Investigation of nonlinear dynamic responses of sandwich FGM cylindrical shells containing fluid resting on elastic foundations in thermal environment

The main aim of the present article is to investigate nonlinear dynamic responses of sandwich-FGM circular cylinder shell containing fluid and surrounded by Winkler- Pasternak elastic mediums under mechanical loads in the thermal environment based on classical shell theory.

Research INVESTIGATION OF NONLINEAR DYNAMIC RESPONSES OF SANDWICH-FGM CYLINDRICAL SHELLS CONTAINING FLUID RESTING ON ELASTIC FOUNDATIONS IN THERMAL ENVIRONMENT Khuc Van Phu1, Nguyen Minh Tuan2, Dao Huy Bich1, Le Xuan Doan2* Abstract: The main aim of the present article is to investigate nonlinear dynamic responses of sandwich-FGM circular cylinder shell containing fluid and surrounded by Winkler- Pasternak elastic mediums under mechanical loads in the thermal environment based on classical shell theory Bubnov-Galerkin method and fourthorder Runge-Kutta method are employed to determine nonlinear dynamic buckling of cylindrical shell Effects of temperature environment, foundations, structure's geometrical parameters, material parameters and fluid on the nonlinear dynamic responses of sandwich-FGM circular cylinder shell are investigated Keywords: Sandwich-FGM; Dynamic stability; Cylindrical shell; Thermal-mechanical load; Filled with fluid INTRODUCTION Functionally Graded Material (FGM) is an important material in modern engineering design and more and more extensively used in many industries Researches on nonlinear dynamic stability of these structure has received attentions by scientists, especially shell structure Bich D H et al studied on natural frequencies and dynamic buckling of FG cylinder panels reinforced by eccentrically stiffeners [1] and thin circular cylinder shell [2] based on classical shell theories The Runge– Kutta method and smeared stiffener technique were employed to investigate B Mirzavand et al [3] solved dynamic post-buckling problems of FG cylinder shells with piezoelectric layer on surface under electro-thermal load based on the thirdorder shear deformation shell theory and Sander’s nonlinear kinematic relations Nguyen Dinh Duc et al [4], [5] analyzed nonlinear responses of imperfect eccentrically stiffened thin and thick S-FGM cylinder shells resting on an elastic mediums and subjected to mechanical load in thermal environment Study on full-filled fluid FGM shells, Sheng et al [6] investigated vibration of FG circular cylinder shells containing flowing fluid under mechanical load and surrounded by elastic foundations including effect of thermal environment This study was continuously expanded to investigate dynamic responses of FGM circular cylinder shell containing flowing fluid subjected to mechanical and thermal loads [7] Zafar Iqbal et al [8] analyzed vibration frequencies of full-filled fluid FGM circular cylinder shell Vibration frequencies of shell were examined for various boundary conditions including the effect of fluid Silva et al [9] resolved nonlinear vibration problems of fluid-filled FG cylinder shell subjected to mechanical load By using the Rayleigh-Ritz method, vibration frequencies of FGM cylindrical shell filled with fluid or containing a flowing fluid partially surround by two parameters elastic foundation were examined by Y W Kim et al [10] Hong-Liang Dai et al [11] studied on thermos electro elastic behaviors of a thin FG piezoelectric material cylinder shell filled with fluid and under mechanical and electrical loads in thermal environment According to the classical shell theory Journal of Military Science and Technology, Special Issue, No.60A, 05 - 2019 95 Mechanics & Mechanical engineering and using Galerkin method, Phu Van Khuc et al [12] investigated non-linear responses of circular cylinder shells made of Sandwich-FGM filled with fluid subjected to mechanical load in thermal environment The review of the literature signifies that there is no research on the analytical solution for dynamic stability of full-filled fluid sandwich-FGM circular cylinder shells surrounded by elastic foundations In the present article, nonlinear dynamic equations of full-filled fluid sandwich-FGM circular cylinder shell resting on elastic mediums under mechanical load including the effect of thermal environment are established base on the classical shell theory Bubnov-Galerkin method and Runge-Kutta method are employed to determine nonlinear dynamics responses of circular cylinder shells Dynamic critical loads are defined by applying Budiansky–Roth criterion SANDWICH- FGM CYLINDRICAL SHELL Examine a cylindrical shell made of sandwich-FGM with the thickness, the length and curvature radius of shell are h, L and R, respectively Configuration and Coordinate system of sandwich-FGM cylinder shell are performed in fig In which hc, hm and hx=h-hc-hm are thickness of ceramic layer, metal layer and FGM core layer, respectively The cylindrical shell surrounded by two-parameter elastic mediums with stiffness are: K1 (Nm-3) and K2 (Nm-1) Assume that the cylindrical shell subjected to simply supported at both ends and under pre-axial compression load (N01= -ph) and external pressure which uniformly distributed varying on time q(t) in the thermal environment Suppose that environment's temperature is steadily increased and ΔT is constant Fig Configuration and Coordinate system of sandwich-FGM cylinder shell filled with fluid embedded in elastic foundations We denote that Vm(z) and Vc(z) are metal and ceramic volume fractions, respectively Suppose that volume fraction of Metal and Ceramic are constantly changed and distributed according to the exponential law Ceramic’s volumefraction Vc(z) is expressed as follows 96 K V Phu, …, L X Doan, “Investigation of nonlinear dynamic … in thermal environment.” Research Vc  z      z  0, 5h  hm  Vc  z     h  hc  hm   Vc  z    ,  0, 5h  z    0, 5h  hm  k   ,   0, 5h  hm   z   0, 5h  hc  , k   ,  0, 5h  hc   z  0, 5h (1) For this rule, properties of material Q(z) such as thermal expansion coefficient α, the Young's modulus E, and the mass density ρ, change through the thickness of shell and can be obtained as follows Q  z   QmVm  z   QcVc  z   Qm   Qc  Qm  Vc  z  , (2) The Poisson’s ratio ν is assumed to be constant GOVERNING EQUATIONS According to the classical shell theory, with von Karman-Donnell sense type of geometrical non-linearity, the nonlinear relation of strains and displacement for a circular cylinder shell can be expressed as [17] x  x0  zx ;  y   y0  zy ;  x0  u  w  v w  w  u v w w   ;  ;  xy     ; y     x  x  y R  y  y x x y x    2w ; x (3) 2 Where  xy   xy0  2zxy y    2w ; y  xy    2w ; xy (4) (5) In which: u, v and w are displacements in the x, y and z direction From Eqs (4), the equation of deformation compatibility can be obtained as   x0 y    y0 x 2   xy0  2 w  2 w 2 w 2 w    ;   x y  x y  R x x y (6) Assume that material properties are independent on temperature and temperature in the cylindrical shell is only transmitted in the z-axis direction, Hooke’s law for sandwich-FGM cylindrical shell subjected to thermo-mechanical loads can be written as E z E  z   z  E  z E  z   z  E  z x  ( x   y )  T ; y  ( x   y )  T ; xy   xy (7) 2     1    Where: σx; σy and τxy are stress components in circular cylinder shell Internal forces and moment resultants expressions of circular cylinder shell can be defined as follows N x  A11 x0  A12  y0  B11  x  B12  y  1 M x  B11 x0  B12  y0  D11  x  D12  y   N y  A12  x0  A22  y0  B12  x  B22  y  1 ; M y  B12  x0  B22  y0  D12  x  D22  y   N xy  A66  xy0  B66  xy In which: N x ; N y ; N xy (8) M xy  B66  xy0  D66  xy are internal forces and M x ; M y ; M xy are moment resultants The additional internal forces and moments resultants which are created by the increase in temperature 1, and stiffness coefficients Aij, Bij and Dij (i, j = 1, 2, 6) in Eq (8) are expressed in Appendix I Journal of Military Science and Technology, Special Issue, No.60A, 05 - 2019 97 Mechanics & Mechanical engineering From equation (8) we define the deformation expression and moment components of circular cylinder shell as follows * *  x0  A22 Nx  A12* N y  B11*  x  B12*  y   A22  A12*  1, * *  y0   A12* Nx  A11* N y  B21 x  B22  y   A11*  A12*  1, * *  xy0  A66 Nxy  2B66 xy (9) * * * * * M x  B11 N x  B21 N y  D11  x  D12*  y  B11  A22  A12  B12  A11*  A12*  1  , * * * * * * M y  B12 N x  B22 N y  D21  x  D22  y  B12  A22  A12  B22  A11*  A12*  1  , * * M xy  B66 N xy 2 D66  xy , In which: Aij* , Bij* and D ij* (10) (i, j=1, 2, 6) are explained in Appendix II Based on the classical shell theory [17], motion equations of circular cylinder shell filled with fluid resting on an elastic mediums subjected to pre-axial compression load p and an external pressure varying on time q(t) can be expressed as Nx Nxy   1  u2 , x y t N y Nxy   1  v2 , y x t 2 2 M xy N  2 w  2 M x 2 M y    Nx  w  Ny  w  y  Nxy  w  K1w  K2   w  q 2 2 x y xy x y R xy  x y  (11)  1  w  21 w  PL , t t Where:  - the coefficient of damping    m h   cm hc   cm h x (12) k 1 PL - dynamic pressure of fluid exerting on the inner surface of shell and be determined by expression [12]: pL  M L 2w t in which: M L   L R.I n ( m ) m I n' ( m ) and ( m  m R ) (13) L ML- mass of fluid corresponding to shell’s vibration Substituting equation (13) into equations (11) then applying Volmir’s assumption [18] (because of u, v