Surkay Akbarov 1) and *Mahir Mehdiyev 2)

Transcription

1 On the influence of initial stesses on the citical velocity of the moving ing load acting in the inteio of the hollow cylinde suounded by an infinite elastic medium Sukay Akbaov 1) and *Mahi Mehdiyev ) 1) Depatment of Mechanical Engineeing, Faculty of Mechanical Engineeing, Yildiz Technical Univesity, Yildiz Campus, 34349, Besiktas, Istanbul,Tukey; ) Depatment of Theoy of Elasticity and Plasticity, IMM ANAS, St. B. Vahabzade, 9, AZ1141, Baku, Azebaijan 1) akbaov@yildiz.edu.t; ) mahimehdiyev@mail.u ABSTRACT The bi-mateial elastic system consisting of the pe-stessed hollow cylinde and pestesses suounding infinite elastic medium is consideed and it is assumed that the mentioned initial stesses in this system ae caused with the compessing o stetching unifomly distibuted nomal foces acting at infinity in the diection which is paallel to the cylinde's axis. Moeove, it is assumed that on the intenal suface of the cylinde the ing load which moves with constant velocity acts and within these famewoks it is equied to detemine the influence of the afoementioned initial stesses on the citical velocity of the moving load. The coesponding investigations ae caied out within the famewok of the so-called thee-dimensional lineaized theoy of elastic waves in initially stesses bodies and the axisymmetic stess-stain state case is consideed. The moving coodinate system method is used and the Fouie tansfom is employed fo solution to the fomulated mathematical poblem and Fouie tansfomation of the sought values ae detemined analytically. Howeve, the oiginals of those ae detemined numeically with the use of the Sommefeld contou method. The citical velocity is detemined fom the citeion, accoding to which, the magnitudes of the absolute values of the stesses and displacements caused with the moving load appoaches an infinity. Numeical esults on the influence of the initial stesses on the citical velocity and inteface nomal and shea stesses ae pesented and discussed. In paticula, it is established that the initial stetching (compessing) of the constituents of the system unde consideation causes a decease (an incease) in the values of the citical velocity. 1. INTRODUCTION A safe and eliable use of the of moden high-speed undegound tains and othe types of undegound moving wheels equies theoetical investigations of coesponding dynamical poblems, one of which is the poblem elated to the dynamics of the ing moving load acting on the inteio of the hollow cylinde 1) Pofesso ) PhD

2 suounded with elastic o viscoelastic medium. This is because, undegound stuctues into which such high-speed wheels move ae modelled as infinite hollow cylindes suounded by an elastic o viscoelastic medium. In ode to impove the adequacy of the theoetical esults to the eal cases it is necessay to take into account the efeence paticulaities of these systems in these investigations. One of these paticulaities is the initial stesses which appea in the constituents of the system hollow cylinde + suounding elastic medium. Namely, the investigation the influence of the initial stesses which appea unde unidiectional compession (stetching) of the system "hollow cylinde + suounding elastic medium" in the load moving diection on the citical velocity of this load and inteface stesses which also appea as a esult of this load, is the subject of the pesent pape. Fo detemination significance and contibution of the investigations made in the pesent wok we attempt to make a bief eview of the studies elated to the dynamics of the moving load and we begin this eview with the pape by Achenbach et al. (1967). Note that in this pape the dynamic esponse of the system consisting of the coveing laye and half plane to a moving load was investigated unde which the motion of the plate was descibed with the use of the Timoshenko theoy, 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. Late investigations stated in the pape by Achenbach et al. (1967), ae developed in the papes by Dieteman and Metikine (1997) and by Metikine and Vouwenvelde (000) and many othes listed theein. The eview of the elated investigations is also consideed in the pape by Ouyang (011). It should be noted that up to now, a cetain numbe of investigations have also been made fo the dynamics of the moving load acting on initially stessed systems. As an example fo ealie investigation in this type it can be take the pape by Ke (1983) in which the influence of 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. In the pape by Metikin and Dieteman (1999), 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 Eule-Benoulli beam, the initial axial compession of this beam is also taken into consideation. The influence of the initial stesses acting on the statified half-plane on the citical velocity of the moving load which acts on the plate was studied in the papes by Babich et al. (1986, 1988, 008a, b), in which 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 1999, 004). At the same time, the motion of the coveing laye, which does not have any initial stesses, as in the pape by Achenbach et al. (1967), was witten by employing the Timoshenko plate theoy. Futhemoe, in ecent yeas it was made seies investigations on the dynamics of the moving load acting on the layeed systems 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. Now we eview some of them and begin this eview

3 with the pape by Akbaov et al. (007) in which 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. In the pape by Dincsoy et al. (009) the same poblem was studied fo the system consisting of the coveing laye, substate and half-plane. The esponse of the system consisting of the initially stessed othotopic coveing laye and initially stessed othotopic half plane to the moving and oscillating moving load wee investigated in the papes by Akbaov and Ilhan (008, 009), and by Ilhan (01). The pape by Akbaov and Salmanova (009) deals with 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. In the pape by Akbaov et al. (015) the 3D poblems on the dynamics of the moving and oscillating moving point-located load acting on the system consisting of a pe-stessed coveing laye and half-space wee investigated. As a esult of this investigation, 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. The detail consideation and analysis of the foegoing investigations was made in the monogaph by Akbaov (015). Note that up to now thee ae also some investigations on the moving and oscillating moving load acting on the hydo-elastic systems, which do not consideed in the monogaph by Akbaov (015). These investigations wee made in the papes by Akbaov and Ismailov (015, 016a, 016b). Now we conside a eview of the investigations elated to the dynamics of the moving load acting on a cylindical suface which bounds the infinite egion filled with homogeneous o piecewise homogeneous elastic mateials. In the histoical aspect, the fist attempt in this field was made in the pape by Panes (1969) in which a ing load moving with constant velocity in the axial diection along the inteio of a cicula boe in an infinite homogeneous elastic medium is investigated. In this pape the theoetical investigations ae made in the 3D case, howeve coesponding numeical esults on the displacement and stess distibution ae pesented fo the axisymmetic case. The case whee in the inteio of the cylindical cavity a tosional moving load acts is consideed in anothe pape by Panes (1980). Note that, in these papes the question elated to the citical velocity is not consideed. Rathe, the question on the detemination of the citical velocity of the moving load acting on infinite (as in the papes by Panes (1969, 1980)) o semi-infinite mediums does not appea in the cases whee these mediums ae homogeneous. This is because, in the mentioned cases the citical velocity is known befoehand, so that in these cases the citical velocity coincides with the Rayleigh wave popagation velocity in the coesponding medium. Note that the question elated to detemination of the citical velocity elates only to moving load poblems acting on the piece-wise inhomogeneous infinite (fo instance, fo the system consisting of a hollow cylinde suounded with elastic medium) o semiinfinite (fo instance, fo the system consisting of a coveing laye and half-space) bodies. What is moe, the citical velocity in the afoementioned infinite and semi-infinite bodies appeas only in the cases whee the modulus of elasticity of the coveing laye mateial is geate than that of the suounding infinite medium o of the statified semi-

4 infinite medium. Fo instance, investigations caied out in the papes by Chonan (1980), Pozhuev (1981), Abdulkadiov (1981) and othes elate namely to the piecewise inhomogeneous infinite cylindically layeed systems. Studies on the dynamic esponse of a cylindical shell impefectly bonded to a suounding infinite elastic continuum unde action of axisymmetic ing pessue which moves with constant velocity in the axial diection along the inteio of the shell, is made by Chonan (1980). It is assumed that the shell and the continuum ae joined togethe by a thin elastic bond and the axisymmetic poblem is consideed. The motion of the shell is descibed by thick shell theoies and the motion of the suounding elastic medium is descibed by the exact equations of linea elastodynamics. Numeical esults on the citical speed of the moving load and on the adial displacement of the shell fo the subcitical moving load ae pesented. The pape by Pozhuev (1981) studies the moving load poblem fo the system consisting of a thin cylindical shell and suounding tansvesally isotopic infinite medium. A thin shell theoy is employed fo descibing the motion of the cylinde, howeve the motion of the suounding elastic medium is descibed with the exact equations of motion of elastodynamics fo tansvesally isotopic bodies. Numeical esults egading displacements and a adial nomal stess ae pesented, but in this pape thee ae no numeical esults elated to the citical velocity of the moving load. The study of low-fequency esonance axisymmetic longitudinal waves in a cylindical laye suounded by an elastic medium is made in the pape by Abdulkadiov (1981), in which unde esonance waves the cases unde which the elation dc / dk 0 takes place, is undestood, whee c is the wave popagation velocity and k is the wavenumbe. The velocity of these esonance waves coincides with the citical velocity of the coesponding moving load. Some numeical examples of esonance waves ae pesented and discussed. Note that in this pape dispesion cuves ae obtained within the scope the exact equations of linea elastodynamics. Besides of all these, in ecent yeas numeical and analytical solution methods have been developed fo studying the dynamical esponse of tunnel (modelled as a hollow elastic cylinde) + soil (modeled as suounding elastic o viscoelastic medium) systems geneated by the moving load acting on the inteio of the tunnels (see, fo instance, the papes by Foest and Hunt (006), Sheng et al. (006), Hung et al. (013), Hussein et al. (014), Yuand et al. (017) and othes listed theein). Howeve, in these papes the main attention is focused on studying the displacement distibution in soil caused by the moving load. This completes the eview of the investigations the subjects of which elate to the subject of the pesent pape. It follows fom this eview that up to now systematic investigations of the citical velocity of a moving load acting on the inteio of the hollow cylinde suounded with elastic medium ae absent. Moeove, this eview shows that the coesponding investigations elated to the pe-stessed hollow cylinde + pestessed suounding elastic medium ae absent completely. Taking this statement into consideation, in the pesent pape we attempt to investigate the citical velocity of the moving ing load acting on inteio suface of the pe-stessed hollow cylinde suounded with the pe-stessed elastic medium. It is assumed that the mentioned pestesses appea as a esult of the action of the unifomly distibuted nomal foces acting at infinity in the diection of the cylinde axis along which the ing load moves.

5 The investigations ae made with employing, so-called thee-dimensional lineaized equations of elastic wave popagation in initially stessed bodies and the axisymmetic case is consideed. Note that the coesponding foced vibation poblem is investigated in the pape by Akbaov and Mehdiyev (017).. FORMULATION OF THE PROBLEM Conside an infinite body consisting of a hollow cicula cylinde with the thickness h and with the extenal adius R and of a suounded infinite elastic medium. We associate the cylindical O z and Catesian Ox1 xx 3 systems of coodinates (Fig. 1) with cental axis of the cylinde and the position of points of this body we detemine though Lagange coodinates in these coodinate systems. The stess-stain state in this infinite body we pesent as a summation of two states which ae named as an "initial" and a "petubed" states. Assume that in the initial state this body is loaded at infinity with unifomly distibuted nomal foces with intensity q acting in the cylinde's axis diection, and the stess-stain state which appea as esult of this action we take as initial stess-stain state. Below, the values elated to the cylinde and to the suounding elastic medium will be denoted by uppe indices and, espectively. Moeove, the values elated to the initial state will be denoted by the additional uppe index 0. It is assumed that mateials of the cylinde and suounding medium ae homogeneous, isotopic and linea elastic. The values elated to the initial state we detemine within the famewok of the classical linea theoy of elastostatics. Note that, in geneal, in the initial state the stess state in the body unde consideation is inhomogeneous one and this inhomogeneity is caused by the diffeence of the Poisson s atio of the mateials of the cylinde (denote it ) and suounding medium (denote it by ). Howeve, in the cases whee by in the initial state the stess state in the bi-mateial system shown in Fig. 1 is inhomogeneous and is detemined as follows z z 0 0 E 0, 10 k 1,, zz q, zz 1 zz. E Hee and below the conventional notation is used. Thus, in the initial state the stesses ae detemined though the expession given in. In the petubed state, we assume that the body having the foegoing initial stesses is loaded by additional otationally symmetic nomal ing load with intensity P 0 which moves with constant velocity V in the diection of the cylinde's axis, i.e. in the diection of the Oz axis. We assume that P0 q and, accoding to Eingen and Suhubi (1975), Guz (1999, 004), Akbaov (015), the stain-stess state caused by this additional loading we descibe with the following thee-dimensional lineaized equations of elastic waves in initially stessed bodies in the axially symmetic case. Equations of motion: z 1 ( ) z 0 u u zz z t,

6 z zz 1 z z 0 uz uz zz z t. Fig. 1 The sketch of the pe-stessed system consisting of the hollow cylinde and suounding elastic medium Elasticity elations: nn ( zz ) nn, nn ; ; zz, Stain displacement elations: z z z. (3) k uz u z u u u ( ) 1,, zz, ( ). (4) z z The equations, (3) and (4) ae the complete system of field equations of the thee-dimensional lineaized theoy of elastic waves in initially stessed bodies in the case whee the stains in the initial state so small that the coesponding stain-stess state can be detemined within the scope of the classical linea theoy of elastostatics. Now we conside the fomulation of the bounday and contact conditions fo the values elated to the afoementioned petubed state, i.e. fo the values which appea as a esult of the action of the additional load which moves with constant velocity V. 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 ), z 0. (5) Rh Rh Suppose that the contact conditions with espect to the foces and adial displacement ae continuous: R R, z z R R, u u R R. u z uz R R. (6) Besides all these, we assume that the moving load velocity is subsonic, i.e. the condition V min c ; c,, 1, c k (7) occus and accoding to this condition, the following decay condition takes place. zz z u uz ; ; ; ; ; 0, k 1, as ( z Vt). (8) This completes fomulation of the poblem and consideation of the govening field equations.

7 3. METHOD OF SOLUTION We use the well-known, classical Lame (o Helmholtz) decomposition (see, fo instance, Eingen and Suhubi (1975)) fo solution to the system of equations -(4): u, u z. (9) z z Substituting the expessions in (9) into the equations -(4), doing coesponding mathematical manipulations we obtain the following equations fo the functions and. whee ( k ) ( k ) ( k ) ( k c ) 0 1 zz 0, z ( c1 ) t zz 0, z ( c ) t z 1 ( ) and ( k ) ( ) ( ) k k c, (10). Note that in the case whee 0 zz 0 equations in (10) coincide with the coesponding classical equations given, fo instance, in the monogaph by Eingen and Suhubi (1975). We intoduce the moving coodinate system ', z ' z Vt (11) which moves with the loading intenal pessue and by ewiting all the foegoing equations with the coodinates ' and z ', we obtain the following equations fo the potentials and 0 zz V 0 0, zz V 0, (1) ( c1 ) z ( c ) z whee the pimes on the and z have been omitted. As a esult of the coodinate tansfomation (11) the fist condition in (5) tansfoms to the following one: Rh P0 () z. (13) Howeve, the othe elations and conditions in (8) emain valid in the new coodinates detemined by (11). Fo simplicity of the consideation 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, we conside the solution to the consideed bounday value poblem which is educed to the solution to the equations in (1). Using the Fouie tansfomation with espect to the coodinate z and taking the poblem symmety with espect to the point z 0 into consideation, the sought values can be pesented as follows. ( k ) ( ) ( ) ( ) ; k ; k ; k 1 u nn nn (, z) F F nnf nnf ; u ; ; (, s )cos( sz ) ds, 0

8 (, k ) ( ) ( ) ( ) ; k ; k ; k 1 uz z z ; (, z) F zf zf zf nn ; ; zz ; u ; ; (, s )sin( sz ) ds. (14) Afte substituting the expessions in (14) into the foegoing equations, elations and contact and bounday conditions, the coesponding ones fo the Fouie tansfomations of the sought values ae obtained. In this case the thid and fouth elations in (4) and the condition (13) and the elations in (9) tansfom to the following ones: duzf zf suf 1 zzf suzf, ( ), F d P 0, R h df d u F F F F s, uzf s F. (15) d d Moeove, accoding to the above-noted tansfomation, we obtain the following equations fo F and F fom the equations in (1). d 1 d 0 ( ) 1 zz V k s 0 d d F, ( c1 ) d 1 d 0 ( ) 1 zz V k s 0 d d F. (16) ( c ) Howeve, the foegoing othe equations and elations ae also valid as ae fo the coesponding Fouie tansfomations. Thus, conside the solution to the equations in (16) which, accoding to the condition (7), can be pesented as follows: F A I ( q ) A K ( q ), F A K ( q ), F B I ( q ) B K ( q ), F B K ( q ), zz V 0 q s, ( c1 ) 1 zz V q s. (17) ( c ) whee I 0 () x and K 0 ( x ) ae modified Bessel functions fo the puely imaginay aguments of the fist and second kind, espectively with zeoth ode, B 1, B and B ae unknown constants. A 1, A, A, Thus, substituting the solutions (17) into the expessions in in (15) and into the Fouie tansfomations of the expessions in (3) and (4) we obtain the following expessions fo the Fouie tansfomation of the sought values. uf A1 q1 I1( q1 ) A q1 K1( q1 ) F 1 1( 1 ) 1( ), zf B1 q I0( q1 ) B q K0( q1 ), uzf A sk0( q1 ) B q K0( q1 ) zf A1 0.5( q1 ) ( I0( q1 ) I( q1 )) s I0( q1 ) u A q K q B sq K q B sq I ( q ) B sq K ( q ), u A si ( q ) A sk ( q ),

9 A 0.5( q1 ) ( K0( q1 ) K( q1 )) s K 0 ( q 1 ) B1 0.5 s( q ) ( I0( q ) I( q )) sq I0( q ) B 0.5 s( q ) ( K0( q ) K( q )) sq K0( q ), zf A 0.5( q1 ) ( K0( q1 ) K ( q1 )) s K0( q1 ) B 0.5 s( q ) ( K0( q ) K( q )), sq K0( q ) 1 (1 F A )( q1 ) 0.5( I0( q1 ) I( q1 )) 1 ( q I1 ( q1 ) s I0 ( q1 )) A (1 )( q 1 ) 0.5( K0( q1 ) K( q1 )) 1 ( q K1 ( q1 ) s K0 ( q1 )) B1 (1 ) s( q ) 0.5( I0( q ) I( q )) ( sq I1 ( q ) sq I0 ( q )) B (1 ) s( q ) 0.5( K0( q ) K( q )) ( sq K1 ( q ) sq K0 ( q )), (1 F A )( q 1 ) 0.5( K0( q1 ) K( q1 )) 1 ( q K1 ( q1 ) s K0 ( q1 )) B (1 ) s( q ) 0.5( K0( q ) K( q )) 1 0 ( sq K ( q ) sq K ( q )), F A1 (( q1 ) 0.5( I0( q1 ) I( q1 )) 1 0 ( 1 )) (1 ) q s I q I1 ( q1 ) A (( q 1 ) 0.5( K0( q1 ) q K( q1 )) s K0( q1 )) (1 )( 1 K 1( q1 )) B1 0 ( s ( q ) 0.5( I ( q ) sq I( q )) sq I0( q )) (1 ) I 1( q ) B 0 ( s ( q ) 0.5( K ( q ) sq K( q )) sq K0( q )) (1 )( K 1( q ), 1 F A (( q ) 0.5

10 q ( K0( q1 ) K( q1 )) s K0( q1 )) (1 )( 1 K 1( q1 )) B ( s ( q ) 0.5( K0 ( q ) K( q )) sq K0( q )) sq (1 )( K 1( q ), zzf A1 (( q1 ) 0.5( I0( q1 ) I( q1 )) q I ( q )) (1 ) s I ( q )) A q q 1 (( ) 0.5 ( K ( q ) K ( q )) K ( q )) (1 ) s K 0 ( q 1 ) B1 ( s ( q ) sq 0.5( I0( q ) I( q )) I1( q )) 0 (1 ) sq I ( q ) B ( s ( q ) 0.5( K0 ( q ) sq K( q )) K1( q )) 0 (1 ) sq K ( q ), zzf A (( q1 ) ( K0( q1 ) K( q1 )) q K ( q )) (1 ) s K ( q ) B ( s ( q ) 0.5( K0( q ) K( q )) sq K1( q )) 0 (1 ) sq K ( q )). (18) Substituting the expessions in (18) into the Fouie tansfomations of the coesponding bounday and contact conditions in (5), (6) and (13) we obtain the following algebaic equations with espect to the unknown constants A 1, A, A, B 1, B. F P0 11A1 1 A R h B and 13B1 14B 15B1 16B P0, zf 0 1A1 A 3B1 4B 5B1 6B 0, R h R R z z R R R R 31A1 3A B B B B, 41A1 4 A B B B B, u u A A B B B B, FR uz uz z 61A1 6 A 63B1 64B 65B1 66B 0 (19) R R R

11 Note that the coefficients ij in (19), whee i; j1,,3,...,6 can be easily detemined fom the expessions in (18). Thus, afte solving the equations in (19) with espect to the unknowns 1 1 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 (14) and calculating these integals, we detemine the oiginals of the stesses and displacements which ae caused by the action of the moving ing load acting on the inteio of the hollow cylinde. This completes the consideation of the solution method. 4. NUMERICAL RESULTS AND DISCUSSIONS 4.1 On the calculation algoithm Numeical esults on the citical velocity of the moving load and on the influence of the initial stesses in the cylinde and suounding elastic medium on these citical velocities, as well as numeical esults on the influence of the initial stesses on the inteface stesses which appea as a esult of the action of the moving load, ae obtained though the numeical calculation of the integals in (14). Note that the algoithm fo this calculation is based on the Sommefield contou method and is developed in the papes by Akbaov et al. (015), Akbaov and Ismailov (015, 016a, 016b) and othe ones listed theein. Moeove, the mentioned algoithm is also detailed in the monogaph by Akbaov (015). Theefoe hee we do not conside detailed desciption of this algoithm and note that the used Sommefield contou is selected such as in Fig., accoding to which the calculation of the integals in (14) ae educed to the calculation of the following ones. Fig. The sketch of the Sommefeld contou ( k ) ( ) ( ) ( ) ; k ; k ; k 1 ( ) ( ) ( ) ( ) u nn nn (, z) Re k ; k ; k ; k F u F nnf nnf ( k ) ( ) ( ) ( ) ; k ; k ; k 1 uz z z ; (, z) Re ; ; ; (, z)cos(( s1 i ) z) ds1, 0 ( ) ( ) ( ) ( ) F zf zf zf (, z)sin(( s1 i ) z) ds1 0 0 S * 1 ds1. (0) Note that duing calculation of the integals in (0) the impope integal ()ds1 is eplaced with the coesponding definite integal () * and the values of S 0 1 ae detemined fom the coesponding convegence equiement. Moeove, duing the

12 S * 1 calculation of the integal () ds 0 1, the inteval [0, S ] is divided into a cetain numbe (denote this numbe though N ) of shote intevals and within each of these 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 equation (19). 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. Numeical esults pesented in the pesent pape ae obtained in the case whee N 00, S1 9 and unde which these esults have sufficient high accuacy in the convegence sense and in the tustiness sense. 4. Numeical esults elated to the influence of the initial stess on the citical velocity We intoduce the notation ( z) ( R, z) ( R, z). (1) Fist, we note that the citical velocity is detemined fom the following citeion: the citical velocity is the velocity with appoaching to which the absolute values of the inteface stess () z (1) (o any quantities chaacteizing the displacement and stess-stain state of the system unde consideation in the petubed state) incease indefinitely. This citeion is geneal one and can be applied fo the cases whee the mateials of the hollow cylinde and suounding elastic medium ae a viscoelastic one. Numeical esults which will be discussed below ae obtained in the following thee cases: Case 1. Case. E E 0.35, E E 0.05, * * 1 0.1, 0.5. () 0.01, 0.5. (3) Case 3. E E 0.5, 0.5, 0.3. (4) As follows fom the elations () (4) that the Poisson s atio of all the selected pais of mateials ae equal to each othe and theefoe fo these pais the homogeneity of the initial stesses, i.e. the elations in ae satisfied exactly. Note that Case was also consideed in the pape by Abdulkadiov (1981) unde h/ R 0.5 and Case 3 was consideed in the pape by Babich et al. (1986) unde hr 0. Consequently the citical velocity obtained fo Case unde h/ R 0.5 must coincide with the coesponding one obtained in the pape by Abdulkadiov (1981) and the citical velocities obtained fo Case 3 must appoach to the citical velocity obtained by Babich et al. (1986) with deceasing the atio hr. Note that these pedictions elate only to the case whee the initial stesses in the cylinde and suounding elastic medium ae absent. Moeove, note that fo selected pais of mateials the elations c c 3.5 in Case 1, c c 5 in Case,

13 and c c 1.0 in Case 3 (5) takes place and accoding to these elation, it can be concluded that if V / c 1, then the moving velocity of the ing load is subsonic. Thus, we conside the esults elated to the influence of the initial stess on the values of the dimensionless citical velocity cc Vc / c (6) and fo estimation this influence we intoduce the following notation. 0 zz. (7) These esults ae given in Tables 1 (fo Case 1), (fo Case ) and 3 (fo Case 3) which ae obtained fo vaious values of the paamete and the atio hr. In Tables and 3 in paticula cases the coesponding esults which mentioned above and obtained in the papes by Abdulkadiov (1981) and Babich et al. (1986) ae also given. The compaison of the pesent esults with coesponding ones obtained in the papes by Abdulkadiov (1981) and Babich et al. (1986) shows the tustiness of the foegoing pediction and in this way it is poven the validity of the used PC pogams and algoithm which ae used unde obtaining the discussed numeical esults. 0 Table 1. The influence of the dimensionless initial stess on the values of the dimensionless citical velocity c c zz c V c in Case 1 hr Thus, we tun to the discussion of the esults given in Tables 1, and 3, accoding to which, it can be concluded that an initial stesses of the constituents of the

14 system unde consideation causes a decease, howeve an initial compession of these constituents causes an incease in the values of the citical velocities c c (6). 0 Table. The influence of the dimensionless initial stess on the values of the dimensionless citical velocity c c zz c V c in Case hr pest by Abdulkadiov (1981) Table 3. The influence of the dimensionless initial stess on the values of the dimensionless citical velocity c c zz c V c in Case 3 hr pest (by Babich et al. (1986))

15 This conclusion agees in the qualitative sense with the coesponding ones obtained in the papes by Babich et al. (1986), Akbaov et al. (015) and othe ones detailed in the monogaph by Akbaov (015). Also, these esults show that the magnitude of the influence of the initial stesses on the values of the citical velocities incease with deceasing the atio hr. Moeove, the mentioned influence in Case is moe consideable than that obtained in Case 1 and Case 3. So that, in Case 1 and in Case 3 the influence of the initial stess on the citical velocity is not moe than 5-6%, howeve this influence in Case may be geate than 17%. This completes the consideation of the numeical esults elated to the influence of the initial stesses on the citical velocity. 4.3 Numeical esults elated to the influence of the initial stesses on the inteface stess distibution Conside numeical esults which illustate the influence of the paamete (7) (i.e. the influence of the initial stesses) on the esponse of the inteface nomal stess () z c V / c to the dimensionless velocity of the moving load. Fo this pupose we conside the gaphs of the dependence between zz ( zz (0)) and c constucted fo vaious values of the paamete. These gaphs ae given in Figs. 3, 4 and 5 which elate to Case 1 (), Case (3) and Case 3 (4), espectively. Note that in these figues the gaphs gouped by lettes a, b, c and d coespond the cases whee h/ R =0.5, 0., 0.1 and 0.05, espectively. Moeove, note that unde constuction these gaphs it is assumed that c cc, i.e. the velocity of the moving load is less than the coesponding citical velocity. Thus, it follows fom Figs. 3, 4 and 5 that the absolute values of the inteface nomal stess incease (decease) as a esult of the initial compession (of the initial stetching) and the magnitude of this incease (decease) becomes moe consideable as the moving velocity appoaches the citical one. Moeove, these esults show that the magnitude of the influence of the initial stesses on the values of the inteface nomal stess becomes moe significantly with deceasing of the atio h/ R. Fo instance, accoding to Fig. 4d, in the case whee h/ R 0.05 in Case as esult of the initial compession the absolute values of the stess unde consideation may be geate two times than that obtained in the case whee the mentioned initial compession is absent. The compaison the gaphs gouped by the lettes a, b, c and d shows that, as a esult of the decease in the values of the atio h/ R the mentioned influence becomes significantly not only fo the cases whee the moving velocity appoaches to the coesponding citical velocity, but also fo the cases whee moving velocity is nea to the At the same time, the compaison of the esults illustated with each othe shows that in Case the influence of the initial stesses on the values of the inteface nomal stess is moe significantly than that in Case 1 and Case. Consequently, the magnitude of the influence of the initial stesses on the values of the nomal stess depends significantly also on the mechanical popeties of the constituents of the system unde consideation.

16 a b c d Fig. 3 Response of the inteface nomal stess at the point z/ h 0 to the moving load velocity in Case 1 () unde h/ R 0.5 (a), 0. (b), 0.1 (c) and 0.05 (d)

17 a b c d Fig. 4 Response of the inteface nomal stess at the point z/ h 0 to the moving load velocity in Case (3) unde h/ R 0.5 (a), 0. (b), 0.1 (c) and 0.05 (d) Now we conside numeical esults elated to the distibution of the inteface nomal and shea stesses with espect to the z/ h. Epues of these distibutions ae given in Figs. 6 and 7 fo the inteface nomal stess shea stess z () z z ( zr, ) z ( zr, ) (1) and fo the inteface, espectively. In these figues the gaphs

18 gouped by lettes a and b elate to the cases whee c 0.5 and 0.7, espectively and Case 1 unde h/ R 0. is consideed. a b c d Fig. 5 Response of the inteface nomal stess at the point z/ h 0 to the moving load velocity in Case (3) unde h/ R 0.5 (a), 0. (b), 0.1 (c) and 0.05 (d) Epues of the distibution of the inteface nomal stess and inteface shea stess obtained in the case whee h/ R 0.05 ae given in Fig. 8a and Fig. 8b, espectively. Note that unde constuction the gaphs given in Fig. 8 it is also Case 1 is consideed and it is assumed that c 0.7. Thus, it follows fom Fig. 6 that in a cetain distance fom the point at which the moving load acts at behind and ahead of this point the inteface nomal stess becomes stetched one. Note that this moment can play impotant ole in the adhesion stength of the system "hollow cylinde +suounding elastic medium" and the esults given in

19 Fig. 6 shows that the values of this stetching nomal stess incease (decease) with initial compession (with initial stetching) of this system. Moeove, the compaison of the esults given in Fig. 6a with coesponding ones given in Figs. 6b and 8a shows that an incease in the values of the velocity of the moving load and a decease in the values of the atio h/ R cause an incease in the values of the afoementioned stetching inteface nomal stess, as well cause an incease of the magnitude of the influence of the initial stesses on the values of this stess. a b Fig. 6 Distibution of the inteface nomal stess with espect to z / h unde c 0.5 (a) and 0.7 (b) in Case 1 a b

20 Fig. 7 Distibution of the inteface shea stess with espect to z / h unde c 0.5 (a) and 0.7 (b) in Case 1 fo h/ R 0. a b Fig. 8 Distibution of the inteface nomal (a) and shea (b) stesses with espect to z / h R and c 0.7 in Case 1 unde / 0.05 Analysis of the gaphs given in Fig. 7 and in Fig. 8b shows that the influence of the initial stesses on the absolute values of the inteface shea stess is simila in the quantitative sense with that elated to the inteface nomal stess. These gaphs also show that the shea stess has its absolute maximum value in a cetain distance fom the point at which the moving load acts. The compaison of the gaphs given in Fig. 7a with coesponding ones given in Fig. 7b and 8b shows that the influence of the initial stesses on the values of the inteface shea stess inceases with inceasing the load moving velocity and with deceasing the atio h/ R. Finally, we note the following statement. Fo this pupose we ecall that the coodinate z with espect to which the distibution of the inteface stesses is illustated in Figs. 6, 7 and 8, is the coodinate in the moving coodinate system detemined by expessions in (11), i.e. the z / R in these figues is the z'/ R. Consequently, fo moe coect explanation of the esults given in Figs. 6, 7 and 8 we must take into consideation ( z Vt) / R (whee z is a coodinate in the efeence coodinate system) instead of z / R. Accoding to this consideation, if we fix the time t( t*) then these figues illustate the distibution of the stesses with espect to spatial coodinate ( z ' z Vt *) which shows the distance in the cylinde's axis diection fom the point at which the moving load acts. If we fix the spatial coodinate z z* in the efeence coodinate system, then, accoding to z ' z * Vt, the Figs. 6, 7 and 8 illustate the change of the stesses with espect to time at the mentioned fixed point. Consequently, the esults given in Figs. 6, 7 and 8 illustate not only the distibution of h

21 the inteface stesses with espect to the spatial coodinate, but also these gaphs illustate the change of these stesses with espect to time. This completes the consideations and analysis of the numeical esults. 5. CONCLUSIONS Thus, in the pesent pape the poblems elated to the dynamics of the moving nomal ing load acting on the inteio of the pe-stessed hollow cylinde and the pestessed suounding elastic medium is investigated with employing the theedimensional lineaized theoy of elastic waves in initially stessed bodies. It is assumed that the initial stesses in the constituents of the system unde consideation appea as a esult of the action of the unifomly distibuted nomal foces applied at infinity in the cylinde's axis diection which coincides with load moving diection and the case whee the initial stesses ae homogeneous, is consideed. The equations fo the potentials which ente into the classical Lame decomposition fo displacements ae obtained fo the consideed case which coincide with coesponding ones used the classical elastodynamics unde absent of the initial stesses. The Fouie tansfom is employed with espect to the spatial axial coodinate and Fouie tansfomation of sought values ae detemined analytically, howeve oiginals of those ae found numeically fo which coesponding algoithm and PC pogams ae developed and composed by authos. Numeical esults on the influence of the initial stesses on the values of the citical velocity and on the distibution of the inteface stesses ae pesented and discussed. Accoding to these esults, it can be made the following concete conclusions: - The initial stetching (compession) of the hollow cylinde and suounding elastic medium causes a decease (an incease) in the values of the citical velocity; - The magnitude of the afoementioned influence incease with deceasing of the atio h/ R whee h is a thickness and R is an extenal adii of the hollow cylinde. At the same time, the values of the citical velocity decease with the atio h/ R; - The influence of the initial stesses on the values of the citical velocity depends also significantly on the mechanical popeties of the mateials of the cylinde and suounding elastic medium; - Absolute values of the inteface nomal and shea stesses incease (decease) with initial compession (with initial stetching) of the constituents of the system unde consideation; - The magnitude of the influence of the initial stesses on the absolute values of the inteface stesses incease with deceasing of the atio h/ R. At the same time, this magnitude incease significantly in the cases whee the load moving velocity appoaches the coesponding citical velocity; - An initial compession (stetching) of the system unde consideation causes an incease (a decease) of the stetching inteface nomal stess which appea in a cetain distance fom the point at which the moving load acts. The adhesion stength of the consideed system can depend significantly on the values of this nomal stess;

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