Luận án tiến sĩ: Phân tích ứng xử dầm sandwich chức năng dưới tải trọng cơ thủy nhiệt - Nguyễn Bá Duy, ĐH Sư phạm Kỹ thuật TP.HCM

BỘ GIÁO DỤC VÀ ĐÀO TẠO TRƯỜNG ĐẠI HỌC SƯ PHẠM KỸ THUẬT TP HỒ CHÍ MINH NGUYỄN BÁ DUY PHÂN TÍCH ỨNG XỬ CỦA DẦM SANDWICH CHỨC NĂNG CHỊU TÁC DỤNG CỦA TẢI TRỌNG CƠ THỦY NHIỆT LUẬN ÁN TIẾN SĨ KỸ THUẬT Tp. Hồ Chí Minh, Năm 2019 BỘ GIÁO DỤC VÀ ĐÀO TẠO TRƯỜNG ĐẠI HỌC SƯ PHẠM KỸ THUẬT TP HỒ CHÍ MINH NGUYỄN BÁ DUY PHÂN TÍCH ỨNG XỬ CỦA DẦM SANDWICH CHỨC NĂNG CHỊU TÁC DỤNG CỦA TẢI TRỌNG CƠ THỦY NHIỆT Chuyên ngành: CƠ KỸ THUẬT Mã chuyên ngành : 9520101 Phản biện 1: PGS. Đào Đình Nhân Phản biện 2: PGS. Nguyễn Trọng Phước Phản biện 3: PGS.

Ngô Hữu Cường NGƯỜI HƯỚNG DẪN KHOA HỌC: 1. Nguyễn Trung Kiên 2. Võ Phương Thức Tp. Hồ Chí Minh, Năm 2019 ANALYSIS OF FUNCTIONALLY GRADED SANDWICH BEAMS UNDER HYGRO – THERMO – MECHANICAL LOADS DISSERTATION Submitted to Ho Chi Minh City University of Technology and Education in partial fullfillment of the requirements for the degree of Doctor of Philosophy 2019 MAJOR : ENGINEERING MECHANICS Ho Chi Minh City – 2019 ORIGINALITY STATEMENT I hereby declare that this submission is my own work and to the best of my knowledge it contains no materials previously published or written by another person, or substantial proportions of material which have been accepted for the award of any other degree or diploma at Ho Chi Minh City University of Technology and Education (HCMUTE) or any other educational institution, except where due acknowledgement is made in the thesis.

Any contribution made to the research by others, with whom I have worked at HCMUTE or elsewhere, is explicitly acknowledged in the thesis. I also declare that the intellectual content of this thesis is the product of my own work, except to the extent that assistance from others in the project’s design and conception in style, presentation and linguistic expression is acknowledged. ACKNOWLEDGEMENTS My thanks go to many people who provided great support and had an important role in this research. I would like to express my gratitude to my supervisor for their continuous support and valuable guidance throughout this research.

I had also the opportunity to work with people in GACES of HCMUTE. Therefore, my acknowledgments are extended to Prof. Nguyen Hoai Son and Nguyen Ngoc Duong for his technical guidance and training. Nguyen Van Hau is thanked for his comment and discussion on functionally graded materials (FGM).

My thanks also go to Le Quoc Cuong who helped and provided me a useful matlab. Thank you to everyone else who help me with this research. Last but not least, I wish to profoundly thank my parents, my wife, my son and my sister for their unconditional love and unlimited support. Without their encouragement, I would not have been able to overcome many difficulties and challenges during this research.

Contents LISTS OF TABLES. V LISTS OF FIGURES. VIII LISTS OF SYMBOLS. X Abstracts Chapter 1 General Introduction.1 Introduction and Objectives .2 Objective and novelty of the thesis .4 List of publications.

16 Chapter 2 Literature review on behaviors of functionally graded beams in hygro- thermo-mechanical environments.1 Composite and functionally graded materials .2 Homogenized elastic properties of functionally graded beams .3 Hygral and thermal variations in FG beams .1 Uniform moisture and temperature rise .2 Linear moisture and temperature rise .3 Nonlinear moisture and temperature rise.4 Theories for behavior analysis of FG beams .1 Classical beam theory (CBT) .2 First-order shear deformation theory (FSDT) .3 Higher-order shear deformation beam theories .4 Quasi-3D beam theory .5 Review of the shear functions .6 Nonlocal elasticity and modified couple stress beam theories .5 Analytical and numerical methods for analysis of FG beam .2 Differential Quadrature Method (DQM) .4 Finite element method. 38 Chapter 3 Novel higher-order shear deformation theories for analysis of isotropic and functionally graded sandwich beams .2 Novel unified theoretical formulation of higher–order shear deformation beam theories .3 Analysis of static, buckling and vibration of FG beams based on the HSBTs………………………………………………………………………………50 3.4 Analysis of static, buckling and vibration of FG beams based on the Quasi- 3D………………………………………………………………………………….5 A novel three-variable quasi-3D shear deformation theory.1 Displacement, strain, and stresses .1 Ritz method for solution 1 .2 Ritz for solution 2 .7 Numerical results and discussion. 66 Example 1: Vibration and buckling responses of RHSBT1, HSBT2 and quasi- 3D2 FG beams (Type A, S-S). 67 Example 2: Bending, buckling and vibration responses of RHSBT1 FG beams (Type B, S-S).

69 Example 3: Buckling and vibration responses of Quasi-3D0 FG beams (Type B, C)…………………………………………………………………………………79 3. 99 Chapter 4 Hygro-thermo-mechanical effects on the static, buckling and vibration behaviors of FGbeams .2 Novel Ritz-shape functions for analysis of FG beams with various BCs .2 Moisture and temperature distribution .1 A shape functions for Ritz method .2 A new hybrid functions for Ritz method .4 Numerical results and discussions. 128 Chapter 5 Size dependent effects on the thermal buckling and vibration behavior of FG beams in thermal environments .2 Geometry of FG beams .3 Theory of FG micro and nano beams. Kinetic and strain.

Equations of motion. Nonlocal elasticity theory for FG nano beams. Modified couple stress theory (MCST). Variation formulation for MCST.

Ritz method for nonlocal theory. Ritz method for MCST .5 Numerical results and discussions. 145 Example 1: Vibration responses of FSBT and the Eringen’s nonlocal elasticity theory for FG nano beam (Type A, the various BCs). 145 Example 2: Vibration and the thermal bucking responses of HSBT1 and the MCST for FG micro beam (Type A, the various BCs).

155 Chapter 6 A finite element model for analysis of FG beams .2 Finite element formulation .2 Higher-order shear deformation beam theory .5 Governing Equations of Motion .6 Finite Element Formulation .3 Numerical results and discussions. 165 Example: Vibration and the thermal bucking responses of HSBT1 using FEM for analysis FG beam (Type A, various BCs). 169 Chapter 7 Conclusions and Recommendations. 171 References IV LISTS OF TABLES Table 3.1 Unified higher-order shear deformation theories .2 Unified refined higher-order shear deformation theories.3 Kinematic BCs of the beams.4 Non-dimensional fundamental frequency (  ) of FG beams with S-S boundary conditions (Type A).5 Non-dimensional critical buckling load ( N cr ) of FG beams with S-S boundary conditions (Type A).6 Non-dimensional fundamental frequency    of  Al/Al2O3  sandwich beams (Type B, homogeneous hardcore).7 Non-dimensional fundamental frequency    of  Al/Al2O3  sandwich beams (Type B, homogeneous soft core).8 Non-dimensional critical buckling load  Ncr  of  Al/Al2O3  sandwich beams (Type B, homogeneous hardcore).9 Non-dimensional critical buckling load  Ncr  of  Al/Al2O3  sandwich beams (Type B, homogeneous soft core).10 Non-dimensional mid-span transverse displacement  w  of  Al/Al2O3  sandwich beams (Type B, homogeneous hardcore and soft core).11 Non-dimensional axial stress  xx  h / 2   of  Al/Al2O3  sandwich beams (Type B, homogeneous hardcore and soft core).12 Non-dimensional transverse shear stress  xz  0   of  Al/Al2O3  sandwich beams (Type B, homogeneous hardcore and soft core).13 Non-dimensional fundamental frequency (  ) of FG sandwich beams.14 Non-dimensional fundamental frequency (  ) of FG sandwich beams.15 Non-dimensional fundamental frequency (  ) of FG sandwich beams.16 Non-dimensional fundamental frequency (  ) of FG sandwich beams.17 Non-dimensional fundamental frequency (  ) of FG sandwich beams.18 Non-dimensional fundamental frequency (  ) of FG sandwich beams.19 Non-dimensional critical buckling load ( N cr ) of FG sandwich beams .20 Non-dimensional critical buckling load ( N cr ) of FG sandwich beams .21 Non-dimensional critical buckling load ( N cr ) of FG sandwich beams .22 Non-dimensional critical buckling load ( N cr ) of FG sandwich beams .23 Non-dimensional critical buckling load ( N cr ) of FG sandwich beams .24 Non-dimensional critical buckling load ( N cr ) of FG sandwich beams .25 Non-dimensional fundamental frequency (  ) of FG sandwich beams with various boundary conditions (Type C).26 Non-dimensional critical buckling load ( N cr ) of FG sandwich beams with various boundary conditions (Type C).27 The first three non-dimensional frequencies of FG sandwich beams .1: Temperature dependent coefficients for ceramic and metal materials.2 Kinematic BCs of the beams.3 A new hybrid functions for Ritz solution.4 Convergence test for the non-dimensional fundamental frequency (  ) of Si3 N 4 and SUS304 beams under Fourier-law NLTR (Type A, p=1, L/h=20 and ΔT=20, ΔC=0).5 Normalized critical temperatures (  ) of FG beams under UTR .7 Critical temperature (  ) of FG beams under LTR and Fourier-law NLTR119 Table 4.11 Fundamental frequency (  ) of FG beams under LTR .12 Fundamental frequency (  ) of FG beams under Fourier-law NLTR .14 Fundamental frequency (  ) of FG beams under linear moisture and temperature rise .15 Fundamental frequency (  ) of FG beams under sinusoidal moisture and temperature rise .1 Kinematic BCs of nano beams.2 The shape functions.3: Convergence studies for fundamental frequencies of FG nano beams .4 The non-dimensional first natural frequencies with respect to the material distribution and the span-to-height ratio of FG nano beams (Type A, S-S).5 The non-dimensional first natural frequencies with the nonlocal parameter of FG nano beams (Type A, C-F, L/h=100, N=10).6 The non-dimensional first natural frequencies with the nonlocal parameter of FG nano beams (Type A, C-C, L/h=100, N=10).7 Convergence studies for The non-dimensional fundamental frequencies of FG micro beams with various BCs and  / h (Type A, p=1, L/h=5, Si3N4/ SUS304) 151 Table 5.8 Fundamental frequency (  ) of FG micro beams under LTR.9 Fundamental frequency (  ) of FG micro beams under NLTR .1 Ceramic and metal materials.2: Convergence of the non-dimensional fundamental frequency(  ) and the critical buckling load  N cr  of FG beams (Type A, p = 1 and L/h = 5) .3 Comparison of the non-dimensional critical buckling load of FG beams with various boundary conditions (Type A, L/h=5 and 10).4 Comparison of the non-dimensional fundamental natural frequency of FG beams with the various boundary conditions (Type A, L/h=5 and 20).

167 VII LISTS OF FIGURES Figure 1.1: Application of composite materials in engineering .1 Particulate and fiber composite materials .2 Laminated composite and functionally graded materials .3 Potentially applicable fields for FGMs [55].4 An example of FGM application for aerospace engineering [56].5 A discrete and continuous model of FG material [57].6 Geometry and coordinate systems of FG sandwich beams.7 The volume fraction function V  z  for the power-law (Type B).8 The volume fraction function V  z  for the exponential-law .9 The volume fraction function V  z  for the Sigmoid -law .10 Kinematics of the Euler–Bernoulli beam .11 Kinematics of the Timoshenko beam .12 Kinematics of the CBT, FSBT, HSBT .13 The shear stress varies over the height of the cross section .14 Variation of the shear functions and its derivative through the beam thickness .15 Discrete beams into finite elements.16 Linear shape functions for an element of length le .17 Hermite shape functions for one-dimensional finite element .1 Geometry of FG sandwich beams.2 Effect of the power-law index p on the non-dimensional fundamental frequency (  ) of FG sandwich beams (Type B, L/h=5).3 Effect of the power-law index p on the non-dimensional critical buckling load  N cr  of FG sandwich beams (Type B, L/h=5).4 Effect of the power-law index p on the non-dimensional mid-span transverse displacement  w  of FG sandwich beams (Type B, L/h=10).5 Distribution of non-dimensional axial stress  xx  through the height of (1- 2-1) FG sandwich beams (Type B, L/h=10).6 Distribution of non-dimensional transverse shear stress  xz  through the height of.