Summary of doctoral thesis in mechanical engineering and engineering mechanics

The effect of porosities which can be occurred within FGMs during the process of sintering has been investigated recently [16, 17, 18, 19]. Due to FGM beams are often used in thermal environment, investigation on the influence of temperature on free vibration of FGM beams has been studied by several authors [20, 21]. To the author’s best knowledge, to the date only Wang and Wu considered forced vibration of FGM beam in thermal environment under moving loads [22], in which dynamic response of FGM beam under a harmonic moving load has been studied by Lagrange method. It should be emphasized that in the authors [22] studied axially FGM perfect beams (without porosities), and only the case of uniform temperature is considered. Mathematically, the uniform temperature distribution is a special case of nonlinear temperature field, and it is relatively simple from computational point of view. Investigation on vibration of FGM beams in thermal environment thus is important from both the research and practical points of view.

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MINISTRY OF EDUCATION AND TRAINING VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY SCIENCE AND TECHNOLOGY ----------------------------- BUI VAN TUYEN DYNAMIC OF FGM BEAMS WITH POROSITIES IN THERMAL ENVIRONMENT UNDER MOVING LOAD Major: Engineering mechanics code: 9520101 SUMMARY OF DOCTORAL THESIS IN MECHANICAL ENGINEERING AND ENGINEERING MECHANICS Hanoi – 2018 The thesis has been completed at: Graduate University Science and Technology – Vietnam Academy of Science and Technology. Supervisors: 1. Assoc. Prof. Dr. Nguyen Dinh Kien 2. Dr. Tran Thanh Hai Reviewer 1: Prof. Dr. Hoang Xuan Luong Reviewer 2: Assoc. Prof. Dr. Tran Minh Tu Reviewer 3: Assoc. Prof. Dr. Phan Bui Khoi Thesis is defended at Graduate University Science and Technology- Vietnam Academy of Science and Technology ato’clock...’, on . ,., 2018 Hardcopy of the thesis be found at : - Library of Graduate University Science and Technology - Vietnam national library 1 INTRODUCTION 1. Relevance of the thesis topic The effect of porosities which can be occurred within FGMs during the process of sintering has been investigated recently [16, 17, 18, 19]. Due to FGM beams are often used in thermal environment, investigation on the influence of temperature on free vibration of FGM beams has been studied by several authors [20, 21]. To the author’s best knowledge, to the date only Wang and Wu considered forced vibration of FGM beam in thermal environment under moving loads [22], in which dynamic response of FGM beam under a harmonic moving load has been studied by Lagrange method. It should be emphasized that in the authors [22] studied axially FGM perfect beams (without porosities), and only the case of uniform temperature is considered. Mathematically, the uniform temperature distribution is a special case of nonlinear temperature field, and it is relatively simple from computational point of view. Investigation on vibration of FGM beams in thermal environment thus is important from both the research and practical points of view. 2. Thesis objective This thesis aims to develop finite element models for studying vibration of FGM porous beams in thermal environment under moving loads. 3. Scope The thesis focuses on two-phase transverse FGM beams forming from ceramic and metal. The external loads considered in the thesis are the constant speed moving forces and moving harmonic forces. 4. Research methods Both analytical method and finite element analysis are employed in the thesis. The analytical method is used to derive equations of motion for the beam, and the finite element method is then employed to solve the governing equations and to determine the dynamic characteristics of the beams. 5. Thesis organization 2 Beside Introduction section, the thesis contains 4 Chapters, a Conclusion section and a list of publications relevant to the thesis. References cited in the thesis are listed at the end of the thesis. CHAPTER 1. OVERVIEW 1.1. FGM beams Functionally graded materials (FGMs) can be considered as a new type of composite material. These materials are often formed from two or more constituent materials whose volume fraction varies in one or more predefined spatial directions. FGMs overcome the disadvantage problems such as delamination and stress concentration which often seen in conventional composite materials. With such advantages, FGMs have great potential in applications where the operating conditions are severe, including spacecraft heat shields, heat exchanger tubes, biomedical implants, flywheels, and plasma facings for fusion reactors, etc. [24]. FGM beams, the structure considered in this thesis, are assumed to be formed from two phases, metal and ceramic. The volume fraction of constituents is considered to vary in a spatial direction, namely in the beam thickness, by a power-law distribution as [3] , 1 2 , 2 1 2 n c c m h zzV V V h h         (1.1) where Vc, Vm are, respectively, the volume fraction of ceramic and metal; z is the co-ordinate in the thickness direction, and n is material grading index, defining the material distribution of the constituents. In addition to the distribution (1.1), several authors also considered the variation of the material properties in axial or both axial and thickness directions. 1.2. Investigations on FGM beams 1.2.1. Mechanical behavior of FGM beams The traditional analytical methods, especially Galerkin method, are employed by researchers in studying mechanical behavior of FGM beams [35-41]. Finite element method (FEM) is also widely used to study the behaviour of FGM beams. Several finite element beam models for analysis of FGM beams have been proposed in 3 recent years [59-64], for example the works by Alshorbagy et al. [25] Mohanty el al. [66, 67], Gan and Nguyen [70, 71, 72]. Eltaher et al. [73, 74] considered the physical neutral axis position in the derivation of a finite element model for free vibration analysis of macro/nano FGM beams. Jin and Wang [76] used the quadrature method to derive stiffness and mass matrices for free vibration analysis of FGM beams. Based on the first-order shear deformation theory, Frikha et al. [77] developed a mixed finite element formulation for bending analysis of FGM beams. 1.2.2. FGM beams with porosities Porosities can lower the material stiffness, and as a result they reduce the ability to resist external loads of structural components. Wattanasakulpong and Ungbhakorn [18], Wattanasakulpong and Chaikittiratana [19] proposed a simple model for free vibration analysis of FGM porous beams, in which the porosity volume is equally divided to both the ceramic and metal phases. The model has been employed by Ebrahimi and Zia [79] to study nonlinear free vibration of FGM Timoshenko beams. Chen et al. [16] proposed a concept “porosity coefficient” in their study of bending and stability of FGM porous beams. The model in [16] is then extended by the authors to nonlinear vibration of sandwich beams with FGM porous core [80], and free and forced vibration of FGM Timoshenko beams with porosities [81]. Shafiei and Kazemi [82] studied stability of nano/micro FGM porous beams with modification of the poroposity model in [18, 19] by considering non-uniform distribution of porosities in the beam cross sections. The non-uniform distribution of porosity model has also been employed to study vibration of 2D- FGM beams [83]. 1.2.3. FGM beams in thermal environment Chakraborty et al. [84] derived a finite element Timoshenko beam model for studying wave propagation in sandwich beams with FGM core in consideration of a uniform temperature rise. Based on the finite element method, Bhangale and Ganesan [85] investigated the effect of temperature on natural frequency and loss factor of FGM sandwich beams with visco-elastic core. Ching and Yen [86] presented a numerical solution for the themo-mechanical 4 deformation problem of FGM beams. The differential quadrature method (DQM) has been employed by Xiang and Yang [87] in studying vibration of non-uniform layer FGM Timoshenko beams prestress by temperature. Pradhan and Murmu [88] studied free vibration of FGM sandwich beams resting on an elastic foundation. DQM was also employed by Malekzadeh [89], Malekzadeh et al. [90] in free vibration analysis of FGM circular arches and FGM curved beams in thermal environment. Esfahani et al [92] examined the influence of elastic foundation support and temperature rise on the nonlinear stability of FGM Timoshenko beams by the general DQM. Mahi et al. [30] presented an analytical method to evaluate the effect of temperature rise on natural frequencies of shear deformable FGM beams. Wattanasakulpong et al. [21] constructed the governing equations of thermo-mechanical stability and free vibration of FGM beams. Ma and Lee [95] proposed an analytical solution for nonlinear behavior of FGM beam under thermal loading. Analytical method has also been employed by Eroglu in free vibration analysis of FGM beams in thermal environment [96]. Trinh et al. [98] presented an analytical method for vibration and stability analysis of FGM beams under thermo-mechanical loads. With the aid of Runge- Kutta method, Kiani et al. [99] examined the effect of environmental temperature on low velocity impact behavior of FGM beams. Ghiasian t al. [100] studied static and dynamic stability of FGM Euler-Bernoulli beams subjected to uniform temperature rise. Ebrahimi et al. [17] derived equations of motion for studying free vibration analysis of FGM Euler-Bernoulli beams with porosities in thermal environment. 1.2.4. FGM beams under moving loads Lagrange multiplayer method has been employed by Şimşek and his co-workers in studying vibration of FGM beams excited by moving loads [4, 5, 6, 8, 10, 11]. Yang et al. [104] studied vibration of cracked beams under a moving load by assuming an exponential variation of material properties in the thickness direction. The Ritz and differential quadrature methods was used by Khalili et al. [105] to investigate dynamic behavior of FGM beams subjected to a moving mass. Rajabi et al. [7] employed Petrov–Galerkin method to transfer a system of the fourth order differential equations of BFGM 5 beams under a moving oscillator to a system of second order differential equations, and then solving the system by Runge-Kutta method. Wang and Wu [22] employed Lagrange method to examine the effect of uniform temperature rise on dynamic bahaviour of Timoshenko beams formed from axially FGM. Taking the effect of neutral axial position into account, Gan and Nguyen [106] formulated a finite beam element for dynamic analysis of multi-span FGM Timoshenko beams. FEM has also been employed by Gan and his co-worker in studying axially FGM beams under moving loads [26], and FGM beams with an immediate support transverse by moving force [107]. 1.3. Studies on FGM beams in Vietnam Using an analytical m method, Nguyen et al. [111] studied the static bending and vibration of axially loaded FGM Timoshenko beam. Bending and vibration of FGM beams have also been considered by Thai and Vo [112] by different higher-order beam theories. Based on the third-order shear deformation theory, , Vo et al. [113] derived equations of motion for FGM sandwich beams with a homogeneous core, and then employed the finite element method to compute the natural frequencies of the beams. Vo et al [34] developed a finite element model for bending and free vibration analysis of FGM beams. The vibration and diagnosis problems of cracked beams have been considered by Huyen [114], Huyen and Khiem [115], Khiem et al [116, 117] by using analytical methods. Based on the co-rotational approach, Nguyen et al. [118, 119, 120] derived the finite element models for large displacement analysis of tapered FGM beams. The finite element method was also employed by Nguyen and his co-workers to study the large deformation analysis of FGM frames [121], FGM sandwich frames [33]. Recently, the effect of plastic deformation on stability behavior and nonlinear bending of FGM beams has been studied by the finite element method [122, 123, 124]. Vibration of FGM beams under moving loads has been investigated by several authors in recent years. Trung [13] studied vibration of FGM beams due to a moving mass or a moving harmonic load by the finite element method. Ha et al. derived a new finite element model for vibration analysis of multi-span FGM 6 beams under a harmonic load [14], non-uniform FGM beams under multiple moving loads [15]. Nguyen et al. [133] used polynomials derived by Kosmatka to derive a finite element formulation for vibration analysis of non-uniform FGM beams subjected to a variable speed moving load. The Kosmatka polynomials have also been employed by Nguyen et al. [9] in derivation of stiffness and mass matrices for analyzing 2-D FGM beams under a moving load. 1.4. Comments and study orientation As can be seen from the above literature review, the investigation on vibration of FGM beams under moving loads has been considered by very few authors in recent years. In [12], Ha has successfully derived finite element formulations for vibration analysis of FGM beams due to moving loads, but the effect of porosities and temperature has not been examined. The influence of porosities and temperature has been considered by several authors, but only on the free vibration problem. Rigidities and mass moments of FGM beams are altered when considering the effect of porosities. In addition, the beams are not only under thermal loading due to the temperature rise, but the elastic moduli are decreased also. These factors significantly influence dynamic behavior of the beams, and it is necessary to take them into consideration. From the above reason, this thesis investigates the vibration of FGM beams with porosities in thermal environment under moving loads. CHAPTER 2. FGM BEAMS IN THERMAL ENVIRONMENT 2.1. FGM beam under moving loads Fig. 2.1 illustrates a FGM beam with length of L, rectangular cross section with constant width b and height h. The beam is under actions of nF loads, F1, F2, FnF, moving from left to right with a constant speed v. The beam is assumed to be formed from two constituent materials, ceramic and metal, whose volume fraction varies according to a power law as 1 , 1 2 n c c m zV V V h        (2.1) in which Vc , Vm are the volume fraction of ceramic and metal, respectively; z is a co-ordinate along the thickness direction, and n 7 (nonnegative) is a material index, defining the distribution of the constituent materials. x yz bL h F1F2F nF y lç rçng z,w MÆt c¾t ngang dÇm gèm (Ec, Gc, c) kim lo¹i (Em, Gm, m) b h Fig.2.1. FGM beam with porosities under moving loads 2.2. Porosities of FGM beam In the model in [18, 19], the porosity volume fraction V (V<<1) is assumed equally divided into ceramic and metal phases. When the beam is in thermal environment, the material properties of the FGM beam are evaluated according to 1( , ) ( ) ( ) 2 ( ) ( ) ( ) 2 c m m m n c zP z T P T P T h VP T P T P T                     (2.3) where Pc and Pm are, respectively, the property of ceramic and metal, and they depend upon temperature T (K) of the environment; Vα is the porosity volume fraction. 2.3. Temperature in FGM beam Temperature distribution along the thickness direction of FGM beam can be obtained by solving Fourier equation [91, 103] ( ) 0d dTz dz dz      (2.4) with boundary conditions T = Tc at z = h/2 and T = Tm at z = - h/2. In Eq. (2.4), the thermal conductivity κ(z) is assumed to be temperature- independent. Solving Eq. (2.4) gives the temperature distribution along the beam thickness in the form 8 /2 /2 /2 ( ) / ( ) ( ) z h m c m h h dz dzT T T T z z        (2.6) It can be seen from above equation that if Tc = Tm then T = Tc = Tm. In this case, the temperature is the same at every points of the beam, and it is called the uniform temperature rise (UTR). In case Tc ≠ Tm, the temperature is a nonlinear function of the z co-ordinate. The temperature field thus is a nonlinear temperature rise (NLTR). In this thesis, a temperature rise T for the NLTR is defined in accordance with the works in [17, 21], that is T = Tc – Tm = Tc - T0 , with T0 = 300K is the referenced temperature. 2.4. Effect of temperature on material properties Touloukian [130] shows that a property P of a material depends on temperature by a nonlinear relation as 1 2 30 1 1 2 3( 1 )P P P T PT PT PT       (2.18) in which P0, P-1, P1, P2 and P3 the temperature-dependent coefficients. Figs. 2.2 and 2.3 illustrate the influence of the porosity volume fraction Vα and the temperature rise ΔT on Young’s modulus of a FGM beam formed from stainless steel SUS304 and alumina for various values of V and for ΔT = 500K. As can be seen from the figures, Young’s modulus decreases clearly when the effect of porosities is taken into account, for both the case of UTR and NLTR. By comparing Fig. 2.2(b) Fig. 2.3, one can see that the effective Young’s modulus E of the FGM beam decreases more significantly for the case of UTR. 2.5. Governing equations 2.5.1. Displacement field Axial and transverse displacements at any arbitrary point of the beam are given by 0 0 ( , , ) ( , ) ( , ) ( , , ) ( , ) u x z t u x t z x t w x z t w x t    (2.24) 9 in which u0(x,t) and w0(x,t) are, respectively, the axial and transverse displacements of a point on the mid-plane; θ(x,t) is the cross-sectional rotation, and t is the time variable. 2.5.2. Strain and stress fields The normal and shear strains resulted from Eq. (2.24) are as follows , 0, , z , , 0, xx x x x x z x x u u z u w w            (2.25) The notation (..),x in the above equation is used to indicate the derivative with respect to x and (..),z is the derivative with respect to z variable. Based on Hook’s law, the normal and shear stresses corresponding to the strains in (2.25) are given by     0, , 0, ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) xx xx x x xz xz x z T E z T E z T u z z T G z T G z T w               (2.26) in which E(z,T) and G(z,T) are, respectively, the effective Young’s modulus and shear modulus, and ψ is the shear correction factor. -0.5 -0.25 0 0.25 0.5 180 200 220 240 260 280 300 320 340 z/h E (G Pa ) -0.5 -0.25 0 0.25 0.5 180 200 220 240 260 280 300 320 340 z/h E (G Pa ) n=5 n=10 n=0.1 n=0.5 n=10 n=1 n=5 n=1 n=0.5 n=0.1 (a) T=0 K,V=0 (b) T=0 K,V=0.1 Fig.2.2. Influence of porosity volume fraction on the effective Young’s modulus of FGM porous beam 10 -0.5 -0.25 0 0.25 0.5 140 160 180 200 220 240 260 280 z/h E (G Pa ) -0.5 -0.25 0 0.25 0.5 140 160 180 200 220 240 260 280 z/h E( G Pa ) n=0.5 n=1 n=5 n=10 n=0.1 n=0.5 n=5 n=10 (a) NLTR, T=500 K, V=0.1 (b) UTR, T=500 K, V=0.1 n=0.1 n=1 Fig.2.3. Influence of temperature on the effective Young’s modulus of FGM porous beam for UTR and NLTR 2.5.3. Strain energy The strain energy (U ) can be written in the form  22 211 0, 12 0, , 22 , 33 0, 0 1 2 2 L x x x x xU A u A u A A w dx           (2.27) in which V in the beam volume, A is the cross-sectional area; A11, A12, A22 and A33 are, respectively, the axial, axial-bending coupling, bending and shear rigidities of the beam. 2.5.4. Strain energy due to initial thermal stress Assuming the beam is free stress at the reference temperature T0 and it is subjected to thermal stress due to the temperature change. The initial thermal stress resulted from a temperature T is given by [30, 91] ( , ) ( , )Txx E z T z T T    (2.29) The strain energy caused by the initial thermal stress has the form [17, 30] 2 20, 0, 0 1 1( , ) ( , ) 2 2 L T x T x V U E z T z T Tw dV N w dx     (2.30) where NT is the axial force resultant due to the initial thermal stress. 11 2.5.5. Kinetic energy The kinetic energy of the FGM with porosities has the form 2 2 211 0 12 0 22 0 0 1 ( ) 2 2 L I u I u I dxw           (2.32) in which I11, I12 and I22 are the mass moments. 2.5.6. Potential of external loads The external loads considered in the present thesis a
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