Tài liệu 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 2 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 1.1 Overview ................................................................................................................ 1 1.1.1 Composite material – Functionally Graded Materials .............................. 1 1.1.2 FGM classification ........................................................................................ 2 1.1.3 Blast load ........................................................................................................ 5 1.2 Research objectives .............................................................................................. 6 1.3 The layout of the thesis ........................................................................................ 6 CHAPTER 2: LITERATURE REVIEW...................................................................... 8 2.1 Structures ............................................................................................................... 8 2.2 Porosity .................................................................................................................. 9 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] ........................... 2   Table 4.1. Comparison the natural frequencies s 1 of homogenous plates with a / b  1, a / h  20, k1  0, k2  0 and T  0. ..............................................................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. ......................................... 5 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). 1 Table 1.1. Properties of component materials of FGM material [3] Properties Material 2 E (N /m )   (o C 1 ) K (W / mK )  (kg / m3 ) 70.0 109 0.30 23.0 106 204 2707 Ti  6 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 )  1  Vm ( z ),  2h  N (1.1) where Vm ,Vc : the volume fractions of metal and ceramic, respectively N : the volume distribution (0  N  ) 2 The effective properties Peff of the P- FGMs are established using the modified mixed rules as follows [7]: (1.2) Peff ( z )  Prc Vc ( z )  Prm Vm ( z ), in which Pr denotes a specific property of the material such as elastic modulus E , thermal expansion coefficient  or density  , thermal conduction K . Sigmoid-law distribution S-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 Sigmoid-law function as:  2 z  h  N   , h/2 z 0  h  Vm ( z )   , Vc ( z )  1  Vm ( z ). N   2 z  h    h  , 0  z  h / 2  (1.3) The effective properties Peff of the S- FGMs are established using the modified mixed rules as follows: Preff  z   PrcVc  z   PrmVm  z   2 z  h  N   , h/2 z 0  h   Prc   Prm  Prc   . N   2 z  h    h  , 0  z  h / 2  (1.4) An exponential-law distribution E-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 exponential-law function as: E  z   Ae B  z  h / 2 (1.5) , where 3 1 E A  Et , B  ln  b h  Et  ,  (1.6) Et is the elastic modulus of structure on the top  z  h / 2 . Eb is the elastic modulus of structure on the bottom  z  h / 2 . FGM sandwich material: The multi-layered sandwich structure is a particularly important type of structure in the aerospace industry as well as in a number of other industries such as ships, automobiles, construction .... Sandwich structure consists of 3 main layers: core layer and two face-sheets. The core layer is made of lightweight material, low hardness between two face-sheets made of very high hardness material. The great advantage of sandwich structure is that it increases the stiffness and bending resistance of the structure while ensuring a small volume, because the core layer is made of light material that can be made with a large thickness that will have an effect to transfer the two face-sheets away from the neutral axis. To avoid the phenomenon of flaking between the layers as well as the phenomenon of stress caused as with conventional multi-layer structures, it was thought that FGM sandwich material with ceramic or metal core layer and two facesheets made of FGM material. 4 a. Sandwich FGM- Metal-FGM b. Sandwich FGM- Metal-FGM Fig.1.1. The distribution types of FGM sandwich material . The effective properties of this materials vary according to the extended Sigmoid distribution law as follows: kt   2z  h   Pri    Pri    , h / 2  z  h / 2  ht ,   2ht    Preff     Pr j  ,  h / 2  ht  z  h / 2  hb ,  kb  2 z  h    , h / 2  hb  z  h / 2,  Pri    Pr ji   2 h b    (1.7) with the case of type 1-1a: i  c, j  m, and the case of type 1-1b: i  m, j  c. 1.1.3 Blast load Blast load: In recent years, the safety of important buildings and infrastructure around the globe has become more fragile by extreme dynamic loads due to the increase in terrorist activities, explosions... The damage from such events cannot be determined, not just economically because many of these ones are symbolic and important heritage, significant architectures and the spirit of the times. Nowadays, considerable efforts in architecture and structural engineering in recent years are often 5 focused towards optimal design and economic efficiency in construction. It is essential to guarantee the safe and secure protection of important infrastructure for the present and future. Explosion loads usually act in a very short time (usually in milliseconds) but transmit very high pressure pulses ( 101  103 kPa). As a result, damage to structural systems can take many forms, such as damage to the outer surface and structural frame of a building; collapse of walls and bearing columns; blow debris of concrete, glass windows and furniture; and damaging safety systems. Most existing buildings are not designed to withstand such extreme dynamic loads, so a comprehensive understanding of the explosion phenomena and the dynamic response of structures is required to be essential for the scientific basis improving the design and material improvement in a feasible manner, in order to improve explosion resistance and ensure the safety of structures. Therefore, the investigation of the effects of explosive loading on structures should be focused. 1.2 Research objectives The research objective of this thesis is to investigate the effects of porosity on vibration and nonlinear dynamic response of multi-layered FGM subjected to blast load.  Studies the effects of porosity to FGM sandwich plates and comparison with different cases: FGM plates without porosity, porous -I FGM plates and porous-II FGM plates.  Investigations on nonlinear dynamic analysis on the structure in FGM plates on elastic foundation subjected to blast load. In numerical results, the effects of the material properties, geometrical parameters, blast load… on the nonlinear dynamic response will be analyzed. 1.3 The layout of the thesis The thesis includes an introduction, five chapters, conclusions, references and appendices. The main contents of the chapters involve: 6  Chapter 1: Introduction The thesis presents an overview of FGM materials. Porosities are also mentioned in this chapter.  Chapter 2: Literature review Chapter 2 presents some studies which have been reported to this thesis’s field. In those publications, I also pointed out their main outstanding results obtained from their research as well as those research’s limitation.  Chapter 3: Methodology Chapter 3 introduces the analytical method by using high-order shear deformation to approach and solve problems…  Chapter 4: Numerical results and discussion The numerical results are presented in this chapter for a FGM sandwich plate on elastic foundation in terms of natural frequencies, effects of geometrical parameters, materials properties on nonlinear dynamic response.  Chapter 5: Conclusions Chapter 5 summarize the main results obtained from this thesis. 7 CHAPTER 2: 2.1 LITERATURE REVIEW Structures Because of their remarkable properties, in recent years, sandwich FGM structure has been attracted a lot of attention of scientists. Among those, Zhaobo Chen et.al [8] presented the free vibration of the functionally graded material sandwich doubly-curved shallow shells under simply supported conditions according to a new shear deformation theory with stretching effects. The wave propagation of FGM sandwich plates with porosities putting on viscoelastic foundation was studied by Chen Liang and Yan Qing Wang [9] based on a quasi-3D trigonometric shear deformation theory. Singh and his co-authors [10] used a semi-analytical approach to analyze thermo-mechanical of porous sandwich S-FGM plate for different boundary conditions using Galerkin Vlasov's method. Furthermore, the nonlinear vibration of imperfect sandwich plates with FGM face sheets also investigated by Kitipornchaii et al. [11] basing on a semi-analytical approach. Hoang Van Tung [12] are analyzed nonlinear bending and post buckling behavior of FGM sandwich plates under thermomechanical loading by using the first order shear deformation theory. The effect of time constant, temperature, mid radius to thickness ratio and time on transient thermo-elastic behavior of sandwich plate with the core as FGM are taken into consideration by Alibeigloo [13]. In his analysis, the sandwich plate’s time dependent response is built from generalized coupled thermo-elasticity when applying the Lord-Shulman expression. Moreover, Xia and Shen [14] introduced an analytical using higher-order shear deformation and a general von Kármán-type function to obtain small- and large-amplitude vibration of compressive and thermal post-buckling sandwich plates with FGM face sheets under uniform and non-uniform temperature fields. Behzad Mohammadzadeh [15] combined higher-order shear deformation with Hamilton’s principle to analyze nonlinear dynamic responses of sandwich plates with FGM faces on elastic foundation subjected to blast loads. Basing on a new four-variable shear deformation plate theory, Mohammed Sobhy 8 [16] evaluated the hydrothermal vibration and buckling of various types of FGM sandwich plates resting on elastic foundations exposed moisture condition, rising temperature, Winkler–Pasternak foundation coefficients and power- law distribution index. Chien et al. [17] used isogeometric approach to investigate static, free vibration and buckling analysis of FGM isotropic and sandwich plates. Tao Fu et al. [18] adopted the space harmonic approach and virtual work principle to describe analytically sound loss when transmitting through two types of porous FGM sandwich structures. 2.2 Porosity The effects of porosities generated during actual manufacturing process to the vibration characteristics of FGM structures have been studied by several authors. However, the number of researches in terms of the mechanical behaviors of porous materials is still limited. The most recent investigations on structures with porosity are listed in the following. Ashraf M.Zenkour [19] used a quasi-3D shear deformation theory to investigate the bending responses of porous functionally graded single-layered and multi-layered thick rectangular plates. By taking Galerkin Vlasov's method into account in thermo-mechanical analysis of sandwich S-FGM plate with three different types of porosity for diverse boundary conditions, Singha and co-author [20] obtained the approach for bending and stress under the thermal environment. They deduced that the deflection and stress escalate significantly for even porosity distribution (P-1) to bring into comparison with uneven symmetric (P-2) or uneven non-symmetric (P-3) porosity distribution; and the effects of temperature on transverse shear stresses of the multi-layered plates. Mojahedin [21] employed higher order shear deformation theory to investigate the buckling of functionally graded porous circular plates. Polat et al. [22] utilized an atmospheric plasma spray system to obtain functionally gradient coatings from five layers which were prepared on Ni substrates from Y2O3 stabilized ZrO2 (YSZ) and NiCoCrAlY powders. In their 9 research, they found that an escalation in porosity ratio of layers lead to the decrease of residual stresses. Chen et al. [23] employed Chebyshev-Ritz method to analyze buckling and bending loads of a novel functionally graded porous plates. Wang et al. [24] focused on effects of parameters on the vibrations of functionally graded material rectangular plates with two types of porosity, namely, even and uneven distributed porosity, and transferring in thermal environment. Based on a sinusoidal shear deformation theory in combination with the Rayleigh–Ritz method, Yuewe Wang et al. [25] depicted the effects of porosity, boundary conditions, and geometrical parameters on free vibration of the functionally graded porous cylindrical shell. An isogeometric finite element model and the nonlocal elasticity were introduced by Phung-Van et al. [26] to investigate the transient responses of functionally graded nanoplates with porosity. Small size effects, nonlocal parameters, and porosity distributions, volume index, the characteristics of dynamic load have considerably influenced on the plate nonlinear transient deflections. Cong et al. [27] acquired closed-form expression in regard to critical bucking loads and post-buckling paths of a porous functionally graded plates on elastic foundations subjected to the coupling of mechanical and thermal loads by applying Reddy's higher-order shear deformation plate theory in conjunction Galerkin method. Analytical solutions and numerical results revealed that porosity I (evenly distribution) behaves better than porosity II (unevenly distribution) according to the static buckling investigations. Chien et al [28] adopted the first-order shear deformation theory taking the out-of-plane shear deformation into account to calculate the fundamental frequencies and nonlinear dynamic responses of porous functionally graded sandwich shells with double curvature under the influence of thermomechanical loads. This study proved that porosities help the shell structures stiffen to some extent. 2.3 Blast load In recent years, explosive loads and their impacts on the safety and efficiency of building and structures have received considerable attention. Tuan et al. [29] presented the results of an empirical investigation conducted in Woomera, Southern 10 Australia, in May 2004 on the explosion-resistance of concrete-panel created by ultrahigh-strength concrete material. A finite-element method was used to analyze concrete structures under blast and impact loading. In the study conducted by Tin and co-authors [30] , they proposed using the explicit finite element software LS-DYNA to induce stress wave propagation and the impacts on structural responses of precast concrete segmental columns subjected to simulated blast loads. Balkan et al. [31] examined the effects of sandwich stiffeners on the dynamic response of laminated composite plates under the non-uniform blast loading. Moreover, the dynamic behavior of stiffened plates exposed to confined blast loads are carried out by Zhao et al. [32] through experimental and numerical studies. Geretto et al. [33] analyzed a series of experiments of square monolithic steel plates to assess the effects of the degrees of confinement of the deformation to blast loads. Asoylar et al. [34] studied the transient stability analysis metal-fiber laminated composite plates under no-ideal explosion load by experiment and finite element methods. In addition, Uybeyli and colleagues [35] used SiC reinforced functionally gradient material via powder metallurgy to investigate the impact of armor piercing projectile. Bodaghi et al. [36] studied non-linear active control of dynamic response of functionally graded beams with rectangular cross-section in thermal environments under blast loadings. Based on meticulous investigations in the available literature, it can be concluded that there are few free vibration and nonlinear dynamic behaviors of porous functionally graded sandwich plates resting on elastic foundations regardless of the high demand for understanding. In particular, literature review indicates lack of investigations on effects of porosity on this structure exposed to blast loads. This study has been implemented to meet the demand. 11 CHAPTER 3: 3.1 METHODOLOGY Configurations of analyzed models The geometry configuration of the rectangular FG sandwich plate with two FGM face-sheets and the core as ceramic resting on elastic foundations under blast load are as follows (Figs 3.1 and 3.2). The plate is referred to a Cartesian coordinate system x, y, z , where xy is the mid-plane of the plate and z is the thickness coordinator, h / 2  z  h / 2 . a, b : the length and width of the plate. h, hc , h f : thickness of the total plate, the core and the face-sheets. z b shear layer y a h x Fig. 3.1. FGM sandwich plate resting on elastic foundation. 12