect to the cylinde cental axis. Within this famewok, we investigate the axisymmetic stess-stain state in this system by employing the exact equations of the linea theoy of elastodynamics within the scope of the piecewise homogeneous body model. Below, the values elated to the cylinde and to the suounding elastic medium will be denoted by uppe indices () and (), espectively.

7 The Influence of the Impefectness of Contact Conditions 09 Figue : The sketch of the elastic system unde consideation We suppose that the mateials of the constituents ae homogeneous and isotopic. We wite the field equations and contact conditions as follows. Equations of motion: ( ) ( ) ( ) k z k ( ( k) ( k) ) ( k) u k z t ( k) ( k) ( k) z zz ( k) ( k) u z z z t Elasticity elations: ( ) ( ) ( ) ( k) k k ( k ( k) ) ( k) ( k) nn zz nn, nn ; ; zz, ( k) ( k) ( k) z z () Stain-displacement elations: ( k) ( k) u ( k) ( k) u ( k), ( k) u ( k) ( k), z zz ( k) u, ( z u z ) (3) z z Note that the Eq. (), () and (3) ae the complete system of the field equations of the linea theoy of elastodynamics in the case unde consideation and in these equations conventional notation is used. Conside the fomulation of the bounday and contact conditions. Accoding to the foegoing desciption of the poblem, the bounday conditions on the inne face suface of the cylinde can be fomulated as follows. () P0 ( z Vt ) Rh, () z 0 Rh (4) We assume that the contact conditions with espect to the foces and adial displacement ae continuous and can be witten as follows:, ()

8 0 Copyight 08 Tech Science Pess CMC, vol.54, no., pp.03-36, 08 () (), R R () () z z, R R u () () u (5) R R At the same time, we assume that shea-sping type impefection occus in the contact conditions elated to the axial displacements and, accoding to Akbaov [Akbaov (05)] and othes listed theein, these conditions ae fomulated by the following equation: () () FR u () z uz () z (6) R R R The dimensionless paamete F in (6) chaacteizes the degee of the impefection and the ange of change of this paamete is F. Note that the case whee F 0 coesponds to complete contact, but the cases whee F coespond to full slipping contact conditions. Note that the main contibution of the pesent investigation is caused with the condition (6). We will also assume that () () V min c ; c, ( n) c ( n) ( n), n,, (7) i.e. we will conside the subsonic moving velocity. Accoding to (7), we can wite that () () ; ; () ; () ; () ; () zz z u uz 0 as and z fo each [0, ] () () ; ; () ; () ; () ; () zz z u uz 0 as z fo each [0, ] (8) This completes fomulation of the poblem and consideation of the govening field equations. 3 Method of solution Fo solution to the poblem fomulated above we use the well-known, classical Lame (o Helmholtz) decomposition (see, fo instance, Eingen et al. [Eingen and Suhubi (975)]): ( k) ( k) ( k) ( k) ( k) u ( k) ( k), u z, (9) z z z ( k) ( c ) t ( k) ( k) whee and satisfy the following equations: ( k) ( k) ( k) 0 ( k), 0, ( k) ( c ) t ( k) ( c ) t, (0) z

9 The Influence of the Impefectness of Contact Conditions ( k) whee c ( ) ( ) ( ) ( k k ) k ( k) and c ( k) ( k). We use the moving coodinate system ', z ' z Vt () which moves with the loading intenal pessue and by ewiting the Eq. (0) with the coodinates ' and z ', we obtain: ( k) ( k) V 0, ( k) ( c ) z ( k) ( k) V 0 () ( k) ( c ) z whee the pimes on the and z have been omitted and V in Eq. () and () is the moving velocity of the intenal pessue. Afte coodinate tansfomation () the fist condition in (4) tansfoms to the following one: () P0 ( z ) (3) Rh but the othe elations and conditions in (-8) emain valid in the new coodinates detemined by (). Below we will use the dimensionless coodinates / h and z z / h instead of the coodinates and z, espectively and the ove-ba in and z will be omitted. Thus, the solution of the consideed bounday value poblem is educed to the solution to the equations in (). Fo this pupose we use the Fouie tansfomation with espect to the coodinate z and by taking the poblem symmety with espect to the point z 0 into consideation, the functions ( k) and ( k), can be pesented as follows. ( k) ( k) ( k) ( k) ( k) ( k) ( k) ( k) ; u ; nn ; nn (, z) F ; uf ; nnf ; nnf (, s)cos( sz) ds, 0 nn ; ; zz ( k) ( k) ( k) ( k) ( k) ( k) ( k) ( k) z z z F zf zf zf ; u ; ; ; (, z) ; u ; ; (, s)sin( sz) ds (4) 0 Substituting the expessions in (4) into the foegoing equations, elations and contact and bounday conditions, we obtain the coesponding ones fo the Fouie tansfomations of the sought values. Note that afte this substitution, the elation (), the fist and second elation in (3), the second condition in (4) and all the conditions in (5) and (6) emain as fo thei Fouie tansfomations. Howeve, the thid and fouth elation in (3) and the condition (3) and the elations in (9) tansfom to the following ones: ( k) ( k) ( k) ( k) du ( ) zzf suzf, ( k zf zf su ) d F, () F P 0 R h

10 Copyight 08 Tech Science Pess CMC, vol.54, no., pp.03-36, 08 ( k) ( k) ( k) df d u F ( k) ( k) V ( k) F s, u ( ) d d zf sf s ( k) F (5) ( c ) Moeove, afte the afoementioned substitution we obtain the following equations fo ( k ) and ( k) fom the equations in (). F d d V ( k) s ( ) 0 d ( k) ( F, d c ) d d V ( k) s ( ) 0 d ( k) ( F d c ) Accoding to the condition (7) and to the fist ow of conditions in (8) fom which () () () () () () follows that ; ; zz ; z ; u ; uz 0 as unde z const, the solution to the equations in (6) we find as follows: () () () () () () () () F A I0 ) A K0 ), F A K0 ) () () () () () () () () F B I0 ) B K0 ), F B K0 ) ( k) V q s ( ), ( k) ( c ) (6) ( k) V q s ( ) (7) ( k) ( c ) Hee I 0 () x and K 0 ( x ) ae modified Bessel functions fo the puely imaginay aguments of the fist and second kind, espectively with the zeoth ode. Thus, using the expessions (), (3), (9), (5) and (7) we obtain the following expessions of the Fouie tansfomations of the sought values. () () () () () () () () () () () () () uf A q I ) A q K ) B sq I ) B sq K ) () () () () () () () uf A q K ) B sq K ) () () () () () () () () () () () uzf A si0 ) A sk0 ) B q I0 ) B q K0 ) () () () () () () uzf A sk0 ) B q K0 ) () () () () () () () zf A 0.5 ) ( I0 ) I )) s I0 ) () () () () () A 0.5 ) ( K0 ) K )) s K0 )

11 The Influence of the Impefectness of Contact Conditions 3 () () () () () () B 0.5 s ) ( I 0 ) I )) sq I 0 ) () () () () () () B 0.5 s ) ( K0 ) K )) sq K0 ) () () () () () () () zf A 0.5 ) ( K0 ) K )) s K0 ) () () () () () () B 0.5 s ) ( K0 ) K )) sq K0 ) () () () () () () () ( F A () ) ) 0.5( I0 ) I )) () () () () () I ) s I0 )) () () () () () A ( ) () ) 0.5( K0 ) K )) () () () () () K ) s K0 )) () () () () () B ( ) s () ) 0.5( I0 ) I )) () () () () () () ( sq I ) sq I0 )) () () () () () () 0 B ( ) s ) 0.5( K ) K )) () () () () () () ( sq K ) sq K0 ))

12 4 Copyight 08 Tech Science Pess CMC, vol.54, no., pp.03-36, 08 () () () () () () () ( F A () ) ) 0.5( K0 ) K )) () () () () () K ) s K0 )) () () () () () B ( ) s () ) 0.5( K0 ) K )) () () () () () () ( sq K ) sq K0 )) () () () () () () () F A () ( ) 0.5( I0 ) I )) () () () () 0 ( )) ( () ) q s I q I ) () () () () () () A ( () ) 0.5( K0 ) K )) s K0 )) () () q () ( )( K () )) () () () () () () () B ( s () ) 0.5( I0 ) I )) sq I0 )) () () sq () ( ) I () ) () () () () () () () B ( s () ) 0.5( K0 ) K )) sq K0 ))

13 The Influence of the Impefectness of Contact Conditions 5 () () sq () ( )( K () ) () () () () () () () () F A () ( ) 0.5( K0 ) K )) s K0 )) () () q () ( )( K () )) () () () () () () () B ( s () ) 0.5( K0 ) K )) sq K0 )) () () sq () ( )( K () ) () () () () () () () zzf A () ( ) 0.5( I0 ) I )) () q () () () ( )) ( I q () ) s I0 )) () () () () () () q () A ( () ) 0.5( K0 ) K )) K )) () () ( ) s K () 0 ) () () () () () B ( s () ) 0.5( I0 ) I )) () sq () () () () ( )) ( I q () ) sq I0 )

14 6 Copyight 08 Tech Science Pess CMC, vol.54, no., pp.03-36, 08 () () () () () () sq () B ( s () ) 0.5( K0 ) K )) K )) () ( () () () ) sq K0 ) () () () () () () () zzf A () ( ) 0.5( K0 ) K )) () q () () () ( )) ( K q () ) s K0 ) () () () () () B ( s () ) 0.5( K0 ) K )) () sq () () () () ( )) ( K q () ) sq K0 )). (8) Thus, using the expessions in (8) we attempt to satisfy the Fouie tansfomations of the conditions (3) and (4-6), accoding to which, the following expessions can be witten. () () () () () () () F P0 A A 3B 4B 5B 6B P0 R h () () () () () () () zf 0 A A 3B 4B 5B 6B 0 R h () () () () () () () () 3A 3A 33B 34B 35B 36B 0 R R () () () () () () () () z z 4A 4A 43B 44B 45B 46B 0 R R () () () () () () () () u u 5A 5A 53B 54B 55B 56B 0 R R () () FR () () () () uz uz () z 6A 6A 63B R R R () () () 64B 65B 66B 0. (9)

15 The Influence of the Impefectness of Contact Conditions 7 The coefficients ij, whee i; j,,3,...,6 expessions in (8). can be easily detemined fom the Thus, solving the equations in (9) with espect to the unknowns () () () () A, A, B, B, () A and () B we detemine completely the Fouie tansfomations of all the sought values and, substituting these values into the integals in (4) and calculating these integals, we detemine the oiginals of the stesses and displacements in the system unde consideation caused by the action of the extenal moving load. This completes the consideation of the solution method. 4 Numeical esults and discussions 4. The citeia and algoithm fo calculation of the citical velocity Fist we conside the citeion fo detemination of the citical velocity unde which the values of the stesses and displacements become infinity and esonance-type behavio occus. Fo this pupose, we detail the expessions of the unknown constants () () () () () A, A, B, B, A and () B which ae obtained fom the Eq. (9) and can be pesented as follows: () () A () () () () () () ; A ; B ; B ; A A A ; B det ij ;det ij ; det ij B () () () () B A B ij ij ij ij det ;det ;det det () A () whee the matices A () ij, B () ij, B () ij, A ij, ij and () B ij ae obtained fom the matix ( ij ) by eplacing the fist, second, thid, fouth, fifth and sixth columns with the column P 0,0,0,0,0,0 T, espectively. At the same time, fo the selected value of the load moving velocity V, the equation det ij 0 () has oots with espect to the Fouie tansfomation paamete s and as a esult of the solution of the equation (), the elation V V () s is obtained. It is obvious that if the ode of this oot is one, then the integals in (8) at the vicinity of this oot can be calculated in Cauchy s pincipal value sense. Howeve, if thee is a oot the ode of which is two, then the integals in (8) have infinite values and namely the velocities (0)

16 8 Copyight 08 Tech Science Pess CMC, vol.54, no., pp.03-36, 08 coesponding to this oot ae called the citical velocity. In othe wods, the citical velocity is detemined as the velocity coesponding to the case whee dv 0 ds. () It should also be noted that if we ename the Fouie tansfomation paamete s with the wavenumbe and the load moving velocity with the wave popagation velocity, then the Eq. () coincides with the dispesion equation of the coesponding nea-suface wave popagation poblem. So, the elation V V () s obtained fom the solution of the Eq. (), which is made by employing the well-known bisection method, is also the dispesion elation of the nea-suface wave. Theefoe, in some investigations (fo instance in the pape by Abdulkadiov [Abdulkadiov (98)] the citical velocity is enamed as esonance waves and is detemined fom the dispesion cuves of the coesponding wave popagation poblem. This completes the consideation of the citeia and algoithm fo detemination of the citical velocity unde which the esonance-type phenomenon takes place. 4. Algoithm fo calculation of the integals in (8) Fist of all, we note that the integals in (8) and simila types of integals ae called wavenumbe integals, fo calculation of which, special algoithms ae employed. These algoithms ae discussed in the woks by Akbaov [Akbaov (05)], Jensen et al. [Jensen, Kupeman, Pote et al. (0)], Tsang [Tsang (978)] and othes listed theein. Accoding to these discussions, in the pesent investigation we pefe to apply the Sommefeld contou integation method. This method is based on Cauchy s theoem on the values of the analytic functions ove the closed contou, and accoding to this theoem the contou 0, is defomed into the contou C (Fig. ), which is called the Sommefeld contou in the complex plane s s is and in this way the eal oots of the Eq. () ae avoided unde calculation of the wavenumbe integals. Figue : The sketch of the Sommefeld contou Thus, accoding to this method, the integals in (8) ae tansfomed into the following ones.

17 The Influence of the Impefectness of Contact Conditions 9 ( k) ( k) ( k) ( k) ( k) ( k) ( k) ( k) nn nn F F nnf nnf ; u ; ; (, z) Re ; u ; ; (, s)cos( sz) ds, C nn ; ; zz ( k) ( k) ( k) ( k) ( k) ( k) ( k) ( k) z z z F zf zf zf ; u ; ; ; (, z) Re ; u ; ; (, s)sin( sz) ds. (3) C Taking the configuation of the contou C given in Fig., we can wite the following elations. f ( s)cos( sz) ds f ( s i)cos( s i) ds i f ( is)cos( is) ds, C 0 0 f ( s)sin( sz) ds f ( s i)sin( s i) ds i f ( is)sin( is) ds C 0 0. (4) Assuming that, we can neglect the integals with espect to s in (4) and obtain the following expessions fo calculation of the integals in (3). ( k) ( k) ( k) ( k) ( k) ( k) ( k) ( k) nn nn F F nnf nnf ; u ; ; (, z) Re ; u ; ; (, s i )cos(( s i ) z) ds 0 ( k) ( k) ( k) ( k) ( k) ( k) ( k) ( k) z z z F zf zf zf ; u ; ; ; (, z) Re ; u ; ; (, s is)sin(( s i ) z) ds 0 Unde calculation of the integals in (5) the impope integal ()ds 0 is eplaced with * S the coesponding definite integal 0 () * ds and the values of S ae detemined fom the coesponding convegence equiement. Moeove, unde calculation of the integal * S 0 () ds, the inteval [0, S * ] is divided into a cetain numbe (denote this numbe though N ) of shote intevals and within each of these shote intevals the integals ae calculated by the use of the Gauss algoithm with ten integation points. The values of the integated functions at these integation points ae calculated though the solution of the Eq. (9) and it is assumed that in each of the shote intevals the sampling inteval s of the numeical integation must satisfy the elation s min, z. All these pocedues ae pefomed automatically in the PC by use of the coesponding pogams constucted by the authos of the pesent pape in MATLAB., (5)

18 0 Copyight 08 Tech Science Pess CMC, vol.54, no., pp.03-36, Numeical esults elated to the citical velocity All numeical esults which will be consideed in the pesent subsection ae obtained though the solution to the Eq.. Fist, we conside the numeical esults obtained in the following two cases: Case : Case : E () E () E () E () 0.35, () () 0.5, 0.05, () () 0.5, () () () () 0. ; (6) 0.0. (7) Note that these cases wee also consideed in the pape by Abdulkadiov [Abdulkadiov (98)] within the assumption that h/ R 0.5 and the existence of the citical velocity was obseved in Case (in Case ) unde F (unde F 0 ) in (6), i.e. unde full slipping impefect (pefect) contact conditions between the hollow cylinde and suounding elastic medium. In the pesent investigation we obtain numeical esults not only fo the cases whee F 0 and F, but also fo the cases whee F 0 and 0 F. The gaphs of the dependencies between the dimensionless citical velocity () Vc c and the impefection paamete F constucted unde h/ R 0.5 ae given in Fig. 3 and 4 fo Case and Case, espectively. It follows fom these gaphs that the esults obtained in the pesent pape in these paticula cases coincide with the coesponding ones obtained in the pape by Abdulkadiov [Abdulkadiov (98)] and in addition, these esults show that thee exist the following elations: Vc V F c V F 0 c and V 0 F 0 c Vc as F, (8) F Vc V F F ' 0 c V F0 c and V 0 F c Vc as F. (9) F Note that the afoementioned coinciding of the pesent esults obtained in the paticula cases with the coesponding ones given in the pape by Abdulkadiov [Abdulkadiov (98)] poves the eliability of the calculation algoithm and PC pogams used in the pesent investigation. Now we attempt to explain the meaning of F ' which is in the elation (9) and fo this pupose, as an example, we conside the coesponding dispesion cuves obtained in Case fo the fist lowest mode and given in Fig. 5. To pevent misundestandings, we note that unde constuction of the gaphs given in Fig. 5, the cases whee () () V min c ; c ae also consideed and in these cases in the solution of the coesponding equations consideed above, the functions In() x and Kn( x ) in (7) ae changed with Jn( x ) and Yn( x ) espectively, whee Jn( x ) and Yn( x) ae Bessel functions of the fist and second kind of the n th ode. Howeve, the citical velocities

19 The Influence of the Impefectness of Contact Conditions ae detemined fom the esults which elate to the subsonic velocity of the moving load, V min c () ; c (). i.e. to the cases whee Thus, we tun to the analyses of the esults given in Fig. 5 which show that in the case whee F 0, the dispesion cuves ae limited to the dispesion cuves obtained in the case whee F 0 (uppe limit) and in the case whee F (lowe limit). Howeve, in the case whee F 0, the dispesion cuve of the fist mode has two banches. The fist banch has cut off values ( sr ) cf.. afte which the wave popagation velocities in this banch ae less than those obtained in the pefect contact case, i.e. by denoting the wave popagation velocity fo the fist banch though V I we can wite the following elation: VI V F 0 unde sr ( sr) cf.. (30) Howeve, the second banch of the dispesion cuve has no cut off value of sr and the wave popagation velocity of this banch (denote it by V II ) is less than that obtained in the pefect contact case, i.e. fo the second banch we can wite the following elation: VII V F 0 fo each sr 0. (3) Moeove, it follows fom Fig. 5 that the point at which () d( V / c ) d( sr) 0 on the fist banch of the dispesion cuves appeas afte a cetain value of the paamete F (denote this value by F ' ). Fo instance, in the case unde consideation, it can be seen that 0. F ' 0.3. Namely, this value of the paamete F entes into the elation (9). We continue the discussion of the dispesion cuves given in Fig. 5 and note that the point at which () d( V / c ) d( sr) 0 on the second banch of the dispesion cuves appeas in each value of the paamete F unde F 0, and the values of the citical velocities (i.e. the velocities coesponding to the point at which () d( V / c ) d( sr) 0 ) ae geate than those obtained in the pefect contact case and incease with F, while afte a cetain value of F they poceed to the subsonic egime, which is not consideed in the pesent pape. Howeve, the citical velocities obtained on the fist banches of the dispesion cuves ae less than those obtained in the pefect contact case. Theefoe, all the esults given in the pesent subsection and egading the case whee F 0 ae detemined accoding to the fist banches of the dispesion cuves. We also note the meanings of the cases whee F 0 and F 0 whee unde selected foms of the contact condition (6), the case whee F 0 means that if () () z z 0 R R ( () () z z 0 ) and uz () 0 R R, uz () 0, then the displacement of the hollow R R cylinde along the Oz axis at the inteface suface is geate (less) than that of the suounding elastic medium. Analogically, the case F 0 means that unde satisfaction of the foegoing conditions, the displacement of the hollow cylinde along the Oz axis on the inteface suface is less (geate) than that of the suounding elastic medium. It is

20 Copyight 08 Tech Science Pess CMC, vol.54, no., pp.03-36, 08 evident that each of these cases has eal application fields and theefoe the esults obtained in the cases whee F 0 also have a eal physico-mechanical meaning. We ecall that the foegoing esults wee obtained in the case whee h/ R 0.5. It should be noted that simila types of esults ae also obtained in the othe values of the atio h/ R. As an example of this, in Fig. 6 and 7 the gaphs of the dependence between () Vc / c and the paamete F constucted unde h/ R 0. ae given fo Case (6) and Case (7), espectively. It follows fom these gaphs that the foegoing conclusions on the chaacte of the influence of the paamete F on the values of the citical velocity do not depend on the atio h/ Rin the qualitative sense. Now we conside in moe detail the esults illustating the influence of the atio h/ R on the values of the citical velocity obtained in the cases whee F 0 and F. These esults, i.e. the values of () Vc / c ae given in Tab. and fo Case and Case, espectively unde vaious values of the atio h/ R. It follows fom these tables that the values of the citical velocity decease monotonically with deceasing of the atio h/ R, and the magnitude of this decease in Case is moe consideable than in Case. Accoding to physico-mechanical consideations, the values of the citical velocity obtained fo the system unde consideation must appoach the coesponding ones obtained fo the system consisting of the coveing laye and half-plane with deceasing of the atio h/ R. Fo illustation of this, we conside the values of the citical velocity obtained in the case whee () / () 0.5, () () 0.3 and E () E () 0.5, which ae given in Tab. 3, although this consideation is also poven with the data given in Tab. and. Note that in this case, the values of the citical velocity fo the coesponding system consisting of the coveing laye and half-plane wee consideed in the pape by Akbaov et al. [Akbaov, Gule and Dincoy (007)] and in the pape by Babich et al. [Babich, Glukhov and Guz (986)] unde F 0. Thus, it follows fom Tab. 3 that the values of the citical velocity obtained fo the system consisting of the hollow cylinde and suounding elastic medium appoach the coesponding ones obtained fo the system consisting of the coveing laye and half-plane. This situation confims again the eliability of the calculation algoithm and coesponding PC pogams which ae used to obtain the numeical esults discussed in the pesent subsection. As noted above, the influence of the atio h/ R on the values of the citical velocity in Case is moe consideable than in Case. We think that the consideable influence of h/ R on the citical velocity in Case is caused by the atio E () E (), the value of which is significantly less than in Case. This explanation is also poven with the data given in Tab. 4 which show the values of the dimensionless citical velocity () V / c in the cases whee () / () E () E () and () () 0.3 unde F 0 and F fo vaious values of h/ Rand E () E (). Thus, it follows fom Tab. 4 that a decease in the values of h/ R and E () E () causes a decease in the values of the citical velocity, and c

21 The Influence of the Impefectness of Contact Conditions 3 the magnitude of the influence of h/ R on the values of this velocity become moe consideable with deceasing of the atio () () E E. This completes consideation of the esults elated to the citical velocity. Note that the esults detailed in the foegoing eight paagaphs and coesponding numeical esults used in this detailing can be taken as a pat of the main contibution of the pesent pape into the elated investigations. The othe pat of this contibution is detailed in the next subsection 4.4 Numeical esults elated to the stess and displacement distibutions The esults discussed in the pesent subsection ae obtained within the scope of the algoithm developed in the subsection 4.. Fist of all we conside convegence of the values of the integals in (5) with espect to N and S in the cases whee V Vc. Fo simplicity of the consideation, we intoduce the notation () c V / c and conside the dependence between h / P 0, whee ( ) () (, ) () z R z ( R, z), (3) and c at the point z/ h 0. All numeical esults which will be consideed in the pesent subsection ae obtained in the pefect contact case, i.e. in the case whee F 0 in (6). Accoding to the esults given in Fig. 5, in the cases whee V Vc unde F 0 the integated expessions do not have any singulaity ove an abitay integated inteval * 0, S. Howeve, in this case the integated expessions have the fast oscillating tems- cos( sz ) o sin( sz ), and this also causes difficulties in the convegence sense of the integals in (5). This difficulty can also be pevented by the use of the Sommefeld contou integation method which is also used in the pesent study. As a esult of the coesponding numeical investigations, it is established that in the accuacy and convegence senses it is enough to assume that in the integals in (5). Note that the values of the integals calculated fo each value of the paamete selected fom the inteval 0 300,0.0 coincide with each othe with accuacy Unde obtaining the numeical esults consideed below, it is assumed that

22 4 Copyight 08 Tech Science Pess CMC, vol.54, no., pp.03-36, 08 Figue 3: The dependence of the citical velocity on the impefection paamete F in Case (6) unde hr 0.5 Figue 4: The dependence of the citical velocity on the impefection paamete F in Case (6) unde hr 0.5 Table : The values of the dimensionless citical velocity () Vc c obtained fo vaious hr unde () () 0., () () 0.5 and E () E () 0.35 in the cases whee F 0 (uppe numbe) and F (lowe numbe) h/ R It should also be noted that unde using the Sommefeld contou integation method in the cases whee V Vc thee is no difficulty in convegence of the integals in (5) with espect to the values of N and the esults obtained fo the case whee N coincide with the esults obtained fo each case whee N. We ecall that N shows the numbe of the shote integation intevals, the summation of which gives the inteval * 0, S. Thus, in the cases unde consideation, thee is no meaning to the convegence of the numeical esults with espect to the numbe N. Theefoe, hee we conside only the examples illustating the convegence of the numeical esults with espect to the length of the integation inteval, i.e. with espect to the values of S *. These esults fo Case (fo Case ) unde h/ R 0. ae given in Fig. 8 (Fig. 9). It follows fom these esults, that the esults obtained in the cases whee S * 9 coincide with accuacy with those obtained in the case whee S * 9. Taking the foegoing situations into

23 The Influence of the Impefectness of Contact Conditions 5 consideation, unde obtaining all the numeical esults which will be discussed below, we assume that N 0 and S * 0. Table : The values of the dimensionless citical velocity () V c obtained fo vaious hr unde () () 0.0, () () 0.5 and F 0 (uppe numbe) and F (lowe numbe) h/ R c () () E E 0.05 in the cases whee Thus, we analyze the numeical esults obtained within the scope of the foegoing assumptions and calculation algoithm. Fist we conside the gaphs of the dependence between the dimensionless nomal stess h / P0 calculated at point z/ h 0 (whee is detemined though the expession (3)) and dimensionless load moving velocity c. These gaphs fo Case (6) and Case (7) ae given in Fig. 0 and, espectively. The gaphs of the same dependence fo the dimensionless shea stess z () z, whee ( ) () (, ) () z z z R z z ( R, z), (33) calculated at point z/ h 0.5 fo Case and Case ae given in Fig. and 3. Table 3: The values of the dimensionless citical velocity () V c obtained fo vaious c () () hr unde () () 0.5, () () 0.3 and E E 0.5 in the cases whee F 0 (uppe numbe) and F (lowe numbe) h/ R ( F 0 ) (by Akbaov et al. [Akbaov, Gule and Dincoy (007)]; (by Babich et al. [Babich, Glukhov and Guz (986)])

24 6 Copyight 08 Tech Science Pess CMC, vol.54, no., pp.03-36, 08 Table 4: The influence of the atios () () E E and h/ R on the values of the dimensionless citical velocity Vc c unde E E, in the cases whee F 0 (uppe numbe) and F (lowe numbe). () h/ R () () E E () () () () () () Figue 5: The influence of the impefection paamete F on the dispesion cuves in Case (7) unde hr 0.

25 The Influence of the Impefectness of Contact Conditions 7 Figue 6: The dependence of the citical velocity on the impefection paamete F in Case (6) unde hr 0. Figue 7: The dependence of the citical velocity on the impefection paamete F in Case (6) unde hr 0. Figue 8: Convegence of the esults with espect to the integation inteval S * in Case Figue 9: Convegence of the esults with espect to the integation inteval S in Case *

26 8 Copyight 08 Tech Science Pess CMC, vol.54, no., pp.03-36, 08 Figue 0: Dependence of the inteface nomal stess on the dimensionless load moving velocity c in Case Figue : Dependence of the inteface nomal stess on the dimensionless load moving velocity c in Case Figue : Dependence of the inteface nomal stess z on the dimensionless load moving velocity c in Case Figue 3: Dependence of the inteface nomal stess z on the dimensionless load moving velocity c in Case

27 The Influence of the Impefectness of Contact Conditions 9 Now we conside the esults elated to the distibution of the stesses and displacements on the inteface suface with espect to the dimensionless coodinate zh. The diagams of these distibutions fo the stess ( z ) ae given in Fig. 6 and 7 (in Fig. 8 and 9) in Case and Case, espectively fo the vaious values of the atio hr. The same distibutions fo the displacement u (= u () (, ) () R z u ( R, z) (fo the displacement uz (= u () (, ) () z R z uz ( R, z) ) ae given in Fig. 0 and (in Fig. and 3) also in Case and Case, espectively unde vaious hr. It follows fom these figues that the absolute values of the stesses and displacements afte some insignificant oscillations ae dicey. Thus, it follows fom Fig. 0-3 that both in Case and in Case the absolute values of the stesses incease monotonically with the load moving velocity c. It should be noted that in Case the absolute values of the stesses incease moe apidly than those in Case as c cc. Analyses of these and othe numeical esults allow us to conclude that the chaacte of the foegoing dependencies does not depend not only on the atio h/ R but also on the atio E () E (). In connection with this we conside the gaphs given in Fig. 4 and 5 which illustate the foegoing dependence fo the stesses and z, espectively fo vaious values of E () E () unde h/ R 0.5, () () 0.3 and () () () () E E. Thus, it follows fom Fig. 4 and 5 that the dependence between the stesses and load moving velocity is monotonic fo each selected value of the atio () () E E. Moeove, it follows fom these esults that the absolute values of the nomal and shea z stesses decease with deceasing of the atio E () E (). Note that these esults agee in the qualitative sense with the coesponding esults elated to the dynamics of the moving load acting on the layeed half-space and discussed in the monogaph by Akbaov [Akbaov (05)] and in othe woks listed theein. Moeove, these esults agee also with the well-known mechanical and engineeing consideations and can be taken as validation in a cetain sense of the used calculation algoithm and PC pogams.