Effects of porosity on free vibration and nonlinear dynamic response of multi layered functionally graded materials subjected to blast loads

VIETNAM NATIONAL UNIVERSITY, HANOI VIETNAM JAPAN UNIVERSITY DO THI THU HA EFFECTS OF POROSITY ON FREE VIBRATION AND NONLINEAR DYNAMIC RESPONSE OF MULTI-LAYERED FUNCTIONALLY GRADED MATERIALS SUBJECTED TO BLAST LOAD MASTER’S THESIS Ha Noi, 2020 VIETNAM NATIONAL UNIVERSITY, HANOI VIETNAM JAPAN UNIVERSITY DO THI THU HA EFFECTS OF POROSITY ON FREE VIBRATION AND NONLINEAR DYNAMIC RESPONSE OF MULTI-LAYERED FUNCTIONALLY GRADED MATERIALS SUBJECTED TO BLAST LOAD MAJOR: INFRASTRUCTURE ENGINEERING CODE: 8900201.04QTD RESEARCH SUPERVISOR: Dr TRAN QUOC QUAN Ha Noi, 2020 ACKNOWLEDGEMENT First of all, I would like to express my sincere appreciation to my supervisor , Dr Tran Quoc Quan who has guided and created favorable conditions and regularly encouraged me to complete this thesis Thank you for all your thorough and supportive instructions, your courtesy and your enthusiasm Without your dedicated guidance, I absolutely have not conducted this research well Secondly, I would like to express my great thankfulness to Master’s Infrastructure Engineering Program for their wonderful supports, especially Prof.Sci Nguyen Dinh Duc, Prof Kato, Prof Nagayama, Dr Phan Le Binh, Dr Nguyen Tien Dung and Mr Bui Hoang Tan Their encouragement and assistance has facilitated me a lot during years studying in the VietNam – Japan University I also want to give my special thanks to all lecturers and staffs at The University of Tokyo for their warmly welcome and supports me in the internship time at Japan Thirdly, I would like to thank all the members at the Advanced Materials and Structural Laboratory, University of Technology- VNU, especially for Mr Vu Dinh Quang, Mr Vu Minh Anh, Mr Pham Dinh Nguyen spending their precious time to point out for me which theories and methodology should I use and give me advices to improve my thesis Finally, there are my family and my friends, who always stay by my side, motivate and encourage me from the beginning until the end of my studying I TABLE OF CONTENTS ACKNOWLEDGEMENT I LIST OF TABLES III LIST OF FIGURES IV NOMENCLATURES AND ABBREVIATIONS V ABSTRACT VI CHAPTER 1: INTRODUCTION 1.1 Overview 1.1.1 Composite material – Functionally Graded Materials 1.1.2 FGM classification 1.1.3 Blast load 1.2 Research objectives 1.3 The layout of the thesis CHAPTER 2: LITERATURE REVIEW 2.1 Structures 2.2 Porosity 2.3 Blast load .10 CHAPTER 3: METHODOLOGY .12 3.1 Configurations of analyzed models 12 3.2 Methodology 17 3.3 Theoretical formulation 18 3.4 Solution procedure 24 3.5 Vibration analysis .25 3.5.1 Dynamic response problem 25 3.5.2 Natural frequency 27 CHAPTER 4: NUMERICAL RESULTS AND DISCUSSION 28 4.1 Validation of the present results .28 4.2 Natural frequency .30 4.3 Dynamic response .33 CHAPTER 5: CONCLUSIONS 40 APPENDIX 41 II LIST OF TABLES Table 1.1 Properties of component materials of FGM material [3]   Table 4.1 Comparison the natural frequencies s 1 of homogenous plates with a / b  1, a / h  20, k1  0, k2  and T  28 Table 4.2 Comparison of natural fundamental frequency parameters  of simply square FGM plates with other theories ( h / b  0.1 ) .29 Table 4.3 The effects of porosity ratio on natural frequency of FGM sandwich plates 31 Table 4.4 Influences of temperature increment, elastic foundations and the volume fraction index on natural frequencies of the FGM sandwich plate with porosity I 32 III LIST OF FIGURES Fig 1.1 The distribution types of FGM sandwich material Fig 3.1 FGM sandwich plate resting on elastic foundation 12 Fig 3.2 FGM-ceramic- FGM model 13 Fig 3.3 Porosity – I: evenly distributed, Porosity – II: unevenly distributed .14 Fig 3.4 Blast pressure function 17 Fig 4.1 Influences of power law index N on the nonlinear dynamic response of the FGM sandwich plates with porosity I 33 Fig 4.2 Influences of power law index N on nonlinear dynamic response of the FGM sandwich plates with porosity II 33 Fig 4.3 Influences of porous ratio on nonlinear dynamic response of the FGM sandwich plates with porosity I 34 Fig 4.4 Influences of type of porosity on nonlinear dynamic response 35 Fig 4.5 Influences of a / b ratio on nonlinear dynamic response of the FGM sandwich plates with porosity I 36 Fig 4.6 Influences of a / h ratio on nonlinear dynamic response of the FGM sandwich plates with porosity I 37 Fig 4.7 Influences of Pasternak foundation on nonlinear dynamic response of the FGM sandwich plates with porosity I 37 Fig 4.8 Influences of Winkler foundation on nonlinear dynamic response of the FGM sandwich plates with porosity I .38 Fig 4.9 Effect of parameter characterizing the duration of the blast pulse Ts on nonlinear response of the FGM sandwich plate with porosity I under blast load .39 IV NOMENCLATURES AND ABBREVIATIONS FGM h hc Functionally Graded Material The length of plate The width of plate The thickness of plate The thickness of the core layer hf The thickness of FGM face-sheet k1 The Winker foundation k2 The Pasternak foundation GPa m, n GigaPascal = 109 Pascal Numbers of half waves in x, y direction a b V ABSTRACT The effects of porosity ratio on free vibration and nonlinear dynamic response of FGM sandwich plates with two FGM face-sheets and a homogeneous core as ceramic resting on elastic foundations subjected to blast load are investigated in this thesis by implementing the third-order shear deformation theory Two types of porosity are proposed, namely evenly distributed porosity and unevenly distributed porosity Assumption that the material properties of multi-layered FGM plate to be changed in the thickness direction accord with a simple-power law distribution with regard to the volume proportion of the components This study obtains numerical results by using the Galerkin method and fourth-order Runge-Kutta method illustrating the significant effects of porous fractions, geometrical parameters, the elastic foundation, blast loads on the nonlinear dynamic response of FGM sandwich plates Key words: Porosity, Functionally graded sandwich plate, Blast loading, The third-order shear deformation theory VI CHAPTER 1: 1.1 INTRODUCTION Overview 1.1.1 Composite material – Functionally Graded Materials Composite material is a material composed of two or more different types of component materials in order to achieve superior properties such as light weight, high stiffness and strength, ability of heat resistance and chemical corrosion resistance, good soundproofing, thus it plays a crucial role in advanced industries in the world that are extensively applied across wide range of fields such as: aviation, aerospace, mechanics, construction, automotive [1] [2] However, this material has a defect as a sudden change of material properties at the junction between the layers is likely to generate large contact stresses at this surface One of the solutions to overcome this disadvantage of layered composite material is to use Functionally Graded Material (FGM) which is a material made up of two main component materials as ceramic and metal, in which the volume ratio of each component varies smoothly and continuously from one side to the other according to the thickness of the structure so the functional materials avoid the common disadvantages in composite types such as the detachment between layers material, fibers breakage and high stress in the surface, which can cause material destruction and reduce the efficiency of the structure, especially in heat-resistant structures Due to the high modulus of elasticity E , the thermal conduction coefficient K and the very low coefficient of thermal expansion  , the ceramic composition makes the material highly variable with high hardness and very good heat resistance While the metal components make the modified materials more flexible, more durable and overcome the cracks that may occur due to the brittleness of ceramic materials when subjected to high temperature (Table 1.1) Table 1.1 Properties of component materials of FGM material [3] Properties Material E (N /m )   (o C 1 ) K (W / mK )  (kg / m3 ) 70.0 109 0.30 23.0 106 204 2707 Ti  Al  4V 105.7 109 0.298 6.9 106 18.1 4429 Ceramic: Zirconia ( ZrO2 ) 151109 0.30 10 106 2.09 3000 320 109 0.26 7.2 106 10.4 3750 Aluminum ( Al ) Aluminum oxide 1.1.2 FGM classification Depending on the power law of the volume ratio of component materials, we can classify different types of FGM Each of these FGM materials is characterized by different mechanical and physical properties by a function that determines the material properties (effective properties), and the value of the function varies with thickness Mathematical functions of material properties used to classify materials [4] Specifically, there are three main types of FGM A power-law distribution P-FGM: is a type of material having a volume fractions of ceramic and metal components which is assumed to vary according to thickness of structure and conforming to the power-law function [5, 6]:  2z  h  Vm ( z )    ,Vc ( z )   Vm ( z ),  2h  N (1.1) where Vm ,Vc : the volume fractions of metal and ceramic, respectively N : the volume distribution (0  N  ) The influences of porous ratio on the deflection amplitude- time curves of the FGM sandwich plates with porosity I is examined in Fig 4.3 The sandwich plate with the lower porosity volume fraction behave better when being subjetcd to blast load.As can be detected, the higher porous ratio is, the higher deflection amplitude of the FGM sandwich plate is The dynamic deflection of the FGM sandwich plate in case of without porosity   is smallest Thus, the existence of porosity deteriorates the dynamic response of the FGM sandwich exposed to blast load Fig 4.4 Influences of type of porosity on nonlinear dynamic response of the FGM sandwich plate The effects of Porosity I, Porosity II and without porosity on the nonlinear dynamic response of the FGM sandwich plate under blast load are depicted in Fig 4.4 The existence of porosity reduces the dynamic performance of FGM sandwich plate In addition, the dynamic deflection of porosity I is larger than this ones of porosity II as considering the same porosity value 35 The influences of the plate length to width a / b ratio and length to thickness a / h ratio on the deflection amplitude – time curves of the multi-layered FGM plates subjected to blast loading are shown in Fig 4.5 and Fig 4.6, respectively It can be seen that the a / b ratio or a / h ratio amplitudes leads to increase of the deflection amplitude of plate From these scrutiny, it could be deduced that the stiffness of the sandwich plate becomes weaker when a / h ratio or a / b ratio is increased Fig 4.5 Influences of a / b ratio on nonlinear dynamic response of the FGM sandwich plates with porosity I 36 Fig 4.6 Influences of a / h ratio on nonlinear dynamic response of the FGM sandwich plates with porosity I Fig 4.7 Influences of Pasternak foundation on nonlinear dynamic response of the FGM sandwich plates with porosity I 37 Fig 4.8 Influences of Winkler foundation on nonlinear dynamic response of the FGM sandwich plates with porosity I The beneficial effects of Winkler foundation with stiffness coefficient k1 and Pasternak foundation with the modulus k on the deflection amplitude – time curves of the FGM sandwich plate are described in Fig 4.7 and Fig 4.8 Obviously, an increase of coefficients k1 or k results in a reduction of the deflection amplitude It can be seen that the stiffness of sandwich plate becomes stronger with the support of elastic foundations 38 Fig 4.9 Effect of parameter characterizing the duration of the blast pulse Ts on nonlinear response of the FGM sandwich plate with porosity I under blast load Fig 4.9 depicts the effect of parameter characterizing the duration of the blast pulse on nonlinear response of the multi-layered FGM plate with porosity for three cases Ts   0.005,0.01,0.02 From this figure, the value of the parameter characterizing the duration of the blast pulse increase results in the amplitude of vibration increases and vice versa 39 CHAPTER 5: CONCLUSIONS Based on higher order shear deformation, thesis investigates the effects of porosity on nonlinear dynamic response and free vibration of the multi-layered FGM plate subjected to blast load Thesis obtains main results as follows: - The differential motion equations analyze nonlinear dynamic response of porous FGM sandwich plates on elastic foundations subjected to blast load and natural frequency - Generally, the existence of porosity reduces the stiffness of FGM sandwich plate Consequently, the deflection amplitude increases and the natural frequency decreases as the porosity fraction rises However, the natural frequency is not affected by the porosity I fraction of porous FGM sandwich plate Besides, the dynamic deflection of porosity I sandwich plate is larger than this ones of porosity II as considering the same porosity value - The elastic foundations have positive impacts on dynamic behavior of the sandwich plate, and the Pasternak foundation have more beneficial effect than the Winkler one - Temperature increment significantly effects on the nonlinear vibration of the porous sandwich plate Temperature increment is considered as external impact which is disadvantage to natural frequency and deflection amplitude of the sandwich plate - The effects of geometrical parameters ( a / b, a / h ratios), the volume ratio N and parameter characterizing the duration of the blast pulse Ts on the nonlinear dynamic responses of the FGM sandwich plates with porosity are remarkable 40 APPENDIX Appendix A I  m h  I3  cm h N 1 ; I1  cm Nh 2( N  1)( N  2) ; I2   m h3    cm h3   , 12  4( N  1) ( N  2)( N  3)  cm h   3   ,  N   4( N  2) ( N  3)( N  4)   m h5 cm h5   12    ,  80 N  16 2( N  2) ( N  2)( N  3) ( N  2)( N  4)( N  5)   h7  h7 30 I  m  cm [    448 N  64 32( N  2) 16( N  2)( N  3) 15 90  ( N  2)( N  3)( N  4) ( N  2)( N  3)( N  4)( N  5) 360  ], ( N  2)( N  3)( N  4)( N  6)( N  7) I4   Appendix B E2 E4   E    E1 x c1    x  E1  3c1E3  c2  E3  3c1E5   H12 x    2(1   ) x    E4   x  c1  E7   E1        E2 E    c1   E      c1   E1         v   41 E42    3x  ,    E7  E1   xy  E2 E4    E5  E   y 1 c1    y  E1  3c1E3  c2  E3  3c1E5   H13  y   2(1   ) y    E42   y  c1  E7   E1      v  E2 E   E42     y  v c1   E   ,   c1     E7   E1     E1   x 2y          P  x, f   2 f 2w 2 f 2w 2 f 2w   , y x xy xy x y  2 w 2 w   E1  3c1 E3  c2  E3  3c1 E5      H11  w   2(1   )  y   x c12    4 w  2 w 2 w  E42  4 w 4 w     k1 w  k2    ,   2 +   E7  E1  x y y   y   x  x H 21  w   c1   E1  3c1 E3    w  c1 E42 E2 E    w 3 w  c E   E    ,      E1 E1   xy x  2(1   ) c2  E3  3c1 E5   x    2 x   2 x   E22 E E H 22 x     E   2c1  E5  2  2(1   ) y   E1 E1     x  E1  3c1 E3   c2  E3  3c1 E5   x ,  2(1   )  H 23  y     E22 E E E   2c1  E5   1     E1 E1  H 31  w   c1  H 32 x     E22 E E E   2c1  E5   1  v   E1 E1   2 E42  c E   1 E1    2 E42    c E    1 E1       y ,    xy  w  E42 E2 E4    w  w   E1  3c1 E3 c E  c  E  ,   1     E1 E1   y x y  2(1   )   c2  E3  3c1 E5   y   2 E42    2x  c E  ,   1 E  x  y       2 y  2 y    E22 E2 E4   E42   H 33  y     E   2c1  E5     c1  E7     y 2(1   ) x   E1 E1  E1        E1  3c1 E3  c2  E3  3c1 E5    y ,  2(1   )  42 Appendix C   m 2  n 2   E1  3c1 E3  c2  E3  3c1 E5     h11      a   b   2(1   )    c2     m 2  n      m   n   E42      k1  k2    +    E7      , a b E a b               h12   m  E1  3c1 E3  c2  E3  3c1 E5    2(1   ) a   c1  m   E2 E4 E42    c1  E7      E5  E1 E1     a     EE      c1   E    c1    E1         v   h13   n  E1  3c1 E3  c2  E3  3c1 E5    2(1   ) b   c1  n   E2 E E42   E   c E   1    E1 E1     b     EE  v   v c1   E    c1    E1            n1   E42   m  n   E      , E1   a  b   4 E1   m   n        , 16   a   b    I 32    m   n  n2  I  c  I        I    a   b   2   ,  43 E42    m  n  E  ,     E1    a  b  2 c1 m   m   n    c1 E42 E E  + c E   E5          E1 E1   a   a   b    m  E1  3c1 E3   c2  E3  3c1 E5   ,  2(1   ) a  h21     m 2  E22 E2 E4   E42    n    h22      2c1  E5     c1  E7      E3      a  2(1   )  b    E1 E1  E1       E1  3c1 E3   c2  E3  3c1 E5   ,  2(1   )  h23     E22 E E  E  mn   2c1  E5    c12  E7    ,  E3  1    ab  E1 E1  E1     2 c1 n  E42 E2 E4    m   n   h31    E5  c1 E7  c1      E1 E1    a   b    b  n  E1  3c1 E3  c2  E3  3c1 E5   , 2(1   ) b    E22 E2 E4   E42    2c1  E5   E3   ,   c1  E7  E1 E1  E1        n   E22 E2 E4   E42    m    h33      2c1  E5     c1  E7      E3      b  2(1   )  a    E1 E1  E1       E1  3c1 E3  c2  E3  3c1 E5   ,  2(1   )  mn h32   1  v  ab   m 2  n    E2  c1 E4     v     a   b   m  h14  , a mn 1  v    m 2  n    E2  c1 E4    v     a   b   n  h15  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and nonlinear dynamic response of multi- layered FGM subjected to blast load  Studies the effects of porosity to FGM sandwich plates and. .. NATIONAL UNIVERSITY, HANOI VIETNAM JAPAN UNIVERSITY DO THI THU HA EFFECTS OF POROSITY ON FREE VIBRATION AND NONLINEAR DYNAMIC RESPONSE OF MULTI- LAYERED FUNCTIONALLY GRADED MATERIALS SUBJECTED TO. .. foundation on nonlinear dynamic response of the FGM sandwich plates with porosity I .38 Fig 4.9 Effect of parameter characterizing the duration of the blast pulse Ts on nonlinear response of