PECULIARITIES OF THE WAVE FIELD LOCALIZATION IN THE FUNCTIONALLY GRADED LAYER

Transcription

1 Materials Pysics and Mecanics (5) 5- Received: Marc 7, 5 PECULIARITIES OF THE WAVE FIELD LOCALIZATION IN THE FUNCTIONALLY GRADED LAYER Т.I. Belyankova *, V.V. Kalincuk Soutern Scientific Center of Russian Academy of Sciences, st. Ceova, 4, Rostov-on-Don, 446, Russia * tbelen45@mail.ru Abstract. Witin te framework of te linearized teory of elasticity, as exemplified by te problem of te sear armonic oscillations of te pre-stressed functionally graded layer, te influence of te type of inomogeneity and of te caracter of te initial stressed state on te distribution of te displacements wit dept is investigated. Te initially deformed state is assumed to be omogeneous, te inomogeneity of te initial stresses is caused by te cange of te material properties. Te transformation of te displacements wit dept for different oscillation frequencies of te layer under te conditions of te equal and arbitrary intensity of te cange of properties is demonstrated. Te possibility to dampen te amplitude of displacements at te definite frequencies by means of canging te initial actions is establised.. Introduction A wide application of composite materials in geomecanics, construction industry and ig tecnology equipment production resulted in te necessity of investigating pysical, tecnological and strengt properties of new materials depending on te modes and conditions of teir operation. It stimulated te extensive experimental, fundamental and applied researc. Efficiency of composite materials and coatings is closely connected wit teir internal state, i.e. wit te effects arising in te vicinity of te materials inomogeneity. Te complexity of te dynamic problems arising from te investigation is in te fact tat it is impossible to obtain te analytical solution for te semi-bounded media wit te properties tat cange in space or for te structures coated wit te similar materials. In scientific papers it is often assumed tat all te properties of material cange according to one law wit one space variable wit te same intensity [-7]. Wen modeling te functionally graded material, it is often divided into te layered elements in wic te material properties are te linear [6] or te quadratic functions of te tickness; and easily differentiable functions and polynomials [, 5] are used as functional dependencies. As a rule, te cange of te material properties in te models is considered eiter wit respect to one «basic» material or wit respect to two materials [4, 7]. Te assumption of te equivalent cange of all te properties of material allows to obtain te analytical solution tat is important wen estimating te results of a more complex numerical and numerically- analytical modeling; but to investigate te material properties it is effective only in some special cases. In tis paper in te framework of te linearized teory of elasticity for te pre-stressed semi-bounded bodies we use te suggested in [8] and improved in [9 ] numerically analytical model of te prestressed functionally graded medium. It is based on reducing te solution of te linearized equations of motion, i.e. of te system of te partial differential equations of te second order wit te variable coefficients, to te solution of te system of te ordinary differential equations (ODE) of te first order wit te boundary and initially boundary conditions wit 5, Institute of Problems of Mecanical Engineering

2 6 Т.I. Belyankova, V.V. Kalincuk respect to te components of te displacement vector and te normal components of te stress vector. In its turn te solution of te ODE system wit te variable coefficients is obtained by Runge-Kutta s metod wit Merson s modification wic allows to control te calculation error. Using as a reference material Murnagan model takes into account te elastic constants III order, and tus more adequately describe te influence of te initial actions on te properties of te material even wit large initial deformation.. Te problem formulation We assume tat in te layer x, x, x under te initial mecanical actions te omogeneous initial deformation is induced: T R rλ, G ΛΛ, Λ vrr, v const. () i j i i j i Here Rr, are te radius-vectors of te medium point in te initially deformed and natural states respectively, v, are te relative lengtenings of te fibres directed i i i in te natural configuration along te axes ai, i,,, coinciding wit Cartesian coordinates, ij is Kronecker delta. We assume tat te layer properties cange wit dept and tat te model of te initially isotropic material wit te elastic potential in Murnagan s form is used as te basic one. In some domain on te surface layer tere is a source of armonic oscillations. Te lower bound of te layer is rigidly coupled. In te context of tese assumptions te boundaryvalue problem in Euler coordinates connected wit te initially deformed state (IDS) is formulated by te linearized motion equations [8] Θ u. () wit te boundary conditions on te surface O O O: O : N Θ= qe -iωt, () O : u=u *, (4) on te lower bound: u, x x. (5) Here is Hamilton s operator in IDS; N is te vector of te external normal to te medium surface in IDS; u, q are te vectors of displacements and stresses determined in te Euler s coordinate system; is te density of te medium material in IDS. We use a linearized tensor of stresses wic in case of te initially isotropic elastic material is of te form [8] as tensor Θ: k m Tu 4J εu FεuF VkmF F ε u. (6) k m T in (6) is te tensor of initial stresses in IDS. It is determined by te formula T J I F F. (7) J is te metric factor; F is te measure of Finger s deformation; εu is te linear

3 Peculiarities of te wave field localization in te functionally graded layer tensor of te deformation of te perturbation state; I is te unit tensor. I k I k F are te invariants of Finger s measure. Te form of coefficients k, Vkm is given in [8]. As te elastic potential, we use te elastic potential in Murnagan s form: 9 n n n ( ) I ( ) I ( ) I I I ( ) I (I ). 4 l l m m m l m 6 (8) In te context of assumptions wit respect to (7), (8) te representation of te components of tensor Θ is in te form: u s l k l k s p, xp M N l k MN l k J M N s v s s. (9) l k s p l s k p l k k s l p l l k l k s p l s sl k vl vk, s l k vl vk, J J 4 s V v v. Let te layer make armonic oscillations under te action of te sear loading i t qx e distributed in te region x a, x, te cange of te pysical parameters of te layer is determined by formulae: x f x x f x x f x,,.,,, l x l f x m x m f x n x n f x l m n In tis case in Fourier s transforms te boundary-value problem () (5) wit respect to (9), () is of te form: '' ' ' U U U () wit te boundary conditions: U Q, U. () ' x x Here, U, Q are Fourier s transforms of te components of te stress tensor, of te displacement vector, of te preset loading, is te transformation parameter. It sould be noted tat te components of tensor Θ (), (), (6), te tensor of initial stresses T (7), te function of te specific potential energy (8) are te smoot functions of te coordinate x. Te cange of te caracteristics is determined bot by te functional dependence (), and by te caracter of te applied initial stresses. According to [8 ], te formulations of te boundary-value problems as well as te solution are given in te dimensionless parameters.. Te boundary-value problem solution Te solution of te boundary-value problem on te sear oscillations of te pre-stressed functionally gradient layer is of te form []: u x, x k x, x, q d, (4) a a 7 () ()

4 8 Т.I. Belyankova, V.V. Kalincuk i s,,,,, K, x, y, x, y,, k s x K x e d ere yjk, x, conditions,, j k, (5) are te linear independent solutions of Caucy s problem wit te initial y for te equation:, x Y M Y, j k M, x. (6) 4. Numerical results Te sear armonic oscillations of te functionally gradient elastic layer wit a rigid inclusion (Fig. a) and wit a soft inclusion (Fig. b) caused by te action of te point source are considered. As a basic material we use te transversal isotropic material wit parameters [8]: ρ = kg/m, μ =.84 N/m, λ =. N/m, l =.5 N/m, m = - 6. N/m, n = 8.4 N/m. Te results of te numerical investigations are presented in dimensionless parameters. x x m= m=9 m= m= - 9 f(x ) f(x ) а) b) Fig.. Functional dependences of te cange of properties. In Figures а) d) and а) d) te transformation of te function of te distribution of te displacement amplitude u (,x ) and of te function of te relative cange of te displacement amplitude u, x u, x in IDS in dept for te dimensionless frequencies ω =.6, 4.9 respectively is given. 9-9 m= u а) b) m= -9 c) d) Fig.. Te influence of te uniaxial IDS on te displacement amplitude for te layer wit different types of inclusions. f, ω =.6.

5 9 Peculiarities of te wave field localization in te functionally graded layer In tis case all te properties of te medium cange according to one law wit equal intensity f, f l m n. Numbers in fig. а) denote te curves corresponding to te functional dependences in fig.. Numbers,, in fig. b) d) correspond to uniaxial extensions х, х, х (vi=.) u uσ а) b) m= -9 c) d) Fig.. Te influence of te uniaxial IDS on te displacement amplitude for te layer wit te different types of inclusions. f, ω = 4.9. Figure 4 а) d) illustrates te influence of te intensity of te density cange on u(,x) and for te rigid inclusion (а)) and for te soft inclusion (b)). ρ m= - 9 ρ ρ= f ρ=.5f ρ= f ρ=/f u u а) b) ρ=/f ρ=/f uσ c) d) Fig. 4. Te influence of te density cange intensity on te displacement amplitude at different dimensionless frequencies witout taking te pre-stresses into account а) - ω = 4.9, b) - ω = 7; c), d) at uniaxial IDS, ω = 4.9. It follows from te figures tat te presence of te rigid inclusion in a layer leads to te localization of displacements at te surface of te layer. Te amplitude of oscillations can increase wen it approaces te resonance frequency. In te case of te soft inclusion, te localization of displacements is not only at te surface of te layer but also in te middle part of it. Canging IDS, it is possible to dampen or to increase te amplitude of oscillations on te surface of te layer or inside it.

6 Т.I. Belyankova, V.V. Kalincuk Acknowledgements Tis work is performed wit te financial aid of te Russian Scientific Foundation (project No ). References [] I.V. Ananev, V.A.Babesko // Mecanics of Solids (978). [] G.A. Maugin // Annales de l'institut Henri Poincaré A 8() (978) 55. [] I.V. Ananev, V.V. Kalincuk, I.B. Poliakova // Journal of Applied Matematics and Mecanics 47 (98). [4] G.R. Liu, J. Tani, T. Oyosi, K. Watanabe // Transactions of te Japan Society of Mecanical Engineers 57A (99). [5] V.A. Babesko, E.V. Gluskov, Z.F. Zincenko, Dynamics of non-uniform linearlyelastic environments (Nauka Publising, Moscow, 989) (in Russian). [6] X. Cao, F. Jin, I. Jeon // NDT&E International 44 () 84. [7] A. Kojaste, M. Raimian, R.Y.S. Pak // International Journal of Solids and Structures 45 (8) 495. [8] V.V. Kalincuk, T.I. Belyankova, Dynamic contact problems for prestressed semi-infinite bodies (Nauka Publising, Moscow, ) (in Russian). [9] V.V. Kalincuk, T.I. Belyankova // Izvestiya vuzov. Severo-kavkazskii region. Natural sciences S (4) 47. (in Russian). [] V.V. Kalincuk, T.I. Belyankova, A.S. Bogomolov // Ecological Bulletin of Researc Centers of te Black Sea Economic Cooperation (6) 6. [] T.I. Belyankova, A.S. Bogomolov, V.V. Kalincuk // Ecological Bulletin of Researc Centers of te Black Sea Economic Cooperation 4 (7). [] V.V. Kalincuk, T.I. Belyankova, Dynamics of a surface of inomogeneous media (Fizmatlit Pablising, Moscow, 9) (in Russian).