Plane Thermoelastic Waves in Infinite Half-Space Caused FACTA UNIVERSITATIS Series: Mechanical Engineering Vol. 12, N o 2, 2014, pp. 95 - 106 ON INFLUENCE OF BOUNDARY CONDITIONS AND TRANSVERSE SHEAR ON BUCKLING OF THIN LAMINATED CYLINDRICAL SHELLS UNDER EXTERNAL PRESSURE  UDC 539.3 Gennadi I. Mikhasev, Ihnat R. Mlechka Belarusian State University, Belarusian State University, Department of Bio and Nanomechanics, Mechanics Mathematics Faculty Minsk, Belarus Abstract. Buckling of a thin cylindrical sandwich shell composed of elastic isotropic layers with different elastic properties under normal external pressure is the subject of this investigation. Differential equations based on the assumptions of the generalized kinematic hypothesis for the whole sandwich are used as the governing ones. Two variants of the joint support conditions are considered at the shell edges: a) there are the infinite rigidity diaphragms inhibiting relative shears of layers along the shell edges, b) the diaphragms are absent. Using the asymptotic approach, the critical pressure and buckling modes are constructed in the form of the superposition of functions corresponding to the main stress- strain state and the edges integrals. As an example, a three-layered cylinder with the magnetorheological elastomer (MRE) embedded between elastic layers under different levels of magnetic field is studied. Physical properties of the magnetorheological (MR) layer are assumed to be functions of the magnetic field induction. Dependencies of the buckling pressure on the variant of boundary conditions and the intensity of applied magnetic field are analyzed. Key Words: Sandwich Cylindrical Shell, Buckling Pressure, Diaphragms, Magnetorheological Elastomer 1. INTRODUCTION Thin multi-layered shells are used in many engineering structures, such as airborn/ spaceborne vehicles, underwater objects and cars [1]. Application of new materials with different physical properties allows one to design sandwich structures fulfilling up-to- date requirements such as good buckling resistance and noiselessness. Buckling as well Received June 11, 2014 / Accepted July 26, 2014  Corresponding author: Gennadi I. Mikhasev Belarusian State University, Department of Bio- and Nanomechanics, Nezavisimosti Avenue 4, 220030, Minsk, Belarus E-mail: Mikhasev@bsu.by Original scientific paper 96 G.I. MIKHASEV, I.R. MLECHKA as vibroprotection of thin–walled structures are of great practical interest for mechanical engineers who develop and model similar structures. It is generally known that the critical value of an external load resulting in buckling of a thin shell depends on a great number of factors such as geometrical characteristics of a structure (thickness, radius, and length), physical properties of a material (Young’s and shear module, Poisson’s ratio), the boundary conditions, the way of loading (if it is combined) [2, 3]. If a laminated shell is assembled from layers having different properties, this dependence becomes more complex. One of the principal characteristics of the sandwich shell is shear compliance (or the reduced shear rigidity for a whole sandwich). It is influenced by a correlation between thicknesses and mechanical properties of layers composing the sandwich. As shown in [4, 5], taking into account the transverse shears results in an appreciable decrease of both the natural frequencies and the critical axial load of thin laminated shells. A less noticeable effect of transverse shears on the buckling external pressure of a three-layered cylindrical shell has been revealed in paper [6]. In these and other papers, the authors consider as a rule the simplest variant of boundary conditions - the joint support conditions when the edges have infinite rigidity diaphragms inhibiting relative shears of layers along the shell edges. Within the scope of the accepted kinematic hypothesis for laminated shells [4], this type of boundary conditions does not permit us to include the edge effects generated by shears, but gives us a possibility to find the buckling modes and the critical load in the explicit form. The present paper mainly aims at studying the influence of shears on the critical pressure for a thin sandwich cylinder when its simply supported edges do not have diaphragms. Because a similar variant of the boundary conditions does not allow us to find an exact value of the critical pressure, the asymptotic approach is applied to predict the shell buckling. The specific goal defined herein is to analyze the influence of a magnetic field on the buckling pressure of a three-layered cylinder with the magnetorheological elastomer (MRE) embedded between the elastic bearing layers. The MREs belong to the group of active materials whose physical properties (viscosity, Young’s and shear module) can vary when subjected to different magnetic field levels [7-10]. The application of an external magnetic field is expected to permit us not only to suddenly change the rheological properties of the MRE-based sandwich and suppress its vibrations [11] but to considerably increase the total stiffness of the structure and prevent its buckling. 2. SETTING A PROBLEM A thin middle length cylindrical sandwich shell consisting of N transversely isotropic elastic layers characterized by length L, thickness hk, Young’s modulus Ek, and Poisson’s ratio k is considered, where k = 1,2,...N. The middle surface of any fixed layer with radius R is taken as the original surface. Coordinate system 1, 2, 3 is illustrated in Fig. 1, where 1, 2, are axial and circumferential coordinates, respectively. The sandwich shell is assumed to be under normal external pressure Qn. Two variants of the joint support conditions are considered at shell edges 1 = 0, L: (a) there are infinite rigidity diaphragms inhibiting relative shears of layers along the shell edges, (b) diaphragms are absent at the edges. The problem is to find buckling conservative pressure Q * n at different variants of the boundary conditions. Buckling of Thin Laminated Cylindrical Shells under Pressure 97 Fig. 1 Thin cylindrical sandwich shell and a curvilinear co-ordinate system We accept here the unique kinematic hypotheses of Timoshenko for the whole package of the sandwich stated in [4]. From these hypotheses the principal ones relate to the distribution laws of the transverse shear stresses ),()(),()(),,( 21 )( 21 )0( 021 )(  k iki k i zfzfz , (1) and tangential displacements ),()(),(),(),,( 21212121 )(  iii k i zgzuzu (2) across the thickness of the kth layer, where f0(z), fk(z) and g(z) are continuous functions of coordinate 3 = z introduced as follows 0 0 12 2 1 1 ( ) ( )( ), ( ) ( )( ) N k k k k f z z z f z z z h h            , 0 0 ( ) ( ) z g z f x dx  , (3) k is the distance between the original surface and the upper bound of the kth layer, h is the total thickness of the sandwich, ui (k) are the tangential displacements of points of the kth layer, i are the angles of rotation of the normal n about the vectors ei (see Fig. 1), i = 1, 2; k = 1,2,..., N, and the functions i (0) , i (k) may be found in paper [4]. The remaining two hypotheses concerning normal deflection w and normal stresses are the same as in the classic Kirchhoff–Love hypothesis. We note that at g  0 hypothesis (2) turns into the Kirchhoff–Love ones. Based on the foregoing hypotheses, the system of five differential equations with respect to w, ui, i have been derived in book [4]. If buckling occurs with formation of large number of hollows although in one direction at the shell surface, these equations may be essentially simplified. Introducing functions i appearing in (2) by 1 ,1 ,2 2 ,2 ,1 ,a a       (4) where a and  are the shear functions defined from equations 2 2 2 1 1 , 2 h h a               , (5) the following compact system 98 G.I. MIKHASEV, I.R. MLECHKA 0 1 1 )1(12 2 2 2 0 22 12 2 2 2 3 3                      w T R hEh , 0 12 2     w R Eh ,             2 1 h w (6) is obtained [4] with respect to the displacement and force functions  and , respectively. In Eqs. (6), T2 0 = RQn is the hoop membrane stresses, Qn is the external normal pressure. Other parameters appearing in (5) and (6) are introduced as follows: 2 3 2 1 1 1 k k k k E h E h       , 1 3 3 2 2 11 1 1 kk k k k k kk k k E h E h                , kk hh  , n nh  , 3 1 2 1 1 12 1 3 k k k k c        , 3 1 2 2 13 12 1 3 k k k k c c        , 3 2 2 3 1 13 1 4 ( 3 ) 3 k k k k k c          , 2 2 1 3 1      , 1 3 2 2 11 1 k k k k k k k k E h E h              , 3 1 12 3 1 k k k k c        , 3 13 1 1 ( ) k k k k c        , 2 2 3 0 2 1 3 0 44 2 3 1 10 1 k k k kk k k k kkk k k k kk q G G                               , 2 44 1 12(1 ) q Eh      , 2(1 ) k k k E G    , (7) 0 1 ( ) k k k f z dz       , 1 ( ) ( ) k kn k n k f z f z dz       , 3 21 1 ( ) 12 k k k h g z dz       , 3 2 1 ( ) 12 k k k h zg z dz       . In Eqs. (5) - (7), the magnitudes E,  are the reduced Young’s modules and Poisson’s ratio of the sandwich, and parameters , , i characterize the reduced shear stiffness of the whole packet. At  1  0, Eqs. (6) degenerate into the well-known equations of the semi-moment theory of isotropic shells (see, for instance in book [3]). In terms of the functions , , the joint support conditions at the shell edges are written as 0,0 1 2     at L,0 1  , (8) 0 at L,01  (9) for the case when there are the infinite rigidity diaphragms inhibiting relative shears of layers along the shell edges. If these diaphragms are absent, then the boundary conditions for  become more complicated: 2 2 2 2 1 1 0, 1 0 h h                      at 1 0, L  (10) Buckling of Thin Laminated Cylindrical Shells under Pressure 99 2 2 2 2 2 2 2 2 2 2 1 2 1 21 2 1 2 2 0, (1 ) 0                                      at 1 0, L  , (11) while the boundary conditions for force function  are the same as in Eqs. (9). We call the boundary-value problems (5), (6), (8), (9) and (5), (6), (9) - (11) as the BV problems “A” and “B”, respectively. The problem is to find the minimum absolute value of hoop stress T2 0 for which every from these problems has a nontrivial solution. Note that the second Eq. (5) has a solution like the edge effect being a rapidly damped function from the shell edges. When considering the BV problem “A”, the shear function  is found from the second Eq. (5) and the last boundary condition (8). To estimate the influence of this function on the critical value of T2 0 , it is necessary to consider the complete system of five differential equations with respect to w, ui, i [4]. So, solving the BV problem “A” based on the simplified equations (5), (6), we can assume  = 0. As concerns the BV problem “B”, the boundary conditions (11) show that functions  and  are linked. 3. SOLUTION OF THE BV PROBLEM “A” The form of a solution of Eqs. (5), (6) depends on the shell sizes. Here we consider a middle length sandwich cylinder for which L ~ R . If the shell edges have diaphragms (the BV problem “A”), then the buckling mode is the same as for an isotropic one-layered simply supported shell (without taking into account transverse shears) [3] and may be found in the explicit form [4-6] 0 0 1 2 ( , ) ( , ) sin( / ) sin( / )n L m R       , (12) where n and m are natural numbers, and 0, 0 are constants. Substituting (12) into (6) results in the following equation for hoop stresses T2 0 : 0 0 8 4 2 2 2 ( , ) nm T T n m hEm       , (13) 4 2 2 2 4 4 4 2 2 1 , 1 nm nm nm nm nm K n h K K R                    , 2 2 2 2nm n m m          , 2 8 3 2 2 , 12(1 ) h L RR        . The minimization of 0 2 T over n and m gives the buckling pressure RTQ n * 2 *  , where * 0 0 0 * 2 2 2 2 , min ( , ) min (1, ) (1, ) n m m T T n m T m T m   . (14) 4. SOLUTION OF THE BV PROBLEM “B” If the simply supported edges do not have diaphragms, Eqs. (5), (6) do not admit a solution in the form (12). However, the introduction of additional assumptions for 100 G.I. MIKHASEV, I.R. MLECHKA parameters appearing in the governing equations permits us to apply the asymptotic approach. Let  be a small parameter. As shown in [12], for a three layered cylinder with the core made of the MRE, parameters K /  2 , K /  2 are also small and their orders depend on the correlation between thicknesses of the layers composed a cylindrical sandwich and its radius. In this study, it is assumed that  4242 , KK at 0 , (15) where ,   1. Let us introduce dimensionless coordinates x,  and load parameter  as follows: 0 6 1 2 2 , ,Rx R T Eh        . (16) Functions ,  and  are sought in the form: 1 4 2 1 1 ( ) sin( ), ( ) sin( ), ( ) sin( )RX x p EhR F x p RS x p                , (17) where p is a wave number, and X(x), F(x), S(x) are unknown functions of x. The substitution of (15) – (17) into Eqs. (5), (6) gives the governing equations written in the dimensionless form: 2 4 4 2 2 4 2 (1 ) (1 ) 0 d F X p X dx                , 2 4 2 4 2 (1 ) 0 d F X dx          , (18) 41 2 S S       , (19) where  = d 2 / dx 2   2 p 2 is the differential operator. The boundary conditions (9)-(11) at x = 0, l = L / R are rewritten as follows: 4 (1 ) 0,X    2 4 2 (1 ) 0 d X dx      , (20) 2 2 2 2 (1 ) 0 d dS p X p dxdx              , 2 2 2 2 2 0 dX d S p p S dx dx     , (21) 0F  , 2 2 2 2 0 d F p F dx    . (22) As seen, Eqs. (18), (19) are singularly perturbed equations. The solution of the boundary- value problem (18)-(22) may be presented in the form [3]: )()()()( , edbsedbs FFFXXX  , (23) where the superscripts (bs) and (ed) denote functions describing the main stress-strain state and the integrals of the edge effects, respectively. It is well known that under buckling of shells of zero Gaussian curvature under external pressure the following asymptotic estimates are valid [3]: )()()()( ~,~ bsbsbsbs FxFXxX  at   0. From Eq. Buckling of Thin Laminated Cylindrical Shells under Pressure 101 (19), it may be seen that SxS 2 ~   at   0. Let 1~, )()( bsbs FX . Then the asymptotic analysis of Eqs. (18), (19) permits us to determine the estimates ( ) ( ) 2 4 ( ) 2 ( ) ( ) 2 ( ) , ~ , ~ , ~ , ~ ed ed ed ed ed ed X F S X x X F x F          at 0  . (24) 4.1. Edge effect solutions We assume that (X (ed) , F (ed) ) = 2 4 ˆˆ ˆ( , ),X F S S  , where ˆˆ ˆ, , ~ 1X F S . When taking into account these correlations as well as above estimates for functions with the superscripts (bs), one obtains the following equations describing the edge effects: 2 4 4 8 2 4 ˆ ˆ ˆ 0, d X d X F dx dx     2 2 4 4 2 2 ˆ ˆ ˆ 0 d F d X X dx dx      , (25) 2 4 2 2 2 ˆ 2 ˆ 0 (1 ) d S m S dx             . (26) Eqs. (25), (26) have the following solutions 2 2 2 2 3 3( ) ( ) 2 2 1 1 ˆ ˆe e , ( 1) e e i i i i x l x x l x i i i i i i i i X A B F A B                                        , (27) 4 4 2 2 ( ) 1 2 ˆ e e x l x S C C          , 2 2 4 2 (1 ) m       (28) decreasing far from the shell edges. Here Ai, Bi, C1, C2 are complex constants which will be determined from the boundary conditions below, and 1, 2, 3 are complex roots (having positive real parts) of the algebraic equation 6 4 2 1 0      . (29) If ,   0, then the system of equations (25) degenerates into a simple edge effect equation, and Eq. (29) has two complex conjugate roots. So, transverse shears may distort the edge effect integrals. 4.2. Main stress state solution We will construct functions X (bs) , F (bs) , and load parameter  in the form of series: ( ) 2 ( ) 2 0 2 0 2 , , bs bs X X X F F F       (30) 2 0 2       (31) The substitution of (23), (24), (30), (31) into Eqs. (18) and the boundary conditions (20)-(22) results in the consequence of the boundary-value problems. 102 G.I. MIKHASEV, I.R. MLECHKA In the first-order approximation, one gets the following boundary-value problem 4 8 60 0 04 ( ) 0 d X p p X dx    , 2 0 2 40 1 dx Xd p F  , (32) 0 0 0X F  at 0,x l . (33) It has the simple solution 0 sin( )X A x l (34) if 0(p) = p 2 +  4 / p 6 , where  =  / l. Minimizing this function, one obtains the zero- order approximation of the load parameter 2/18/104/30 00 0 0 3,34)()(min   ppp p . (35) It should be noted that parameter  0 0 does not take into account transverse shears in the shell. Eqs. (35) are the same as in the classical shell theory [3]. In the second-order approximation, one has the non-homogeneous differential equation 4 0 8 0 0 6 02 0 2 0 24 [( ) ( ) ] ( , , ) d X p p X AZ p dx      (36) with the non-homogeneous boundary conditions 2 1 2 2 2 3 2 4 (0) , ( ) , (0) , ( )X X l X X l        , (37) where A is an arbitrary constant, the prime denotes differentiation over variable x, and 0 8 0 10 2 0 6 0 2 6 0 2 4 0 6 0 2 ( ) ( ) 2 ( ) 2( ) ( ) ( )Z p p p p p p              , (38) 3 3 2 2 1 2 1 1 ( 1) , ( 1) , i i i i i i A B           3 3 0 4 2 2 0 4 2 2 3 3 1 1 ( ) (1 ) , ( ) (1 ) i i i i i i i i p A p B             . (39) To determine constants Ai, Bi appeared in Eqs. (27), (39), we substitute (23), (27), (34) into the boundary conditions (21) and the second condition (20). Collecting coefficients at  2 , one gets the following non-homogeneous system of six algebraic equations 3 3 3 3 2 2 4 4 1 1 1 1 0, 0, 0, 0, i i i i i i i i i i i i A B A B                  3 1 3 1 , i ii i ii ABAA (40) with respect to constants Ai, Bi (i = 1,2,3) It may be seen that Ai, Bi and i from (39) are directly proportional to A. Hence, without loss of generality, one can set A = 1 in the boundary-value problem (36), (37). For the non-homogeneous boundary-value problem (36), (37) to have a solution the following equation Buckling of Thin Laminated Cylindrical Shells under Pressure 103 0 0 2 0 2 0 1 0 2 0 3 0 4 0 0 ( , , ) (0) ( ) (0) ( ) l Z p X dx X X l X X l            (41) is required to hold. From this equation, we obtain the correction for the load parameter 1/ 4 1/ 4 4 2 1 2 3 4 2 3 / 4 2 [4 3 3 3 ( )] 2 ( ) 2( ) 3                     , (42) which takes into account the transverse shears (the parameters ,  from Eqs. (5), (6)). It should be noted that function SS ˆ 4  , with Ŝ defined by (28), does not affect parameter 2. To find an appropriate correction for  introduced by function S it is required to consider the next approximation. However, the order of an expected correction equals h / R and are the same as the order of an error of the shell theory applied here. Thereby, if the shear parameters satisfy estimates (15), then the influence of shear function  on the buckling pressure is negligibly small. So, in the BV problem “B” as well as in the problem “A”, we can set  = 0. 5. EXAMPLE: BUCKLING OF THREE-LAYERED CYLINDER WITH MRE CORE As an example, we have considered a three-layered cylinder with MRE core of length L = 1m and radius R = 0,5m. We have performed calculations of Q * n vs. induction B of the magnetic field for the case when the external supporting surfaces having thicknesses h1 = h3 = 0,5m are made of the ABS-plastic SD-0170 with parameters E1 = E3 = 1510 9 Pa, 1 = 3 = 0,4, and the internal layer of the thickness h2 = 8mm is the MRE with Poisson’s ratio 2 = 0,42 and Young’s modules E2 specified by equation [11] kPa040,45230,13)( 2 BBE  . (43) This equation are valid at B < 200 mT. It approximates the experiment date obtained for MRE by means of the rheometer Physica MCR 301 (Anton Paar) at the frequency of the external impact 10 Hz [9]. Now we can analyze the influence of the magnetic field induction on the buckling pressure for two variants of simply supported edges. If the edges have the diaphragms, critical pressure Q * n will be calculated by Eqs. (13), (14). If the diaphragms are absent, then the following asymptotic formula * 6 1 0 2 4 0 2 [ ( )] n Q EhR O         (44) with the parameters  0 0, 2 defined from (35), (42) will be applied. It should be noted that in our example parameters K, , , ,  appearing in Eqs. (13), (29), (42) are functions of the magnetic field induction. Table 1 demonstrates the dependence of dimensionless parameters 2, ,  and the critical pressure upon various factors: the variant of boundary conditions, transverse shears and the intensity of a magnetic field. It may be seen that assumption   1 (see Eqs. (15)) holds weakly at a very low intensity of a magnetic field (at least for B  20 mT). But at B > 20 mT, the constructed solutions (17), (23), (30), (31) are asymptotically correct. 104 G.I. MIKHASEV, I.R. MLECHKA As follows from the asymptotic constructions, wave parameter p 0 and zero approximation  0 0 are not influenced by the transverse shears and the magnetic field induction. For geometrical parameters and materials accepted above, one has p 0 = 1,437,  0 0 = 2,756. The negative values of 2 at 0  B  60 mT point out that accounting transverse shears results in decreasing load parameter  (see Eq. (31)). But at a very high intensity of the magnetic field the influence of shears on correction 2 decreases. However, in spite of the complex dependence of the load parameter on a magnetic field, the increase of its intensity results in an increase of both the reduced Young’s modules for whole sandwich and the critical pressure. Comparing data in the last two columns, one can conclude: the buckling pressure for the cylindrical sandwich shell having diaphragms on both edges is higher than the critical pressure for the shell without diaphragms. However, it is necessary to bear in mind that the data in the last column are obtained as the result of the exact solution of BV problem “A”, whereas the critical pressures for the shell without diaphragms are the asymptotic estimates which are probably slightly understated. Table 1 Dimensionless parameters 2, ,  and critical pressure Q * n vs. magnetic induction B for two variants of the boundary conditions B, mT 2   Critical pressure Q * n, Pa Edges without a diaphragm Edges with a diaphragm 0 11,598 1,664 4,298 4174 7937 10 7,462 2,054 3,260 5160 8935 20 4,961 2,290 2,628 5762 9697 30 3,286 2,448 2,203 6168 10300 40 2,088 2,560 1,898 6463 10789 50 1,188 2,645 1,668 6687 11195 60 0,487 2,710 1,489 6865 11538 70 0,074 2,763 1,345 7009 11752 80 0,533 2,806 1,227 7129 11906 90 0,915 2,842 1,129 7232 12042 100 1,239 2,872 1,045 7320 12162 6. CONCLUSIONS Governing equations, based on the assumptions of the generalized kinematic hypothesis of Timoshenko’s, are used for the prediction of buckling of a thin composite sandwich cylindrical shell under external normal pressure. Two variants of the joint support conditions have been taken into consideration. If the shell edges have diaphragms preventing shears in the edge plane, the approximate solutions of the governing equations and the estimate for buckling pressure are found by using the asymptotic method; when the diaphragms are absent, the appropriate solution and the buckling pressure are determined in the explicit form. Buckling of Thin Laminated Cylindrical Shells under Pressure 105 As an example, the three-layered cylinder with MRE core has been considered. It is observed that the presence of the diaphragms at the shell edges leads to increasing the critical pressure. Other conclusions of this study concern the influence of the applied magnetic field and transverse shears on the shell buckling. When the induction of the magnetic field does not exceed 60 mT, accounting transverse shears results in decreasing dimensionless load parameter . However, for both variants of the boundary conditions, applying the magnetic field with the induction varying in the interval from 0 to 100 mT gives an increase of both the reduced Young’s modules and the buckling pressure. REFERENCES 1. Qatu, M.S., Sullivan, R.W., Wang, W., 2010, Recent research advances on dynamic analysis of composite shells: 200-2009, Composite Structures, 93, pp. 14-31. 2. Librescu, L, Hause, T., 2000, Recent developments in the modeling and behavior of advanced sandwich constructions: a survey, Composite Structures, 48, pp. 1-17. 3. Tovstik, P.E, Smirnov, A.L., 2001, Asymptotic methods in the buckling theory of elastic shells. Singapore: World Scientific. 4. Grigolyuk, E.I, Kulikov, G.M., 1988, Multilayer Reinforced Shells: Calculation of Pneumatic Tires. Moscow: Mashinostroenie, 288 p. [in Russian]. 5. Korchevskaya, E., Mikhasev, G., Marinkovich, D., Gabbert, U., 2003, Buckling and vibrations of composite laminated cylindrical shells under axial load, In: Kasper R., editor. Entwicklungsmethoden und Entwicklunsprozesse im Maschinenbau: 6 Magdeburger Maschinenbau-Tage. Berlin: Logos-Verlag, pp. 183-189. 6. Mikhasev, G., Seeger, F., Gabbert, U., 2001, Local buckling of composite laminated cylindrical shells with oblique edges under external pressure: Asymptotic and finite element simulation, Technische Mechanik, 21(1), pp. 1-12. 7. Deng, H-X, Gong, X-L, 2008, Application of magnetorheological elastomer to vibration absorber, Communications in Nonlinear Science and Numerical Simulation,13, pp.1938–1947. 8. Gibson, R.F., 2010, A review of recent research on mechanics of multifunctional composite materials and structures, Composite Structures, 92, pp. 2793- 2810. 9. Korobko, E.V, Mikhasev, G.I, Novikova, Z.A., Zurauski, M.A., 2012, On damping vibrations of three- layered beam containing magnetorheological elastomer. Journal of Intelligent Material Systems and Structures, 23(9), pp. 1019–1023. 10. Yeh, J-Y., 2013, Vibration analysis of sandwich rectangular plates with magnetorheological elastomer damping treatment. Smart Material Structures, 22, 035010. 11. Mikhasev, G.I., Botogova, M.G., Korobko, E.V., 2011, Theory of thin adaptive laminated shells based on magnetorheological materials and its application in problems on vibration suppression. In: Altenbach H, Eremeyev VA, editors. Shell-like Structures. Advanced structured materials, vol. 15. Berlin: Springer, pp. 727–750. 12. Mikhasev, G.I., Altenbach, H., Korchevskaya, E., 2014, On the influence of the magnetic field on the eigenmodes of thin laminated cylindrical shells containing magnetorheological elastomer, Composite Structures, 113, pp. 186-196. O UTICAJU GRANIČNIH USLOVA I POPREČNOG SMICANJA NA IZVIJANJE TANKIH LAMINIRANIH CILINDRIČNIH PANELA POD SPOLJNIM PRITISKOM Predmet ovog istraživanja je izvijanje tankog cilindričnog sendvič panela koji se sastoji od elastičnih izotropskih slojeva sa različitim elastičnim svojstvima pod normalnim spoljnim pritiskom. Kao vodeće jednačine koriste se diferencijalne jednačine zasnovane na pretpostavkama o uopštenoj kinematskoj pretpostavki za ceo sendvič. Dve varijante zajedničkih uslova potpore se razmatraju na ivicama panela: a) da postoje dijafragme beskonačne krutosti u relativnim smicanjima 106 G.I. MIKHASEV, I.R. MLECHKA slojeva duž ivice panela, i b) da su dijafragme odsutne. Uz pomoć asimptotskog pristupa, kritični pritisak i modusi izvijanja se konstruišu u obliku superpozicija funkcija koje odgovaraju glavnom stanju naprezanja/deformacije i integrala ivica. Daje se primer troslojnog cilindra sa magnetoreološkim elastomerom (MRE) umetnutim između elastičnih slojeva pod različitim nivoima magnetnog polja. Fizička svojstva magnetorološkog sloja (MR) se uzimaju kao da su funkcije indukcije magnetnog polja. Zavisnosti pritiska izvijanja na varijantu graničnih uslova se proučavaju kao i intenzitet primenjenog magnetnog polja. Ključne reči: sendvični cilindrični panel, pritisak izvijanja, dijafragme, magnetoreološki elastomer