STATIC AND DYNAMIC ANALYSIS OF FUNCTIONALLY GRADED FLAT PANELS A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Bachelor of Technology In Mechanical Engineering By Bikash Kumar Majhi Roll No.:109ME0159 DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL INSTITUTE OF TECHNOLOGY ROURKELA-769008 MAY 2013 STATIC AND DYNAMIC ANALYSIS OF FUNCTIONALLY GRADED FLAT PANELS A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Bachelor of Technology In Mechanical Engineering By Bikash Kumar Majhi Roll No.:109ME0159 Under the guidance of Prof. S. K. Panda DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL INSTITUTE OF TECHNOLOGY ROURKELA-769008 MAY 2013 NATIONAL INSTITUTE OF TECHNOLOGY ROURKELA CERTIFICATE This is to certify that the work in this thesis entitled “Static and Dynamic analysis of Functionally Graded Flat Panels” by Bikash Kumar Majhi, roll no 109ME0159 has been carried out under my supervision in partial fulfilment of the requirements for the degree of Bachelor of Technology in Mechanical Engineering during session 20122013 in the Department of Mechanical Engineering, National Institute of Technology, Rourkela. To the best of my knowledge, this work has not been submitted to any other University/Institute for the award of any degree or diploma. Dr. Subrata Kumar Panda Date: 08/05/2013 (Supervisor) Assistant Professor Department of Mechanical Engineering National Institute of Technology, Rourkela ACKNOWLEDGEMENT I wish to express my sincere gratitude to Dr. Subrata Kumar Panda for his inspiring encouragement, guidance and efforts taken throughout the entire course of this work. His constructive criticism, timely help, and efforts made it possible to present the work contained in this Thesis. I am grateful to Prof. S.K. Sarangi, Director, and Prof K.P Maity, Head of the Department, Mechanical Engineering, for their active interest and support. I would like to thank Mr. Girish Kumar Sahu, M.Tech and Mr. Pankaj Katariya, M.Tech (Res) , Department of Mechanical Engineering, National Institute of Technology, Rourkela for their constant help in understanding of the technical aspects of the project. I will also be grateful to Ph.D scholar Mr. Vishesh Ranjan Kar, for his constant help in the successfully carrying out the new results. I express my deep sense of gratitude and reverence to my beloved parents for their blessings, patience and endeavour to keep my moral high at all times. Last but not the least, I wish to express my sincere thanks to all those who directly or indirectly helped me at various stages of this work. Bikash Kumar Majhi 109ME0159 CONTENTS Pages 1. INTRODUCTION 1-3 1.1 Background 2 1.2 Application of FGM 3 1.3 Objective of the work 3 2. LITERATURE REVIEW 4-7 2.1 Static analysis 5 2.2 Dynamic analysis 6 3. FINITE ELEMENT FORMULATION 8-12 4. EFFECTIVE MATERIAL PROPERTY 13-16 4.1 Exponential law 14 4.2 Power law 14 4.3 FE modelling of FGM plates 16 5. STATIC ANALYSIS 17-26 5.1 FG model (ANSYS) 18 5.2 Convergence and validation 18 5.3 Numerical results 19 5.3.1 Aluminium/Zirconia plate 20 5.3.2 Different boundary conditions 21 5.3.3 Aluminium/Alumina plate 23 5.3.4 Silicon Nitride/Stainless steel plate 25 6. DYNAMIC ANALYSIS 27 6.1 FG model (ANSYS) 28 6.2 Numerical results 28 7. CONCLUSION 31 7.1 Conclusion 32 7.2 Future scope of work 32 8. REFERENCE 33 9. APPENDIX-A 36 ABSTRACT Functionally graded materials have received a lot of interest in recent days by their diversified and potential applications in aerospace and other industries. They have high specific mechanical properties and high temperature capabilities which makes them special over all the exiting advanced materials. The present work investigated static and dynamic analysis of functionally graded plate. The material properties vary continuously from metal (bottom surface) to ceramic (top surface). The effective material properties of functionally graded materials for the plate structures are assumed to be temperature independent and graded in the plate thickness direction according to a power law distribution of the volume fractions of the constituents. The present model is developed using ANSYS parametric design language code in the ANSYS platform. An eight noded isoparametric quadrilateral shell element is used to discretise the present model for both static as well as dynamic analysis. A convergence test has been done with different mesh refinement and compared with published results. The parametric study indicates that the power-law indices, thickness ratios, aspect ratios, support conditions and different material properties have significant effect on non-dimensional mid-point deflection. i LIST OF FIGURES Figure No Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 Figure 14 Pages Geometry and dimensions of the FG plate Variations of volume fraction of ceramic through non-dimensional thickness coordinate Variations of volume fraction of metal through non-dimensional thickness coordinate FGM plate modelled in ANSYS 13.0 using SHELL281 element Convergence study Variation of non-dimensional mid-point deflection with different mesh size in ANSYS SHELL 181 model for simply supported FGM plate Static analysis of Aluminum/Zirconia plate Variation of non-dimensional mid-point deflection with different mesh size in ANSYS SHELL 181 model for simply supported FGM plate Variation of non-dimensional mid-point deflection with different thickness ratio ( a/h ratio ) in ANSYS SHELL 281 model for simply supported FGM plate 9 15 Variation of non-dimensional mid-point deflection with different aspect ratio ( a/b ratio ) in ANSYS SHELL 281 model for simply supported FGM plate Different boundary conditions Non-dimensional deflection vs. Length to thickness ratio square FGM plate with simply supported (SSSS) boundary condition Non-dimensional deflection vs. Length to thickness ratio square FGM plate with clamped (CCCC) boundary condition Non-dimensional deflection vs. Length to thickness ratio square FGM plate with clamped-simply supported (CSCS) boundary condition Non-dimensional deflection vs. Length to thickness ratio square FGM plate with clamped-simply supported (SSCC) boundary condition Aluminum/Alumina plate Variation of non-dimensional mid-point deflection with different mesh size in ANSYS SHELL 281 model for simply supported FGM plate Variation of non-dimensional mid-point deflection with different thickness ratio ( a/h ratio ) in ANSYS SHELL 281 model for simply supported FGM plate 21 ii 15 16 19 20 20 21 22 22 23 23 24 LIST OF FIGURES Figure No Figure 15 Figure 16 Figure 17 Figure 18 Figure 19 Figure 20 Figure 21 Figure 22 Figure 23 Pages Variation of non-dimensional mid-point deflection with different aspect ratio ( a/b ratio ) in ANSYS SHELL 281 model for simply supported FGM plate Silicon Nitride/Stainless steel plate Variation of non-dimensional mid-point deflection with different mesh size in ANSYS SHELL 281 model for simply supported FGM plate Variation of non-dimensional mid-point deflection with different thickness ratio ( a/h ratio ) in ANSYS SHELL 281 model for simply supported FGM plate 24 Variation of non-dimensional mid-point deflection with different aspect ratio ( a/b ratio ) in ANSYS SHELL 281 model for simply supported FGM plate Dynamic analysis of Aluminum/Zirconia plate Deflection of mid-point of simply supported FG flat panel with n=0 Deflection of mid-point of simply supported FG flat panel with n=1 Deflection of mid-point of simply supported FG flat panel with n=2 Deflection of mid-point of simply supported FG flat panel with n= ∞ Deflection of mid-point of simply supported FG flat panel with n=0 in time interval of 0.0001 to 0.012s 26 iii 25 25 28 29 29 30 30 LIST OF TABLES Table No Table 1 Table 2 Pages Properties of the FGM plate constituents Convergence study of non-dimensional mid-point deflection of simply supported FG (Al2O3/ZrO2) flat panel with a/b = 1 and a/h = 5) iv 16 19 NOMENCLATURE a Length of plate b Width of plate h Height of plate a/h Thickness ratio a/b Aspect ratio u, v, w Displacement field in x, y, z direction u0, v0, w0 Mid-plane displacement in x, y, z direction x , y , z Rotational displacement in x, y, z direction xx , xy , xz Lateral strain xz , xy , yz Shear strain Rx, Ry, Rz Radius of curvature in x, y, z direction E(z) Young’s modulus of material Poisson’s ratio Displacement vector Strain vector [B] Strain-displacement matrix [D] Rigidity matrix [K] Stiffness matrix [M] Mass matrix J Jacobian matrix F(t) Time-dependent force Ec Young’s modulus of ceramic Em Young’s modulus of metal c Density of ceramic m Density of metal c Poisson’s ratio of ceramic m Poisson’s ratio of ceramic v Chapter I Introduction CHAPTER-1 INTRODUCTION 1.1 Background Laminated composites have received a lot of interest in recent days by diversified and potential applications in automotive and aerospace industry due to their strength to weight, stiffness to weight ratio, low fatigue life and toughness and other higher material properties. These are made from two or more constituent materials which have different chemical or physical properties and produced a material having different behaviour from the individual. These are used in buildings, storage tanks, bridges etc. Each layer is laminated in order to get superior material properties. The individual layer has high strength fibres like graphite, glass or silicon carbide and matrix materials like epoxies, polyimides. By varying the thickness of laminas desired properties (strength, wear resistance, stiffness) can be achieved. Although these materials have superior properties, their major drawback is the weakness of laminated materials. This is known as delamination phenomenon which leads to the failure of the composite structure. Residual stresses are present due to difference in thermal expansion of the matrix and fibre. It is well known that at high temperature the adhesive being chemically unstable and fails to hold the lamination. Sometimes due to fibre breakdown it also prematurely fails. Functionally Graded Material (FGM) is combination of a ceramic and a metal. A material in which its structure and composition both varies gradually over volume in order to get certain specific properties of the material hence can perform certain functions. The properties of material depend on the spatial position in the structure of material. The effect of inter-laminar stress developed at the laminated composite interfaces due to sudden change of material properties reduced by continuous grading of material properties. Generally microstructural heterogeneity or non-uniformity is introduced in functionally graded material. The main purpose is to increase fracture toughness, increase in strength because ceramics only are brittle in nature. Brittleness is a great disadvantage for any structural application. These are manufactured by combining both metals and ceramics for use in high temperature applications. Material properties are varies smoothly and continuously in one or many directions so FGMs are inhomogeneous. FGM serves as a thermal barrier capable of withstanding 2000K surface temperature. Fabrication of FGM can be done by different processing such as layer processing, melt processing, particulate processing etc. FGM has the ability to control shear deformation, corrosion, wear, buckling etc. and also to remove stress 2 concentrations. This can be used safely at high temperature also as furnace liners and thermal shielding element in microelectronics and thermal protection systems for spacecraft, hypersonic and supersonic planes and in combustion chamber also. 1.2 Application of FGM i. Engineering Application a. Shafts b. Engine parts c. Blades of turbine ii. Aerospace Engineering a. Rocket engine components b. Aerospace parts and skins iii. Electronics a. Sensor b. Actuator c. Integrated chips d. Semiconductor iv. Biomaterials a. Artificial bones b. Drug delivery system 1.3 Objective of the work The objective of the present study is to analyse the static and dynamic behaviour of FG flat panel under different volume fraction indices, aspect ratios, thickness ratios and different support conditions. ANSYS parametric design language (APDL) code is use to develop the model in ANSYS13.0 platform and solve the problem using appropriate algorithm. The present model is discretised by using an eight noded isoparametric quadrilateral shell element (SHELL281), as defined in the ANSYS element library. Three types of FG flat panels are used in this study namely, Aluminium/Alumina, Aluminium/Zirconia and Silicon Nitride/Stainless steel. The effects of different parameters on non-dimensional mid-point deflection are studied. 3 Chapter II Literature review CHAPTER-2 LITERATURE REVIEWS FG material plates have created revolution in aerospace industry for its thermal properties, multifunctionalities. It also provides opportunities to take the benefits of different material system. Its static and dynamic analysis is necessary to estimate the properties of flat panels. Many researchers reported static and dynamic behaviour of functionally graded plates based on different theories and developed new methods of solutions. 2.1 Static and vibration analysis Talha and Singh [1] investigated the free vibration and static analysis of rectangular FGM plates using higher order shear deformation theory with a special modification in the transverse displacement in conjunction with finite element models. Neves et al. [2] studied the static deformations analysis of functionally graded plates by collocation with radial basis functions, according to a sinusoidal shear deformation formulation for plates. Aragh and Hedayati [3] studied the characteristics of free vibration and static response of a 2-D FGM open cylindrical shell. Formulations are done by 2-D generalized differential quadrature method (GDQM). Ferreira et al. [4] studied static deformations of functionally graded square plates of different aspect ratios using meshless collocation method, the multiquadric radial basis functions and a third-order shear deformation theory. Reddy[5] studied static and dynamic analysis of FGM plates using third-order shear deformation theory. Navier solutions are obtained for a simply supported square plate. Abrate[6] investigated static, buckling and free vibration deflections of FGM plates by using classical plate theory, FSDT model and HSDT model. Zenkour[7] studied the static behaviour of a rectangular FG plate under simply supported condition and subjected to uniform transverse load. Ferreira et al. [8] studied static deformations of simply supported functionally graded plate by using HSDT and multiquadric radial basis functions. Vel and Batra[9] investigated the exact 3-D elasticity solutions of simply supported rectangular FG plates under thermo-mechanical load. The author has assumed power law for material volume fractions. The exact solutions of displacements and stresses are used to find out the accuracy of the solutions. Qian et al.[10] investigated plain strain static thermostatic deformations of simply supported thick rectangular FG elastic plate. Displacement and stress are computed and validated from the 3D exact solutions of the problem. Ramirez et al.[11] studied static analysis of 3D, elastic, anisotropic FG plates. The 5 author has taken simply supported graphite/epoxy material for analysis. Zenkour [12] further studied the static response of FG plates using shear deformation plate theory using power law for grading. Bhangale and Ganesan[13] investigated static analysis of simply supported FG plates which are exponentially graded in the thickness direction. Aghdam et al.[14] studied static analysis foe bending of FG clamped thick plates. The solutions are compared with the solutions of finite element code ANSYS, power law is used for grading the properties in thickness direction. Neves et al.[15] investigated the static deformations of FG square plates using radial basis function. Talha and Singh [16] investigated the static and free vibration analysis using C0 finite element with 13 degrees of freedom per node and formulated by HSDT. Nguyen-Xuan et al.[17] studied the static, free vibration and mechanical/thermal buckling problems of FG plates by Reisnner/Mindlin plate theory. 2.2 Dynamic Analysis Yang and Shen [18] studied dynamic response of FGM thin plates under initial stress and partially distributed impulsive lateral loads. The author used silicon nitride/ stainless steel rectangular plates, assumed temperature dependent material properties clamped on two opposite edges, used power law for grading and used Modal superposition method for transient response. In 2001 Yang and Shen [19] studied free and forced vibration analysis for the same plate and found functionally graded plate with material properties intermediate to isotropic material do not necessarily have intermediate natural frequency if thermal effects are considered. Liew et al. [20] investigated dynamic stability of symmetrically laminated FGM rectangular plates under uniaxial plane load. Formulation is done by Reddy’s thirdorder shear deformation theory and material is silicon nitride and stainless steel. [21]Kim studied vibration characteristics of rectangular FGM plate under initial stress. Third-order shear deformation plate theory is adopted and Rayleigh-Ritz procedure is applied for getting frequency equation. Lanhe et al. [22] investigated dynamic stability of thick FGM plate under aero-thermo-mechanical loads and used novel numerical solution technique. The equations for dynamic analysis are derived by Hamilton’s principle. For different parameters dynamic instability regions are studied. Ansari and Darvizeh [23] investigated vibrational behaviour of functionally graded shells, based on first-order shear deformation shell theory. The grading functions are power law, sigmoid and exponential distribution. Behjat et al. [24] studied dynamic response, static bending of functionally graded piezoelectric material plate (PZT4/PZT-5H), formulated by using potential energy and Hamilton’s principle. Effects of material composition and boundary conditions on dynamic response are also studied. Sladek 6 et al. [25] investigated dynamic analysis of FG plates by MPLG method. For displacement field author used Reissner-Mindlin plate bending theory. Simply supported and clamped boundary conditions are taken in to consideration. Wen et al. [26] studied 3-D analysis of isotropic and orthotropic FG plates with simply supported edge under dynamic loads. The equations formulated is based on state-space approach in Laplace transform domain and solved by RBF method. Grading has done by exponential method as well as volume fraction law. Shariyat [27] studied the vibration and dynamic buckling response of rectangular FG plates under thermo-mechanical loading. A nine noded second-order formulation has done and graphs are studied under temperature dependent material properties. From the above study it has been seen that very few researcher studied the dynamic analysis of FG plates. Since most of the practical cases deals with transient or dynamic load, its responses has to be analysed with different parameters like volume fraction index. This work analysed dynamic responses of aluminium/zirconia flat panel under step load with different volume fraction index (n=0, 1, 2, ∞); 7 Chapter III Finite element formulation CHAPTER-3 FINITE ELEMENT FORMULATION In the present analysis, a FG plate of uniform thickness h with rectangular base of sides a and b is established through APDL code and shown in the Figure 1. The FG plate model is developed in ANSYS based on the inbuilt FSDT kinematics. The displacements field u, v and w at any point along x, y and z axes can be written as follows: u ( x, y, z ) u 0 ( x, y ) z x ( x, y ) v( x, y, z ) v 0 ( x, y ) z y ( x, y ) (1) w( x, y, z ) w0 ( x, y ) z z ( x, y ) where, u, v and w denote displacements; u 0 , v 0 and w0 are the mid-plane displacements in x, y, z axes respectively and , and are the shear rotations. z a y b Ceramic rich x +h/2 -h/2 Metal rich Figure 1: Geometry and dimensions of the FG plate The linear stains corresponding to the displacement field is xx u w , x Rx yz y w v , y Ry yy v w , y Ry xz x w u , y Rx zz z , xy For the flat panel i.e. plate, Rx = Ry = Rxy = 9 u v 2w y x Rxy (2) It can be rewritten as , yy , zz , yz , xz , xy , T xx k x' xx x0 ' 0 ky yy y 0 k z' zz z l 0 z ' k yz yz yz 0 k ' xz xz xz' 0 xy xy k xy 0 where x (3) (4) w0 u 0 w0 v 0 0 u 0 w0 0 v 0 w0 , z 0, yz0 y , xz x , , y y Ry y Ry y Rx x Rx y u 0 v 0 2w0 ' x z ' y z ' , kx , k y z , k yz' z , , ky y x Rxy y Ry y Ry x Rx y z ' k xz' z x , k xy x y x Rxy y Rx xy0 The linear constitutive relations are xx Q11 Q12 yy Q21 Q22 zz Q31 Q32 0 yz 0 xz 0 0 0 xy 0 Q13 Q23 Q33 0 0 0 0 0 0 Q44 0 0 0 0 0 0 Q55 0 0 xx 0 yy 0 zz 0 yz 0 xz Q66 xy (5) where, E ( z )(1 v 2 ) (1 3v 2 2v3 ) E ( z )v(1 v) Q12 Q13 Q23 (1 3v 2 2v3 ) E( z) Q44 Q55 Q66 2(1 v) Q11 Q12 Q13 The modulus of elasticity E(z) and elastic coefficients Qij vary through the plate thickness. For the implementation of finite element method, the developed model is discretised by using an eight noded isoperimetric quadrilateral shell element (SHELL281), as defined in the ANSYS element library. This element is suitable for analysing thin to moderately-thick shell structures with six degrees of freedom.(three translations and three rotations) at each 10 node in the x, y and z directions. The displacements are expressed in terms of interpolation functions (Ni) 8 N i i 1 (6) i T where, i u 0 , v0 , w0 , x , y , z . The interpolation functions for eight noded isoperimetric quadrilateral shell element in natural (ξ-η) coordinates are given as [15] 1 N1 (1 )(1 )( 1) , 4 1 N 2 (1 )(1 )( 1) , 4 1 N3 (1 )(1 )( 1) , 4 1 N 4 (1 )(1 )( 1) , 4 1 N5 (1 2 )(1 ) , 2 N6 1 1 1 2 , 2 1 1 2 1 , 2 N8 1 1 1 2 2 N7 (7) The strain vector in terms of nodal displacement vector can be written as B (8) where, B is the strain-displacement matrix containing interpolation functions and derivative operators and { } is the nodal displacement vector. The generalized stress-strain relation with respect to its reference plane can be written as { } D{ } (9) where x y z xy yz xz and x T y z xy yz xz are T the linear stress and strain vector, respectively and D is the rigidity matrix. [Q] [T ]T [Q][T ] [ D] h /2 h /2 [Q]dz (10) (11) The elemental stiffness matrix K e and the mass matrix [M]e are integrated by using Gauss-quadrature integration over the domain to obtain the global stiffness and mass matrices and this can be conceded as K e M e 1 1 1 1 1 B D B J d d T 1 1 N m N J d d 1 11 (12) T (13) where, J is the determinant of the Jacobian matrix and [N] is the interpolation function matrix. The jacobian is used to map the domain from natural coordinate to the general coordinate. The governing equation of static analysis of FG plate under force F can be expressed as follows: K { } F 0 (14) where, [K] is the global stiffness matrices. Eqn. (8) is a generalized eigenvalue problem and non-dimensional central point deflection can be found from this equation. The governing equation of dynamic analysis of FG plate under dynamic load F (t) can be expressed as follows: K { } M { } F (t ) 12 (15) Chapter IV Effective material properties CHAPTER-4 EFFECTIVE MATERIAL PROPERTIES The effective material properties of the FGM plate are assumed to be varying continuously along their thickness direction as discussed earlier and are obtained by using a simple powerlaw distribution or exponential law which counts the volume fraction of each constituent. 4.1 Exponential law Exponential law of grading FGM states that for a FGM structure of uniform thickness ‘h’, the material properties ‘P(z)’ at any point located at ‘z’ distance from the mid-plane surface is given by: P( z ) Pe t 2z 1 h , where, 1 P ln t 2 Pb (16) P( z ) denotes material property like Young’s modulus of elasticity (E), shear modulus of elasticity (G), Poisson’s ratio ( ), material density ( ) of the FGM structure. Pt and Pb are the material properties at the top (z=+h/2) and bottom (z=-h/2) surfaces. is the material grading indexes which depend on the design requirements. 4.2 Power law The power-law distribution of a panel considered from the mid-plane reference plane can be written as z 1 Vf h 2 n (17) where, n is the power-law index, 0 n . The variations of volume fraction of the ceramic and metal phase through the non-dimensional thickness coordinate are plotted in Figure 2 and 3 for five different values of power-law indices (n = 0.2, 0.5, 1, 2 and 10). The functionally graded material with two constituents and their properties such as, Young’s modulus E and the mass density ρ have been obtained using the following steps. n z 1 E Ec Em Em h 2 (18) n z 1 c m m h 2 (19) n z 1 c m m h 2 14 (20) In the present work, the power- law distribution is used for the continuous gradation of material properties in thickness direction. The effective material properties are calculated based on the Eqns. (18), (19) and (20), when z = - h/2, E= Em , ρ = ρm and = m similarly, when z = + h/2; E= Ec , ρ = ρc and = c i.e., the material properties vary continuously from metal at the bottom surface to ceramic at the top surface. The Poisson’s ratio ν is assumed to be constant throughout the thickness of the shell panel. The properties of the FGM constituents at room temperature (27oC) are used for the analysis and presented in Table 1. The different material properties are used to analyse the responses for throughout the study. Volume fraction of ceramic 1.0 0.8 n = 0.2 n = 0.5 n=1 n=2 n = 10 0.6 0.4 0.2 0.0 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 Non-dimensional thickness coordinate Figure 2: Variations of volume fraction of ceramic through non-dimensional thickness coordinate 1.0 Volume fraction of metal 0.8 0.6 0.4 0.2 0.0 -0.6 n = 0.2 n = 0.5 n=1 n=2 n = 10 -0.4 -0.2 0.0 0.2 0.4 0.6 Non-dimensional thickness coordinate Figure 3 Variations of volume fraction of metal through non-dimensional thickness coordinate 15 Table 1: Properties of the FGM plate constituents Properties Young's Modulus E (GPa) Poisson's Ratio Density (Kg/m3) Aluminium (Al) 70 0.3 2707 Alumina (Al2O3) 380 0.3 3800 Zirconia (ZrO2) 151 0.3 3000 Silicon Nitride (Si3N4) 348.43 0.28 2370 Steel (SUS304) 201.04 0.28 8166 Materials 4.3 FE Modelling of FGM plates: FGM plates with different length to thickness ratio( ) , aspect ratio (a/b) are analysed in this experiment. The analysis is performed in commercially available software (ANSYS 13.0). The loading conditions are assumed to be static. The element chosen for this analysis is SHELL281, which is a layered version of the 8-node structural shell model. This is suitable for analysing thin to moderately-thick shell structures. This shell element has six degrees of freedom at each node namely three translations and three rotation in the nodal x, y and z directions respectively. The FGM plate is modelled in ANSYS 13.0 as shown in the fig 4. Figure 4: FGM plate modelled in ANSYS 13.0 using SHELL281 element 16 Chapter V Static analysis CHAPTER-5 STATIC ANALYSIS 5.1 FG model (ANSYS) The static responses of the FG plates are analysed using ANSYS 13.0 under static surface load for simply supported boundary condition for Aluminium/Zirconia FG flat panel. The computed results are validated and compared with those available in the literature. The analysis is carried out for thickness ratio (a/h) = 5, aspect ratio (a/b=1) with different volume fraction indices. APDL code has been developed in ANSYS 13.0 for analysing the above panel. The following non-dimensional parameters are used: Central deflection ̅ Load parameter (21) 5.2 Convergence and validation: The static analysis of FG plates is analysed by ANSYS 13.0 using APDL program. The aluminium/zirconia material is analysed for different mesh size under simply supported boundary condition. These boundary conditions are there to reduce the no of unknowns from the final equation and in order to avoid rigid body motion. Simply-supported (SSSS): v0 w0 y z 0 at x=0 and a u 0 w0 x z 0 at y = 0 and b For validation of obtained data and the efficiency of present finite element model, the results obtained using FG flat panel model is compared with the published literature. The nondimensional mid-point deflection ̅ of square simply supported FG flat panel (a/h =5) are computed for five different power-law indices (n=0, 0.5,1,2, ∞) and tabulated in Table. It is observed from the results that convergence satisfies at a (16 x 16) mesh and the differences between present and published results are negligible. 18 Table 2: Convergence study of non-dimensional mid-point deflection of simply supported FG (Al2O3/ZrO2) flat panel with a/b = 1 and a/h = 5) Mesh size n Ref.[4] 6x6 8x8 10x10 12x12 14x14 16x16 18x18 Ceramic 0.024875 0.024855 0.024845 0.02484 0.024835 0.024835 0.024835 0.0247 0.5 0.031245 0.031215 0.0312 0.031195 0.03119 0.031185 0.03118 0.0313 1 0.03459 0.034555 0.03454 0.03453 0.03452 0.034515 0.034515 0.0351 2 0.037955 0.03792 0.0379 0.03789 0.03788 0.037875 0.03787 0.0388 Metal 0.05366 0.053615 0.053595 0.053585 0.053575 0.05357 0.05357 0.0534 0.055 Ceramic n=0.5 n=1 n=2 Metal Non dimensional deflection 0.050 0.045 0.040 0.035 0.030 0.025 6x6 8x8 10 x 10 12 x 12 14 x 14 16 x 16 18 x 18 Ref [4] Mesh size Figure 5: Variation of non-dimensional mid-point deflection with different mesh size in ANSYS SHELL 181 model for simply supported FGM plate 5.3 Numerical results: In this section some new problems have been solved and new data, graphs are given for different parameters and responses are discusses. The variations of non-dimensional midpoint deflection of FG plate for simply supported boundary condition (SSSS) with different mess size in ANSYS SHELL 181 element are plotted in figure 6. The results are obtained using other geometric properties i.e., thickness ratio, aspect ratio for five fraction indices (n = 0, 0.5,1,2 and ∞). And also there are results for different material properties and different mesh size in ANSYS. Different material properties of FGM plates are given in table 1. 19 Graphs varying mesh size, thickness ratio and aspect ratio are plotted for aluminium/ zirconia, aluminium/ alumina, silicon nitride/stainless steel respectively. And also graphs are plotted for different boundary conditions for aluminium, zirconia FGM plates. Figure 7 and figure 8 shows the variation of thickness ratio and aspect ratio and central deflection varied. Figure 9-12 shows the central deflection varied under different boundary conditions for aluminium/zirconia plate. Figure 13-15 shows the behaviour of mid-point deflection if mesh size, thickness ratio and aspect ratio varies respectively for aluminium/alumina plate. For aluminium/stainless steel flat panel, the same behaviours are studied and plotted in figure 1618. 5.3.1 Aluminium/Zirconia plate: 0.055 Ceramic n=0.5 n=1 n=2 Metal Non dimensional deflection 0.050 0.045 0.040 0.035 0.030 0.025 6x6 8x8 10 x 10 12 x 12 14 x 14 16 x 16 18 x 18 Mesh size Figure 6: Variation of non-dimensional mid-point deflection with different mesh size in ANSYS SHELL 181 model for simply supported FGM plate 0.30 0.28 Ceramic n=0.5 n=1 n=2 Metal 0.26 Non dimensional deflection 0.24 0.22 0.20 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0 5 10 15 20 a/h ratio Figure 7: Variation of non-dimensional mid-point deflection with different thickness ratio ( a/h ratio ) in ANSYS SHELL 281 model for simply supported FGM plate 20 Ceramic n=0.5 n=1 n=2 Metal 500 450 Non dimensional deflection 400 350 300 250 200 150 100 50 0 -50 0 5 10 15 20 a/b ratio Figure 8: Variation of non-dimensional mid-point deflection with different aspect ratio ( a/b ratio ) in ANSYS SHELL 281 model for simply supported FGM plate 5.3.2 Different boundary conditions: Non Dimesional Center Deflection SSSS 0.045 n = 0.5 n=1 0.040 n=2 n=5 0.035 0.030 0.025 0 20 40 60 80 100 Length to thickness ratio Figure 9: Non-dimensional deflection vs. Length to thickness ratio square FGM plate with simply supported (SSSS) boundary condition 21 CCCC Non Dimesional Center Deflection 0.020 n = 0.5 n=1 0.015 n=2 n=5 0.010 0.005 0 20 40 60 80 100 Length to thickness ratio Figure 10: Non-dimensional deflection vs. Length to thickness ratio square FGM plate with clamped (CCCC) boundary condition CSCS Non Dimesional Center Deflection 0.030 n = 0.5 n=1 0.025 n=2 n=5 0.020 0.015 0.010 0 20 40 60 80 100 Length to thickness ratio Figure 11: Non-dimensional deflection vs. Length to thickness ratio square FGM plate with clamped-simply supported (CSCS) boundary condition 22 SSCC 0.030 Non Dimesional Center Deflection n = 0.5 n=1 0.025 n=2 n=5 0.020 0.015 0.010 0 20 40 60 80 100 Length to thickness ratio Figure 12: Non-dimensional deflection vs. Length to thickness ratio square FGM plate with clamped-simply supported (SSCC) boundary condition 5.3.3 Aluminium/Alumina plate: Ceramic n=0.5 n=1 n=2 Metal 0.055 Non dimensional deflection 0.050 0.045 0.040 0.035 0.030 0.025 0.020 0.015 0.010 (6 x 6) (8 x 8) (10 x 10) (12 x 12) (14 x 14) (16 x 16) (18 x 18) Mesh size Figure 13: Variation of non-dimensional mid-point deflection with different mesh size in ANSYS SHELL 281 model for simply supported FGM plate 23 Ceramic n=0.5 n=1 n=2 Metal 0.30 Non dimensional deflection 0.25 0.20 0.15 0.10 0.05 0.00 0 5 10 15 20 a/h ratio Figure 14: Variation of non-dimensional mid-point deflection with different thickness ratio ( a/h ratio ) in ANSYS SHELL 281 model for simply supported FGM plate 500 Ceramic n=0.5 n=1 n=2 Metal Non dimensional deflection 400 300 200 100 0 0 5 10 15 20 a/b ratio Figure 15: Variation of non-dimensional mid-point deflection with different aspect ratio ( a/b ratio ) in ANSYS SHELL 281 model for simply supported FGM plate 24 5.3.4 Silicon Nitride/Stainless steel plate: 0.055 Non dimensional deflection 0.050 Ceramic n=0.5 n=1 n=2 Metal 0.045 0.040 0.035 0.030 0.025 0.020 (6 x6) (8 x 8) (10 x 10) (12 x 12) (14 x 14) (16 x 16) (18 x 18) Mesh size Figure 16: Variation of non-dimensional mid-point deflection with different mesh size in ANSYS SHELL 281 model for simply supported FGM plate Ceramic n=0.5 n=1 n=2 Metal 0.30 0.28 Non dimensional deflection 0.26 0.24 0.22 0.20 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 0 2 4 6 8 10 12 14 16 a/h ratio Figure 17: Variation of non-dimensional mid-point deflection with different thickness ratio ( a/h ratio ) in ANSYS SHELL 281 model for simply supported FGM plate 25 Ceramic n=0.5 n=1 n=2 Metal Non dimensional deflection 500 400 300 200 100 0 0 2 4 6 8 10 12 14 16 a/b ratio Figure 18: Variation of non-dimensional mid-point deflection with different aspect ratio ( a/b ratio ) in ANSYS SHELL 281 model for simply supported FGM plate 26 Chapter VI Dynamic analysis CHAPTER-6 DYNAMIC ANALYSIS 6.1 FG model (ANSYS) Rectangular simply supported Aluminium/Zirconia FG flat panel has been developed in ANSYS13.0 platform. Time dependant step load has been taken for transient dynamic analysis. Step type loading has been taken in to consideration. From time 0 to 0.001s force is zero and from 0.001 to 0.002s force is 10kN. APDL code has been developed in ANSYS 13.0 for analysing the above panel. 6.2 Numerical results: The analysis is carried out for different volume fraction indices (n=0, 1, 2, ∞). Dynamic behaviour of FG flat panel can be seen in figure 19-22. By time step of 0.0001s analysis has been performed and displacement has been plotted. . An enlarged view of dynamic response has been shown in figure 23 for ceramic flat panel. Figure 19: Deflection of mid-point of simply supported FG flat panel with n=0 28 Figure 20: Deflection of mid-point of simply supported FG flat panel with n=1 Figure 21: Deflection of mid-point of simply supported FG flat panel with n=2 29 Figure 22: Deflection of mid-point of simply supported FG flat panel with n=∞ Figure 23: Deflection of mid-point of simply supported FG flat panel with n=0 in time interval of 0.0001 to 0.012s 30 Chapter VII Conclusion CHAPTER-7 CONCLUSION 7.1 Conclusion: In this study, static and dynamic responses of FGM plates are analysed. The effective material properties of functionally graded materials for the plate structures are assumed to vary continuously through the plate thickness and are graded in the plate thickness direction according to a volume fraction power law distribution. Various boundary conditions have been considered to check the efficacy of ANSYS model. Convergence tests and comparison studies have been carried out with the commercially available software (ANSYS). An eight noded layered shell element (SHELL281) is used throughout the problem. The obtained results have illustrated a good agreement with those available in the literature for different volume fraction indices, thickness ratios, aspect ratios and different support conditions. The following points revealed the concluded remarks for thin to thick FGM plates are: For all the boundary conditions, the non-dimensional central deflection increases as the volume fraction index increases. For all the boundary conditions, the non-dimensional central deflection increases as the aspect ratio increases For all the boundary conditions, the non-dimensional central deflection increases as the thickness ratio increases For simply supported boundary condition vibration amplitude increases as the volume fraction index increases. 7.2 Future Scope of work Different geometric structures can be modelled such as cylindrical, spherical, conical, hyperboloid etc. Temperature dependent material property can be considered. Different type of analysis like buckling, post buckling, free vibration, forced vibration etc. can also be performed using the presented model. 32 REFERENCES [1] Mohammad Talha, B.N. Singh, Static response and free vibration analysis of FGM plates using higher order shear deformation theory, Applied Mathematical Modelling 34 (2010) 3991–4011 [2] A.M.A. Neves et al. Bending of FGM plates by a sinusoidal plate formulation and collocation with radial basis functions. 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Compos Struct 2009;88:240–52. 35 Appendix-A 36 APPENDIX-A u ( x, y, z ) u 0 ( x, y ) z x ( x, y ) v( x, y, z ) v 0 ( x, y ) z y ( x, y ) w( x, y, z ) w0 ( x, y ) z z ( x, y ) xx u0 w u w z x 0 z z ( 0 0 ) z( x z ) x x Rx Rx x Rx x Rx yy y w0 y z u0 v w z z z ( 0 0 ) z( ) x y Ry Ry y Ry y Ry zz z yz y y y w0 v w v z z 0 z ( y 0 0 ) z ( z ) y y Ry Ry y Ry y Ry xz x w0 u w u z z 0 z x ( x 0 0 ) z ( z x ) y y Rx Rx y Rx y Rx xy y 2w0 y z v y y v0 2w0 z x 0 z z z ( ) z( x ) u0 y x x Rxy Rxy u0 x Rxy y x Rxy 1 0 0 l 0 0 0 0 0 0 0 0 z 0 0 0 0 1 0 0 0 0 0 z 0 0 0 0 1 0 0 0 0 0 z 0 0 0 0 1 0 0 0 0 0 z 0 0 0 0 1 0 0 0 0 0 z 0 0 0 0 1 0 0 0 0 0 37 x0 0 y z0 0 0 yz 0 0 xz 0 0 xy 0 k x' 0 k y' z k z' ' k yz k xz' ' k xy x 0 0 0 x 0 0 y z0 1 0 yz Rx 0 xz xy0 y ' kx 0 k' y' kz 0 ' k yz k xz' 0 ' k xy 0 0 0 Q11 Q 21 Q31 0 0 0 Q zQ 11 zQ21 zQ 31 0 0 0 0 1 Rx 0 0 y 1 Ry 0 0 0 0 0 0 1 Ry y 0 1 0 y 1 0 x 2 Rxy 0 0 0 0 x 0 0 0 0 y 0 0 0 0 0 0 0 0 0 0 0 1 Ry 1 Rx 0 y x 0 0 1 0 0 u0 v 0 0 . w0 1 x Rx y 1 z Ry 0 y y 1 Rxy Q12 Q13 0 0 0 zQ11 zQ12 zQ13 0 0 Q22 Q23 0 0 0 zQ21 zQ22 zQ23 0 0 Q32 Q33 0 0 0 zQ31 zQ32 zQ33 0 0 0 0 Q44 0 0 0 0 0 zQ44 0 0 0 0 Q55 0 0 0 0 0 zQ55 0 0 0 0 Q66 zQ12 zQ22 zQ32 0 zQ13 zQ23 zQ33 0 0 0 0 zQ44 0 0 0 0 0 0 0 0 z Q12 2 2 22 zQ 2 2 z Q11 z Q21 z Q31 0 0 2 0 2 z Q32 0 0 0 0 2 z Q13 0 0 2 0 0 0 0 z Q23 2 z Q33 0 2 z Q44 0 2 0 0 0 zQ55 0 0 0 0 0 z Q55 0 0 0 0 zQ66 0 0 0 0 0 38 0 0 0 0 0 zQ66 0 0 0 0 0 z 2 Q66

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