VIBRATION ANALYSIS AND DAMPING CHARACTERISTICS OF Hybrid COMPOSITE PLATE USING FINITE ELEMENT ANALYSIS A Thesis submitted in partial fulfillment of the Requirements for the degree of Master of Technology In Mechanical Engineering Specialization: Machine Design and Analysis By SHIVAPRASAD BAAD Roll No. : 212ME1272 Department of Mechanical Engineering National Institute of Technology Rourkela Rourkela, Odisha, 769008, India June 2014 VIBRATION ANALYSIS AND DAMPING CHARACTERISTICS OF Hybrid COMPOSITE PLATE USING FINITE ELEMENT ANALYSIS A Thesis submitted in partial fulfillment of the Requirements for the degree of Master of Technology In Mechanical Engineering Specialization: Machine Design and Analysis By Shivaprasad Baad Roll No. : 212ME1272 Under the Guidance of Prof. Tarapada Roy Department of Mechanical Engineering National Institute of Technology Rourkela Rourkela, Odisha, 769008, India June 2014 Dedicated to…, My parents , my sister, My maternal aunt and uncles DEPT. OF MECHANICAL ENGINEERING NATIONAL INSTITUTE OF TECHNOLOGY, ROURKELA ROURKELA – 769008, ODISHA, INDIA Certificate This is to certify that the work in the thesis entitled VIBRATION ANALYSIS AND DAMPING CHARACTERISTICS OF HYBRID COMPOSITE PLATE USING FINITE ELEMENT ANALYSIS by Shivaprasad Baad is a record of an original research work carried out by her during 2013 - 2014 under my supervision and guidance in partial fulfillment of the requirements for the award of the degree of Master of Technology in Mechanical Engineering (Machine Design and Analysis), National Institute of Technology, Rourkela. Neither this thesis nor any part of it, to the best of my knowledge, has been submitted for any degree or diploma elsewhere. Place: NIT Rourkela Professor Date: 02 June 2014 Dr. Tarapada Roy DEPT. OF MECHANICAL ENGINEERING NATIONAL INSTITUTE OF TECHNOLOGY, ROURKELA ROURKELA – 769008, ODISHA, INDIA Declaration I certify that a) The work contained in the thesis is original and has been done by myself under the general supervision of my supervisor. b) The work has not been submitted to any other Institute for any degree or diploma. c) I have followed the guidelines provided by the Institute in writing the thesis. d) Whenever I have used materials (data, theoretical analysis, and text) from other sources, I have given due credit to them by citing them in the text of the thesis and giving their details in the references. e) Whenever I have quoted written materials from other sources, I have put them under quotation marks and given due credit to the sources by citing them and giving required details in the references. Shivaprasad Baad 2nd June 2014 ACKNOWLEDGEMENTS It is my immense pleasure to avail this opportunity to express my gratitude, regards and heartfelt respect to Prof. Tarapada Roy, Department of Mechanical Engineering, NIT Rourkela for his endless and valuable guidance prior to, during and beyond the tenure of the project work. His priceless advices have always lighted up my path whenever I have struck a dead end in my work. It has been a rewarding experience working under his supervision as he has always delivered the correct proportion of appreciation and criticism to help me excel in my field of research. I also express my sincere gratitude to Prof. (Dr.) K. P. Maity, Head of the Department of Mechanical Engineering for valuable departmental facilities. I would like to make a special mention of the selfless support and guidance I received from my senior Ashirbad Swain, Ph.D. scholar, Department of Mechanical Engineering, NIT Rourkela during my project work. Last but not the least; I would like to express my love, respect and gratitude to my parents, younger sister and my maternal aunt and uncles, who have always supported me in every decision I have made, guided me in every turn of my life, believed in me and my potential and without whom I would have never been able to achieve whatsoever I could have till date. SHIVAPRASAD BAAD [email protected] i ABSTRACT Hybrid composite is a composite which consists of nanoparticles to enhance the strength as compared to conventional composites. A model has been proposed to determine the elastic properties of hybrid composite. The hybrid composite consists of conventional fiber and nanocomposite as matrix. The first step here is to determine the properties of nanocomposite which is done by using Mori – Tanaka method. The CNTs are considered as cylindrical inclusions in polymer matrix in Mori – Tanaka method. Assuming perfect bonding between carbon fibers and nanocomposite matrix, the effective properties of the hybrid composite has been evaluated using mechanics of materials approach. An 8 noded shell element has been used for the finite element analysis having 5 degrees of freedom each node u, v, w, x , y . A 10 10 finite element mesh has been used to model the shell element. The shell coordinates which are in Cartesian form are converted into parametric form using two parameters 1 , 2 . These parameters are again mapped into isoparametric form , . A 16 layered laminate with stacking sequence [0 -45 45 90]2S has been used for vibration analysis of simply supported shell element. The dynamic equations of shell are derived using Hamilton’s principle. As the damping characters of the dynamic system are not available, for further investigation damping ratio of first mode and last active mode are assumed. Using Rayleigh damping the damping ratios of intermediate modes can be calculated. The time decay of the system from maximum amplitude to 5% of the maximum amplitude has been used as a parameter to study various shell structures by varying the volume fraction of CNTs in nanocomposite and by varying carbon fiber volume fraction. ii CONTENTS ACKNOWLEDGEMENTS.............................................................................................. I ABSTRACT .................................................................................................................. II CONTENTS ................................................................................................................. III NOMENCLATURE....................................................................................................... V LIST OF FIGURES .................................................................................................... VII LIST OF TABLES ..................................................................................................... VIII 1 INTRODUCTION ...................................................................................................... 1 2 LITERATURE REVIEW AND MOTIVATION ............................................................ 5 2.1 Literature Review ...................................................................................................... 5 2.1.1 Material property – Hybrid composite ..................................................................... 5 2.1.2 Material Property – Hybrid composite ..................................................................... 6 2.1.3 Vibration analysis of plate........................................................................................ 6 2.1.4 Damping in composites ............................................................................................ 6 2.2 Motivation .................................................................................................................. 6 2.3 Objective ..................................................................................................................... 7 3 MATERIAL MODELING ........................................................................................... 8 3.1 CNT bases nanocomposite modeling ....................................................................... 8 3.2 Hybrid Composite modeling ................................................................................... 11 4 FINITE ELEMENT FORMULATION ....................................................................... 14 iii 4.1 Geometry of mid-surface of shell ........................................................................... 14 4.1.1 Isoparametric mapping ........................................................................................... 16 4.1.2 Transformation matrix used for isoparametric mapping ........................................ 17 4.2 Strain displacement relations ................................................................................. 18 4.2.1 In-plane/bending strain-displacement matrix ......................................................... 18 4.2.2 Transverse strain displacement matrix ................................................................... 20 4.2.3 ABBD matrix ......................................................................................................... 20 4.3 Equation of motion .................................................................................................. 22 4.3.1 Static finite equations ............................................................................................. 22 4.3.2 Dynamic finite equations ....................................................................................... 24 4.4 Calculation of non-dimensional frequencies ......................................................... 26 4.5 State space method for impulse response .............................................................. 26 5 RESULTS AND DISCUSSION ............................................................................... 28 6 CONCLUSION ........................................................................................................ 35 REFERENCES: .......................................................................................................... 37 iv NOMENCLATURE CNT Carbon nanotubes SWCNT Single walled carbon nanotubes DWCNT Double walled carbon nanotubes MWCNT Multi walled carbon nanotubes nc, m, hc Nanocomposite, matrix, hybrid composite Cp11, Cp12, Cp22, Cp23, Cp55 Elastic properties of constituent phase nr , kr , pr , lr , mr Hills elastic constant , Euler angles for CNT orientation , Stress and strain A Strain concentration tensor I Identity matrix S Eshelby tensor K Bulk modulus G Shear modulus E Young’s modulus vCNT Volume fraction of CNT vm Volume fraction of matrix vf Volume fraction of fiber 1 , 2 Curvilinear coordinates s 1 , s 2 Isoparametric curves r1 , r2 Tangent to isoparametric curves A1 , A2 Lame’s parameters R1 , R2 Normal curvatures of shell mid-surface R12 Twist curvatures of shell mid-surface a Length of shell along x-axis N Shape function u0i , v0i , w0i Displacements of ith node in 1 , 2 and z directions , v Isoparametric coordinates J Jacobian matrix Shear strain k Curvature of the mid-surface Bbe In-plane strain displacement matrix Bse Transverse strain displacement matrix N, M Resultant force and moment per unit length A In-plane stiffness matrix B Coupling stiffness matrix D Bending stiffness matrix z Thickness of lamina h Thickness of laminate Q Reduced stiffness matrix K Shear correction factor Total potential energy U Strain energy T Kinetic energy W Work done by external force L Lagrangian M uue Elemental mass matrix K uue Elemental stiffness matrix de Displacement vector de Acceleration vector t Time Density Natural frequency ** Non-dimensional natural frequency i Modal coordinates for ith degree of freedom i Modal damping ratio 1..... 2n State vector vi LIST OF FIGURES Figure 3-1 Ref: heshmati and yas (2012) ............................................................................. 8 Figure 3-2 Hexagonal RVE ................................................................................................ 11 Figure 4-1 Geometry of shell structure in Cartesian coordinates ..................................... 14 Figure 4-2 Isoparametric mapping .................................................................................... 16 Figure 4-3 Stacking sequence in laminate ......................................................................... 21 Figure 5-1 Variation of C11 w.r.t variation of carbon fiber and CNT volume fraction ..... 30 Figure 5-2 Variation of C12 w.r.t variation of carbon fiber and CNT volume fraction ..... 31 Figure 5-3 Variation of C23 w.r.t variation of carbon fiber and CNT volume fraction ..... 31 Figure 5-4 Variation of C22 w.r.t variation of carbon fiber and CNT volume fraction ..... 32 Figure 5-5 Variation of C55 w.r.t variation of carbon fiber and CNT volume fraction ..... 32 Figure 5-6 impulse response of cfrp composite for thick plate .......................................... 33 Figure 5-7 Impulse response of cfrp composite for thin plate ........................................... 33 Figure 5-8 Decay time for thick plate by varying the cnt volume fractions for different volume fractions of carbon fiber ...................................................................... 34 Figure 5-9 Decay time for thin plate by varying the cnt volume fractions for different volume fractions of carbon fiber ...................................................................... 34 vii LIST OF TABLES Table 5-1 Non-dimensional frequency [Ref:23] .................................................................. 29 Table 5-2 Non-dimensional frequency for the present formulation ..................................... 29 Table 5-3 Material properties of various constituents in hybrid composite ........................ 30 viii 1 INTRODUCTION As long as there is development in the field of aerospace, automobile, healthcare, electronics and consumer industry the demand for new materials will never seize. The demand for new materials has led to continuous research and development of new techniques to satisfy the needs. 1.1. Nanocomposites Nanocomposites consist of reinforcements of nanoscale spread evenly or randomly in polymer matrix. The commonly used polymeric matrix materials are: Epoxy Polystyrene Nylon Polyimide PEEK – Polyether ether ketone Carbon nanotubes – SWCNTs and Nanotitanium oxide MWCNTs Nanosilica The commonly used nano fillers are: Nanoaluminium oxide The reinforcements can be particles or fibres of size of few nanometers. The nanocomposite has a wide range of materials from 3-D metal matrix composites, 2-D laminated composites and nano-wires of small dimension representing variations of nano reinforcements. Using nanoscale reinforcements was introduced by Usuki et al [1] who 1 built a nanocomposite using polyimide and organophilic clay. The nanocomposite formed had twice the tensile modulus as compared to neat polyimide with just 2% volume fraction of nano reinforcement. Nanocomposites have gained a wide popularity among researchers. Researchers have discovered that the properties of the nanocomposite are better when compared to the individual components of the composite. Properties such as increased tensile strength, increased thermal conductivity are observed. 1.2. Hybrid Composites The important properties that are desired from any composite are strength, stiffness, ductility, toughness, damping, energy absorption, thermal stability and low weight. With conventional materials it is not possible to get all the desired properties, but with composite materials we can tailor the properties of material as per our needs. By using reinforcements of nanoscale in polymer composites there has been tremendous increase in mechanical properties as compared to neat polymer matrix. Hybrid composites are new type of three phase composites which have reinforcements of nanoscale in addition to conventional reinforcing fiber in matrix or by growing reinforcements of nanoscale on the surface of fiber. R.C.L. Dutra et al [2] defined hybrid composites as composites consisting of different fibers. The main purpose of using hybrid composites is it increases the matrix dominated properties. 1.3. Classification of Plates [3] Plates can be classified into two types: Thin plates A thin plate can further be classified as: Plate with small deflection Plates with large deflection 2 Thick plates 1.3.1.1. Plates with small deflection If the deflection of a plate when subjected to loading is less than or nearly equal to thickness, then the plate is said to have small deflection. The necessary assumptions for developing theory for plates having small deflection are: The middle plane does not deform on loading. Point on the plate initially normal to middle plane remain normal to middle plane even after loading. The stresses in the thickness direction can be neglected. 1.3.1.2. Thin plates with large deflection If the deflections of the plate are large when subjected to lateral loading as compared to thickness, then the plate is said to have large deflection. When plate is loaded mid plane strains are developed. In plates with small deflection it is normally neglected, as a result the stresses are also neglected. But if deflections are large as compared to thickness, the strains developed are large. So stresses cannot be neglected. In this case we obtain nonlinear equations and analysis becomes complicated. 1.3.2. Thick plate The above approximations for thin plates are not applicable. Thick plate theory must be used. The thick plate theory involves 3-D theory of elasticity and calculation of stresses is quite complicated. 1.3.3. Difference between plate and shell The major difference between plates and shells can be observed under the action of loading. When a plate member is subjected to lateral load, equilibrium is possible by the action of bending and twisting moments. In shells, when it is subjected to lateral loading, equilibrium is possible by membrane stresses which act parallel to tangential plane at a point on middle surface and are distributed uniformly over the thickness of shell. 1.4 Plates are plane member and shells are curved structural members. Damping in composites Damping is a very important parameter for vibration control, noise reduction, stability of system, fatigue and impact resistance [4]. The damping in fibre reinforced composites is different from that of metals. The various forms of energy dissipation are [5]: Damping behaviour of matrix material. The major contribution to damping is from matrix, the damping of fiber must also be included for calculation of damping. Damping behaviour of interphase Interphase is the region between matrix and fiber. The type of interphase plays an important role in damping. The interphase can be weak or strong. 3 Damping due to damage a) Frictional damping due to delamination. b) Damping due to energy dissipation of broken fibers or cracks in matrix. Viscoplastic damping At higher amplitudes of stresses, there is non-linear damping due to presence of high stress and strain. 1.5. Impulse response of linear time invariant system (LTIS) Impulse force is a force which acts on the system for very short amount of time. Knowing the impulse response of LTIS we can obtain by superposition the response of the same system to any input provided the input conditions are zero in all cases. Unit impulse input has very short intervals of time but very large amplitude and hence the effect of the behaviour of the system under study is not negligible. Ex: Ball hitting the cricket bat, the ball is acted upon by very large force for a short duration of time. 4 2 LITERATURE REVIEW AND MOTIVATION 2.1 Literature Review Jian Ping Lu [6] Elastic properties of SWCNTs, MWCNTs and nanoropes are investigated using force constant model. Material property – Hybrid composite 2.1.1 Raifee et al [7] estimated mechanical properties of epoxy based nanocomposite with SWCNT, MWCNT and graphene platelets were compared for weight fractions of 0.1%. The material properties measured were Young’s modulus, fracture toughness, ultimate tensile strength. The tensile strength of graphene based nanocomposites showed better properties as compared to CNT based nanocomposites. F.H. Gojny et al [8] observed mechanical properties resulted in an increase in Young’s modulus, strength at weight fractions of 0.1%. There was good agreement between experimental observed data and results from modified Halpin-Tsai relation. Florian H [9] proposed choosing appropriate type of CNTs (SWCNTs or DWCNTs or MWCNTs) has been a problem ever since they are being used in composites. They evaluated the properties of nanocomposite for different nano fillers. The nanocomposites exhibited greater strength, stiffness and fracture toughness. They found that DWCNT based nanocomposite exhibited greater fracture toughness. Seidel et al [10] estimated effective elastic properties of composites consisting of aligned SWCNTs or MWCNTs using Mori-Tanaka method. The effects of an interphase layer between CNTs and the polymer is also investigated using a multi-layer composite cylinders approach. Liu and Chen [11] estimated effective elastic properties of the nanocomposite are evaluated using continuum modelling and finite element method. The extended rule of mixture is used to determine the properties of the continuum model. 5 Material Property – Hybrid composite 2.1.2 Dutra et al [2], made a hybrid composite consisting of carbon fiber and Polypropylene fiber and mercapto-modified polypropylene blend fiber (PPEVASH). They found that hybrid composites showed better impact resistance than CFRP composite. Mathur et al [12] CNTs were grown on unidirectional carbon fiber. These fibers were used as reinforcements in matrix material. They found that the mechanical properties improved with increase in amount of CNT deposition as compared to neat CFRP composite. Garcia et al [13] CNTs were grown on alumina fiber cloth. These fibers were used as reinforcements in matrix material. The growth of CNTs led to an increase in inter-laminar shear properties of the order of 69% as compared to alumina cloth composite. Kundalwal and Ray [14], they evaluated the elastic properties of FFRC (Fuzzy fiber reinforced composite) using mechanics of materials approach and Mori-Tanaka method considering with and without the interphase between CNT and polymer. 2.1.3 Vibration analysis of plate Roy and Chakraborty [15] formulated layered shell finite element model for coupled electromechanical analysis of curved smart composite structure. 2.1.4 Damping in composites Gibson et al [4] used vibration used modal vibration response measurements to characterize the mechanical properties of laminated structures. They showed that vibration in either first mode or multiple modes can be used to determine the elastic properties and damping ratios. Modal testing was done by impulse excitation methods. Kyriazoglou and Guild [17] found damping ratio using experimental methods and by FEM. The FEM uses Rayleigh damping method and particularly mass proportional damping. R. Verdejoan et al. [18] found that even a small volume fraction of CNT can increase the sound absorption capabilities. 2.2 Motivation Hybrid composites are new type of three phase composites which increase the matrix dominated properties. The hybrid composite that is to be modeled here consists of nanocomposite matrix and continuous long carbon fibers. The nanocomposite is made up of randomly distributed CNTs and polymer matrix. The nanocomposite is modeled using Mori-Tanaka method. The hybrid composite can be modeled using mechanics of materials approach. As the mobility of system goes on increasing, modeling damping for such systems becomes complicated. Rayleigh damping model has been used to model such multi degree of freedom systems. Further investigation has been carried out by assuming suitable damping ratios for first mode and last significant mode where 6 mass is proportional to damping. Impulse response of the system has been carried out and a comparative study has been made to know effect of damping in systems by varying the volume fractions of carbon fiber and CNTs. 2.3 Objective Material modelling and material characterization 1. Nanocomposite has been modelled using Mori – Tanaka method. 2. Hybrid composite consisting of carbon fiber and nanocomposite matrix has been modelled using mechanics of materials approach. 8 noded shell element formulation 1. Mindlin theory of plates and shells has been used to model shell. Modelling damping and Impulse response 1. Rayleigh damping has been used to model the damping of MDOF system. 2. Impulse response of the system has been carried out using the state space model. 7 3 MATERIAL MODELING The material modeling is divided in two phases: CNT based nanocomposite modeling. Hybrid composite modeling. The nanocomposite consists of randomly distributed straight MWCNT as reinforcements and epoxy as matrix. As the CNTs are randomly distributed the nanocomposite can be modeled as isotropic material. The property of nanocomposite is evaluated using the Mori – Tanaka method. Assuming perfect bonding between fiber and nanocomposite the hybrid composite can be modeled similar to conventional composite using mechanics of materials approach. 3.1 CNT bases nanocomposite modeling FIGURE 3-1 REF: HESHMATI AND YAS [18] Fig. 3.1 shows a RVE of randomly distributed CNT in epoxy matrix. The Mori – Tanaka method was used to estimate the elastic properties of the randomly distributed MWCNT in matrix. The procedure to determine the isotropic properties of randomly oriented MWCNTs dispersed in epoxy matrix is as follows: 8 The Hill’s elastic constants for the MWCNT can be obtained by equating the stress – strain matrix of MWCNT with the Hill’s elastic matrix. C11CNT CNT C21 C31CNT CCNT 0 0 0 nr l r l CCNT 0r 0 0 C12CNT CNT C22 C32CNT 0 0 0 lr kr mr kr mr 0 0 0 C13CNT CNT C23 C33CNT 0 0 0 CNT C44 0 0 lr kr mr kr mr 0 0 0 0 0 0 mr 0 0 0 0 0 0 0 0 0 C55CNT 0 0 0 0 0 pr 0 CNT C66 0 0 0 0 0 0 0 0 0 0 pr (1) (2) Eqn (1) and eqn. (2) are the stress – strain matrix of MWCNT and Hill’s elastic matrix. The orientation of the CNT can be specified by two Euler angles and . The base vectors ei and ei of the global 0 x1 x2 x3 and local co-ordinates can be related by the relation, ei gij e j (3) cos Where gij sin 0 cos sin cos cos sin sin sin sin cos cos As the CNTs are randomly distributed in matrix it can be characterized by two Euler angles and . The orientation distribution of CNTs in the composite is characterized by the probability density function p , satisfying the normalization condition and is given by the equation. 2 /2 p , sin (d )(d ) 1 0 For completely randomly oriented CNTs, p , (4) 0 1 2 (5) From Mori – Tanaka method one can relate stress CNT , and strain CNT , of the CNT to the stress in the matrix m by, CNT , CCNT A , m CCNT A , Cm1 m And 9 (6) CNT , A , m A , Cm1 m (7) Where strain concentration tensor is given by, A I S (Cm )1 (CCNT Cm ) 1 (8) Where S corresponds to Eshelby Tensor and is given by Li and Dunn [19] for cylindrical inclusion. The CNT average stress and strain for the randomly oriented CNTs 2 /2 p , CCNT A , Cm1 sin d d m 0 0 2 /2 CNT 0 given by, (9) p , A , sin d d is (10) m 0 Using rule of mixture one can get the stresses and strains in the nanocomposite. As the CNTs are randomly distributed in the matrix, the nanocomposite behaves like an isotropic material. The bulk modulus, shear modulus, Young’s modulus of the nanocomposite [20] are given by, K nc K m vCNT ( CNT 3K m CNT ) 3 vm vCNT CNT Gnc Gm vCNT (CNT 2Gm CNT ) 2 vm vCNT CNT Enc (11) 9 K nc Gnc 3K nc Gnc Where, CNT 3( K m Gm ) kCNT lCNT 3(Gm kCNT ) 2 Gm 3K m Gm Gm 3K m 7Gm 4Gm 1 4Gm 2kCNT lCNT 5 3(Gm kCNT ) Gm pCNT Gm 3K m Gm mCNT 3K m 7Gm CNT 1 3 CNT nCNT 2lCNT 2kCNT lCNT 3K m 2Gm lCNT Gm kCNT 8Gm mCNT 3K m 4Gm 2 kCNT lCNT 2Gm lCNT 8G p 1 2 nCNT lCNT m CNT 5 3 Gm pCNT 3K m mCNT Gm Gm 7mCNT Gm 3 Gm kCNT CNT (12) vCNT vm 1 (13) vCNT and vm are volume fractions of CNT and matrix material, nc represents nanocomposite. 10 3.2 Hybrid Composite modeling F IGURE 3-2 H EXAGONAL RVE Fig. 3.2 shows a hexagonal RVE of hybrid composite consisting of carbon fibers distributed in nanocomposite matrix. Using the above calculated nanocomposite properties, the properties of the transversely isotropic hybrid composite can be evaluated by the formulation of Kundalwal and Ray for fuzzy fiber [14]. Assuming perfect bonding between carbon fiber and nanocompostie, the normal strains in hybrid composite, carbon fiber and nanocomposite are equal along the fiber direction and the transverse stresses in the same phase are equal along the direction transverse to the fiber from isofield conditions. Using rules of mixture one can express the longitudinal and transverse stresses and strains in terms of volume fractions of nanocomposite and carbon fiber. Using isofield conditions and rule of mixture we can write, v f vNC 1 (14) Where v f and vNC are volume fractions of carbon fiber and nanocomposite. 1f 1NC 1HC f NC HC 2 2 2 3f 3NC 3HC f NC HC 23 23 23 13f 13NC 13HC f NC HC 12 12 12 (15) HC represents hybrid composite. 1f 1NC 1HC f NC HC 2 2 2 f NC HC v f 3f vNC 3NC 3HC 23 23 23 13f 13NC 13HC f NC HC 12 12 12 11 (16) There the stress and strain in the hybrid composite lamina is given by, C C (17) V V (18) HC f NC 1 HC 2 f NC 1 2 But from iso-field conditions, C3 f C4 NC 0 (19) By solving above equations we get, C HC HC HC (20) C HC is the effective elastic matrix of the proposed Carbon fiber is reinforced polymer and is given by, C HC C1 V3 C2 V4 1 1 (21) The various matrices appearing in the above equations are given below. C11f C12f 0 0 0 0 C1 v f 0 0 0 0 0 0 vNC C11NC NC C12 C12NC C2 0 0 0 1 C f 12 C13f C3 0 0 0 12 0 C22f C23f 0 0 0 C12f 0 0 0 0 0 vNC C12NC C11NC C12NC 0 0 0 0 C23f C22f 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 vNC C12NC C12NC C11NC 0 0 0 0 0 0 C44p 0 0 0 0 0 0 C55p 0 (22) 0 0 0 C44NC 0 0 0 0 0 0 0 C55p 0 0 0 0 C44NC 0 0 0 0 0 0 C44NC (23) (24) 1 C NC 12 C12NC C 4 0 0 0 0 C11NC C12NC 0 0 0 0 0 0 v f 0 0 V1 0 0 0 0 0 0 0 0 vf 0 0 0 1 0 0 v NC 0 0 V2 0 0 0 0 0 0 0 C12NC C11NC 0 0 0 0 0 0 vf 0 0 0 0 vNC 0 0 0 0 0 0 C44NC 0 0 0 0 0 0 vf 0 0 0 0 vNC 0 0 0 0 0 0 C44NC 0 0 0 0 0 0 v f 0 0 0 0 vNC 0 0 0 0 0 0 C44NC (25) (26) 0 0 0 0 0 vNC (27) V3 V1 V2 C4 C3 (28) V4 V2 V1 C3 C4 (29) 1 1 13 4 FINITE ELEMENT FORMULATION 4.1 Geometry of mid-surface of shell F IGURE 4-1 G EOMETRY OF SHELL STRUCTURE IN C ARTESIAN COORDINATES The shell geometry used in the present formulation has been developed using an orthogonal curvilinear coordinate system with the mid-plane of the shell assumed to be the reference surface as shown in Fig.4.1. The shell mid-surface in the Cartesian rectangular coordinate system has been first mapped into a parametric domain through the suitable exact parameterization. Two independent coordinates (1 , 2 ) in the parametric space have been considered as the mid-surface curvilinear coordinates of the shell. The normal direction coordinate to the middle surface of the shell has been represented by z. The reference surface or the shell mid-surface can be described in the global Cartesian coordinates in terms of the position vector as, r (1 , 2 ) X (1 , 2 ) i Y (1 , 2 ) j Z (1 , 2 ) k (30) Where, i , j and k are unit vectors along the X, Y and Z axis, respectively. The tangent to the isoparametric curves s 1 and s 2 respectively are r1 r ; 1 r2 r 2 (31) The vector joining two points on the middle surface (1 , 2 ) and (1 d1 , 2 d 2 ) is given as 14 ds r1d1 r2 d 2 (32) Scalar product of ds, ds.ds (r1.r1 )d12 (r2 .r2 )d 22 (33) Lame’s parameters can be defined as, A1 r1.r1 (34) A2 r2 .r2 Eqn. (4) can be written as ds 2 A12 d12 A22 d 22 (35) Since the 1 and 2 are independent coordinates ds 2 ds21 ds2 2 Where, (36) ds 1 A1d1 ds 2 A2 d 2 The unit tangent vectors to the isoparametric curve s 1 and s 2 can be expressed respectively as, e1 r1 ; A1 e2 r2 A2 (37) The unit normal vector to the tangent plane of any point on the reference surface can be expressed as en r1 r2 r1 r2 (38) The normal curvatures and twist curvatures of the mid-surface of shell can be expressed as: e .r 1 n 211 R1 A1 e .r 1 n 222 R2 A2 (39) e .r 1 n 12 R12 A1 A2 Where R1 , R2 are the normal curvatures of the midsurface of the shell and R12 is the twist curvature of the mid-surface of the shell. 15 4.1.1 Isoparametric mapping F IGURE 4-2 I SOPARAMETRIC MAPPING Fig.4.2 shows Cartesian coordinates are converted into curvilinear coordinates and it is mapped into isoparametric form. The shell midsurface in the rectangular cartesian coordinate system has been mapped into the parametric space (1 , 2 ) and the midsurface in the parametric space has been divided into required number of quadrilateral elements or sub-domains. The reference coordinates ( , ) map the quadrilateral element in the curvilinear coordinates (1 , 2 ) into the reference coordinates that is a square as shown in Fig. 4.2. Any point within an element in the parametric space has been approximated by the isoparametric mapping. Hence the curvilinear coordinates (1 , 2 ) of any point within an element may be expressed as nd 1 N i1i i 1 (40) nd 2 N i 2i i 1 (1 , 2 ) is the coordinate of midsurface at i th node in curvilinear coordinate system. u0i , v0i and w0i are the deflection of midsurface at i th node in 1 , 2 and z directions respectively. 1i is the rotation of normal at i th node about 2 axis and 2i is the rotation of normal at i th node about 1 axis. The displacement components on the shell midsurface at any point within an element may be expressed as 16 u0i u0 v v 0i 0 nd w N i wi i 1 1 1i 2 2i (41) Where, nd is the number of nodes in an element, N i is the shape function corresponding to the i th node and shape functions of 8 noded serendipity element are given below 1 N1 (1 )(1 )(1 ) 4 1 N 2 (1 2 )(1 ) 2 1 N 3 (1 )(1 )(1 ) 4 1 N 4 (1 )(1 2 ) 2 1 N 5 (1 )(1 )(1 ) 4 1 N 6 (1 2 )(1 ) 2 1 N 7 (1 )(1 )(1 ) 4 1 N8 (1 )(1 2 ) 2 (42) 4.1.2 Transformation matrix used for isoparametric mapping Since the integration is to be done in natural coordinates ( , ) , the element is mapped into the isoparametric space ( , ) using the isoparametric shape functions. The transformation matrix used is given below. The relation between the shape function derivatives in parametric space (1 , 2 ) isoparametric space ( , ) are given as N i N i 1 1 J N i N i 2 17 (43) and in The jacobian matrix can be expressed as 1 J 1 4.2 2 2 (44) J a J* (45) d J d d (46) Strain displacement relations Neglecting normal strain component in the thickness direction, the five strain components of a doubly curved shell may be express as xx xx0 k xx 0 k yy yy yy xy xy0 z k xy 0 yz yz 0 0 xz xz 0 (47) Where xx0 , yy0 and xy0 is the in-plane strains of the midsurface in the cartesian coordinate system and k xx , k yy and k xy are the bending strains (curvatures) of the midsurface in the cartesian coordinates system. After incorporating the effect of transverse stain in Koiter’s shell theory, inplane and transverse strain-displacement relations may be expressed as described in the following subsections. 4.2.1 In-plane/bending strain-displacement matrix The strain components on the midsurface of shell element are xx0 yy0 xy0 kxx k yy k xy T (48) By using isoparametric 8-noded shell element, the displacement component on the shell midsurface at any point within an element can be expressed as u0 v 0 e w N d 1 2 18 (49) The mid-surface strains and curvatures from Koiter’s shell theory are: xx0 1 u v A1 w A1 1 A1 A2 2 R1 (50) yy0 1 v u A2 w A2 2 A1 A2 1 R2 (51) xy0 1 v 1 u u A1 v A2 2w A1 1 A2 2 A1 A2 2 A1 A2 1 R12 (52) k xx A1 1 1 1 1 v 1 u u A1 v A2 2 A1 1 A1 A2 2 2 R12 A1 1 A2 2 A1 A2 2 A1 A2 1 (53) k yy A2 1 2 1 1 v 1 u u A1 v A2 1 A2 2 A1 A2 1 2 R12 A1 1 A2 2 A1 A2 2 A1 A2 1 (54) 1 A1 2 A2 1 2 1 1 A A A A A A 1 1 2 2 1 2 2 1 2 1 k xy 1 1 1 1 v 1 u u A1 v A2 2 R1 R2 A1 1 A2 2 A1 A2 2 A1 A2 1 (55) By using 8-noded isoparametric shape functions from Eqn. (13), the strain components at any point on the shell midsurface Ni A1 1 Ni A1 1 A1 A2 2 Ni A2 1 Ni A1 A2 1 A2 2 1 Ni Ni A1 Ni A2 1 Ni A2 2 A1 A2 2 A1 1 A1 A2 1 8 1 1 1 N N A 1 1 1 N N A i i i i 1 2 i 1 2 R12 A2 2 A1 A2 2 2 R12 A1 1 A1 A2 1 1 1 1 Ni Ni A1 1 1 1 Ni Ni A2 2 R A A A 2 R A 12 2 2 A1 A2 2 12 1 1 1 2 1 A A N N N N 1 1 i i i i 1 2 C0 C0 A A A A A A 2 2 1 2 2 1 1 1 2 1 can be expresse Ni 0 0 R1 Ni 0 0 R2 2 Ni 0 0 R12 0 1 Ni A1 1 0 Ni A2 A1 A2 1 0 Ni A1 1 Ni A2 2 A1 A2 2 Ni A1 A2 2 1 Ni A2 2 Ni 1 Ni A1 1 A1 A2 (56) Bbe d e 1 1 1 Bbe is the element in plane stain- displacement matrix and Co 2 R1 R2 19 A1 (57) u0i v0i wi 1i 2i A2 1 4.2.2 Transverse strain displacement matrix According to the FSDT, the transverse shear strain vector of a doubly curved shell element may be expressed as 1 2 A2 yz xz 1 1 A1 w u v 2 R12 R2 w u v 1 R1 R12 (58) And hence the transverse shear strain at any point on the shell mid surface can be expressed as Ni R nd yz 12 xz k 1 Ni R 1 N i R2 1 Ni A2 2 N i R12 1 Ni A1 1 u0i 0 Ni v0i wi Ni 0 1i 2i yz e e Bs d xz (59) (60) Bse is the element transverse stain- displacement matrix. 4.2.3 ABBD matrix The resultant force per unit width is N N xx N xy N yy T (61) The resultant moment per unit width is M M xx M yy M xy T (62) In plane strains of mid-surface is 0 0 xx yy0 xy0 T (63) Curvatures of mid-surface is k kxx k yy kxy T (64) The strain at any point on an element ( xy ) 0 z k (65) The resultant force per unit width can be expressed as h /2 N ( xy ) h /2 dz N A 0 B k 20 (66) The resultant moment per unit width M h /2 ( xy ) zdz (67) h /2 M B o D k Eqs. (37) and (38) can be expressed in matrix form as N A M B B 0 D k (68) A B B D (69) F IGURE 4-3 Qb k Q11 Q22 Q12 Q11 Q12 Q12 Q22 0 0 S TACKING SEQUENCE IN LAMINATE 0 0 Q66 (70) E1 (71) 12 21 E1 E2 (75) E2 (72) 12 2 1 (76) (73) 21 1 2 (77) 1 12 21 1 12 21 12 E2 E 21 1 1 12 21 1 12 21 Q66 G12 (74) The transformation matrix cos 2 1 RT k sin 2 sin cos sin 2 cos 2 sin cos 2sin cos 2sin cos cos 2 sin 2 (78) The transformed elasticity matrix for bending Qb RT 1 Qb RT T k k k k 21 (79) KG23 0 0 KG13 QS (80) Where K denotes shear correction factor The transformation matrix cos sin sin cos Rs (81) The transformed elasticity matrix for transverse shear Q s Rs T QS Rs (82) n D s Q s ( Z k Z k 1 ) k 1 (83) The extensional stiffness matrix n A Qb Z k Z k 1 k 1 (84) k The extensional-bending coupling stiffness matrix B Qb Zk2 Zk21 2 1 n k 1 (85) k The bending stiffness matrix D Qb Z k3 Z k31 3 1 n k 1 4.3 (86) k Equation of motion The elastic finite element formulation has been derived for static and dynamic analysis. 4.3.1 Static finite equations The total potential energy of the element is given by U W (87) U and W are the strain energy of the entire structure and work done by the external force. Strain energy of the shell structure is given by, U 1 T dV 2V U 1 T C dV 2V T T 1 U d e Bue C Bue dV d e 2 V U 1 e T d Kuue d e 2 Work done by external force is given by, 22 (88) W d f x, y dA e T e s A1 W d e T N f x, y dA T e s (89) A1 W d F e T e Substituting the values of U and W in total potential energy Eqn.58, we have Kuue d e F e (90) Eqn. (61) is the elastic equation for one element. Where, e Kuu is the element structural stiffness matrix is given by Kuue Bue C Bue dV (91) e e Kuu Kbb K sse (92) Bbe 0 B 0 Bse (93) D 0 C 0b D s (94) T V e u Db is the in-plane/bending constitutive matrix and Ds is the transverse shear constitutive matrix. e Kbb , is the element in plane/bending stiffness matrix. Kbbe Bbe T Db Bbe dV V A K B B e T b e bb B Be d D b (95) K sse , is the element transverse shear stiffness matrix K sse Bse T Ds Bse dV V K sse Bse D s Bse d T (96) F , is the element external mechanical force vector e F N f x, y dA T e e s A Hence all the element stiffness matrices can be expressed in its final form as 23 (97) 1 1 K e K he J d d (98) 1 1 After assembling the elemental stiffness matrices, the global set of elastic equations is given by Kuu d F (99) 4.3.2 Dynamic finite equations The dynamic finite element formulation has been derived by using Hamilton’s principle as t2 L W dt 0 (100) t1 where, the Lagrangian, L T U T is the kinetic energy of the system, U is the elastic strain energy, W is the external work done by the force on the structure. t2 T U W dt 0 (101) t1 where t1 and t2 defines the time interval. Kinetic energy T is given by, 1 T d 2 V d dV T (102) is the mass density. Individual parts of the Hamilton equation can be written as follows: Part 1: The kinetic energy of the system t2 t2 t1 t1 V 1 d 2 Tdt t2 t2 t1 t1 V t2 t2 t1 t1 t2 t2 t1 t1 d dVdt T Tdt d d dVdt T V e T e Tdt = d N N dV d dt T Tdt d e T M uue d e dt (103) Part 2: The total internal energy of the system t2 t2 t2 t2 t1 t1 t1 t1 m e e e Udt V dt Udt d Kuu d dt T Part 3: The work done by the external forces 24 (104) t2 t2 e We dt d t1 t1 A1 t2 t2 T e We dt d t1 t1 e s N f x, y dAdt T T t2 f x, y dAdt e s A1 W dt d F e T e (105) e t1 Substituting eqns. (74), (75), (76) in eq. (71) gives t2 d e T t1 M d K d F dt 0 e uu e e uu e e (106) Since d e can be any arbitrary values, d e 0 and therefore the eq. (43) is zero only if M uue d e Kuue d e F e (107) Eqn. (78) is dynamic finite element equations of one element. Where the structural mass matrix is given by, M uue N T N dV (108) V The displacement at any point in the laminate cane be expressed as u u0 z x v v0 z y w w u nd N i v 0 w i 1 0 (109) 0 0 zN i Ni 0 0 Ni 0 0 u0i 0 v0i zN i wi 0 xi yi (110) u e v N d w (111) M uue N T N dV (112) V M uue h 2 N N dzd h 2 25 T (113) Mass matrix in its final form can be expressed as 1 1 h /2 M uue N N dz J d d T (114) 1 1 h /2 4.4 Calculation of non-dimensional frequencies In order to verify the above formulation for spherical shell having mesh 10 10 , the various nondimensional frequencies of the present formulation are compared with existing ones. ** a2 h E2 (115) Where ** is the non-dimensional frequency, is the natural frequency, a is the length of shell structure along x-axis, h is the thickness of the laminate, is the density of composite, E2 is the transverse modulus of the lamina having 00 orientation. 4.5 State space method for impulse response The equation of motion of system considering damping in modal form is given as: i 2 ii i i2 i fi (116) Where i , i , i and fi are the modal coordinates, modal damping ratio, natural frequency and force vector in modal form for i=1, 2, 3…., dof. For damped systems which are modeled using Rayleigh proportional damping, it is difficult to determine the Rayleigh constants. Calculating Rayleigh damping coefficients for large degree of freedom system has been provided with detail in Chowdhury and Dasgupta [21]. Assuming 3% of the total modes to be active modes or which participate in mass proportional damping. As there is no data regarding the damping ratios for the above formulated hybrid composite we proceed the investigation by assuming the damping ratio of the first mode to be 0.01 and damping ratio of the last active mode to be 0.05. The constants and can be calculated using the damping ratios and natural frequencies of the first mode and the last active mode. The damping ratio for the i th intermediate mode i can be calculated once the value of and are known. Thus i i 2i 26 (117) The state space and output equations in modal form for n number of modes can be written as, 1 0 n n 2 n n n 2n 1 T 0 dof 1 dof dof Fdof 1 (118) T 1 2 nn n n dof dof M dof dof 2n I n n 1 1 T I 2 n 2n 02 n 2 n dof dof Fdof 1 2n 2n (119) Where 0 n n is null matrix, I n n is the identity matrix, n2 matrices, 1..... 2n are the state vectors. 27 n n and 2 nn n n are diagonal 5 RESULTS AND DISCUSSION Summary of the above chapters Hybrid composite is a composite which consists of nanoparticles to enhance the strength as compared to conventional composites. A model has been proposed to determine the elastic properties of hybrid composite. The hybrid composite consists of conventional fiber and nanocomposite as matrix. The first step here is to determine the properties of nanocomposite which is done by using Mori – Tanaka method. The CNTs are considered as cylindrical inclusions in polymer matrix in Mori – Tanaka method. Assuming perfect bonding between carbon fibers and nanocomposite matrix, the effective properties of the hybrid composite has been evaluated using mechanics of materials approach. An 8 noded shell element has been used for the finite element analysis having 5 degrees of freedom each node u, v, w, x , y . A 10 10 finite element mesh has been used to model the shell element. The shell coordinates which are in Cartesian form are converted into parametric form using two parameters 1 , 2 . These parameters are again mapped into isoparametric form , . A 16 layered laminate with stacking sequence [0 -45 45 90]2S has been used for vibration analysis of simply supported shell element. The dynamic equations of shell are derived using Hamilton’s principle. As the damping characters of the dynamic system are not available, for further investigation damping ratio of first mode and last active mode are assumed. Using Rayleigh damping the damping ratios of intermediate modes can be calculated. The time decay of the system from maximum amplitude to 5% of the maximum amplitude has been used as a parameter to study various shell structures by varying the volume fraction of CNTs in nanocomposite and by varying carbon fiber volume fraction. 28 Validation of formulation Free vibration analysis is done to validate the above formulation. A 2, 3 and 4 layered cross ply laminate has been used to carry out free vibration analysis by varying the a/h ratio and R/a ratio of the spherical shell. Reference [23] Panel a/h R/a 1 2 5 10 20 50 100 500 0/90 10 14.481 10.749 9.2302 8.9841 8.9212 8.9035 8.9009 8.9001 0/90 100 125.93 67.369 28.826 16.706 11.841 10.063 9.7825 9.6873 0/90/0 10 16.115 13.382 12.372 12.215 12.176 12.165 12.163 12.162 0/90/0 100 125.99 68.075 30.993 20.347 16.627 15.424 15.244 15.183 0/90/90/0 10 16.172 13.447 12.437 12.280 12.240 12.229 12.228 12.226 0/90/90/0 100 126.33 68.294 31.079 20.38 16.638 15.426 15.245 15.184 T ABLE 5-1 N ON - DIMENSIONAL FREQUENCY [R EF :23] Present Formulation Panel a/h R/a 1 2 10 20 50 100 500 0/90 10 0/90 100 136.0635 73.90925 31.89584 18.20593 12.12838 9.353029 8.72118 8.358714 0/90/0 10 10.75471 0/90/0 100 142.7383 76.47821 32.7317 0/90/90/0 10 15.51217 11.1521 5 15.5545 8.906713 8.373873 8.166252 8.061078 8.027878 7.990367 12.22585 11.01655 10.82998 10.78238 10.76802 10.7647 19.15716 13.78834 11.85253 11.54031 11.43834 14.8861 11.99612 10.95903 10.79977 10.75936 10.74652 10.74316 10.73102 0/90/90/0 100 134.838 72.67357 31.29011 18.50931 13.52047 11.74193 11.45988 11.35508 T ABLE 5-2 29 N ON - DIMENSIONAL FREQUENCY FOR THE PRESENT FORMULATION Material Properties Properties of various constituents of Hybrid composite Constituent C11(GPa) C12(GPa) C22(GPa) C23(GPa) C55(GPa) Density(kg/m3) Carbon 236.4 10.6 24.8 10.7 25 1800[24] 1180 146 411 133 189 1740[25] 13.4615 5.7692 13.4615 5.7692 3.8462 1150 fiber[22] (25,25), 5 walled CNT[6] Epoxy[18] T ABLE 5-3 M ATERIAL PROPERTIES OF VARIOUS CONSTITUENTS IN HYBRID COMPOSITE C11 F IGURE 5-1 V ARIATION OF C 11 W . R . T VARIATION OF CARBO N FIBER AND CNT VOLUME FRACTION From Fig. 5.1, it can be seen that as the carbon fiber volume fraction increases the longitudinal elastic properties increase. At lower volume fractions of carbon fiber (10%) it can be observed that for CFRP composite the elastic modulus is around 35GPa, but with the increase in volume fraction of CNT from 1% to 5% the elastic modulus has increased from 38GPa to 47.3GPa. With the increase in carbon fiber volume fraction, the volume fraction of nanocomposite goes on decreasing; as a result the elastic properties almost converge at higher volume fractions of carbon fiber. 30 C12 and C23 F IGURE 5-2 V ARIATION OF C 12 W . R . T VARIATION OF CARBO N FIBER AND CNT VOLUME FRACTION F IGURE 5-3 V ARIATION OF C 23 W . R . T VARIATION OF CARBO N FIBER AND CNT VOLUME FRACTION From Fig. 5.2 and Fig. 5.3, similarly as C11, the elastic properties along 1-2 direction also increase with the increase with in volume fraction of carbon fiber. At lower volume fractions (10%) it can be observed that CFRP has elastic modulus of 6Gpa, but with increase in CNT volume fraction from 1% to 5%, the elastic properties have increased from 6.8Gpa to 10Gpa. With the increase in carbon fiber volume fraction, it can be observed that composites having lower CNT volume fractions show steep increase in elastic properties than compared to composites having higher CNT volume fractions. 31 C22 F IGURE 5-4 V ARIATION OF C 22 W . R . T VARIATION OF CARBO N FIBER AND CNT VOLUME FRACTION From Fig. 5.4, it can be seen that the transverse elastic properties have improved with the addition of CNT in matrix material for lower volume fractions of carbon fiber. This increase in elastic properties can be attributed to the randomly distributed CNTs. With the increase in carbon fiber volume fraction the composites having lower volume fractions of CNT have shown an increase in transverse elastic modulus, but for composites having higher volume fractions of CNT with the increase in carbon fiber volume fraction there is decrease in transverse elastic modulus. C55 F IGURE 5-5 V ARIATION OF C 55 W . R . T VARIATION OF CARBO N FIBER AND CNT VOLUME FRACTION From Fig. 5.5, it can be observed that the in-plane shear properties have increased with increase in volume fractions of CNT and carbon fiber. Composites normally fail due to shear. So in order avoid failure it is advantageous to use hybrid composite in place of conventional CFRP composites. 32 Impulse response F IGURE 5-6 F IGURE 5-7 IMPULSE RESPONSE OF CFRP COMPOSITE FOR THICK PLATE I MPULSE RESPONSE OF CFRP COMPOSITE FOR THIN PLATE Fig. 5.6 and Fig. 5.7 indicate the response of thick plate and thin plate in modal coordinates for the first mode of vibration. 33 F IGURE 5-8 D ECAY TIME FOR THICK PLATE BY VARYING THE CNT VOLUME FRACTIONS FOR DIFFERENT VOLUME FRACTIONS OF CARBON FIBER F IGURE 5-9 D ECAY TIME FOR THIN PLATE BY VARYIN G THE CNT VOLUME FRACTIONS FOR DIFFERENT VOLUME FRACTIONS OF CARBON FIBER Fig. 5.8 and Fig. 5.9 show the decay time for thick and thin plates. At lower volume fractions carbon fiber, the decay time of the system goes on decreasing with increase in CNT volume fraction. As the volume fraction of carbon fiber increases the decay time also decreases. 34 6 CONCLUSION This chapter presents important observations on the material properties, Rayleigh damping in composite materials, impulse response, and decay time. Conclusions The hybrid composite has been modeled using Mori-Tanaka method and mechanics of materials method. It is found that 1. The longitudinal properties C11 of the hybrid composite increase with the increase in volume fraction of CNT at lower volume fractions of carbon fiber. As the volume fraction of carbon fiber goes on increasing the longitudinal modulus tends to converge because the volume fraction of CNT goes on decreasing. 2. There is tremendous increase in elastic properties C12 and C23 of the hybrid composite with the increase in volume fraction of CNT at lower volume fractions of carbon fiber. As the volume fractions of carbon fiber goes on increasing there is slow increase in composites having higher volume fractions of CNT as compared to composites having lower volume fractions of CNTs. 3. The transverse modulus C22 of the hybrid increases with the increase in CNT volume fraction but as the volume fraction of carbon fiber increases the composites having lower CNT volume fractions show increase in transverse modulus and composites having higher CNT volume fraction show gradual decrease in transverse modulus. 4. The in-plane shear modulus C55 increases with increase in CNT and carbon fiber volume fraction. 5. The amplitude goes on decreasing with increase in damping ratio. 6. As the volume fraction of CNT increases the decay time goes on decreasing. 7. The decay time of thick plate is less than the decay time of thin plate. 35 Future Scope Estimate temperature dependent and hygrothermal properties. Buckling analysis of hybrid composite laminated shell structure. Active vibration control of the laminated shell structure. Delamination analysis. Nonlinear analysis of laminated shell structure. 36 REFERENCES: [1]. [2]. Usuki et al, Inorganic Polymeric Nanocomposites and Membranes Advances in Polymer Science Volume 179, 2005, pp 135-195. R.C.L. Dutra, B.G. Soaresb, E.A. Campos, J.L.G. Silva, Hybrid composites based on polypropylene and carbon fiber and epoxy matrix, Polymer 41, 2000, pp 3841–3849. [3]. Timoshenkho and Krieger, Theory of Plates and Shells, McGraw Hill Publication. [4]. Ronald F. Gibson, Modal vibration response measurements for characterization of composite materials and structures, Composites Science and Technology 60, 2000, pp: 2769-2780 [5]. R. Chandra 1, S.P. Singh, K. Gupta, Damping studies in fiber-reinforced composites - a review, Composite Structures 46, 1999, pp 41-51 [6]. J.P. Lu, Elastic properties of carbon nanotubes and nanoropes, Physical Review Letters vol. 79, 1997, pp. 1297–300 [7]. Mohammad A. Rafiee, Javad Rafiee, Zhou Wang, Huaihe Song, Zhong-Zhen Yu, and Nikhil Koratkar, Enhanced Mechanical Properties of Nanocomposites at Low Graphene Content, American Chemical Society – NANO, Vol. 3, No. 12, 2009, pp 3884–3890 [8]. F.H. Gojny, M.H.G. Wichmann, U. Koepke, B. Fiedler, K. Schulte, Carbon nanotube-reinforced epoxy-composites: enhanced stiffness and fracture toughness at low nanotube content, Composites Science and Technology 64, pp: 2363–2371, 2004 [9]. Florian H. Gojny 1, Malte H.G. Wichmann, Bodo Fiedler, Karl Schulte, Influence of different carbon nanotubes on the mechanical properties of epoxy matrix composites – A comparative study, Composites Science and Technology 65, pp: 2300–2313, 2005 [10]. Gary D. Seidel, Dimitris C. Lagoudas, Micromechanical analysis of the effective elastic properties of carbon nanotube reinforced composites, Mechanics of Materials 38, pp: 884–907, 2006 [11]. Y.J. Liu, X.L. Chen, Evaluations of the effective material properties of carbon nanotube-based composites using a nanoscale representative volume element, Mechanics of Materials 35,pp: 69– 81, 2003 [12]. R.B. Mathur, Sourav Chatterjee, B.P. Singh, Growth of carbon nanotubes on carbon fibre substrates to produce hybrid/phenolic composites with improved mechanical properties, Composites Science and Technology 68, pp: 1608–1615, 2008 [13]. Enrique J. Garcia, Brian L. Wardle, A. John Hart, Namiko Yamamoto, Fabrication and multifunctional properties of a hybrid laminate with aligned carbon nanotubes grown In Situ, Composites Science and Technology 68, pp: 2034–2041, 2008 [14]. S. I. Kundalwal, M. C. Ray, Micromechanical analysis of fuzzy fiber reinforced composites, Int J Mech Mater Des 7,pp: 149–166, 2011 37 [15]. Tarapada Roy, DebabrataChakraborty, Optimal vibration control of smart fiber reinforced composite shell structures using improved genetic algorithm, Journal of Sound and Vibration 319, pp: 15–40, 2009 [16]. C. Kyriazoglou, F.J. Guild, Finite element prediction of damping of composite GFRP and CFRP laminates – a hybrid formulation – vibration damping experiments and Rayleigh damping, Composites Science and Technology 67, pp: 2643–2654, 2007 [17]. R. Verdejo, R. Stämpfli, M. Alvarez-Lainez, S. Mourad, M.A. Rodriguez-Perez, P.A. Brühwiler,M. Shaffer, Enhanced acoustic damping in flexible polyurethane foams filled with carbon nanotubes, Composites Science and Technology 69,pp: 1564–1569, 2009 [18]. M.H. Yas, M. Heshmati, Dynamic analysis of functionally graded nanocomposite beams reinforced by randomly oriented carbon nanotube under the action of moving load, Applied Mathematical Modelling 36, pp: 1371–1394, 2012 [19]. JiangYu Li and Martin L. Dunn, Anisotropic coupled field inclusion and inhomogeneity problems, PHILOSOPHICAL MAGAZINE A, 1998, VOL. 77, and No. 5, 1341 – 1350. [20]. Dong-Li Shi, Xi-Qiao Feng, Yonggang Y. Huang, Keh-Chih Hwang, Huajian Gao, The Effect of Nanotube Waviness and Agglomeration on the Elastic Property of Carbon Nanotube- Reinforced Composites, JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY, Vol. 126, JULY 2004, pp: 250 – 257 [21]. Indrajit Chowdhury, Shambhu P. Dasgupta, Computation of Rayleigh Damping Coefficients for Large Systems, Department of Civil Engineering, Indian Institute of Technology. [22]. Kuniaki Honjo, Thermal stresses and effective properties calculated for fiber composites using actual cylindrically-anisotropic properties of interfacial carbon coating, Carbon 45, 2007, pp:865– 872 [23]. Tarapada Roy, P.Manikandan , DebabrataChakraborty, Improved shell finite element for piezothermoelastic analysis of smart fiber reinforced composite structures, Finite Elements in Analysis and Design 46, pp:710–720, 2010 [24]. Akihiro Mizutani, Yoshitake Nishi, Improved Strength in Carbon Fiber Reinforced Plastics due after Electron Beam Irradiation, Materials Transactions, Vol. 44, No. 9, pp. 1857 to 1860, The Japan Institute of Metals, 2003. [25]. S.H. Kim, G.W. Mulholland, M.R. Zachariah, Density measurement of size selected multiwalled carbon nanotubes by mobility-mass characterization, Carbon (47), pp:1297-1302, 2009 38

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement