JOURNAL OF MECHANICS OF MATERIALS AND STRUCTURES Journal of Mechanics of Materials and Structures Volume 1 No. 2 February 2006 Journal of Mechanics of Materials and Structures 2006 Vol. 1, No. 2 Volume 1, Nº 2 February 2006 mathematical sciences publishers JOURNAL OF MECHANICS OF MATERIALS AND STRUCTURES http://www.jomms.org EDITOR-IN-CHIEF Charles R. Steele ASSOCIATE EDITOR Marie-Louise Steele Division of Mechanics and Computation Stanford University Stanford, CA 94305 USA SENIOR CONSULTING EDITOR Georg Herrmann Ortstrasse 7 CH–7270 Davos Platz Switzerland BOARD OF EDITORS D. BIGONI University of Trento, Italy H. D. BUI Ecole´ Polytechnique, France J. P. CARTER University of Sydney, Australia R. M. CHRISTENSEN Stanford University, U.S.A. G. M. L. GLADWELL University of Waterloo, Canada D. H. HODGES Georgia Institute of Technology, U.S.A. J. HUTCHINSON Harvard University, U.S.A. C. HWU National Cheng Kung University, R.O. China IWONA JASIUK University of Illinois at Urbana-Champaign B. L. KARIHALOO University of Wales, U.K. Y. Y. KIM Seoul National University, Republic of Korea Z. MROZ Academy of Science, Poland D. PAMPLONA Universidade Catolica´ do Rio de Janeiro, Brazil M. B. RUBIN Technion, Haifa, Israel Y. SHINDO Tohoku University, Japan A. N. SHUPIKOV Ukrainian Academy of Sciences, Ukraine T. TARNAI University Budapest, Hungary F. Y. M. WAN University of California, Irvine, U.S.A. P. WRIGGERS Universitat¨ Hannover, Germany W. YANG Tsinghua University, P.R. China F. ZIEGLER Tech Universitat¨ Wien, Austria PRODUCTION PAULO NEY DE SOUZA Production Manager SILVIO LEVY Senior Production Editor NICHOLAS JACKSON Production Editor ©Copyright 2007. Journal of Mechanics of Materials and Structures. All rights reserved. mathematical sciences publishers JOURNAL OF MECHANICS OF MATERIALS AND STRUCTURES Vol. 1, No. 2, 2006 DYNAMIC RESPONSE OF MULTILAYER CYLINDERS: THREE-DIMENSIONAL ELASTICITY THEORY ALEXANDER SHUPIKOV AND NATALIYA DOLGOPOLOVA We suggest an analytic-numerical approach to solving the problem of vibrations of multilayer cylinders under impulse loading. The behavior of the cylinder is described by dynamic equations of three-dimensional elasticity theory. The number of layers in the pack and the thickness and mechanical characteristics of each layer are selected arbitrarily. The possibilities of the approach are proposed, and the validity of results obtained is demonstrated by numerical examples. 1. Introduction Structural elements in the form of multilayer plates and shells are used extensively in different branches of machine building and civil engineering. The stressed-strained state (SSS) of real objects can be described most simply using the finite-element method, but the development of analytic and hybrid calculation methods is the focus of much current work. In works where analytic and hybrid methods are used, the SSS is investigated most often by using different 2-dimensional discrete, continuous and discrete- continuous theories [Grigoliuk and Kogan 1972; Grigoliuk and Kulikov 1988; Reddy 1989; 1993; Noor and Rarig 1974; Noor and Burton 1989; Noor and Burton 1990a; 1996; Smetankina et al. 1995; Shupikov and Ugrimov 1997; Shupikov et al. 2004]. Using these theories, obtaining numerical results is relatively straightforward. In the process, one is faced with the complex problem of determining the limits within which these theories adequately describe the behavior of the object being investigated with a prescribed accuracy. At impulse and other nonstationary short- time actions, the solution of this problem is yet more challenging. The behavior of a multilayer structure can be investigated most effectively in terms of three-dimensional elasticity theory. Pagano [1969] was one of the first to investigate this problem. He studied the cylindrical bending of a simply supported orthotropic infinite laminated strip under static loading. Later the exact solution of the bending problem for a finite-dimension plate under static loading was obtained [Pagano 1970a; 1970b; Little 1973; Noor and Burton 1990b]. Keywords: three-dimensional theory, elasticity theory, cylindrical shell, multilayer shells, dynamics, equations of motion. 205 206 ALEXANDER SHUPIKOV AND NATALIYA DOLGOPOLOVA Several other works concerned with the behavior of cylindrical shells under static loading can be mentioned. Thus, Ren [1987] studied the plane strain deformation of an infinitely long cylindrical shell subjected to a radial load changing harmonically along the circumference. In terms of three-dimensional elasticity theory, solutions were obtained for finite-length, cross-ply cylindrical shells, simply supported at both ends and subjected to transverse sinusoidal loading [Varadan and Bhaskar 1991], and for shell panels [Ren 1989] subjected to a transverse load that changes both in the axial and circumferential directions. Subsequent works [Bhaskar and Varadan 1993; 1994; Bhaskar and Ganapathysaran 2002; 2003] dealt with the three-dimensional analysis of cylindrical shells subjected to different kinds of loads acting both in the axial and circumferential directions. Liu [2000] presented static analyses of thick rectangular plane-view laminated plates, carried out in terms of the three-dimensional theory of elasticity using the differential quadrature element method. Attention has also been given to the study of vibrations of multilayer cylindrical shells in terms of three-dimensional elasticity theory. Noor and Rarig [1974] obtained equations of free vibrations of a simply supported laminated orthotropic circular cylinder based on linear three-dimensional elasticity theory. Kang and Leissa [2000] suggested a three-dimensional analysis method for determining the free vibrations and the form of the segment of a variable-thickness spherical shell. The displacement components in the meridional, normal, and circumferential directions were taken to be sinusoidal with respect to time, periodic in the circumferential direction, and were expanded into algebraic polynomials in the meridional and normal directions. Weingarten and Reismann [1974] gave solutions for nonaxisymmetrical nonsta- tionary vibrations of a uniform finite-length cylinder. They considered vibrations of uniform cylindrical shells within the framework of the three-dimensional theory of elasticity, and compared the results obtained with those given by other theories of shells. They showed that none of the two-dimensional theories could describe satisfactorily the wave-like character of the initial strain phase. Philippov et al. [1978] solved a similar problem, providing an analytic solution to the problem of axisymmetrical vibrations of a uniform infinite-length cylinder subjected to an impulse load. Shupikov and Ugrimov [1999] have suggested an analytic-numerical method for solving the three-dimensional problem in the elasticity theory of nonstationary vibrations of multilayer plates subjected to impulse loads. The displacements in the tangential direction are expanded into a double Fourier series, and the partial derivatives in the transverse coordinate are replaced by their finite-difference version. As a result of these transformations, the problem of nonstationary vibration of a DYNAMICRESPONSEOFMULTILAYERCYLINDERS 207 multilayer plate under the action of an impulse load is reduced to integrating a system of ordinary differential equations with constant coefficients. The objective of this work is to develop and generalize further the approach suggested by [Shupikov and Ugrimov 1999], and to investigate the vibrations of a thick multilayer closed cylindrical shell subjected to an impulse load. 2. Problem formulation We consider a multilayer cylindrical shell of finite length A and radius R0, which is composed of I uniform constant-thickness isotropic layers. The shell is referenced to the right-hand system of orthogonal curvilinear coordinates z, θ, r. The coordinate surface is linked to the outer surface of the first layer, R0 is the radius of inner surface of the shell (Figure 1). Contact between layers prevents their delamination and mutual slipping. The behavior of each layer is described by Lame’s´ equations [Novatsky 1975]: ui 2 ∂u ∂ 1 ∂ 1 ∂ui ∂ui ∂2ui µi ∇2ui − r − θ +(λi +µi ) (rui )+ θ + z −ρi r = 0, r r 2 r 2 ∂θ ∂r r ∂r r r ∂θ ∂z ∂t2 ui 2 ∂u µi ∇2ui − θ + r θ r 2 r 2 ∂θ 1 ∂ 1 ∂ 1 ∂ui ∂ui ∂2ui +(λi +µi ) (rui )+ θ + z −ρi θ = 0, r ∂θ r ∂r r r ∂θ ∂z ∂t2 ∂ 1 ∂ 1 ∂ui ∂ui ∂2ui µi ∇2ui +(λi +µi ) (rui )+ θ + z −ρi z = 0, z ∂z r ∂r r r ∂θ ∂z ∂t2 (1) q + (θ, z, t) I h 3 u h2 ξ r ξ2 3 h1 ξ1 z θ q − (θ, z, t) 1 uθ 2 3 I A I –I Figure 1. Multilayer cylindrical shell. 208 ALEXANDER SHUPIKOV AND NATALIYA DOLGOPOLOVA where ∇2 = ∂2/∂r 2 + (1/r)∂/∂r + (1/r 2)∂2/∂θ 2 + ∂2/∂z2. This system of equations is solved with the boundary conditions on the external surfaces of the first and I -th layers, namely 1 1 1 − σzr = σr = 0, σrr = −q for r = R0, θ i (2) I I I + I i X j σzr = σrθ = 0, σrr = −q for r = R0 + ξ , ξ = h ; j=1 the boundary conditions at the ends, i i i σzz = ur = uθ = 0 for z = 0, L, i = 1, I ; (3) the contact conditions at adjacent layers, i i+1 i i+1 i i+1 ur = ur , uθ = uθ , uz = uz , (4) i i+1 i i+1 i i+1 i σrr = σrr , σrθ = σrθ , σrz = σrz for r = R0 + ξ , i = 1, I − 1; and the initial conditions ∂ui (r, θ, z, 0) ui (r, θ, z, 0) = = 0 for i = 1, I . (5) ∂t Here i is the layer number, λi , µi are Lame’s´ coefficients, ρi is the specific density, i i i i and u = {ur , uθ , uz} is the displacement vector of a point in the i-th layer. The stress tensor components are calculated from i i i i i σ jk = 2µ ε jk + λ δ jk1 , (6) where δ is Kronecker’s delta, i i i i 1 = εrr + εθθ + εzz for i = 1, I , and the components of the strain tensor have the form i θ i i i i i ∂ur i 1 ∂ui i i ∂uz i 1 1 ∂ur ∂uθ uθ εrr = , εθθ = + ur , εzz = , εrθ = + − , ∂r ri ∂θ ∂z 2 ri ∂θ ∂r ri i i i i i 1 ∂uθ 1 ∂uz i 1 ∂uz ∂ur εθz = + , εrz = + , for i = 1, I .