This is the published version of a paper published in Mechanics of composite materials. Citation for the original published paper (version of record):

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1 This is the published vesion o a pape published in Mechanics o coposite ateials. Citation o the oiginal published pape (vesion o ecod): Kaii. Zaanov A. () tability o a ibe suounded by two shells i an elastic atix. Mechanics o coposite ateials 49(5): 7-76 Access to the published vesion ay equie subsciption. N.B. When citing this wok cite the oiginal published pape. Peanent link to this vesion:

2 Mechanics o Coposite Mateials Vol. 49 No. 5 Novebe (Russian Oiginal Vol. 49 No. 5 eptebe-octobe ) stability o A ibe suounded by two shells in an elastic atix A. D. Zaanov * and. M. oltanova Keywods: subcitical deoation ibe shell atix Bessel unction stability buckling odes chaacteistic equation Based on the odel o a piecewise hoogeneous body with the use o the thee-diensional lineaized theoy o elastic stability o sall subcitical deoations the stability o a single ibe suounded by two shells in an ininite elastic atix is investigated. Duing stateent o the poble the hoogeneity o the subcitical stess state is substantiated. Expessions ae deived o eleents o the chaacteistic deteinant. tateent o the Poble Let us conside an ininite elastic body containing an ininite ectilinea ibe suounded by two shells. The adius o coss section o the ibe is R the coss sections o the ist and second shells have the intenal and extenal adii R and R o the ist (I) shell and R and R o the second (II) shell (Figs. and ). hell I diectly touches the ibe while shell II is in contact with the atix. The values eeing to the ibe ae aked by the supescipt those to the ist and second shells by the supescipts and espectively and those elated to the atix have the supescipt. The ibe is eeed to the Lagange ectangula ( x x x ) and cylindical ( θ x ) systes o coodinates. We assue that the ibe axis coincides with the Ox axis (see Fig. ) the ateials o the ibe shell and atix ae hoogeneous and tansvesely isotopic and the isotopy axis coincides with the Ox axis. In addition we assue that the ininite body is copessed at ininity by unioly distibuted noal oces o intensity P acting along the Ox axis. We will also pesue that the extenal load is applied so that it ensues the sae shotening o the ibe shell and atix along the Ox axis. Theeoe o the subcitical state we can wite the elations [-] Institute o Matheatics and Mechanics National Acadey o ciences o Azebaijan Baku Azebaijan * Coesponding autho; e-ail: a_zaanov@ail.u Tanslated o Mekhanika Kopozitnykh Mateialov Vol. 49 No. 5 pp eptebe-octobe. Oiginal aticle subitted May // pinge cience+business Media New Yok 48

3 p x R R R x R R R x p Fig.. cheatic sketch o an ininite ectilinea ibe. Fig.. cheatic sketch o coss section o the ibe. ε = ε = ε = ε = ε ε = ε ε = ε ε = ε. () Based on the odel o a piecewise hoogeneous body with the use o the thee-diensional lineaized theoy o stability o elastic bodies we will investigate the buckling in the stuctue o the ininite body [ 4]. The subcitical state in the body is deteined accoding to the geoetically linea theoy o elasticity. We will assue that this state in the ibe shells and atix is hoogeneous and in addition to Eqs. () the conditions σ = const σ = const σ = const σ = const σ = σ = σ = σ = σ = σ = σ = σ =. () ae ulilled. Conditions () ae ulilled exactly i the coeicients o tansvese expansion o the ibe shells and atix coincide. Fo the exact equations ij σ = i it is easy to ind that in this case the subcitical state is hoogeneous and can be ound o oulas () and (). Othewise (i.e. in the cases whee the Poisson atios o any two coponents do not coincide) condition () will not be stictly satisied but will be equal to the dieence o Poisson atios o ateials o the ibe shell and atix. Hence the inhoogeneity o the stess state in deteining the subcitical state in the ininite body can be neglected wheeas the subcitical state can be descibed by elations () and (). Let us assue that on inteaces o the ibe the ist and second shells and the atix the conditions o coplete adhesion ae ulilled. Then o the poble consideed the conditions o the vecto o oce and displaceent can be witten in the o θ θ = R = R = R = R = R = R P = P P = P P = P U = U Uθ = Uθ U = U () = R = R = R = R = R = R θ θ = R = R = R = R = R = R P = P P = P P = P 484

4 θ θ = R = R = R = R = R = R U = U U = U U = U P = P Pθ = Pθ P = P () = R = R = R = R = R = R θ θ = R = R = R = R = R = R U = U U = U U = U. We assue that the buckling odes o the body along the Ox axis ae peiodic and intoduce the daping conditions at θ P P P Also we pesue that U Uθ U. (4) σ ij Ui < <. (5) We also assue that the extenal copessive oces o intensity P opeating at ininity ae dead and accoding to [ 5 6] will solve the poble by using the Eule appoach. Choice o the Buckling Mode and Deivation o the Chaacteistic Equation The equations o thee-diensional lineaized theoy o elastic stability in displaceents can be witten as whee L U = = =. (6) α α α L α = ω ijα x x i olutions to syste (6) ae given in the o + + ijα i α i i iα i α i i α. ω = δ δ A + δ δ δ δ δ G δ δ σ ( j) et Ls ( j) uα Φ j α =. (8) L = ( jα ) The unctions Φ ( j) in Eqs. (8) ae ound o the elation Accoding to [4] the geneal solutions o Eq. (8) ae u = u det L Φ = s =. (9) s j u = θ ψ x χ θ ψ χ θ x = ( ω + ω ) ω + ω χ. () x 485

5 The unctions ψ and c ae ound o the dieential equations + ξ ψ = + ξ + ξ x x χ = () x obtained o Eq. (9). The solution o Eqs. () o the ibe the ist and second shells and the atix has the o U n = i n π l i sin π exp ( θ ) x. cos () Inseting Eq. () into Eqs. () with account o Eqs. (4) and (5) and consideing the assuption that ξ > i = we have o the unctions ψ and c π π π ψ = A I ξ θ l l x sin sin ψ χ χ = B I ξ π π π ξ θ l + C I l l x cos cos = π π π A I ξ A K ξ θ l + sin sin π l x sk sk sk sk sk = B π π π I ξ B K ξ C I ξ + + sk sk sk sk sk sk sk χ π + π Csk K ξ sk cos θ cos l x k = () ψ = π π π A K ξ θ l x sin sin = π π π B K ξ C K ξ θ + l x cos cos. Hee I ( x) is the Bessel unction o a puely iaginay aguent; K ( x) is the McDonald unction; A B C A sk Ask Bsk Bsk Csk Csk (k = ) A Band C ae unknown constants. Thus taking into account Eqs. () () and () and inseting the coesponding expessions o coponents o the vecto o oces and displaceents into contact elations () o the condition o existence o nontivial solutions as in [8 9] we coe to the chaacteistic equation in the geneal o o the cases whee the ateials o the ibe shells and atix ae tansvesely isotopic det = i j = (4) i j Let us wite expessions o the quantities i j which ente into Eq. (4): ω ( ξ )= ω + ω ( + k k = ω ω χ R χξ χξ R χξ ( ω ( χξrk ( χξ ) χξ R χξ χr χξ + ) k k (5) 486

6 + ω χ R ( χξ ) ω χ ξ R ( χ ξ ) k k k k + = ω χ ξ R χ ξ ω χ R χ ξ ( ξ ) ω ω 4 χ ξ χ χξ = ( )( R i ( )+ Ri ( ξ χ )) ω ( ξ )= ω χ ξ ξ Ri χ ( ( ) χξ R χξ χ R χ ξ ω + ω 5 ( i i + R ) + R + ω χ ξ χ ω χ ξ χ ξ ω ω χ ξ ( χ ξ )) = ( ξ ) R i ) i i i χξ R ξ χ ( + 7 = ω ω χrk χξ χξr k χξ ) ω ( ξ )= ω + ω 8 ( k + ) + ) + ( k k ω χ ξ R χ ξ χ ξ R χ ξ k χ R χ ξ ω χ R χ ξ ω χξ R k χξ + ωχξ k ( χξ ) ωχ k( χξ)) R R (5) = ( ξ ) = = = 9 8 = = = = = = = = = = = = = = = = ( ξ )= ω s s s s i s s ( ωχ ξqi ( χξ)+ ω χξ Qi χ + ω ) = ω Q χξ ω χ Q χξ ξ 8 ( ξ )= ω 8 4 s s s s k i 8 8 = s s ( ωχ ξqk ( χξ)+ ω χξ Qk χξ + ω ) = ω Q χξ ω χ Q χξ ξ 86 ( ξ )= ω ω k = ( ωχ ξ Qi ( χξ ) + ωχξ Q χξ i ω i ( χξ ) ωχ i ( χξ )) 88 = 87 ( ξ ) Q Q. 487

7 Hee the ollowing designations have been assued: π R π R π R χ = χ = χ = l l = = = = = = Q χ H χ R χ Q χ H χ R χ K χ k k k k k k = = = = = = Q χ H χ R χ Q χ H χ k k k k k Q χ H χ R χ Q χ H χ k k k k k Rk χ ( K χ + K( χ )) χ 4 χ χ ) = = = = = Rk = K+ K( = = = = = = Q χ H χ R χ Q χ H χ k k k k k Rk χ ( K + K+ K4 ) 8 χ 4 χ χ (6) = = = = = = Q χ H χ R χ Q χ H χ R χ I χ i i i i i i = = = = = Ri = I+ I Q χ H χ R χ Q χ H χ i i i i i Q χ H χ R χ Q χ H χ i i i i i χ (( χ χ ) χ 4 χ χ = = = = = Ri = ( I + I) = = ( χ)= ( χ)= ( χ ) = Ri = ( I+ I+ I4 ) Q X H X R Q H i i i i i χ 8 χ 4 χ χ. s s We should note that in Eqs. (5) the values o ξi ξi ξi ξi (i = ) and ωijα ωijα ωijα ωij α i j α= ae calculated as in []. Analysis o Nueical Results The chaacteistic equation (4) is solved nueically o the case whee the ateials o the ibe shells and atix s s s s ae hoogeneous and isotopic with the elastic chaacteistics E v E v E v E and v (hee Е is Young s s s odulus). It is assued that v = v = v = v =.. The subcitical condition is hoogeneous: E σ ε ε ε ε ε E p P = = = = = = E σ = p σ = E σ E p = E E p (7) ξ = ξ ξ = ξ ξ = ξ ξ = ξ. Let us intoduce the designations h = R R h = R R (8) whee h and h ae thicknesses o the shells o extenal adii R and R espectively (see Fig. ). Taking into account Eqs. (8) we have o cc and c χ = πr/ l χ = χ + h χ h h. (9) ( + + ) We will conside only the cases whee the ollowing inequalities ae valid: E > E > E > E. () 488

8 TABLE h E /E E /E E /E /R (.4) (.4) (.4) (.4) (.45) (.45) (.5) (.55) (.5).476 (.).59 (.5).876 (.4).599 (.5).478 (.).6 (.5).877 (.4).6 (.5).479 (.).6 (.5).64 (.5).48 (.).6 (.5).886 (.4).66 (.5).48 (.).64 (.5).885 (.4).66 (.5).48 (.).64 (.5).64 (.5).489 (.).68 (.5).9 (.4).65 (.5).49 (.).69 (.5).9 (.4).66 (.5).49 (.).69 (.5).64 (.5).59 (.).8 (.5).947 (.4).64 (.5).5 (.).8 (.5).945 (.4).64 (5).59 (..8 (.5).66 (.5).56 (.).946 (.5).987 (.45).66 (.5).58 (.).95 (.5).986 (.4).666 (.5).59 (.).96 (.5).74 (.5).564 (.).4 (.5).67 (.45).76 (.5).567 (.).4 (.5).67 (.45).77 (.5).567 (.).49 (.5).85 (.4).649 (.5).4768 (.).9 (.5).89 (.4).647 (.5).479 (.).8 (.45).8 (.5).6478 (.5).479 (.5).887 (.4).7 (.5).58 (.).65 (.55).876 (.4).79 (.5).5 (.).65 (.5).877 (.4).748 (.5).5 (.) Let us intoduce the designation ε = ε [with ε ound o Eqs. (7)] and exaine elations between the quantities ε and c at dieent values o poble paaetes. In addition we intoduce the designation { εc = in ε ( χ )} () and denote the quantity c coesponding to ε c as c c. Condition () ensues the ulillent o Eq. () at χ c. In othe wods condition () ensues the eegence o intenal buckling in the body with the stuctue shown in Fig.. Let us analyze the nueical esults o the quantity ε c pesented in Tables - (the values o c c ae given in paentheses) o the cases whee h / R = 5.. and. (in Tables and espectively). Fo data o the tables it ollows that o all h / R and h / R an incease in the atios E /E and E / E causes a onotonical decease in ε c. At ixed h / R and gowing h / R as well as at ixed h / R and gowing h / R the values o ε c incease onotonically. The values o ε c obtained o the cases h / R = (designated as ε c. ) and h / R = (designated as ε c. ) coincide with the coesponding values o ε c deived in []. The values o ε c at dieent h / R satisy the inequality ε < ε < ε () c. c c.. Thus the pesence o a second shell at an insigniicant distinction between the elastic oduli o the shell and atix (o exaple at E / E = ) leads to a consideable gowth in the extenal copessive beaking load. Now we will analyze elations between the quantities c c and paaetes o the poble. As seen o the given tables at athe sall values o E / E h / R E / E with gowing h / R the values o c c onotonically incease. 489

9 TABLE h E /E E /E E /E /R (.5) (.5) (.5) (.4) (.45) (.7) (.7) (.7) (.7) (.75) (.5).6 (.).46 (.5).74 (.).8 (.).658 (.5).48 (.5).69 (.).799 (.5).65 (.5).49 (.5).9 (.).79 (.).55 (.5).6 (.).94 (.).74 (.5).54 (.5).75 (.5).88 (.5).79 (.5).54 (.5).4 (.).4 (.).747 (.5).8 (.).4 (.5).4 (.5).778 (.5).65 (.5).48 (.5).9 (.5).775 (.).487 (.4).69 (.5).845 (.).8 (.5).55 (.).6 (.5).899 (.5).8 (.5).49 (.5).75 (.5).887 (.5).555 (.5).4 (.45).9 (.5).8 (.5).664 (.5).9 (.5).985 (.).97 (.5).568 (.5). (.5).984 (.5).58 (.55).67 (.45).9449 (.4).4 (.5).78 (.45).96 (.4).7 (.5). (.5).679 (.5).9 (.9).55 (.).589 (.55).8 (.5).959 (.4).74 (.55).754 (.5).4 (.45).65 (.4).4 (.4).7 (.4).47 (.5).94 (.5).594 (.55).84 (.5).967 (.4).88 (.6).76 (.5).4 (.45).75 (.4).6 (.45).755 (.4).457 (.4). (.5).596 (.55).87 (.5).979 (.4).94 (.6).77 (.5).45 (.45).8 (.4).56 (.5).769 (.45).47 (.4).7 (.5).598 (.55).89 (.5).977 (.45).97 (.6).774 (.5).44 (.45).88 (.4).65 (.5).777 (.45).48 (.4).6 (.4) a b Fig.. Relation ε = ε( χ) o the cases E / E = (a) and 5 (b) at h / R =. ().5 ().5 () 4.5 (4) 5.5 (5) 6.5 (6) 7.5 (7) 8.5 (8) (9) and (). h / R =.8 E E / = and E / E =. Howeve the pictue is dieent at athe geat values o E / E and h / R. The sae can be said in analyzing the dependence ε = ε( χ ) o elatively geat values o h / R.(Fig. ). It is seen that at athe geat values o h / R the dependence ε = ε( χ ) has a coplicated chaacte. The sae coplicated chaacte o this atio is also obseved at athe geat values o E / E. Howeve with gowth in E / E and E / E the chaacte o ε( χ)becoes siple (Fig. 4). 49

10 TABLE h E /E E /E E /E /R (.4) (.4) (.4) (.4) (.45) (.45) (.5) (.55) (.5).487 (.).66 (.5).97 (.4).64 (.5).497 (.).74 (.5).9 (.4).69 (.5).5 (.).76 (.5).67 (.5).499 (.).75 (.5).9 (.4).64 (.5).55 (.).79 (.5).98 (.4).67 (.5).56 (.).8 (.5).6 (.5).5 (.).78 (.5).99 (.4).69 (.5).58 (.).8 (.5).96 (.4).64 (.5).5 (..8 (.5).65 (.5).56 (.).87 (.5).97 (.4).658 (.5).5 (.).9 (.5).968 (.4).66 (.5).55 (.).9 (.5).675 (.5).55 (.).4 (.5).5 (.4).685 (.5).54 (.).45 (.5). (.4).687 (.5).544 (.).47 (.5).78 (.5).575 (.).48 (.5).6 (.4).74 (.5).584 (.).44 (.5). (.4).74 (.5).586 (.) 46 (.5).85 (.4).658 (.5).486 (.).6 (.45).85 (.5).67 (.5).495 (.).59 (.45).854 (.5).676 (.5).498 (.).895 (.4).78 (.5).54 (.).6 (.5).97 (.4).75 (.5).55 (.). (.5).9 (.4).79 (.5).557 (.) a b Fig. 4. Relation ε = ε( χ) o the cases E / E = (a) and (b) at h / R =. ().5 () 4.5 () 5.5 (4) 6.5 (5) 7.5 (6) 8.5 (7) (8) and (9). h / R =.5 E E / = and E / E =. Thus o the esults pesented it ollows that an incease in the nube o tansition zones o the ibe to atix leads to a gowth in the citical copessive oce i these zones can be odeled as shells ade o dieent ateial which suounding the ibe ollow one ate anothe. 49

11 REFERENCE.. D. Akbaov tability o two ibes in a atix at sall deoations Pikl. Mekh. 7 No (98)... D. Akbaov A. N. Guz tability o two ibes in an elastic atix at sall deoations Pikl. Mekh. No. - (985).. A. N. Guz Fundaentals o the Thee-Diensional Theoy o tability o Deoable Bodies pinge (999). 4. A. N. Guz Factue Mechanics o Coposite Mateials in Copession [in Russian] Naukova Duka Kiev (99). 5. I. Yu. Babich and A. N. Guz Applicability o the Eule appoach to investigating the stability o deoations o anisotopic nonlinea elastic bodies at inite subcitical deoations Dokl. Akad. Nauk R No (97). 6. A. N. Guz tability o Elastic Bodies unde Finite Deoations [in Russian] Naukova Duka Kiev (97). 7.. G. Lekhnitskii Elasticity Theoy o Anisotopic Bodies [in Russian] Nauka Moscow (977). 8. I. Yu. Babich I. N. Gaashchuk and A. N. Guz tability o ibes in an elastic atix in an inhoogeneous subcitical state Pikl. Mekh. 9 No. -7 (98).. 9. A. D. Zaanov On the stability o hollow cylindical bodies in hydostatic copession in the case o sall subcitical deoations Izv. Akad. Nauk Azeb. R. e. Phiz.-Tekhn. Mate. No (986).. A. D. Zaanov and. M. oltanova On the stability o an elastic body einoced with single staight ibes with a shell in: Tans. Inst. Mate. Mekh. Akad. Nauk Azeb. VI (XIV) (997) pp