4. Tomtat_La_Vu_Van_Tham (Tieng Anh).Pdf

Chuyên đề 2 Dao động tự do và uốn tấm composite lớp MINISTRY OF EDUCATION AND TRAINING NATIONAL UNIVERSITY OF CIVIL ENGINEERING VU VAN THAM STATIC AND VIBRATION ANALYSIS OF PIEZOELECTRIC FUNCTIONALLY[.]

MINISTRY OF EDUCATION AND TRAINING NATIONAL UNIVERSITY OF CIVIL ENGINEERING VU VAN THAM STATIC AND VIBRATION ANALYSIS OF PIEZOELECTRIC FUNCTIONALLY GRADED CARBON NANOTUBE-REINFORCED COMPOSITE PLATE AND DOUBLY CURVED SHALLOW SHELL MAJOR: Engineering Mechanics CODE: 9520101 SUMMARY OF DOCTORAL DISSERTATION Ha Noi - 2021 The work was completed at: NATIONAL UNIVERSITY OF CIVIL ENGINEERING (NUCE) Academic supervisor: Assoc Prof Dr Tran Huu Quoc – NUCE Prof Dr Tran Minh Tu – NUCE Peer reviewer 1: Prof Dr Nguyen Dong Anh Peer reviewer 2: Prof Dr Dao Huy Bich Peer reviewer 3: Prof Dr Nguyen Thai Chung The doctoral dissertation will be defended at the level of the University Council of Dissertation Assessment's meeting at the National University of Civil Engineering at hour .', day month year 2021 The dissertation is available for reference at the libraries as follows: - National Library of Vietnam; - Library of National University of Civil Engineering; INTRODUCTION The necessity of the topic Functionally graded carbon nanotube-reinforced composite (FG-CNTRC) materials provide new advantages for composite materials In FG-CNTRC, CNTs are designed to grade with specific rules along with desired directions within an isotropic matrix to enhance the mechanical properties of the structures The addition of CNTs improves the mechanical, electrical, and thermal properties of the structures In addition, the integration of laminated composites with piezoelectric materials provides structures with superior mechanical properties of composite materials and the capability to sense and adapt their static and dynamic responses However, there have been a limited number of studies regarding the static and vibration of FG-CNTRC structures with integrated piezoelectric layers To enrich the studies on mechanical behaviors of FG structures in both sides: computational models and methods, the subject of the dissertation is chosen as "Static and vibration analysis of piezoelectric functionally graded carbon nanotube-reinforced composite plate and doubly curved shallow shell" Aims and content of the research  Improvement and development of the four-variable shear deformation refined theory for both plate and shell  Using the four-variable shear deformation refined theory to develop a finite element model and analytical solution for static and vibration analysis of piezoelectric functionally graded carbon nanotube-reinforced composite plate and doubly curved shallow shell  Build a Matlab's code to investigate the effect of material parameters, geometric dimensions, and boundary conditions on deflection, stress field, and dynamic behavior of FG-CNTRC plate and doubly-curved shells Object and scope of the research  Object: FG-CNTRC plate and doubly-curved shells  Scope: Linear analysis – predicting deflection, stress components, frequencies, and displacement responses of FG-CNTRC doubly-curved shell with various boundary conditions Scientific basis of the thesis  Inheritance and development of related studies  Verification to prove the accuracy, reliability of the model and the solution Research methodology  Analytical method: Using the four-variable shear deformation refined theory to develop the governing equations, algorithm, and Matlab's codes for analyzing the static and vibration response of piezoelectric FG-CNTRC plates and doubly-curved shells with simply supported boundary conditions  Finite element method: Develop an algorithm, finite element model and Matlab's codes to analyze the static and dynamic response of piezoelectric FGCNTRC plates and doubly-curved shells with various boundary conditions Significant contributions of the dissertation  Refine and develop the four-variable shear deformation refined theory for both piezoelectric FG-CNTRC plates and doubly-curved shells  Using the four-variable shear deformation refined theory to build the governing equations and analytical solution for static and vibration analysis of piezoelectric functionally graded carbon nanotube-reinforced composite plate and doubly curved shallow shell  Develop an algorithm, finite element model for static and vibration analysis of piezoelectric functionally graded carbon nanotube-reinforced composite plate and doubly curved shallow shell based on four-variable shear deformation refined theory and C1 four-node rectangular element  Build a Matlab's code to investigate the effect of material parameters, geometric dimensions, and boundary conditions on deflection, stress field, and dynamic behavior of FG-CNTRC plate and doubly-curved shells  Some useful comments and conclusions are drawn Outline of the dissertation The outline of dissertation including Introduction, Four chapters, Conlusions, Research proposal, references and Appendix CHAPTER OVERVIEW OF THE RESEARCH TOPIC This chapter gives a brief introduction about the functionally graded carbon nanotube-reinforced material - the mechanical properties of materials and piezoelectric material; the piezoelectric FG-CNTRC structures and applications; Overview of national and international researches on static and dynamic analysis of PFG-CNTRC plates and shells This chapter also reviews the plates and shell theory The review study shows that the classical plate theory (CPT) only provides accurate results for thin plates but becomes not suitable for moderately thick or thick plates and shells; the firstorder shear deformation theory (FSDT) is not convenient to use because it is not evident to determinate the correct value of shear correction factor, the higher-order shear deformation plate theory is complicated A class of the four-variable refined plate theories was introduced to overcome the limitations of CPT, FSDT, and HSDT There is no work available on the static analysis of laminated FG-CNT reinforced composite plates integrated piezoelectric layers based on the fourvariable refined plate theory From the overview study, the scope of the dissertation is determined: static and vibration analysis of piezoelectric functionally graded carbon nanotubereinforced composite plate and doubly curved shallow shell CHAPTER STATIC AND VIBRATION ANALYSIS OF PIEZOELECTRIC FUNCTIONALLY GRADED CARBON NANOTUBE-REINFORCED COMPOSITE PLATE AND DOUBLY CURVED SHALLOW SHELL – ANALYTICAL SOLUTION 2.1 Introduction In this chapter, the four-variable shear deformation theory is refined and applied for static and vibration analysis of piezoelectric functionally graded carbon nanotube-reinforced composite plates and doubly curved shallow shells using an analytical solution Hamilton's principle and Navier's technique are used for establishing and solving the governing equations of simply supported rectangular plates and doubly curved shallow shells 2.2 Analytical solution for PFG-CNTRC plates and shells 2.2.1 Piezoelectric FG-CNTRC doubly curved panel Consider a laminated FG-CNTRC doubly curved panel (FG-CNTRC-DCP) with two perfectly bonded piezoelectric layers at the top and bottom surfaces, as shown in Figure 2.1 Figure 2.1 Schematic of laminated PFG-CNTRC doubly curved shell panel The panel with constant principal curvatures is referred to as an orthogonal curvilinear coordinate system (x, y, z) The length, width, and two radii of principal curvatures of the middle surface of the panel are denoted by a, b, Rx and R y , respectively The thickness of the core and each piezoelectric layer are also denoted by hc and hp, respectively 2.2.2 Piezoelectric FG-CNTRC materials In this dissertation, for each layer, five types of distributions of CNT are considered UD represents the uniform distributions, and the other four types of functionally graded distributions of CNT are denoted by FG-A, FG-V, FG-X, and FG-O According to the distributions of CNTs, the CNT volume fractions VCNT  z  for each FG-CNTRC layer are given as [83]: * UD : VCNT ( z )  VCNT z  zk (2.1) * FG-V : VCNT ( z )  2VCNT zk 1  zk FG- * : VCNT ( z )  2VCNT zk 1  z zk 1  zk z  zk  zk 1  *  : VCNT ( z )  2VCNT 1   zk 1  zk    z  zk  zk 1  * FG-X : VCNT ( z )  2VCNT    zk 1  zk  wCNT *  where VCNT wCNT   CNT  m    CNT  m  wCNT FG-O in which, wCNT is the mass fraction of the CNT,  m and CNT are mass densities of the matrix and CNT, respectively 2.3 Theoretical formulation 2.3.1 Assumptions - The transverse normal strain  z is negligible - The shell is considered only to be subjected to forces perpendicular to the surface of the shell - The stress-strain relationship is linear according to Hooke's law - The transverse displacement w includes two components of bending wb and shear ws - The in-plane displacements u and v consist of extension, bending, and shear components - The shear stresses xz, yz are satisfied to be zeros at the top and bottom faces of the shells 2.3.2 Approximation on the mechanical displacement The following equations give the displacements field for doubly curved shell based on the four-variable shear deformation refined theory:  z  w ( x, y, t ) w ( x, y, t ) u ( x, y , z , t )     u0 ( x, y , t )  z b  f z s x x  Rx   (2.2) z  w ( x, y, t ) w ( x, y, t ) v ( x, y , z , t )    v0 ( x, y, t )  z b  f z s   R  y y y   w( x, y, z , t )  wb ( x, y, t )  ws ( x, y, t ) where u0 and v0 are the in-plane displacements in the directions of x and y; wb and ws represent the bending and shear components of the transverse displacement, respectively 2.3.3 The strain fields The strains at any point in the shell space associated with the displacement field in Eq (2.2):  xx   yy   y0  z yb  f  z   ys  ;  z Ry  xy   yx   x0  z xb  f  z  xs  ;  z Rx  xy0  z xyb  f  z  xys  ;  z Rx (2.3) b  yx0  z yx  f  z   yxs  ;  z Ry  xz   w s  w s   1 f '  z ;  yz   f '  z       z Ry  y   z Rx  x    where:  u0 wb ws  b  u0  wb  s  ws     ; kx    ; kx   ;   x R R R  x  x x x x    x   v w w   v0  wb  s 2w  y0    b  s  ;  yb     ; k y   2s ;  y R R   R y y  y y y    y  v0  wb  s v  ws u0  xy0  ;  xyb    ;    ;  yx  ;  R x xy  xy x  x  y  y  y   u0  wb  s  ws b  yx    ;  xz  g  z   xsz ;  ;  yx   xy  z Rx  Rx y xy  w w  xzs  s ; g  z   1  f '  z   ;  yz  g  z   yzs ;  yzs  s x  z Ry y x (2.4) The polynomial shape function is used as:   z 2  f  z   z      (2.5)   h   2.3.4 Approximation of the electric potential - The electric potential distribution in the transverse direction of each piezoelectric layer is approximated by the linear function as follows: t  t  (x, y,z, t)   z  h /  h  (x, y, t) p    b (x, y,z, t)    z  h /   b (x, y, t)  hp h/2  z  h/2 + h p (2.6)  h/2  h p  z  h/2 - The electric field is related to electric potential by the following relation:  t  t  t  t t t  Ex    z R x ; E y    z R y ; Ez   z x y  (2.7)  b b b    Eb   ; E yb   ; Ezb    x  z Rx x  z Ry y z  2.3.5 Constitutive equations - The stress-strain relation for the CNT layer is expressed as:  xkx   Q11k ( z ) Q12k ( z ) 0 Q16k ( z ) Q16k ( z )   xx   k   k   k 0 Q26k ( z ) Q26k ( z )   yy   yy  Q12 ( z ) Q22 ( z )   yzk   0 Q44k ( z ) Q45k ( z ) 0   yz  (2.8)    k  k k   0 Q ( z ) Q ( z ) 0 xz xz 45 55       k k   xyk  Q16k ( z ) Q26k ( z ) 0 Q66 ( z ) Q66 ( z )  xy     k   k k k k  Q ( z ) Q ( z ) 0 Q ( z ) Q ( z )    yx  26 66 66  yx   16 - For the piezoelectric layers, the stress components and electric displacement relations are given as:  xpkx   C11k C12k  pk   k k  yy  C12 C11  yzpk   0 C55k   pk   0  xz   pk  xy   0  pk   0  yx   0 0 0 C55k 0 k C1122 k C1122   xx        yy   0   yz        xz  e15k k   xy   C1122    k C1122   yx   0 k e24 0 e31k   e31k   Ex     Ey     Ez  0   (2.9) and  Dxk   0  k  k  Dy    0 e24  D k  e k e k  z   31 31 e15k 0  xx    yy 0     p11k   yz   0      xz 0      xy     yx  p11k 0   Ex     Ey  p33k   Ez  (2.10) 2.3.6 Governing equation Hamilton's principle is applied herein to obtain the equations of motion of laminated PFG-CNTRC doubly curved shell panels: u N xx N yx Qxb w b  w s    I 0u0  I1     J1 x y Rx x  x  Rx (2.11)  v0 w b  N yy N yx Qyb w s    I 0v0  I1     J1 R y x Ry  y y  y  N xx N yy Qxb Qyb     pz   I  w b  w s  Rx Ry x y N xx N yy Qxs Qys     pz   I  w b  w s  Rx Ry x y t Dxt Dy   Dzt  qt  0; x y b Dxb Dy   Dzb  qb  x y where  N x , N y , N xy , N yx  ,  M xb , M yb , M xyb , M yxb  ,  M xs , M ys , M xys , M yxs  , and  Qxs , Qys  are stress resultants 2.3.7 Boundary conditions - Simply supported with cross-ply lamina (SS-1): v0  w b  w s  w b , y  w s , y  0;  b s t b (x=0, x=a)  N xx  M xx  M xx       u0  w b  w s  w b , x  w s , x  0; N  M b  M s  t  b  (y=0, y=b) yy yy  yy - Simply supported with angle-ply lamina (SS-2): u0  w b  w s  w b , y  w s , y  0;  b b s t b  N xy  M xy  M xx  M xx      (x=0, x=a)  v0  w b  w s  w b , x  w s , x  0;  N  M b  M b  M s   t   b  (y=0, y=b) yx yy yy  yx (2.12) (2.13) 2.3.8 Solution procedure Following the Navier solution procedure, the expansion displacements of the u0 , v0 , ws , wb , t and  b are chosen to satisfy the boundary conditions: - Simply supported with cross-ply lamina (SS-1):   u0 ( x, y, t )  U mncos m x sin  n y; m 1 n 1   v0 ( x, y, t )  Vmn sin  m xcos n y; m 1 n 1   w b ( x, y, t )  Wbmn sin  m x sin  n y; m 1 n 1   w s ( x, y, t )  Wsmn sin  m x sin  n y; m 1 n 1 (2.14)    t ( x, y, t )    t mn sin  m x sin  n y; m 1 n 1    b ( x, y, t )    b mn sin  m x sin  n y; m 1 n 1 m n ,  ; U mn , Vmn , Wbmn ,Wsmn ,  t mn ,  b mn are unknowns; m, n a b denote the number of haft-waves in the x and y directions, respectively - Simply supported with angle-ply lamina (SS-2): where:     u0 ( x, y, t )  U mnsin m xcos n y; m 1 n 1   v0 ( x, y, t )  Vmncos m xsin n y; m 1 n 1   w b ( x, y, t )  Wbmn sin  m x sin  n y; m 1 n 1  (2.15)  w s ( x, y, t )  Wsmn sin  m x sin  n y; m 1 n 1    t ( x, y, t )    t mn sin  m x sin  n y; m 1 n 1    b ( x, y, t )    b mn sin  m x sin  n y; m 1 n 1 - Uniform distributed load qz ( x, y ) are also expanded as:   p ( x, y, t )   pmn sin  m x sin  n y; m 1 n 1  (2.16)  q ( x, y, t )   qmn sin  m x sin  n y m 1 n 1 Substituting the expansion displacement functions into governing equations, one obtained:  m11 m  12  m13   m14 0  0 m12 m13 m14 m22 m23 m24 m23 m33 m34 m24 m34 m44 0 0 0 0  U mn   k11   0  Vmn   k12 0  Wbmn   k13    0  Wsmn   k14 0   tmn   k15    0   bmn   k16 k12 k13 k14 k15 k22 k23 k24 k25 k23 k33 k34 k35 k24 k34 k44 k45 k25 k35 k45 k55 k26 k36 k46 k56 k16  U mn  0  k26  Vmn  0  k36  Wbmn   pmn     k46  Wsmn   pmn  t  k56   tmn  qmn  b   b  k66   mn  qmn  (2.17) The Eq (2.17) can be rewritten in the short form as:  M uu 0 d   Kuu Ku  d   Fu        0     K      u K     F   The equation for static analysis can be obtained: (2.18) 10 3.2 Shell and element  a  b The shell with the surface equation z   x   y  Rx   Ry  2 is discretized into m×n rectangular C1, as shown in figure 3.1 Figure 3.1 The meshing of PFG-CNTRC shell 3.3 Finite element formulation 3.3.1 Interpolation functions In this study, a rectangular nonconforming bending element was used The generalized displacements u0 and v0 are C interpolated over an element Two components of the transverse displacement wb and ws are C interpolated by the following expression 3.3.2 Displacement Fields In local coordinate, the flat element can be considered as a plate Rxl=Ryl=) Thus, the displacement HSDST-4 is expressed: u  xl , yl , zl , t   u0 ( xl , yl , t )  z wb ( xl , yl , t ) v  xl , yl , zl , t   v0 ( xl , yl , t )  z wb ( xl , yl , t ) xl yl w  xl , yl , zl , t   wb ( xl , yl , t )  ws ( xl , yl , t ) Eq (3.1) can be written in matrix form as: d    H d   f z ws ( xl , yl , t )  f z ws ( xl , yl , t ) xl yl ; ; (3.1) (3.2) where d   u v w T (3.3) 11 1 0  z 0  f ( z )   f ( z)  H   0 0  z 0  0 0  d   u v0 wb wb, x wb, y ws ws , x ws , y  T 3.3.3 Strain fields ˆ111   Bu 1132 de 321 where  Bu    Bu1 Bu Bu Bu  is the deformation matrix 3.3.4 Approximation of electric potentials The electric potential at a point can be calculated as: 4 i 1 i 1 (3.4)  t   Ni  , i t ;  b   Ni  , i b ; (3.5) The electric potential is also interpolated by C interpolation: E61   B 68 e81 (3.6) where  B    B1 B B B  is the matrix for calculating the electric field 3.3.5 Governing equations of motion The dynamic equations of the hybrid panel can be derived using Hamilton's variational principle: t2   T e  U e  W e dt  (3.7) t1 where T , U , and W are the kinetic energy, strain energy, and work done by the applied forces, respectively a) The strain energy: T T T U           dV   E DdVp (3.8)     V Vp b) The kinetic energy: T T     d e  d e dV  V  c) The work done by the external forces T T T W   d e   p   e  q t    e  q b dA   A d) The global equations of motion:  M d      K uu    K u   K   Ku  d   Fu    K u   K  1 (3.9) (3.10) 1 F  q (3.11) 3.4 Comments In this chapter, a finite element based on four-variable shear deformation refined theory is established for static and dynamic analysis of piezoelectric FGCNTRC shell Some major results are included as: 12 A finite model using a 4-node element with Lagrange and Hermite interpolation based on four-variable shear deformation refined theory is established The finite model considers the effects of mechanical and electrical loadings The finite model can calculate natural frequencies of PFG-CNTRC shell in cases of open-circuit and closed-circuit The Matlab codes are written based on the established finite model CHAPTER EFFECTS OF GEOMETRIC AND MATERIAL PARAMETERS ON STATIC AND DYNAMIC RESPONSE OF PFG-CNTRC PLATES AND SHELLS 4.1 Introduction In this chapter, numerical results are conducted to investigate the effect of geometric and material parameters on the static and dynamic response of PFG-CNTRC plates and shells 4.2 Validation examples Some comparison studies were performed to prove the convergence and accuracy of the analytical solution and finite-element model 4.3 Bending of PFG-CNTRC plates and shells Considered the PFG-CNTRC shallow shell [p/0/90/0/p], two PFRC layers are bonded at the top and bottom surface of the shell, Rx, Ry are radius in x and ydirection The shells are subjected to mechanical and electrical sinusoidal loads Thickness of each FG-CNT layer is hCNT = 1mm, each piezoelectric layer is hp = 0,25 mm Material properties òFG-CNT and PFRC are: - Material properties of FG-CNTRC [28, 116]: Polyme matrix: Em = 2.5 GPa, m = 0.34 Carbon nano-tube: E11CNT = 5.65 TPa, E22CNT = 7.08 TPa, G12CNT = 1.95 TPa 1 = 0.149, 2 = 0.934 and 3 = 2 for the case V*CNT = 11%; 1 = 0.150, 2 = 0.941 and 3 = 2 for the case V*CNT = 14%; 1 = 0,149, 2=1,381, 3=2 for V*CNT = 17% - Piezoelectric material PFRC [53]: C11 = 32,6 GPa; C12 = C21 = 4,3 GPa; C13 = C31 = 4,76 GPa; C22 = C33 = 7,2 GPa; C23 = 3,85 GPa; C44 = 1,05 GPa; C55 = C66 = 1,29 GPa; e31 = -6,76 C/m2; p11 = p22 = 0,037e-9 C/Vm; p33 = 0,037e-9 C/Vm Some dimensionless parameters are used in this investigation: y x  a b  100 E2 a b h a b h w  , ,0  = w;  , ,  = ;  , ,  = ; xx yy     2 q0 S h 2   2  q0 S  2  q0 S   h h   a  b h  xy  0,0,   = xy ;  yz  ,0,   = yz ;  xz  0, ,   = xz  q0 S  q0 S  2   q0 S (4.1) Fig 4.1 and 4.2 show the effect of the different input voltage on the normalized displacement w of the FG-CNTRC shell It is observed that when the PFRC layer is subjected to a larger applied voltage, the deflection of the plate also becomes larger Under the vertically downward mechanical load, the PFRC layer causes the reversal of the transverse displacements of the plate if the polarity of the voltage changes It can be concluded that the activated PFRC layer has a significant effect 13 on the bending behavior of the FG-CNTRC plates Figs also depict the variation of transverse displacement w concerning aspect ratio a/b of the FG-CNTRC shell with and without applied electric voltages at the top of the PFRC actuator surface The figure shows that as the aspect ratio a/b increases from to 2, the transverse displacement w increases to reach the biggest values and then decreases when aspect ratio a/b becomes higher 4.3.1 Effect of electrical load on the displacement Figure 4.1 Effect of applied voltage Vt on Figure 4.2 Effect of applied voltage Vt, Vb on the w of the FG-CNTRC spherical shell w of the FG-CNTRC spherical shell integrated with the PFRC layer integrated with the PFRC layer 4.3.2 Effect of applied voltage on stresses Four types of CNT distribution are considered: UD, FG-V, FG-X, FG-O Other parameters: SSSS, a/h=20; R/a=5; FG-X, V*CNT=17% Figure 4.3 Distribution of normal stress  xx across the thickness of the FG-CNTRC with and without applied voltage to the PFRC layer 14 Figure 4.4 Distribution of normal stress  yy across the thickness of the FGCNTRC with and without applied voltage to the PFRC layer Figure 4.5 Distribution of shear stress  xy across the thickness of the FGCNTRC with and without applied voltage to the PFRC layer 15 Figure 4.6 Distribution of shear stress  xz across the thickness of the FGCNTRC with and without applied voltage to the PFRC layer Figure 4.7 Distribution of shear stress  yz across the thickness of the FGCNTRC with and without applied voltage to the PFRC layer 16 It can be seen that the stresses vary linearly with the thickness coordinate of each layer in UD CNTRC plates and they vary nonlinearly with the thickness coordinate of each layer in FG-X, FG-O and FG-V CNTRC plates For FG-V CNTRC plates, the axial stress is close to zero at the bottom, where the concentration of CNTs and the same case exists at the top and bottom of for FG-O CNTRC plates In all of the numerical examples, the stress reversal also takes place due to the change in polarity of the voltage applied to the PFRC layer 4.3.3 Effect of boundary conditions Considered an PFG-CNTRC spherical shell with FG-X distribution, [p/45/45/-45/45/-45/p], a=b, a/h=20; R/a=10; V*CNT=17%, qmax=-40 N/m2 subjected to electrical sinusoidal voltage Vt = -100 (Volt) The shell under different boundary conditions: "C" denotes clamped; "S" denotes simply supported; "F" means free Figure 4.8 shows the effect of boundary conditions on the displacement of the shell As expected, the shells with the clamped condition have the smallest displacement, and the shell with free edges have the largest displacements Figure 4.8 Effect of boundary condition on w(a 2, b 2) of PFG-CNTRC shells 4.4 Free vibration of PFG-CNTRC plates and shells Considered the PFG-CNTRC shallow shell [p/0/90/0/p], two PFRC layers are bonded at the top and bottom surface of the shell, Rx, Ry are radius in x and ydirection The shell projection is a × b = 0.4 × 0.4 (m) Material properties of FGCNT and PFRC are: - Material properties of FG-CNTRC [65] : Polyme matrix: Em = (3.52 – 0.0034T) GPa, m = 0.34 m =1.15 g/cm3 at 300°K Carbon nano-tube: E11CNT=5.64 TPa, E22CNT = 7.0800 TPa, G12CNT=1.9455 TPa, 12CNT=0.175 CNT=1.4 g/cm3 And three parameters: 1=0.137, 2=1,022 and3=0.72 for V*CNT=12%; 1=0.142, 2=1.626 and 3 =0,72 for V*CNT=17%; 1=0,141, 2=1,585, 3=0,72 for V*CNT=28% Shear module: G13=G12; G23=1,2G12 - Piezoelectric material properties (PZT-5A) [36]: E = 63 GPa, G=23,3 GPa, =0,35, =7750 kg/m3, e31=e32= 7,209 C/m2, e33=15,12 C/m2, e15=e24=12,322 C/m2, p11=p22=1,53×10-8 F/m p33=1,5×10-8 F/m 17 4.4.1 Effect of CNT distribution type Figure 4.9 Effect of CNT distribution Figure 4.10 Effect of V*CNT đến tần số type on frequency  (Hz) PFG-CNTRC  (Hz) on frequency  (Hz) PFGspherical panels for different  CNTRC spherical panels for different  It is observed from these Figs 4.9 that the distribution types of CNT have a significant effect on the stiffness of the panel In detail, FG-O panels have the lowest value of frequencies, while the FG-X panels have the highest ones Fig 4.10 indicates that the fundamental frequencies of the PFG-CNTRC doubly curved shell panels strongly increase with the increase of CNT volume fractions 4.4.2 Effect of electrical boundary conditions Figure 4.11 Effect of electrical boundary condition on frequency  (Hz) of PFG-CNTRC shells Figure 4.12 Effect of electrical boundary condition on frequency  (Hz) of PFG-CNTRC shells for different Rx/Ry The natural frequency of laminated cross-ply FG-CNTRC doubly curved shell panels couped with close and open piezoelectric layers are shown in Fig 4.11 and 4.12 The parameters of the panels in this example are set by a/h=20; a/b=1; Rx/a=5; FG-X; V*CNT=28% Fig 4.12 shows that the FG-CNTRC panels couped with the open-circuit always vibrate with the higher value of frequencies in comparison with the FG-CNTRC panels couped with the closed-circuit because the open-circuit converts electric potential to mechanical energy while the closedcircuit does not 18 4.4.3 Effect of curvature of the shell panels Figure 4.13 Effect of curvature on  Figure 4.14 Effect of curvature on  (Hz) of PFG-CNTRC cylindrical (Hz) of PFG-CNTRC spherical shell shell panels panels Figures from Fg 4.13 to 4.15 show the effect of curvature on the natural frequencies of the SPH, CYL, and HPR, respectively It can be seen from these figures that with the increase of Rx a ratio, the frequencies of SPH and CYL decrease while the frequencies of HPR increase The frequencies of all three types of shell panels are approximately equal to those of the plate with corresponding input parameters when the value of Rx a ratio reaches 20 This observation once again confirms that the opposite curvature will reduce the stiffness of the shells Figure 4.15 Effect of curvature on  Figure 4.16 Effect of number of layers (Hz) of PFG-CNTRC hyperbolic on  (Hz) of PFG-CNTRC spherical paraboloid shell panels shell panels 4.4.4 Effect of number of layers The influence of the number of layers (n is a couple of layers (0/90)) on natural frequencies of the shell are depicted in Figures 4.16 Here, the geometrical *  0.17 , FG-X, dimensions of the panels are taken as a/b = 1, a/h = 20, and VCNT (0/90)n, open-circuit As the Figures show, with a fixed value of total thickness, the non-dimensional frequencies and the percentage change of frequency of laminated FG-CNTRC panels are strongly affected by the number of layers, changing from one layer to two layers However, these two dimensionless parameters vary very 19 slightly for the number of layers greater than three This is compatible with the investigations of Reddy [72], for conventional fiber-reinforced composites 4.4.5 Effect of piezoelectric layer thickness In order to investigate the effect of piezoelectric layer thickness on the free vibration response of the composite shell panel, the variation of percentage difference in natural frequency  is defined as:   Closecircuit  (%)  Opencircuit 100% (4.1) Closecircuit Figure 4.17 Effect of piezoelectric layer Figure 4.18 Effect of piezoelectric thickness on  of PFG-CNTRC layer thickness on  of PFG-CNTRC spherical shell for hp / h spherical shell for V*CNT Figure 4.17 shows the effect of piezoelectric layer thickness on  of PFGCNTRC spherical shell [p/0/90/0/p] for hp / h The figure indicates that the frequencies of the shell in case of open-circuit and closed-circuit are more different when the piezoelectric layer thickness increase Figure 4.18 shows that the piezoelectric layers have more effect on the shells which have lower stiffness 4.4.6 Effect of boundary conditions Figure 4.19 Effect of boundary on  Figure 4.20 Effect of boundary on  (Hz) of PFG-CNTRC shell (Hz) of PFG-CNTRC shell [p/-45/45/[p/0/90/0/p] 45/p] Consider a PFG-CNTRC shell panel with lamina configuration [p/0/90/0/p] and [p/-45/45/-45/p] The shell parameters are: a=b; a/h=20; Rx=10a; V*CNT=28% Figures 4.19÷ 4.20 show the effect of boundary conditions on the frequencies of the 20 PFG-CNTRC shell panel As expected, the natural frequencies of the shell panel in the case of CCCC are higher than those of the other considered boundary conditions, while the cantilever panel has the lowest frequencies It can be concluded that more constraints always lead to more stiffness 4.5 Dynamic response of PFG-CNTRC shells In this section, active vibration control results were obtained to illustrate the effectiveness of the type of load and feedback control gains on the dynamic behavior of the PFG-CNTRC spherical shell panels 4.5.1 Dynamic vibration control of PFG-CNTRC spherical shell panels A fully clamped (CCCC) PFG-CNTRC spherical panel, with lamination sequence [a/0/90/0/s], where a and s represent the piezoelectric actuator and sensor layers made of PZT-5A, bonded on the upper and lower surfaces, respectively, is considered The substrate is made of material The side dimension is a = 0.4 m, the thickness of the substrate h = 0.04 m, the thickness of each piezoelectric layer hp  0.004 m , and radius R = m The panel is subjected to sinusoidally distributed transverse loads expressed as: x   y  q  q0 sin  (4.2)  sin   F t   a   b  where F (t ) is defined as:  1  t  t1 Step load  t>t1  0  1 t / t Triangular load  t  t1   F t    (4.3)  0 t>t1   t  t  t1 Explosive blast load  e   t>t1  in which q0  104 N/m ,   330 s-1 , t1  0.002 s , and F (t ) is plotted as shown in Fig 4.21 The transient response of the shell panel is solved by the Newmark-β direct integration method [1], and the parameters  N and  N are taken to be 0.5 and 0.25, respectively All the calculations for transient response were performed using a time step of 0.0005 s, and the initial modal damping ratio was assumed to be 0.8% [2] The effects of the velocity feedback gain, CNT volume fraction, and CNT distribution type on the transient response of the center point A (a/2, a/2) of the hybrid panel were investigated First, the active vibration control effect of the velocity feedback controller for the PFG-CNTRC spherical panel was studied The transient responses of the panel with and without the velocity feedback gain are displayed in Figs 4.22–2.24 The figures show that, in the case of a velocity feedback gain of zero, the transient 21 responses of the panel decrease with time because of structural damping These figures also indicate that increasing the velocity feedback gain causes the active damping to become stronger, resulting in a smaller amplitude of the centre point deflection and faster suspension vibration of the hybrid panel Moreover, the panel is in the free vibration state after the load is removed Figure 4.21 Types of load: step load, Figure 4.22 Central deflection of the triangular load, and explosive load PFG-CNTRC spherical panel subjected to step load Figure 4.23 Central deflection of Figure 4.24 Central deflection of the PFG-CNTRC spherical panel the PFG-CNTRC spherical panel subjected to triangular load subjected to explosive blast load * 4.5.2 Effects of the volume fraction V CNT a) b) c) Figure 4.25 Central deflection of the PFG-CNTRC spherical panel subjected to a) to step load; b) triangular load; c) explosive blast load 22 The effects of the volume fraction of CNTs and the distribution type of CNTs on the transient response of the PFG-CNTRC spherical panel were investigated The effect of the CNT volume fraction is indicated for the uniform distribution (UD) of CNTs with velocity feedback control gain Gv = 5e−5 Figures 4.25 illustrate that, in all the study cases, the vibration is suppressed faster with an increase in the volume fraction of CNTs This is compatible with the previous conclusion that addition of CNTs leads to more stiffness of the panels and results in a smaller deflection amplitude 4.5.3 Effect of the CNT distribution type a) b) c) Figure 4.26 Effect of CNT distribution type to central deflection of the PFGCNTRC spherical panel The effect of the CNT distribution type is indicated for the CNT volume * fraction VCNT  28% with velocity feedback control gain Gv = 1.3e−5 Figure 4.26 depict the transient responses of the panel associated with stepping load, triangular load, and explosive blast load, respectively Among the five possible graded patterns of CNTs, FG-O has the largest central deflection, while FG-X has the smallest one Therefore, the above numerical results demonstrate that, through the proper use of the velocity feedback gain, the vibration of the PFG-CNTRC spherical panels can be controlled as expected 4.6 Comments In this chapter, numerical results were conducted:  Investigation of the effect of material parameters, geometrical parameters and boundary conditions on displacement, stresses of PFG-CNTRC shell under mechanical and electrical loadings;  Investigation of the effect of material parameters, geometrical parameters, and boundary conditions on natural frequencies of PFG-CNTRC shell are conducted CONCLUSIONS AND PROPOSAL The new contributions of the dissertation 1) Proposing the function f(z) to represent the law of shear stress distribution according to thickness direction to enrich the layer of four-variable shear deformation refined theories The function f(z) helps the shear stresses satisfy to 23 be zeros at the upper and lower surfaces of the structure, and the calculated results are more accurate while the number of equations and numbers of the unknown is only four, smaller than other higher shear strain theories 2) Based on the four-variable shear deformation refined theory with proposed f(z), governing equations of FG-CNT plates and shells integrated with piezoelectric layers are established Navier's solution is applied to perform analytical solutions for simply supported shells in both cases of electrical boundary conditions: opencircuit and closed-circuit 3) A finite model using a 4-node element with Lagrange and Hermite interpolation based on four-variable shear deformation refined theory is established The finite model considers the effects of mechanical and electrical loadings, and it can calculate natural frequencies of PFG-CNTRC shell in cases of open-circuit and closed-circuit 4) Computational programs for the analytical solution and finite element model are coded in Matlab Comparison results confirm the accuracy and convergence of the model and programs Some new numerical results are also conducted to investigate the effect of material parameters, geometrical parameters, boundary condition to displacement, stresses, natural frequencies, and dynamic responses of FG-CNTRC plates and shells integrated with piezoelectric layers are conducted From the obtained results, some conclusions can be drawn as: (i) The carbon nano-tube distribution type has a significant effect on the stiffness of the structure Specifically, when carbon nano-tube are more distributed to the top and bottom sides of each layer (FG-X), the structure will have a greater stiffness than when carbon nanotubes are distributed more in the middle of the structure or the center of each material layer (FG-O) (ii) When the number of material layers is increased by more than four, the stiffness of the structure is not significantly increased, so the reasonable number of material layers ranges from three to four (iii) The structure with an open-circuit always vibrates with the higher value of frequencies in compare 5) with structure couped with the closed-circuit because the open-circuit converts electric potential to mechanical energy while the closed-circuit does not (iv) The more stiffness the structure is, the smaller the piezoelectric effect (v) A reasonable feedback circuit can actively control the vibration response of FGCNTRC composite shell integrated with piezoelectric layers Proposal 1) Nonlinear static, buckling and vibration analysis of FG-CNTRC shallow shell integrated with piezoelectric layers based on four-variable shear deformation refined theory 2) Bending of stiffened FG-CNTRC shell integrated with piezoelectric layers subjected to combined mechanical, thermal and electrical loadings based on fourvariable shear deformation refined theory 3) Static, buckling and vibration analysis of FG-CNTRC shallow shell with various shapes, various boundary conditions under various loadings LIST OF PUBLISHED WORKS BY AUTHOR RELATED TO THIS DISSERTATION'S TOPIC The main contents of the dissertation are published 03 papers in ISI journal (Q1: 02 papers, Q2: 01 paper); 01 ESCI journal and 06 paper in other national journal and conferent proceeding Trần Minh Tú, Trần Hữu Quốc Vũ Văn Thẩm (2017) Phân tích tĩnh kết cấu composite có gắn lớp áp điện có kể đến ảnh hưởng nhiệt độ theo tiếp cận giải tích Hội nghị Cơ học tồn quốc lần thứ X ISBN: 978-604-913-235-3 Trần Minh Tú, Trần Hữu Quốc Vũ Văn Thẩm (2018) Phân tích tĩnh composite có lớp áp điện theo lý thuyết biến dạng cắt bậc cao Reddy phương pháp giải tích Tạp chí Khoa học Công nghệ Xây dựng (KHCNXD)-ĐHXD, 12(4): pp 40-50, ISSN: 2615-9058 Trần Hữu Quốc, Trần Minh Tú Vũ Văn Thẩm (2018) Nghiên cứu dao động điều khiển dao động kết cấu FGM có gắn lớp vật liệu áp điện Tuyển tập Hội nghị Khoa học toàn quốc Cơ học Vật rắn lần thứ XIV, ISBN: 978-604-913-832-4 Vu Van Tham, Tran Huu Quoc, and Tran Minh Tu (2018) Optimal placement and active vibration control of composite plates integrated piezoelectric sensor/actuator pairs Vietnam Journal of Science and Technology, 56(1): pp 113, ISSN: 2525-2518 Vũ Văn Thẩm, Trần Hữu Quốc Trần Minh Tú (2019) Phân tích dao động riêng kết cấu composite lớp gia cường ống nano carbon có gắn lớp vật liệu áp điện Tạp chí Khoa học Cơng nghệ Xây dựng (KHCNXD)-ĐHXD, 13(3V): pp 42-54, ISSN: 2615-9058 Tran Huu Quoc, Tran Minh Tu, and Vu Van Tham (2019) Free Vibration Analysis of Smart Laminated Functionally Graded CNT Reinforced Composite Plates via New Four-Variable Refined Plate Theory Materials, 12(22): pp 3675, ISSN: 19961944 ISI (Q2) Vu Van Tham, Tran Huu Quoc, and Tran Minh Tu (2019) Free Vibration Analysis of Laminated Functionally Graded Carbon Nanotube-Reinforced Composite Doubly Curved Shallow Shell Panels Using a New Four-Variable Refined Theory Journal of Composites Science, 3(4): pp 104, ISSN: 2504-477X (ESCI) Tran Huu Quoc, Vu Van Tham, Tran Minh Tu, and Nguyen-Tri Phuong (2019) A new four-variable refined plate theory for static analysis of smart laminated functionally graded carbon nanotube reinforced composite plates Mechanics of Materials: pp 103294, ISSN: 01676636 ISI (Q1) Vũ Văn Thẩm, Dương Thành Huân Chu Thanh Bình (2020) Phân tích tĩnh kết cấu chữ nhật E-FGM có gắn lớp vật liệu áp điện Tạp chí Khoa học Công nghệ Xây dựng (KHCNXD)-ĐHXD 14(4V): p 39-53, ISSN: 2615-9058 10 Tran Huu Quoc, Vu Van Tham and Tran Minh Tu (2021) Active vibration control of a piezoelectric functionally graded carbon nanotube-reinforced spherical shell panel Acta Mechanica: pp 1-19 ISI (Q1)