UFDC Home  Search all Groups  UF Institutional Repository  UF Institutional Repository  UF Theses & Dissertations  Vendor Digitized Files   Help 
Material Information
Subjects
Notes
Record Information

Full Text 
OPTIMIZATION OF MATERIAL DAMPING AND STIFFNESS OF LAMINATED FIBERREINFORCED COMPOSITE STRUCTURAL ELEMENTS BY JIINGKAE WU A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1985 ACKNOWLEDGEMENTS The author wishes to express his deep appreciation to his major professor, Dr. ChangTsan Sun, whose valuable advice brings a nice end to this study. The author is also indebted to Professor Malvern for his invaluable comments and constructive criticisms. Pro found appreciation is also due to Professors Kurzweg, Taylor, and Verma, other committee members, for their worthy comments and for the time they spent in reviewing this dissertation. Much thanks are given to Jenny Sun for her contribu tion in typing this dissertation. And the author grate fully acknowledges support of this research work from the U.S. Air Force Office of Scientific Research under Grant No. AFOSR830154 and AFOSR830156 monitored by Dr. D. R. Ulrich. Finally, the author expresses his greatest apprecia tion to his parents and his wife, MongShya Rau Wu, for encouraging him to study in the U.S.A. and for providing warm family support during the past four years. TABLE OF CONTENTS Page ACKNOWLEDGEMENTS. . . .. ii LIST OF TABLES . .. vi LIST OF FIGURES . ... .. vii ABSTRACT . . xiv CHAPTERS 1 INTRODUCTION . . 1 1.1 Literature Survey . 2 1.2 Scope of This Study . 4 1.3 Material Constants and Ranges of Design Parameters . 5 2 DAMPING . . 8 2.1 Definition of Damping 8 2.2 Damping Mechanism .... 9 2.2.1 Viscous Damping 9 2.2.2 Dry or Coulomb Damping. 10 2.2.3 Material Damping .. 11 2.3 Types of Damping Representation 15 2.3.1 Damping Ratio 16 2.3.2 Logarithmic Decrement 17 2.3.3 Loss Tangant .. .18 2.3.4 Specific Damping Capacity for Cyclic Loading .. 20 3 DAMPING OF UNIDIRECTIONAL SHORTFIBER COMPOSITES . .. 23 3.1 Introduction . 23 3.2 Damping Analysis of Unidirectional Fiber Composites .. .24 3.2.1 ShortFiber Composite Model .. 24 iii 3.2.2 Damping Analysis of Aligned Fiber Composites ...... 30 4 DAMPING OF RANDOMLY ORIENTED SHORTFIBER COMPOSITES .... . .... .40 4.1 Introduction .. 40 4.2 Damping Analysis of InPlane Randomly Oriented ShortFiber Composites ........... .41 5 DAMPING OF LAMINATED FIBER COMPOSITES LAMINATED PLATE THEORY APPROACH ... .45 5.1 Introduction 45 5.2 Damping Analysis of Laminated Fiber Composite Through Laminated Plate Theory Approach . ... .45 6 DAMPING OF LAMINATED FIBER COMPOSITES ENERGY APPROACH ... .. 54 6.1 Introduction ... 54 6.2 Damping Analysis of Laminated Fiber Composites Through Energy Approach .... .55 7 EXPERIMENTAL MEASUREMENT OF DAMPING 61 7.1 Introduction . 61 7.2 Apparatus . .... 62 8 OPTIMIZATION OF DAMPING AND SPECIFIC STIFFNESS FOR FIBER COMPOSITES 69 8.1 Introduction .... 69 8.2 Brief Introduction to Sequential Simplex Method ... .70 8.3 Mathematical Formulation of Design Problems ... .74 9 RESULTS AND CONCLUSIONS .. 85 9.1 Preliminary Remarks . 85 9.2 Damping and Stiffness of Uni directional Fiber Composites 86 9.3 Damping and Stiffness of Randomly Oriented Short Fiber Composites ........ .89 Page CHAPTER CHAPTER APPENDICES A B C D E F HYSTERESIS LOOP OF VISCOELASTIC MATERIALS . . EXPRESSIONS OF 'S . ... AN ALTERNATIVE WAY OF DETERMINING Er and Gr . . INVERSION OF A SYMMETRIC COMPLEX MATRIX . ENERGY EXPRESSION OF DAMPING FOR LAMINATED COMPOSITES . . FORMULATION OF FINITE ELEMENT METHOD . F.I Stiffness Matrix .... F.2 Formulation of Finite Element Method . . REFERENCES .. . . . BIOGRAPHICAL SKECH . . 147 . 151 . 154 . 157 . 160 . 163 . 163 . 166 * 170 . 176 Page 9.4 Damping and Stiffness of Laminated Fiber Composites Laminated Plate Theory Approach . 90 9.5 Damping and Stiffness of Laminated Fiber Composites Energy Approach .. 93 9.6 Experiemntal Results of Damping and Stiffness ... 95 9.7 Optimization of Damping and Specific Stiffness of Fiber Composites . ... 96 9.8 Conclusions . ... 98 LIST OF TABLES Table Page 1.1 Material Properties Data of the Matrix and the Fibers . 7 3.1 Influence of Vi and s/d on tanh (Bs/2) (Bs/2) .. 28 9.1 Experimental Results of 05/905/05 Glass Epoxy Composite Plates . .145 9.2 Influence of Weighting Constants on Optimum Design for Case One .145 9.3 Optimum Design of Cross Ply Composite Plates for Case One. . .. .146 9.4 Optimum Design of Cross Ply Composite Plates for Case Two. . .146 LIST OF FIGURES Figure Page 2.1 Sketch of Hysteresis Loop .. 14 2.2 Sketch of Viscous Damping Model 16 2.3 Sketch of Logarithmic Decrement 18 2.4 RotatingVector Representation of Harmonic Motion . 19 2.5 Hysteresis Loop of an Inelastic Body 21 3.1 ShortFiber Composite Model . 24 3.2 Representative Volume Element of OffAxis ShortFiber Composites .. 30 5.1 Sketch of Laminated Fiber Composite Plate . . 46 7.1 Schematic Drawing of the Experimental Setup . . 66 7.2 Typical Display of Real and Imaginary Parts of Frequency Response Function for a Graphite Epoxy Composite Beam 67 7.3 Enlarged Schematic Drawing of Real Part of Frequency Response Function ...... 68 vii Figure Page 8.1 Rule 1 of Sequential Simplex Method .. 73 8.2 Rule 2 of Sequential Simplex Method .. 73 8.3 Rule 3 of Sequential Simplex Method .. 74 9.1 Plots of Ex /Em vs s/d using 0 as a Parameter for Graphite Epoxy Composites 101 9.2 Plots of E" /E" vs s/d using 0 as a Parameter for Graphite Epoxy Composites 102 9.3 Plots of qx/nm vs s/d using 0 as a Parameter for Graphite Epoxy Composites 103 9.4 Plots of Ex/Em vs 0 using s/d as a Parameter for Graphite Epoxy Composites 104 9.5 Plots of E"/Em vs e using s/d as a Parameter for Graphite Epoxy Composites 105 9.6 Plots of nx/Tm vs e using s/d as a Parameter for Graphite Epoxy Composites 106 9.7 Plots of Ex/E vs s/d using 0 as a Parameter for Kevlar Epoxy Composites 107 9.8 Plots of E"/Em vs s/d using 0 as a Parameter for Kevlar Epoxy Composites 108 9.9 Plots of x/nx vs s/d using 0 as a Parameter for Kevlar Epoxy Composites 109 9.10 Plots of nx/rm, Ex/E and E"/E vs 0 keeping s/d=100 for Kevlar Expoxy Composites . .... 110 viii Figure 9.11 Plots of nx/qm and E/Em vs 8 keeping s/d=100 for Graphite Epoxy Composites and Kevlar Epoxy Composites .. 111 9.12 Plots of Ex/Em vs 0 using Vf as a Parameter for Graphite Epoxy Composites .. .... .112 9.13 Plots of nx/nm vs e using Vf as a Parameter for Graphite Epoxy Composites . . 113 9.14 ThreeDimensional Plots of Ex/Ej vs e and s/d for Kevlar Epoxy Composites 114 9.15 ThreeDimensional Plots of x /nm vs 8 and s/d for Kevlar Epoxy Composites 115 9.16 Contour curves of Ex/E' vs 0 and s/d for Kevlar Epoxy Composites 116 9.17 Contour curves of nx'/m vs 0 and s/d for Kevlar Epoxy Composites 117 9.18 Plots of nx/nm, Er/Em and E"/E" vs s/d for Randomly Oriented Glass Epoxy Composites . . 118 9.19 Plots of nx m, E /E and E"/E" vs s/d for Randomly Oriented Graphite Epoxy Composites . ... 119 9.20 Plots of nx/lm, Er/Em and E/Em vs s/d for Randomly Oriented Kevlar Epoxy Composites .. ... 120 ix Page Figure Page 9.21 Plots of nGr/nGm' G /Gr and G/Gm vs s/d for Randomly Oriented Glass Epoxy Composites . 121 9.22 Plots of GGr/nGm, Gr/Gm and G"/G vs s/d for Randomly Oriented Graphite Epoxy Composites . ... 122 9.23 Plots of rGr/'Gm' Gr/G and G"/G vs s/d for Randomly Oriented Kevlar Epoxy Composites . .... 123 9.24 Plots of nr/nm vs Vf using s/d as a Parameter for Randomly Oriented Glass Epoxy Composite . .... 124 9.25 ThreeDimensional Plots of E_/EA vs s/d and Ej for Randomly Oriented Fiber Composites . ... 125 9.26 ThreeDimensional Plots of ,r/nm vs s/d and Ej for Randomly Oriented Fiber Composites . ... 126 9.27 Contour Curves of Er/Em vs s/d and Et for Randomly Oriented Fiber Composites 127 9.28 Contour Curves of r/ m vs s/d and Ej for Randomly Oriented Fiber Composites 128 9.29 Plots of D1~/Dm and Fpll/nm vs s/d for QuasiIsotropic Graphite Epoxy Composites . ... .129 Figure 9.30 Plots of D66/DGm and Fn66/nGm vs s/d for QuasiIsotropic Graphite Epoxy Composites . . 13 9.31 Plots of Eil/Em vs 8 using s/d as a Parameter for Angle Ply Graphite Epoxy Composites . 131 9.32 Plots of ll1 /nm vs 0 using s/d as a Parameter for Angle Ply Graphite Epoxy Composites . 132 9.33 Plots of E66/Em vs 8 using s/d as a Parameter for Angle Ply Graphite Epoxy Composites . 133 9.34 Plots of ig66/Dm vs 8 using s/d as a Parameter for Angle Ply Graphite Epoxy Composites . ... 134 9.35 Comparisons of DI/Dm vs s/d for Four Kinds of Laminated Graphite Epoxy Composites . . 135 9.36 Comparisons of Fnll/ m vs s/d for Four Kinds of Laminated Graphite Epoxy Composites. . . 136 9.37 Comparisons of D66/DGm vs s/d for Four Kinds of Laminated Graphite Epoxy Composites . . 137 Page Figure Page 9.38 Comparisons of Fn66/nGm vs s/d for Four Kinds of Laminated Graphite Epoxy Composites and Hybrid Fiber Composites 138 9.39 Influence of s/p on the InPlane Longitudinal Damping Through Energy Approach . . 139 9.40 Influence of a/h on the InPlane Longitudinal Damping of Laminated Graphite Epoxy Composites Through Energy Approach . 140 9.41 Influence of a/h on the Flexural Normal Damping of Laminated Graphite Epoxy Composites Through Enery Approach . .141 9.42 Comparison Between Analytical Results and Experimental Results for Unidirectional Discontineous Graphite Reinforced Epoxy Composite Beams .. 142 9.43 Comparison Between Analytical Results and Experimental Results for Off Axis Unidirectional Continuous Graphite Reinforced Epoxy Composites Beams 143 9.44 Contour Curves of Objective Function for Case One vs s/d and s/p for Graphite Epoxy Composite Plate . 144 xii Figure Page A.1 Hysteresis Loop of a Viscoelastic Material . .. 149 F.1 Principal Coordinates of a Fiber Composite Material . 165 xiii Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy OPTIMIZATION OF MATERIAL DAMPING AND STIFFNESS OF LAMINATED FIBERREINFORCED COMPOSITE STRUCTURAL ELEMENTS By JiingKae Wu December 1985 Chairman: C. T. Sun Major Department: Engineering Sciences Analysis of material damping and optimization of both material damping and specific stiffness of laminated, continuous or discontinuous fiber reinforced polymer matrix is the major objective of this study. Two different approaches, laminated plate theory approach and energy approach, are used in conjunction with elasticviscoelastic correspondence principle for the analysis of the material damping for fiber reinforced composites. Damping values obtained through these two approaches are close to each other under certain situa tions. A discontinuous fiber composites model is developed to determine the longitudinal modulus of dis continuous fiber reinforced composites. Aligned or off axis unidirectional fiber composites, inplane randomly xiv oriented fiber composites, and several kinds of laminated fiber composites are considered in damping and stiffness analysis. Experimental results on damping and stiffness by the impulse hammer technique agree with the analytical results for unidirectional fiber composites and for cer tain cross ply fiber composites. In the energy approach, a threedimensional finite element method based on three dimensional elasticity is applied to determine the strain field of an elastic body. Dissipated energy can be deter mined through this strain field and loss moduli. Damping through energy approach depends on the boundary condi tions, the loading conditions, and the geometry (espe cially the dimension in thickness direction) of the body. Sequential Simplex method, laminated plate theory, and an elasticviscoelastic correspondence principle are used to optimize both material damping and specific stiff ness of composites. Minimum flexure deformation design of orthotropic laminated fiber composite plates is obtainable through this optimization procedure. CHAPTER 1 INTRODUCTION Due to the high strengthtoweight and stiffnessto weight ratios, composite materials are ideal for weight sensitive structures such as aircraft, spacecraft, and automotive vehicles. In recent years, with the advent of jet propulsion, particularly with the current increased interest in short takeoff and landing aircraft, it has become necessary to pay increasing attention to the higher frequency motions of such structures. These motions depend strongly on the structure's damping or capability for dissipation of vibratory energy [1]. In addition, a new type of excitation has become more preva lent, random excitation either of mechanical or acoustical origin [2]. For example, jet engine exhaust generally contains a noise spectrum wide enough to excite most of the natural frequencies encountered in aircraft structures [3]. The natural resonance phenomena, so produced, can be very destructive. Since near resonant conditions can no longer be avoided in many types of structures, the maximi zation of damping within a structural system provides a most useful concept in controlling resonance [4]. Unfortunately, as will be shown in later chapters, high damping is mostly coupled with low stiffness, and high stiffness is mostly coupled with low damping. Therefore, the optimization of damping and of the stiffnesstoweight ratio is a practical idea in designing a proper composite material to be used in aircraft and space vehicles. 1.1 Literature Survey Most results from a series of researches on damping beginning in the 1920's [5] indicated that damping is a material property. Kimball and Lovell [5] experimentally showed that for stress cycles of frequency of from two to three a minute, up to fifty a second the frictional loss (a kind of energy loss) is independent of the frequency but is depen dent on the amplitude of strain of the cycles for eighteen different solids, including several metals, glass, cellu loid, rubber and maple wood, when strain was below the elastic limit. Crandall [6] pointed out that the values for material damping which will be introduced in Section 2.3.3 encountered in practice ranged from about 0.00001 to 0.2; however, Lazan [4] pointed out the material damping ranged from 0.001 to 0.1. And the material damping depends on both the amplitude and frequency of the oscillation. If, however, the system is completely linear, then damping is independent of amplitude [6]. Recently, the composite materials got more attention in industrial application. Lazan [4] gave a detailed review on material damping of materials and material composites. Kume, Hashimoto and Maeda [7] used the damping stress function, derived by Lazan [8], to calculate the material damping of cantilever beams. They found that low order modes of a cantilever beam with equal maximum stress amplitude gave almost the same material damping, theoreti cally, and experimental results have the same order of magnitude as the theoretical results when the maximum stress amplitude is less than a certain value. Schultz and Tsai [9] indicated that unidirectional glass fiber reinforced composites exhibit anisotropic, linear viscoelastic behavior when those undergoing small oscillation and that damping increases in magnitude with change in fiber orien tation angle with respect to loading direction in the order 0*, 22.5, 90", 45. Ni, Lin, and Adams [10, 11] used the laminated plate theory and twodimensional energy approach to predict the flexural damping of laminated composites. In their work, damping coefficient was deter mined by freefree flexural modes of vibration [12]. Siu and Bert [13] discussed the vibration of composite plates having material damping. Suarez, Gibson, and Deobald [14] observed the dependence of damping on frequency of fiber reinforced epoxy or polyester. Gibson and Plunkett [15] found that for small strain, damping and stiffness are independent of amplitude of strains, but, once the thres hold strain is exceeded (i.e., failure starts), the resulting increase in damping is much more significant than the corresponding reduction in stiffness. Similar results were also observed by Tauchert and Hsu (16]. Bert and Clary [17, 18] gave a complete review on measurement and analysis of damping and dynamic stiffness for compo sites. The first paper to optimize the damping of the structure was perhaps that of Plunkett and Lee [19]. The damping of a beam is improved by introducing thin con strained viscoelastic layers on the top and bottom of the structure. These viscoelastic layers are then stiffened by properly designed constraining layers. Cox [20] discussed the stress distribution in fibrous materials. Cox's shear lag stress analysis was later on used to analyze the stress distribution of shortfiber composites, as in studies [21, 22]. Photoelasticity [23, 24] and finite element methods [25, 26] were used to investigate the stress concentration in the matrix around fiber tips of shortfiber composites. Strength of short fiber composites was analyzed in several studies [27, 28]. Analysis of complex moduli for such kind of material was presented in studies [29, 30, 31]. High damping of short fiber composites was analytically and experimentally observed in references [31, 32]. Material damping of randomly oriented and unidirectional laminar shortfiber composites has been discussed by Sun, Wu, Chaturvedi, and Gibson [33, 34]. 1.2 Scope of This Study The objectives of this study are to analyze the mate rial damping and to optimize the specific stiffness (the ratio of the stiffness to the density) and material damping of continuous and/or discontinuous fibers rein forced laminated composite structure elements. The work involved in this research is briefly introduced as follows: A. To develop a shortfiber composite model to determine the moduli of shortfiber composites. B. To analyze the stiffness and material damping of unidirectional laminar fiber composites, randomly oriented fiber composites, and certain kinds of laminated fiber composites through classical lami nated plate theory approach. C. To analyze the material damping of laminated fiber composites by an energy approach, where a three dimensional displacement finite element method is used. D. To optimize material damping and the specific stiff ness of laminated composite plates. 1.3 Material Constants and Ranges of Design Parameters In this study, four different kinds of widelyused fiber composites (i.e., glass epoxy, Kevlar epoxy, graphite epoxy, and boron epoxy) are involved; and much interest is concentrated on graphite epoxy and Kevlar epoxy, because, generally speaking, the former has higher stiffness, while the latter has higher damping. In order to compare with the experimental results of Suarez, etc. [35, 36], the material constants used in this study are the same as those of the experimental specimens. Some material constants, which are not given in those experimental data, are obtained from reference [37]. Unless specially specified, the material constants used in this study are given in Table 1.1. The length of fiber is one of the characteristic parameters of shortfiber composites. Due to the existence of a critical fiber length, sc (i.e. the minimum fiber length in which the ultimate strength ofu can be achieved [38]), there is a minimum value for fiber length, s f i=i s fu 1.1 d d 2Ty where Ty is the matrix yield stress in shear, and d is the fiber diameter. The lowest ofu of those four fiber com posites is 2750 MPa, as given in reference [37], for graphite T300. And the matrix yield shear stress is, according to the manufacturer's test results, 97 MPa of AS4/3501 graphiteepoxy tape. Therefore, the fiber aspect ratio, the ratio of fiber length to its diameter, should be s/d Z 14.2 In this study, the fiber aspect ratio is chosen to be between 25 and 10000; while the fiber volume fraction Vf is chosen to be 0.65 for the most cases or 0.5 for randomly oriented fiber composites. Table 1.1: Material Properties Data of the Matrix and the Fibers Constants E1 (Gpa) E2 (Gpa) G12 (Gpa) v12 Matrix Epoxy 3.94 3.94 1.465 0.345 Glass Glass 72.4 13.8 27.6 0.22 Fibers Kevlar 99.8 6.9 13.8 0.376 Graphite Graphite 175.8 13.8 27.6 0.16 k (Gpa) 4.236 n1 0.015 0.0015 0.011 0.0015 0.0019 p(kg/m3) 1220. 2539. 1479. 1760. 2481. C 2 2 2 2 2 1 1 1 1 where &1 and &2 are constants used in HalpinTsai Equation, n1 is the longitudinal damping, k is the bulk modulus, and p is the density. Boron 381.9 35.0 70.0 0.21 I Z CHAPTER 2 DAMPING 2.1 Definition of Damping The process by which vibration steadily diminishes in amplitude is called damping. In many ways, the assumption that systems possess no damping is a mathematical conven ience, rather than a reflection of physical evidence. In fact, if a system is set in motion and allowed to vibrate freely, the vibration will eventually die out; the rate of decay depends on the amount of damping. This reduction in vibrating amplitude occurs because the energy of the vibrating system is dissipated as friction or heat or is transmitted as sound [39]. And this is why damping is also interpreted as any phenomenon within the body of the material where energy is dissipated [40]. The concept of an undamped system serves not only a useful purpose in analysis, but can also be justified in certain circumstances. For example, if the damping is small and one is interested in the free vibration of a system over a short interval time, there may not be sufficient time for the effect of damping to become noticeable. Similarly, for small damping one may not be able to notice the effect of damping in the case of a system with harmonic excitation, provided the driving frequency is not in the neighborhood of any of the natural frequencies of the system [41]. On the other hand, damp ing of a given system should be considered if this system is subjected to vibration near its resonant frequencies because damping has a large influence on the amplitude in the frequency region near resonance [42]. 2.2 Damping Mechanism There are many mathematical models representing damp ing. The mechanism of damping can take any of these forms and often more than one form may be present at a time. Therefore, in order to analyze or predict the damping of a given system, one ideally should take into account all possible damping mechanisms; fortunately, in most practical cases, one or two mechanisms predominate so that one may neglect the effect of all others. Three widely used mathematical models for damping are introduced below: 2.2.1 Viscous Damping The viscous damping force is defined as Fd = CX 2.1 where the constant C (of dimension force per unit velocity) is called the coefficient of viscous damping. This type of damping occurs in lubricated sliding surfaces, dashpots, hydrolic shockabsorbers [43]. The minus sign indicates that this damping force is always opposite to the direc tion of the motion. The work done by damping force, namely, dissipated energy during one cycle of harmonic motion, x = Aosinwt, will be 2n Sdx2 2 (UD)cyc = Fd dx = oC(2)dt = 2CAow 2.2 where w is the circular frequency in radians per unit time. Apparently, dissipated energy due to viscous damping in a cyclic motion is proportional to the frequency and to the square of the amplitude of motion. 2.2.2 Dry or Coulomb Damping This type of damping occurs in the sliding of dry surfaces. The damping force during motion is constant'and is given, according to Coulomb's law, by Fd = fN 2.3 where N is the normal component of the force upon the sur face of contact, and f is the coefficient of dry friction. Damping induced by the joints is mainly because of dry damping. And it is known that damping of builtup structures (i.e. structure made by joining together skins, strings, frames, etc.) could further be caused by the effect of the joint [1]. 2.2.3 Material Damping This kind of damping is also referred to as internal, hysteresis or structual damping. It is caused by the internal friction, the viscoelastic behavior of the material, and the interfacial slip in the material itself. It is well known that an elastic body which is repeatedly stressed becomes hot. If an elastic body is subjected to forced oscillation, a positive work of the exciting force must be spent to keep the amplitude of the oscillations constant in time. The reason for the heating of the body and the expenditure of external work is the internal friction of the material. Although this explanation can be easily accepted in a qualitative way, it is more difficult to translate the problem into mathematical terms. Three principal hypotheses have been proposed to explain the phenomenon of internal friction, i.e., the vis cous theory, the hereditary theory, and the hysteresis loop. Viscous theory. The viscous theory assumes that in solid bodies, there exist some viscous actions which can be compared to the viscosity of fluids. These viscosity effects are assumed to be proportional to the first time derivative of strain. The coefficient of the proportionality (constant for each material at constant temperature) is called the coefficient of viscosity. Based on this assumption, many mathematical models (such as Maxwell model, KelvinVoigt model, and threeparameter models [44, 45]) have been introduced to represent dif ferent materials. For the case of KelvinVoigt model under normal deformation, the relationship between stress a and strain e is expressed by de O = Ee + E d 2.4 where E is Young's modulus and is the coefficient of viscosity of the material. Hereditary theory. The hereditary theory attributes the dissipation of energy due to material damping to the elastic delay by which the deformation lags behind the applied force [43]. According to this theory, the defor mation at a given instant, instead of depending only on the actual applied stress at that time as it would if the materials followed Hooke's law, depends on all the stresses previously applied to the elastic body. The stressstrain relationship is given by t a = Ee + I (t, T) e(T) dT 2.5 Jco where t is the actural time and T is an instant of time between t =  and t = t. The function $(t, T) is called the hereditary kernel or memory function. Hysteresis loop. For a material under a cyclic loading, the stressstrain curve is a closed curve which is called the hysteresis loop. The physical meaning of this hysteresis loop is given in Section 2.3. The area within the hysteresis loop is proportional to the dissipated energy. This area, being a material property, may or may not depend on the frequency. A mathematical model can be used to explain the energy dissipation, when this energy dissipation is independent of frequency in a material loaded by a cyclic force. In this mathematical model, the damping force is assumed to be proportional to velocity and inversely proportional to frequency, i.e. F h k 2.6 d = If an external force Fe is applied just enough to balance the damping force and to maintain a simple harmonic motion, x = Aosinwt, then F = Kx + h 2.7 e W where Kx represents the elastic force of the system; for example, a single spring, w is the circular frequency, Ao is the amplitude, and h is the hysteretic damping constant. The relation ship between Fe and x is given in Equation 2.7, and the plot of external force Fe as a function of displacement is a skewed ellipse, as in Figure 2.1. 2 (x)2 + F Kx)= 2.8 Ao hAo F e x Figure 2.1: Sketch of Hysteresis Loop The work done by damping force (dissipated energy) in one cycle (UD)cyc is 2r (Ucyc = F dx = h()2 dt = hA 2.9 Thus, the energy dissipated in one cycle is proportional only to the square of the amplitude. This expression agrees with the results of experiments of Kimball and Lovell [5] which indicate that for a large variety of materials such as metals, glass, rubber and maple wood, subject to cyclic stress such that the strains remain below the elastic limit, the internal friction is entirely dependent on the rate of strain. Equation 2.9 also agrees with Lazan's notes [46] on dissipated energy. For example, at low amplitudes of stress, the dissipated energy is proportional to the square of the stress amplitude, and the hysterestic loop is elliptical in form. Unlike homogeneous materials, fiber reinforced mate rials have interfaces between matrix and fiber. When a fiber reinforced composite is subject to a tensile strain cycle, the high shear stress may cause the fiber matrix interface to fail so that energy is dissipated by friction as the matrix slides over the fibers [47]. Damp ing is then increased due to interfacial slips between matrix and fiber. In this study, perfect bonding between matrix and fiber is assumed; conseqeuntly, interfacial slip is not considered. 2.3 Types of Damping Representations Many different disciplines have been concerned with damping measurements, and this has further complicated nomenclature. Confusion has been caused not only by the large variety of damping units used, but also by the lack of unique definition for many wellaccepted units. It is, therefore, desirable to review the various damping units currently used and to indicate relationships between them. 2.3.1 Damping Ratio (4) Figure 2.2 shows a single degree of freedom system with viscous damping, excited by force F(t). F(t) Figure 2.2: Sketch of Viscous Damping Model Its differential equation of motion is found to be MX + Cx + Kx = F(t) 2.10 If F(t)=0, one has the homogeneous differential equation whose solution corresponds physically to that of free damped vibration. The general solution to this homogeneous equation is x = e (c/2m)t I (cel + c2ebt) where b = c/2m) k/m1/2 b = [(c/2m) k/m] 2.11 2.12 cl and c2 are constants to be determined by initial conditions. In order to have oscillation, one will expect to have (c/2m)2 < k/m 2.13 Apparently, there exists a critical value cc for c, when (c/2m)2 equals k/m. Damping ratio [38], C, is defined as Sc 2.14 cc where cc = (4mk) 2.15 2.3.2 Logarithmic Decrement (6) A convenient way to determine the amount of damping present in a system is to measure the rate of decay of free oscillations. Logarithmic decrement [38] is defined as the natural logarithm of the ratio of any two successive amplitudes, as in Figure 2.3, of a free vibration. X. 6 = Ln 2.16 Xi+1 Figure 2.3: Sketch of Logarithmic Decrement 2.3.3 Loss Tangent (tan $) It is well known that polymer behaves as viscoelastic material, i.e., combining two material properties, one of which is perfectly elastic, while the second is viscous fluid [43]. Let such a viscoelastic material be subject to a sinusoidal stress experiment at frequency w such that the period 27/w of oscillation is sufficiently large as compared to the transit time of elastic waves through the specimen that stress and strain can be considered uniform throughout the test section. Under these condi tions, the response to a steadystate sinusoidal stress a is a steadystate sinusoidal strain E at the same frequency [44], out of phase by the angle 0, e.g. 0 = 0o sinwt 2.17 e = Eo sin(wt P) 2.18 Both the response amplitude and the phaseshift (or phase angle) D are frequencydependent, but in the linear range eo is proportional to co. The phase relationships are conveniently shown in the rotatingvector representation of simple harmonic motion, as in Figure 2.4. E"e  A 0 EA 0 1 o I B Figure 2.4: RotatingVector Representation of Harmonic Motion The rotating vector OB of magnitude E'co lags behind the stress OA by 4 radians. Stress OA may be resolved into two components, E'eo in phase with strain and E"eo, 7/2 radians out of phase with strain, as in Figure 2.4. Here E' is the storage modulus and E" is the loss modulus. The loss tangent tan 0 is defined as tan Q = E"/E' 2.19 Some authors call tan P the loss coefficient. The ratio E"/E' is a measure of the ratio energy loss to energy stored, as will be shown in Appendix A. For viscoelastic material, the moduli are often expressed in terms of a complex number, called the complex modulus. In this study, the superscript is used to indicate the complex modulus, for example: E* = E' + iE" 2.20 where i is J1. 2.3.4 Specific Damping Capacity for Cyclic Loading (*c) The physical meaning of the hysteresis loop of Section 2.2.3 is considered here. Since materials do not behave in a perfectly elastic manner even at very low stress [46], inelasticity is always present under all types of loading, although in many cases extremely precise measurements are necessary to detect it. Under a cyclic loading condition, inelastic behaviors lead to energy dis sipation. This means that the stressstrain (or load deformation) curve is not a singlevalued function but forms a hysteresis loop. Energy is absorbed by the mate rial system under cyclic load, and the energy absorbed is proportional to the area within the hysteresis loop [46], as in Figure 2.5. F Figure 2.5: Hysteresis Loop of An Inelastic Body Consequently, another measurement of damping called specific damping capacity [46] can be obtained by compar ing the energy dissipated (or absorbed) (UD)cyc of the system in a cycle with the maximum strain energy stored (Us)max in the system during that cycle. 4c = (UD)cyc/(Us)max 2.21 In Appendix A, viscoelastic material is shown to have such a hysteresis loop under cyclic loading, and the same expression for specific damping capacity is obtained. The difference is that specific damping of viscoelastic material is also a function of frequency, as reported in studies [48, 49]. This is because the storage and loss moduli are functions of frequency. 22 For small damping, the relationships between those representations of damping are given in references [50, 51]. 6 2 (UD)cyc 2.22 tan = 2 (U, iT 2 2 (Us max (1 5 ) CHAPTER 3 DAMPING OF UNIDIRECTIONAL FIBER COMPOSITES 3.1 Introduction The objective of this chapter is to determine theo retically the damping of unidirectional fiber reinforced polymer matrix composites. The major damping mechanism of such composites is the viscoelastic behavior of the poly mer and fibers. The analysis is carried out by first applying the concepts of balance of force and equal strain energy on shortfiber composite model to determine the longitudinal modulus of shortfiber composite. Then the elementary mechanics approach is used to find the modulus Ex along the loading direction as a function of the mecha nical properties of the fiber and matrix materials. This is followed by applying the viscoelasticelastic correspondence principle [45, 52] to express the mechani cal properties of the composite, fiber, and matrix; then after the real and imaginary parts of complex modulus are separated, the damping of the composite can be obtained. 3.2 Damping Analysis of Unidirectional Fiber Composites 3.2.1 ShortFiber Composite Model The shortfiber composite model is composed of a finitelength fiber and the polymer matrix, as in Figure 3.1(a). a a f ^T t H T ~I i i 0 HI a a (a) (b) (c) Figure 3.1: ShortFiber Composite Model Figure 3.1(c) is the homogeneous material equivalent to the composite of Figure 3.1(a). Figure 3.1(b) is the front middle longitudinal section view of Figure 3.1(a), where d and s are the diameter and the length of fiber respectively; D and L are the diameter and the length of the composite model, respectively; and P is interpreted as the distance between fiber tips along fiber direction. The ratio of P to s is defined as R and is interpreted as the degree of discontinuity. During the derivation of Young's modulus along the fiber direction, the shortfiber composite model is treated as if it is composed of two materials connectedin series along the fiber direction. One material which is between sections H and H' is the mixture of fiber and matrix having length s, while the other material is just the pure matrix having length P. As in some other analytical work [23, 31, 32] on shortfiber composites, the results of Cox's shear lag stress analysis [20] are used in this study. The expres sion for elastic stiffness of the discontinuous fiber composite is derived from the average of fiber stress based on Cox's fiber stress distribution (in which the longitudinal fiber stress is a function of position). Of = CfEf fi cosh[8(s/2x)]} 3.1 cosh(Bs/2) where x, B, and Os/2 are defined in Appendix B, and ef is the strain of the fiber. In this study, the square pack ing array of fiber composites is considered; therefore, Bs/2 can be written, according to reference [31], as G 1/2 Bs = 2 m 3.2 2 d Ef Ln  4vf The average fiber stress is = s/2 s/2 f f= o af dx 3.3 'f s/ go Substitute Equation 3.1 into Equation 3.3 Of = Ef Ef [1 tanh(s/2)] 3.4 For the composite between sections H and H' in Figure 3.1(a), in order to have static equilibrium, the total longitudinal force q applied to this composite must be q = c Ac = of Af + m Am 3.5 Therefore, ac = Ec Ec = Of Vf + m Vm 3.6 I I where Vf and Vm are the fiber volume fraction and matrix volume fraction within sections H and H' separately. It is assumed that the composite, fiber, and matrix (all between sections H and H') have the same extensional strain e. The longitudinal modulus of material between sections H and H' can be obtained from Equation 3.6. Ec = E Vf [1 tanh(8s/2)] + E V 3.7 c f f 8s/2 I I If Vf and Vm are expressed in terms of Vf, V,, and R, Equa tion 3.7 can be rewritten as E = Ef(Vf + VfR) [1 tanh(Ss/2)] + Em (Vm VfR) os/2 3.8 Alternatively, if the same assumption is used as in conti nuous fiber composites is considered, the fiber stress along longitudinal direction is assumed to be uniform everywhere in fiber, and the longitudinal modulus of material between sections H and H' can be obtained by using rule of mixtures. Ec = Ef (Vf + VfR) + Em(Vm Vf R) 3.9 Equations 3.8 and 3.9 show that for continuous fiber com posites, the longitudinal Young's modulus obtained by the rule of mixtures is higher than that obtained by Cox's analysis. This is because in the rule of mixture approach, uniform longitudinal fiber stress is assumed, while in the Cox's approach, uniform longitudinal fiber stress exists only at the locations far away from the fiber tips, and this longitudinal fiber stress reduces to zero at fiber tips. Finite element stress analyses [25, 26] show that the reduction of longitudinal fiber stress around fiber tips does exist; and the magnitude of this stress is not zero but finite. So it is hard to say which approach (rule of mixtures or Cox's analysis) is more nearly correct. However, Table 3.1 shows the values of tanh (ss/2)/(Ss/2) of graphite epoxy and Kevlar epoxy with VI being 0.7 or 0.4. This table indicates that the modi fication term tanh(Bs/2)/(8s/2) becomes important when fiber volume fraction and fiber aspect ratio are both small. On the other hand, when the fiber volume fraction is greater than 0.4 and the fiber aspect ratio is greater than 100, the effect of tanh(Bs/2) (s/2) could be neglected. Table 3.1: Influence of Vi and s/d on tanh (Bs/2) (Bs/2) Composite Vf s/d tanh (Bs/2)/(Bs/2) Graphiteepoxy 0.4 5. 0.725 Graphiteepoxy 0.7 5. 0.369 Kevlarepoxy 0.4 5. 0.610 Graphiteepoxy 0.4 25. 0.180 Graphiteepoxy 0.7 25. 0.074 Kevlarepoxy 0.4 25. 0.135 Graphiteepoxy 0.4 100. 0.045 When the same external stress a is applied to short fiber composite model and to its equivalent homogeneous materials, the same strain energy density U is presumed for those two materials (Figure 3.1(a) and Figure 3.1(c)) Ua = U 3.10 and 2 2 1 a P +1 s 3.11 U = + a 2 Et L 2 Ec L U 1 a2 3.12 c 2 EL where EL is the longitudinal Young's modulus of the homo geneous material equivalent to shortfiber composite model. After Equations 3.11 and 3.12 are substituted in Equation 3.10, EL can be expressed as E E E c m 3.13 E + E c 1+R m 1+R where Ec can be obtained by Equation 3.8 or Equation 3.9, and R is the ratio of P to s. When R equals zero, EL equals Ec. On the other hand, if R is a very large number, EL will be very close to Em. Since the fiber aspect ratio considered in this study is between 25 and 10,000, unless specially mentioned, the modified Cox's analysis (i.e., Equations 3.8 and 3.13) is used to anaylize the longitudinal Young's modulus of shortfiber composite model. It should be noted that the continuous fiber compo site can be induced either by letting the fiber aspect ratio be a very large number in the modified Cox's analy sis or by letting R be zero in the modified rule of mixture. 3.2.2. Damping of Aligned ShortFiber Composites A typical representative volume element of offaxis, shortfiber composite is shown in Figure 3.2 Fiber Figure 3.2: Representative Volume Element of OffAxis ShortFiber Composites For a continuous aligned composite, the offaxis modulus Ex along the loading direction is given in reference [38] 4 4 V 2 2 1 Ccos + sin + (1 2 LT) sin 8 cos 8 E EL ET GLT EL 3 3.14 where EL and ET represent the moduli along and transverse to the fiber direction respectively, GLT is the inplane shear modulus, and vLT is the major Poisson's ratio. Equation 3.14 can be easily derived from the elementary mechanics approach for offaxis continuous fiber composites. For offaxis aligned shortfiber composites, one can derive a similar expression for Ex from Equation 3.14 by replacing EL, ET, GLT, and vLT by the correspond ing formula for aligned shortfiber composites. The longitudinal modulus can be obtained from Equations 3.8 and 3.13. The transverse modulus ET, inplane shear modulus GLT and the major Poisson's ratio, those material constants are assumed to be independent of length of fiber, can be obtained by using the HalpinTsai Equation [53] and the rule of mixtures, i.e. E =E 1 + 2n V T m 1 nlVf 1+ 2 Vf 3.16 LT G n LT m 1 n2Vf VLT = VfLT Vf + Vm Vm 3.17 where (E /E ) 1 fT m 3.18 1 (EfT/Em) + 2 (GfLT/Gm) 319 2 (GfLT/Gm) + 1 Up to now, all equations presented in this chapter are derived for elastic material. When a viscoelastic material is considered, the elasticviscoelastic correspondence principle [52] can be used to obtain the corresponding relationships of viscoelastic material. For viscoelastic material the basic material properties are redefined as EfL Ef' + i EL E' + i E" ET T + ifL GfLT = G'LT+ i G"LT fLT fLT E* = E' + i E" 320 G* = G' + i G" S = V' + i v" m m m VfLT = fLT E* = E' + i E" 3.21 x x x Where the prime quantities indicate the storage moduli or storage Poisson's, the double prime quantities indicate the loss moduli or loss Poisson's ratio, and the i is defined as JT. In this research, bulk modulus of epoxy matrix Km is assumed to be real and independent of fre quency [15]. For isotropic epoxy matrix, E m 3.22 m 3(1 2vm) in WF~2V While the viscoelastic behavior of epoxy is considered, E' + E" m m K = m 3(1 2v' i 2v") m m The complex form of vm can be obtained by Equation 3.23 E' + i E" V + i V" = 1 m m) m m (1 3k m After separation of the real and imaginary parts of right hand side of Equation 3.24, one will have 3.25 E" v' + i V" = V' + i [ v_ )] m m m E m 2 m Similarily, the complex form of shear modulus, G'm+ G"m of viscoelastic matrix can be expressed as functions of Em' v, and E, i.e., E' E' 9K E" m m m m G' + i G" + i m m 2(1+v') + 2(+v) 9k E Sm m3.26 3.26 3.23 3.24 The complex form of 8s/2 of Equation 3.3, as shown in Appendix B, is G" E' 's + i B"s + i s + Bs ( m fL) 3.27 2 2 2 4 G' E' m fL The reason that the imaginary part of fiber Poisson's ratio is set as zero in the last equation of Equation 3.20 is because first, most fibers are known to be anisotropic materials; therefore, Equation 3.22 is not true for most fibers. Secondly, the corresponding term of E"/E' for most fibers (i.e., E" /Ej), except Kevlar fiber, is much less than E"/E' Due to the lack of the available data m m of transverse damping and shear damping, those two dampings are assumed to be equal to E" /FL . fL fL It should be noted that n and nf are treated as as material properties, and they are defined as E" m 3.28 *m = E m E" fL 3.29 fL By using Equations 3.25 and 3.283.29, the righthand side of Equations 3.20, 3.21, and 3.27 can be rewritten as E' fL EfT G' fLT E' m G' m V' m fLT E' x + i E" = E' (1 fL fL fT fT + i G LT= G LT(1 + i E" = E' (1 m m + i G" = G' (1 fLT fLT + i E" = E' (1 m m + i G" = G' (1 m m + i ) = EL +i ) =E T + i ) = GLT f fLT + in ) E* m m + in ) = G Gm m + i v" = v' + i (v' I) = * m m m m 2 m fLT _ * + i E" = x fLT *k 2'S + i s [1 + i 1 (n 2 2 2 2 Gm where qGm  n ) = * f 2 is defined as 9K G" m m "Gm 9Km Em y m = G 3.31 From Equation 3.31, it is observed that nGm is higher than nm. After using Equations 3.20, 3.27, and 3.30, one can rewrite Equations 3.13 and 3.153.17 for viscoelastic material, as follows 3.30 V (E' + i E") (E' + i E") Et c c m m (EB' + i E") + (E' + E") c c 1i1 m m In E 1 + 2n* V 1 f E* = (E'm + i E")  T m m 1 n Vf G* = (G' + i G") LT m m V* = (v' V + LT fLT f 1 + n2V 1 nf v' V ) + i n (v' )v m m m m 2 m E' + i E" = (E'L + i E L)(V + VfR) [1 tanh(8s/2)] + (Em + iE" )(V VfR) 8 s/2 m tanh(8*s/2) = tanh($s/2) + i Gm f 1 2 2 cosh2 (Bs/2) 3.32 3.33 3.34 3.35 where 3.36 3.37 (ET + i E"T)/(Em + i Em) 1 n1 (EF' (E+ i E)/( + E) + 2 fT fT m m (GILT + i G T)/(G' + G") 1 2 (G LT + i G LT)/(G + +1 After substitute E*, E*, GLT and vLT from Equations 3.323.35 for EL, ET, GT, and VLT, respectively, and Ex+iEx for Ex into Equation 3.14, one obtains 4 4 * 1 cos4 sin 1 LT 2 2 E'+iE" + + ( 2 ) cos sin x x EL ET GLT EL 3.40 Damping of the aligned shortfiber composite along x direction, nx, is then determined by E" x 3.41 x = x x Equations 3.40 and 3.41 show that material damping and stiffness of aligned shortfiber composite are functions of material properties of fiber (i.e., E' E' G' , fL fT fLT VfLT, and fn) and matrix (i.e., E', v', and m), fiber aspect ratio (s/d), fiber volume fraction (V), degree of aspect ratio (s/d), fiber volume fraction (Vf), degree of discontinuity (R), loading direction (6), and packing geometry of fiber. If the four different kinds of pre packed tapes (glassepoxy, Kevlarepoxy, graphiteepoxy, and boronepoxy) are used to make the fiber composite, the design variables utilized to analyze the stiffness and material damping are s/d, 8, R, Vf, Ef ({EIL, E T, GfLT' VfLT f m})' and Em ({F, N, n }), i.e. E' = f (Ef, Em, s/d, 6, R, Vf) 3.42 x 1 f nx = f2 (Ef, Em, s/d, 0, R, Vf) 3.43 A similar approach can be applied to determine damping along y direction, ny, and stiffness along y direction, E . CHAPTER 4 DAMPING OF RANDOMLY ORIENTED SHORTFIBER COMPOSITES 4.1 Introduction The objective of this chapter is to determine analytically the material damping of inplane randomly oriented shortfiber composites. The analysis is carried out by using the extension of the shortfiber com posite model and part of the results obtained in Chapter 3. An averaging procedure is first applied to the six offaxis reduced stiffnesses Qij (i, j = 1, 2, 6) with respect to the angle e between the fiber orienta tion and the applied load. The results of integration show that inplane randomly oriented fiber composites behave like a planar isotropic material. By using the properties of isotropic materials, Young's and shear moduli can be obtained as functions of the reduced stiffnesses Qij (i, j = 1, 2, 6). After the application of the elasticviscoelastic correspondence principle and separation of the real and imaginary parts of the complex Young's and shear moduli, material damping is obtained. 4.2 Damping Analysis of InPlane Randomly Oriented ShortFiber Composites For inplane randomly oriented shortfiber compo sites, no difference caused by different direction paral lel to the planes on which fibers are laid. The averaging procedure is one of the approaches which will lead to the isotrop. Therefore, an averaging by integrating the six moduli of offaxis shortfiber composites with respect to 8 from 8=0 to O=7 should be used. However, from Equation 3.14 for Ex and similar Equations for Ey, Gxy, Vxy, mx and my [38], one finds that it is not convenient to integrate and obtain the average Ex in closed form in terms of EL, ET, GLT, and VLT. Instead of integrating the six engineering moduli, one can integrate the six components of the offaxis reduced stiffness of the plane stress case Qij (i, j = 1, 2, 6) and obtain the average Qij, i.e. Qij = Qij dO i, j = 1, 2, 6 4.1 The expression for Qij (i, j = 1, 2, 6) as a function of 0 can be found in reference [38]. After integrating with respect to 0 from e=o to O=i and then dividing each of the result by 7, one obtains 3 1 1 = 022 = 3 (Q11 + Q22) + 4 (266 + Q12) 4.2 1 1 1 Q66 = (Q11 + Q22) i Ql2 + 1 66 4.3 Q22 1= ( + Q22) + Q2 066 4.4 Q16 Q66 = 0 4.5 It is easy to show from Equations 4.24.4 that the follow ing relation exists Ql2 + 2Q66 11 4.6 Therefore, after integration, there are only two inde pendent material constants, namely, Qr and G,. Qr = Q11 4.7 Gr = Q66 4.8 This implies, as expected, that inplane randomly oriented shortfiber composites behave as planar isotropic mate rials with two independent material constants. The subscript r represents randomly oriented shortfiber composites. For isotropic materials, the following relations exist SEr 4.9 1v S= Er 4.10 r 2 2(1 + v) where v is defined by 012/Q11. Elimination of v from Equations 4.9 and 4.10 yields the expression for Er, the Young's modulus of a randomly oriented short fiber composite as a function of Gr and Qr E = 4 Gr ( r 4.11 Qr Substitution of Equations 4.2, 4.3, 4.7, and 4.8 in Equa tion 4.11 yields Er as a function of the four reduced stiffness Q11, Q22' Q12, and Q66. S= [ (Q + Q22) Q12 + 2Q66] Q11 + Q22 2Q12 + 4Q66 3 (Q + ) + 2 (Q2 + 2Q66) 4.12 Similarly, 1 1 1 Gr = 8 (Ql + Q22) 412 + Q66 4.13 Since Q11, Q22' Q12, and Q66 are directly related to the four basic engineering constants EL, ET, GLT, and vLT (defined in Equations 3.13, 3.15, 3.16, and 3.17), accord ing to Equations 4.12 and 4.13, Er and Gr can be expressed as functions of EL, ET, GLT, and VLT. Next, as in Section 3.2.2 for aligned shortfiber com posites, according to the elasticviscoelastic correspond ence principle, one may replace Er by E = E' + i Er, Gr by G G' + i' EL by E* ET by ,' GT by G VLT by v*T where E E G IT ,and vT are defined in Equations 3.323.35. After separation of the real and imaginary parts, the material damping constants nr and nGr of inplane randomly oriented shortfiber composites can be obtained. Er 'rGr r An alternative way of determining Er and Gr is given in Appendix C. CHAPTER 5 DAMPING OF LAMINATED FIBER COMPOSITES LAMINATED PLATE THEORY APPROACH 5.1 Introduction In this study, laminated plate theory and an energy approach are used to analyze the material damping and stiffness of symmetrically laminated fiber composites. In this chapter, we will discuss all analytical work of laminated plate theory approach, while in Chapter 6 the energy approach will be presented. According to laminated plate theory, the constitutive equations (Equation 5.1) have already been given in refer ences [38, 50], in terms of [A], [B], and [D] (i.e. [A]*, [B]*, and [D]*) matrices. Material damping of laminated composites can then be derived from the expression of [A], [B], and [D]. 5.2 Damping Analysis of Laminated Fiber Composites Through Laminated Plate Theory Approach For a laminated fiber composite plate, as in Figure 5.1, the constitutive equations are given in references [38, 50] as shown in Equation 5.1. M\ xy X, x\ S// M N x xy M / \. N 7 xy 7 N y Figure 5.1: Sketch of Laminated Fiber Composite Plate Nx Ny Nxy Mx My Mxy All A12 A22 A16 A26 A66 Bll B12 B16 Dll B12 B22 B26 D12 D22 Symm B16 B26 B66 D16 D26 D66 Ex Yxy kx ky kxy 5.1 In Equation 5.1, ex, E, and yxy are middle plane strains, kx, ky and kxy are plate curvatures defined in [38, 50], and Aij, Bij, and Dij (i, j = 1, 2, 6) are the equivalent reduced inplane stiffness, coupling stiffness and reduced flexure stiffness, respectively. They are expressed in terms of Qij and total thickness of the plate, h, as follows: Sh/2 Aj = Aij h/2 J h/2 Bij = lh/2 J h/2 Dij = h/2 Qij dz = Z Qij dz 2 Qij dz n k (Qij)k (hk1hk1) k=1 5.2a n 1 2 2 E 7(Qij~k (hkhk1) k=1 5.2b n1 _ k 3(Qij)k (1k k1 k=1 5.2c where Qij can be expressed in vector form 1 1 cos 2 6 cos 4 e 4 Ui (i = 1, 2 5) 1 Ul = 1 U2 = 1 U3 = U3 8 1 U4  1 U5 =8 are defined in [50] as (3Q11 + 3Q22 + 2Q12 + 4Q66) (4Q11 4Q22) (Q11 + Q22 2Q12 4Q66) (Q11 + Q22 + 6Q12 4Q66) (Q11 + 22 2012 + 4Q66) U2 Q11 Q22 Q26. U3 U3 U3 1 U2 1 U2 U3 and 5.3 5.4 [50] Qij (i, j = 1, 2, 6) are known to be, by references [38, 50], S = EL 11 IVLT VTL S ET 22 IVLT VTL 5.5 VLT ET 12 IvLT VTL Q66 = GLT and = LT ET 5.6 TL EL where EL, ET, GLT, and vLT are defined in Equations 3.13, 3.153.17. Interesting relations should be noted that Qij (i, j = 1, 2, 6) of Equation 5.3 will be equal to Qij of Equations 4.24.5, providing U2 and U3 are set as zero. Equation 5.1 can be rewritten in matrix form as N A B Ce = 5.7 M B D k For symmetric laminates, the coupling stiffness matrix B is a zero matrix, and Equations 5.1 and 5.7 can be uncoupled as fAll A12 A12 A22 A16 A26 = [A] {E } A16 Ex A26 I E A66 0 xy D16 rD1I D12 D22 D26 D26 D66 5.8a 5.8b .5.9a {M} = [D] {k} From Equation 5.8a and the definition of Aij, one can express inplane moduli Eij (i, j = 1, 2, 6) as E = 1 1 1 h ij Aij 5.9b 5.10 and D16 Nx < N Nxy M M y MXY kx ky kxy where h is the total thickness of laminated composite and Ai are the elements of the inverse of matarix [Aij], and Ai are determined by the following system of equations. 1 1 1 A1 A12 A16 A11 Al2 A16 1 0 0 1 1 1 A12 A22 A26 Al2 A22 A26 0 1 0 1 1 1 A16 A26 A66 A16 A26 A66 0 0 1 5.11 According to elasticviscoelastic correspondence principle, when the viscoelastic behaviors of fiber composites are considered the elastic material constants EL, ET, GLT, and VLT (defined in Equations 3.13, 3.153.17) should be replaced by the corresponding complex moduli, respectively. Consequently, the complex form of Equations 5.8 and 5.9 can be written as t * N A11 A12 A16 x * SN A12 A22 A26 < 5.12a * Nx A16 A26 A66 y N* = [A*] I{E} 5.12b M D11 D12 D16 kx * Mx, D16 D26 D66 kxy. or {M*} = [D ] {k) 5.13b The corresponding complex form of inplane complex moduli Eij can be expressed as S1 + 5.14 E (= 1 Eij + iEij (Aij) 1 where (A.j) is the element in the ith row and jth column of the inverse of matrix [Aij]. Appendix D shows the detail of the derivation of (Aj) 1 The inplane material damping inij and flexure material damping Fnij are defined as follows: = _Ei i, j = 1, 2, 6 5.15 Iij Ej ij SDi i, j = 1, 2, 6 5.16 F ij Dj ij For the same kind of fiber composite, the result obtained by Equation 5.15 for sixteen plies of unidirec tional laminated composite is same as that obtained by Equation 3.41 for laminar composite with the same offaxis angle 8. This indicates that the approach presented in this chapter is correct, although it may not be convenient because the inverse of a complex matrix is involved. CHAPTER 6 DAMPING OF LAMINATED FIBER COMPOSITESENERGY APPROACH 6.1 Introduction The drawback of damping analysis using laminated plate theory approach is that it does not include the effect of interlaminar stresses (the stresses at the interfaces of laminated composites). It has been shown in study [54] that even when an inplane uniform tension load is applied to the laminated composites, there do exist appreciable interlaminar stresses around the free edges. In this chapter, an energy approach in conjunction with a threedimensional finiteelement method [55, 56] is used to analyze the material damping under certain loading and boundary conditions. It is believed that this model represents a more realistic approach by including the energy dissipated at the interfaces. By using this approach, we improve the approach from a twodimensional, classical, laminated plate theory to a threedimensional elasticity theory. 6.2 Damping Analysis of Laminated Fiber Composites Through Energy Approach The damping of laminated materials in the first mode vibration is determined as n q (q D)cyc n= q= 6.1 n Z 27 qUs 2 U q=1 where n is the total number of the plies, (qUD)cyc is the energy dissipated in the qth layer during a cycle, and qUs is the maximum strain energy stored in the qth layer. Detailed expressions of (qUD)cyc and qU are given in Appendix E in which the energy expressions for a visco elastic material given in reference [57] are used. The analytical expression of the maximum strain energy Us for an elastic body is U = 1 E j Cjk 'k dv 6.2 v (j, k = 1, 2 6) where cj are the maximum strains, and Cjk are the moduli of the elastic body. The expressions of Cjk (j, k = 1, 2 6) for orthotropic elastic material are given by Jones [58]; here k and j axes are material principal axes. In this study, each layer of fiber composite is considered to be transversely isotropic material. Therefore, as presented in Appendix F.1, the moduli of mth layer fiber composite could be defined as functions (Cjk)m = f (EL ET I GLT I VLT VTT')m 6.3 where EL ET GLT and VLT are given in Section 3.1 for shortfiber composites, and VTT, is assumed equal to vm. An analytical expression for VTTi is available in the literature [38], but the assumption that VTTy = Vm' is believed to be accurate enough in our analysis. According to the elasticviscoelastic correspondence principle, when the viscoelastic behaviors of the body are considered, EL, * ET, GLT', LT', TT, should be replaced by EL, ET, GLT, VLT' * and VTT', respectively. For example, TL EL C = TL L6.4 vTLVLT changes into V ' S(TL + i TL) (EL + i L) 12 1(L + iv" ) (VT + )6.5 TL TL LT LT or * C = C + C"2 6.6 12 12 + i 12 After separating the real and imaginary parts and using matrix rotation (if the material principal axes do not coincide with the global axes), the complex moduli C* with respect to the global axes (i.e., x, y, and z axes) of the kth layer of fiber composites are then obtained in the form * [C ]k = [C' ]k + i IC"]k where C' are storage moduli, and C" are loss moduli. three matrices are symmetric matrices. The storage energy qUs and the dissipated energy during a cycle (qUs)cyc in the qth layer of laminated composites can be expressed as 6.7 All 1 1 qUs = vq Vq (qUD) cyc 9 ey3 Ej Cjk k dv j Cjk sk dv correspondingly. Consequently, the material damping of a n layers laminated composite in the first mode vibration and 6.8 6.9 z {cl dv f () [C"i] {e) dv n = q 6.10 q=1 Jvq where strain field {e} depends on the loading and boundary conditions, and the strain field would be determined in this study through a threedimensional finite element method. From Equation 6.10, one may thus arrive that even a highly dissipative modulus cannot contribute significantly to the total loss factor n, if it's associated strain (or stress) does not participate considerably in the total stored energy. The procedures taken in energy approach are briefly described as follows: Step 1: As in the laminated theory approach, the elastic solution is first sought in energy approach. The equation of motion of an elastic body in static case can be derived from the principle of stationary potential energy given in reference [59]. 6(UWe) = 0 6.11 where U and We are the strain energy and the work done by external forces, respectively. Step 2: After the assumed displacement function (as a function of nodal displacement q) is substituted into Equation 6.11, the equations of motion can be expressed as a system of equations (details are given in Appendix F.2). [ki {q} = I?1 6.12 where [k] is the stiffness matrix, and {f} is the nodal forces. Step 3: After the displacement field is determined by sub stituting the solution of Equation 6.12 into the assumed displacement function, the strain field {)e of the elastic body could be obtained through displacementstrain relations. Step 4: Once the strain field of an elastic body is known, the material damping of a viscoelastic body under the same loading and boundary conditions can be determined by Equation 6.10. 6.3 Loading and Boundary Conditions Two sets of loading and boundary conditions are dis cussed in this study. However, other loading and boundary conditions can also be accommodated in this approach. Case 1: Uniform inplane tension load. In this case, only one quarter of the plate is analyzed by finite element method. External stress a is applied at one edge. Case 2: One side clamped plate under Mx load. In this case, the plate is clamped at x=0. The external force F is applied along x=L, and the external force F is applied along x=0.983L. The numerical results are presented in Section 9.5. CHAPTER 7 EXPERIMENTAL MEASUREMENT OF DAMPING 7.1 Introduction Free vibration decay [9], bandwidth method [9], resonantdwell method [60], forcedvibration techniques [61], and impulse techniques [62, 63, 64] are the very popular experimental techniques used to measure the material damping. All of these tests are subject to the air drag [65, 66], if the tests are not conducted under a vacuum condition. In this study, an improved impulse technique approach [64] is utilized to measure the material damping. The impulse technique consists of the application of a force pulse at a point on the test structure and the measurement of the response at another point. The input force and response signal are digitally processed by the analyzer to form the frequency response function or transfer function. Damping and natural frequencies can then be extracted from the output of the analyzer. All measures are performed using a digital signal process ing technique. It involves filtering and sampling the input wave forms. The sampling process converts a voltage (at a certain point in time) into a numerical representation. These numbers are then processed digitally to produce the various calculations performed by the analyzer. The primary output from the Fourier analyzer is the frequency response function, which is a measure of system's characteristics. The analyzer also calculates the coherence function, which ranges from 0.0 to 1.0. The larger the potential measurement noise or the system non linearity, the lower the coherence function will be [67]. A coherence value of unity indicates that the output is completely related to the input. Thus, the coherence function used here is a measure of the "quality" of the data. 7.2 Apparatus The composite plates or beams are fixed by two alumi num blocks at one end. A noncontact probe (KD23103U, Kaman Science Corporation, Colorado Springs, Colorado), known as a motion transducer, is located about 1.5 mm below the tip of each specimen. This motion transducer operates under the principle of eddy current. An aluminum foil target of diameter 20 mm is cemented underneath the tip of the specimen. A force transducer is mounted to the head of the impulse hammer (Model K291A, Piezotronics, Inc., New York), to measure the force input to the speci men. The other tip of the hammer is connected to the fixed spring so that the magnitude, location, and the dwelling time of the impact load can be controlled. To improve the coherence, an octave filter (4302 dual 24db, Ithaco, Inc., New York), which is connected to the force transducer at the hammer tip on one end and to the Fast Fourier Analyzer (FTT, Model 5420, Hewlett Packard) on the other end, is used to magnify the input signal 100 times. The impact point is near the stiffer portion of the speci men. The excitation and response signals are fed into the FFT analyzer, which displays the frequency response func tion and coherence function. Each frequency response and each coherence function are based on a statistical analy sis of an ensemble of six tests. A schematic drawing of the experimental setup is shown in Figure 7.1. The frequency response function is a complex valued function. A typical experimental display of the real and imaginary parts of the frequency response function for a graphiteepoxy composite, according to reference [64], is shown in Figure 7.2 and an enlarged schematic drawing of real part of frequency response function is shown in Figure 7.3. As prescribed in references (62, 63, 64], the peak of the imaginary part determines the resonant fre quency, and then from the corresponding real part (see Figure 7.3) the material damping (loss factor, n) can be calculated by S(f /f) 2 1 7.1 (fa/fb) + 1 In using the impulse technique to measure the mate rial damping, following precautions, as reported in study [64], must be taken. 1. In order to improve the accuracy of experimental data of damping associated with a particular mode of vibration, the values of natural frequency within the frequency range of interest (say 0 to 1,600 Hz) are first approximately determined. Then a zooming technique to increase the resolution of the response in the neighborhood of this particular frequency is used. 2. It is necessary to avoid measurement of response near a nodal point for the modes to be tested. Such meas urements would consist primarily of noise, since the actural response is very small near nodal points. 3. The amplitude of the vibration must be kept below the thickness of the specimen to ensure that air damping (air drag) is negligible, since the air damping is linearly proportional to the amplitudetothickness ratio of the beam specimen [66]. 4. It is important to optimize the Analog to Digital Converter (ADC) range setting on an FFT analyzer before making a measurement, since an optimized ADC range set will increase resolution in the digitizing process [35]. Recently, Suarez and Gibson [35, 36], did some experiments on material damping of shortfiber composite 65 through impulse hammer technique. Some of those experi mental results for unidirectional shortfiber composites presented in this chapter are compared with the analytical results. 66 $4 (Ow, (cn r 4 *44 u O t~"'3 $4~ eCO 2 I1 t $4 44 (Ij 4i 00 44 u to CO4 r 4.4 COC 4) CO I r to Cd r4 0 orr4 r4O C)k k0 rd i (*2.. E U 0u2 0 r.n ,4 /, / r4 U~ .,4 Wa) 4 fU) 0 to In En CLL 41 rz: po 0 r4J p 0H N Z 4j mc~ N4 C14 10 Cd P4 o a)k t44 i 00 cJ~c. ZO 0 41 a. C) 0 r4r CL ~tE 1: 4 Er4 rX4 P4 E k4 (.3 41 44 0 C1 '4 0. Figure 7.3: C. O 0. c 1 fb fr f Enlarged Schematic Drawing of Real Part of Frequency Response Function CHAPTER 8 OPTIMIZATION OF DAMPING AND SPECIFIC STIFFNESS FOR FIBER COMPOSITES 8.1 Introduction Plunkett and Lee [19] pointed out that maximum damp ing can be obtained through properly choosing the length and spacing of the constraining layers. Those layers are used to constrain the viscoelastic layer (or coating) on the structure surface. In the study presented here, the optimization of damping is based on the composite material itself (i.e., choosing the proper fiber length, fiber direction, the longitudinal distance between fiber tips, etc.). It is known that high damping can reduce the displacement at and near the resonant frequencies, while high stiffness can further reduce the displacement at other frequencies. The best way to optimize both damping and stiffness is to keep them as high as possible. Unfor tunately, there exists a general trend that higher damping is mostly coupled with low stiffness and vice versa. Therefore, the idea of the optimization in this study is to try to increase damping without sacrificing the stiffness too much. From the formulations presented in previous chap ters, one can see the complication contained in damping analysis. It will be a simpler approach for the optimi zation analysis if no derivative is needed. Consequently, the socalled Sequential Simplex Method [68, 69] has been selected to analyze the optimization. 8.2 Brief Introduction to Sequential Simplex Method The sequential Simplex Method [68, 69] takes a regu lar geometric figure (known as a simplex) as a base. Thus, in two dimensions (i.e., twodesign variables), one should choose an equilateral triangle; and in three dimen sions (i.e., threedesign variables), one should choose a tetrahedron. Observations (experiments) are located so that the objective function is evaluated at the points formed by vertices of the geometric figure. One vertex is then rejected as being inferior in value to the others. The general direction of search may then be taken in a direc tion away from this worst point, the direction being chosen so that the movement passes through the center of gravity of the remaining points. A new point is then selected along this direction so as to preserve the geome tric shape of the figure, and the function is evaluated anew at this point. The method proceeds with this process of vertex rejection and regeneration until the figure straddles the optimum. When no subsequent moves would lead to further improvement, the last few geometric figures are essentially repeated. There are three rules that govern the whole procedure of this method. These rules are explained in the twodimensional case as follows: 1. Take the point to be rejected where the worst value of the objective function is obtained, and replace it by its reflection in the opposite side of the tri angle. See Figure 8.1, where point B is replaced by point D. 2. No return can be made to points which have just been left; see Figure 8.2, where point A is replaced by point D. If point D is still the worst point of triangle BCD, then point B is replaced by point E. 3. If the best valued vertex remains unchanged for more than M iterations, then the simplex size is reduced by, for example, halving the distance of all other vertices from that vertex. The next stage can then start. For example, as in Figure 8.3, point C is sequentially repeated in five triangles ABC, BDC, DEC, EFC, and FGC; and after the size of the fifth triangle FGC is reduced to triangle F'G'C, the rule 1 is applied at the new triangel F'G'C. The magnitude of M depends on the number of variables. Spendley [68] suggested that M = 1.65n + 0.05 n2 8.1 where n is the total number of design variables. The search can finally be stopped when the simplex is small enough to locate the optimum adequately. It should be noted that Rule 1 and Rule 2 force the simplexes to circle continuously about an indicated opti mum, rather than oscillate over a limited range such as a ridge (see Figure 8.2). If one ensures that at any point violating a constraint, a sufficiently large negative value (or positive value, if one is maximizing) is set for the response at such a point, the system of simplexes will then move along rather than cross the constraints. The advantages of this method are as follows: 1. It is easy to apply. 2. It is useful when analytical or numerical derivatives of the objective function are not available. The disadvantage of this method is that it could not guarantee that the global optimum design is obtained. Therefore, several different initial locations (designs) should be considered to get global optimization. 73 x2 C x Figure 8.1: Rule 1 of Sequential Simplex method x2 *C E D x1 Figure 8.2: Rule 2 of Sequential Simplex method xl Figure 8.3: Rule 3 of Sequential Simplex method 8.3 Mathematical Formulation of Design Problems Energy approach and laminated plate theory approach are both utilized in damping analysis. The main advantage of the energy approach over the laminated plate theory approach is that the interlaminar stresses are included, The disadvantage of the energy approach is that it takes much more computer time than the laminated plate theory approach does. The influence of interlaminar stresses can be neglected, if the thickness of each group of lami nated composite is thin (a detailed discussion is given in Section 9.5). Then laminated plate theory approach for damping analysis is a good way for the optimization analysis for thin structures. Generally speaking, the design variables of continu ous or discontinuous fiber composites include fiber volume fraction Vf, fiber moduli Ef, fiber aspect ratio s/d, the longitudinal distance between fiber tips p, and fiber orientation angle 8 for certain special stacking sequence. Among those six variables, only fiber aspect ratio s/d and fiber tip longitudinal distance p are treated as the design variables in the optimization analysis. Since orthotropic materials are utilized in the optimization analysis, the fiber direction 8 is kept as certain con stants. The fiber volume fraction is also set as a constant (0.65) in this study, since the fiber composites are made from prepreg tape. Stacking sequence is fixed and most of this study is concentrated on graphiteepoxy, Kevlarepoxy and the hybrid of those two composite materials, so that only two sets of fiber moduli are considered. Two different cases are considered in optimization analysis: Case one is to optimize the specific stiffness and material damping of an orthotropic square plate simply supported on all four sides under free vibration. The equation of motion for this case is Dl Wxxxx + 2(D12 + 2D66) W xxyy + D22,yyyy = tt 8.2 where the Dij are derived from laminated plate theory, and W is the flexural deformation. Let W = Amn Fm (x) Hn (y) T(t) 8.3 By separation of variables, the form (8.4) is obtained for T(t); while Fm (x) and Hn (y) satisfy Equation 8.5 T(t) = e (i k t k = 1, 2, 3 . P 8.4 DI F (iV)Hn + 2(D12 + 2D66) FH + D22 FHiV) = kFmHn 8.5 8.5 where Ak is the constant to be determined, and i is J in Equation 8.4. In accordance with the simple support boundary conditions, Fm and Hn functions can be assumed as Fm(x) = sin miy 8.6 m a Hn(y) = sin ni 8.7 where a and b are the lengths of the edges along x and y directions, respectively. By substituting Equations 8.6 and 8.7 into Equation 8.5, one obtains 44 2 24 n44 m Tr m n w n Tr Dl T + 2(D12 + 2D66) 2 2 + D22 7 Xk a ab b 8 8.8 Letting b=a for a square plate and considering only the first mode of vibration (i.e., m=n=l), one can obtain X1 as 4 1 = T (D11 + 2D12 + 4D66 + D22) 8.9 a The natural circular frequency of the first mode, wl, is defined as x1 S= ( 8.10 Then the first mode solution of Equation 8.2 is .wx wy iwlt W = all sin sin el 8.11 11 a a If the viscoelastic behavior of the materials is considered, the complex form of Dij should be included, i.e. S 2 1 h S= 2 (DI + 2D2 + 4D6 + D 2 p 11 12 66 22 a After separating the real and imaginary parts and neglecting the higherorder terms of the binomial expansion of the quantity on the righthand side in Equation 8.12, the following expression is obtained. (1 = W' + i ;" where 8.12 8.13 2 w =  a 2 2 2a  (D1 + 2Di2 + 4DN6 + D2) D" + 2D2 + 4D" + D" 11 12 66 22) p(D' + 2D'i + 4D'6 + D' ) 11I 212 466 D22 Hence, for viscoelastic material, Equation 8.11 should be rewritten as 8.14 8.15 x sy i it I t W = (all sin  sin a e ) e 8.16 Equation 8.16 is similar to the mathematical model of logarithmic decrement [42], and the logarithmic decrement 6 can be approximated by 2 7# 2r  1 1 6 =   2 r 8.17 1 2 1 1 (1) W1 for light damping, i.e. when w"/wi < <1. By using Equation 2.22, the loss factor of this system is then determined as w" D" + 2D" + 4D" + D" 1 11 12 66 22 8.18 n =2 1 D' + 2D' + 4D6 +D2 818 1 11 12 66 22 The value of Amn depends on the stiffness. High stiffness reduces the deflection at resonance; the material damping can further reduce the displacement at resonance. Gibson [611 shows that flexural vibration for a double cantilever beam under forced vibration at resonant frequency depends inversely on the product of material damping, area moment of inertia of cross section and Young's modulus. It should be noted that his result is based on the equation of motion for free vibration and the boundary conditions of forced vibration. For the case considered here, the product of the area moment of inertia and Young's modulus corresponds to the generalized stiffness D, which is defined as D = D' + i D" 8.19 D'= D' + 2D' + 4D' + D" 8.20 12 66 22 S= D + 2D" + 4D" + D" 8.21 11 12 66 22 Equation 8.18 can be rewritten as n = D"/D' 8.22 In order to have small resonant deformation, a high value of the product of stiffness and material damping is required. But high specific stiffness is the major pro perty of fiber composites to be widely used in space vehicle and aircraft. Therefore, the mathematical formu lation of optimization on damping and specific stiffness is to seek the maximum value of the objective function 8.23 f, (P/d, s/d) 81 of two variables p/d and s/d, where fl is assumed in the form fl (p/d, s/d) = T P + T2  D' D' o 0 Po P tlm 8.24 or fl (p/d, s/d) = (T1 + T ) 8.25 P ( 1 2 Tm 0 The ranges for design variables p/d and s/d are 0 < p/d < 0.05 s/d 8.26 25 < s/d 5 10000 8.27 where T1 and T2 are weighting constants, nm is the damping value of epoxy, p is the density of designed composite, D' and n of design composite are defined in equations 8.20 and 8.22, respectively, and D6 and Po are the correspond ing values of D' and density of continuous graphite rein forced epoxy having the same stacking sequence and fiber volume fraction. If high specific stiffness is important in structures, T1 could be chosen as a high value. Alternatively, if small resonant deformation is the major consideration, T2 should be a high value. Case two is to optimize the specific stiffness and material damping of an orthotropic square plate clamped on all four sides under free vibration. The equation of motion is just the same as Equation 8.2. The first mode solution is assumed, according to reference [70] W = All F(x) H(y) T(t) 8.28 where F(x) = 81 cos Xlx 81 cosh X1x + sin X1x sinh Xlx 8.29 H(y) = 1 cos AXy B1 cosh Xly + sin Xly sinh Aly 8.30 sin Xla sinh a 8a S= 8.31 1 cos A a + cosh Ala 1 11 and X1 is the constant to be determined. The natural frequency of first mode vibration of a clamped square plate was given in reference [701 as W I = [5.14D1 + 1.55(2D22 + 4D6g) + 5.14D22] 8.32 8.32 83 By a similar approach to that prescribed in case one, one will obtain the loss factor of the system for case two as 5.14D"1 + 1.55 (2D"2+ 4Dg6) + 5.14D" n = 5.14D'1 + 1.55 (2D' + 4D" ) + 5.14D22 11 12 66 22 8.33 The mathematical formulation of optimization for this case is to seek the maximum value of the objective function f2 (p/d, s/d) of two variables p/d and s/d, where f2 is defined in Equation 8.37 The ranges of design variables are 0 5 p/d 6 0.05 s/d 25 : s/d 5 10000 8.34 8.35 8.36 where f2 (Pl, s/d) = T1 + T2 O' 0 0 0P T IM 8.37 and 84 D' = 5.14D' + 1.55(2D'2 + 4D' ) + 5.14D'2 11 12 66 22 D" = 5.14D' + 1.55(2D2 + 4D"6) + 5.14D" 11 = D"/22 n = "/b 8.38 8.39 8.40 CHAPTER 9 RESULTS AND CONCLUSIONS 9.1 Preliminary Remarks The damping analyses presented in Chapters 3 to 6 are applicable to all kinds of fiber composites, including continuous fiber composites, discontinuous fiber compo sites, symmetrically or unsymmetrically laminated or lami nar composites, randomly oriented fiber composites, etc. The optimization analysis on damping and specific stiff ness presented in Chapter 8 is based on orthotropic material. However, a similar approach is applicable for more general anisotropic materials. Since most widely and practicallyused fiber composites are symmetrically laminated fiber composites, the numerical analysis of this study is concentrated on certain kinds of symmetri cally laminated fiber composites, unidirectional laminar composites, and inplane randomly oriented shortfiber composites. As mentioned before (see Chapters 3 and 5), damping nx is defined as E"/Ex (or D"/Dx). Since both Ex and E (or Dx and D") are functions of Ef, E, s/d, 0, P (the longitudinal distance between fiber tips), Vf and rnf, etc., the variations of nx and Ex (or Dx) may not follow the same pattern. Consequently, in the numerical results, 85 
Full Text 
Package Processing Log
.logFileName { fontsize:xlarge; textalign:center; fontweight:bold; fontfamily:Arial } .logEntry { color:black; fontfamily:Arial; fontsize:15px; } .errorLogEntry { color:red; fontfamily:Arial; fontsize:15px; } .completedLogEntry { color:blue; fontfamily:Arial; fontsize:15px; } Package Processing Log 8/15/2011 12:14:39 PM Error Log for AA00003821_00001 processed at: 8/15/2011 12:14:39 PM 8/15/2011 12:14:39 PM  8/15/2011 12:14:39 PM There are more than one METS file! 8/15/2011 12:14:39 PM  