Optimization of material damping and stiffness of laminated fiber-reinforced composite structural elements


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Optimization of material damping and stiffness of laminated fiber-reinforced composite structural elements
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xv, 176 leaves : ill. ; 28 cm.
Wu, Jiing-Kae, 1954-
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Damping (Mechanics)   ( lcsh )
Fibrous composites   ( lcsh )
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )


Thesis (Ph. D.)--University of Florida, 1985.
Includes bibliographical references (leaves 170-175).
Statement of Responsibility:
By Jiing-Kae Wu.
General Note:
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University of Florida
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The author wishes to express his deep appreciation to

his major professor, Dr. Chang-Tsan 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. AFOSR-83-0154 and AFOSR-83-0156 monitored by

Dr. D. R. Ulrich.

Finally, the author expresses his greatest apprecia-

tion to his parents and his wife, Mong-Shya Rau Wu, for

encourag-ing him to study in the U.S.A. and for providing

warm family support during the past four years.





LIST OF FIGURES . ... .. vii

ABSTRACT . . xiv



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


3.1 Introduction . 23
3.2 Damping Analysis of Unidirectional
Fiber Composites .. .24
3.2.1 Short-Fiber Composite
Model .. 24


3.2.2 Damping Analysis of
Aligned Fiber
Composites ...... 30

COMPOSITES .... . .... .40

4.1 Introduction .. 40
4.2 Damping Analysis of In-Plane
Randomly Oriented Short-Fiber
Composites ........... .41


5.1 Introduction 45
5.2 Damping Analysis of Laminated
Fiber Composite Through
Laminated Plate Theory
Approach . ... .45


6.1 Introduction ... 54
6.2 Damping Analysis of Laminated
Fiber Composites Through
Energy Approach .... .55


7.1 Introduction . 61
7.2 Apparatus . .... 62


8.1 Introduction .... 69
8.2 Brief Introduction to Sequential
Simplex Method ... .70
8.3 Mathematical Formulation of
Design Problems ... .74


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














Er and Gr . .



F.I Stiffness Matrix ....
F.2 Formulation of Finite Element
Method . .



. 147

. 151

. 154

. 157

. 160

. 163
. 163

. 166

* 170

. 176


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


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


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 Rotating-Vector Representation of

Harmonic Motion . 19

2.5 Hysteresis Loop of an Inelastic Body 21

3.1 Short-Fiber Composite Model . 24

3.2 Representative Volume Element of Off-Axis

Short-Fiber Composites .. 30

5.1 Sketch of Laminated Fiber Composite

Plate . . 46

7.1 Schematic Drawing of the Experimental

Set-up . . 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


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



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 Three-Dimensional Plots of Ex/Ej vs e

and s/d for Kevlar Epoxy Composites 114

9.15 Three-Dimensional 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



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 Three-Dimensional Plots of E_/EA vs s/d

and Ej for Randomly Oriented Fiber

Composites . ... 125

9.26 Three-Dimensional 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

Quasi-Isotropic Graphite Epoxy

Composites . ... .129


9.30 Plots of D66/DGm and Fn66/nGm vs s/d for

Quasi-Isotropic 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


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 In-Plane

Longitudinal Damping Through Energy

Approach . . 139

9.40 Influence of a/h on the In-Plane

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


Figure Page

A.1 Hysteresis Loop of a Viscoelastic

Material . .. 149

F.1 Principal Coordinates of a Fiber

Composite Material . 165


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



Jiing-Kae 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

elastic-viscoelastic 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, in-plane randomly


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 three-dimensional 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 elastic-viscoelastic 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.


Due to the high strength-to-weight and stiffness-to-

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 take-off 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 stiffness-to-weight

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 two-dimensional energy

approach to predict the flexural damping of laminated

composites. In their work, damping coefficient was deter-

mined by free-free 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 visco-elastic 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 short-fiber

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 short-fiber 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 short-fiber

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


A. To develop a short-fiber composite model to determine

the moduli of short-fiber 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


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 widely-used

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 short-fiber 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 T-300. And the matrix yield shear stress is,

according to the manufacturer's test results, 97 MPa of

AS4/3501 graphite-epoxy tape. Therefore, the fiber aspect

ratio, the ratio of fiber length to its diameter, should


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


E1 (Gpa)

E2 (Gpa)

G12 (Gpa)






















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 Halpin-Tsai

Equation, n1 is the longitudinal damping, k is the bulk

modulus, and p is the density.








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


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, dash-pots,

hydrolic shock-absorbers [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

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 built-up

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, Kelvin-Voigt model, and three-parameter

models [44, 45]) have been introduced to represent dif-

ferent materials. For the case of Kelvin-Voigt model

under normal deformation, the relationship between stress

a and strain e is expressed by

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

stress-strain relationship is given by

a = Ee + I (t, T) e(T) dT 2.5

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 stress-strain 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.

(x)2 + F Kx)= 2.8
Ao hAo



Figure 2.1: Sketch of Hysteresis Loop

The work done by damping force (dissipated energy) in one

cycle (UD)cyc is


(Ucyc = F dx = h()2 dt = hA


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 well-accepted 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).


Figure 2.2:

Sketch of Viscous Damping Model

Its differential equation of motion is found to be

MX + Cx + Kx = F(t)


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

+ c2e-bt)


b = c/2m) k/m1/2
b = [(c/2m) k/m]



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 = (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


6 = Ln 2.16

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 steady-state sinusoidal stress a

is a steady-state sinusoidal strain E at the same

frequency [44], out of phase by the angle 0, e.g.

0 = 0o sinwt


e = Eo sin(wt P) 2.18

Both the response amplitude and the phase-shift (or phase

angle) D are frequency-dependent, but in the linear range

eo is proportional to co. The phase relationships are

conveniently shown in the rotating-vector representation of

simple harmonic motion, as in Figure 2.4.

E"e ------ A

0 E------------A
0 1

o I B

Figure 2.4: Rotating-Vector 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 J-1.

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 stress-strain (or load-

deformation) curve is not a single-valued 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.


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.


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 )


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 short-fiber composite model to determine the

longitudinal modulus of short-fiber 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 viscoelastic-elastic

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 Short-Fiber Composite Model

The short-fiber composite model is composed of a

finite-length 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: Short-Fiber 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 short-fiber

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

short-fiber 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/2-x)]} 3.1

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 -

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


ac = Ec Ec = Of Vf + m Vm 3.6

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

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


Table 3.1: Influence of Vi and s/d on tanh (Bs/2) (Bs/2)

Composite Vf s/d tanh (Bs/2)/(Bs/2)

Graphite-epoxy 0.4 5. 0.725
Graphite-epoxy 0.7 5. 0.369
Kevlar-epoxy 0.4 5. 0.610
Graphite-epoxy 0.4 25. 0.180
Graphite-epoxy 0.7 25. 0.074
Kevlar-epoxy 0.4 25. 0.135
Graphite-epoxy 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


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 short-fiber composite

model. After Equations 3.11 and 3.12 are substituted in

Equation 3.10, EL can be expressed as

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 short-fiber

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


3.2.2. Damping of Aligned Short-Fiber Composites

A typical representative volume element of off-axis,

short-fiber composite is shown in Figure 3.2


Figure 3.2: Representative Volume Element of Off-Axis
Short-Fiber Composites

For a continuous aligned composite, the off-axis 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

where EL and ET represent the moduli along and transverse

to the fiber direction respectively, GLT is the in-plane

shear modulus, and vLT is the major Poisson's ratio.

Equation 3.14 can be easily derived from the elementary

mechanics approach for off-axis continuous fiber

composites. For off-axis aligned short-fiber 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 short-fiber composites. The

longitudinal modulus can be obtained from Equations 3.8

and 3.13. The transverse modulus ET, in-plane 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 Halpin-Tsai 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


(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 elastic-viscoelastic

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

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 J-T. In this research, bulk modulus of epoxy

matrix Km is assumed to be real and independent of fre-

quency [15]. For isotropic epoxy matrix,

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

After separation of the real and imaginary parts of right-

hand side of Equation 3.24, one will have


v' + i V" = V' + i [- v_ )]
m m m E m 2

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



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

m 3.28
*m = E-

fL 3.29

By using Equations 3.25 and 3.28-3.29, the right-hand

side of Equations 3.20, 3.21, and 3.27 can be rewritten as









+ i E" = E' (1
fL fL

fT fT

+ i G LT= G LT(1

+ i E" = E' (1
m m
+ i G" = G' (1
+ 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


_ *

+ i E" =



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


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 3.15-3.17 for viscoelastic

material, as follows



(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)








(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

3.32-3.35 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


Damping of the aligned short-fiber composite along x

direction, nx, is then determined by

x 3.41
x = x

Equations 3.40 and 3.41 show that material damping and

stiffness of aligned short-fiber 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 (glass-epoxy, Kevlar-epoxy, graphite-epoxy,

and boron-epoxy) 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 .


4.1 Introduction

The objective of this chapter is to determine

analytically the material damping of in-plane randomly

oriented short-fiber composites. The analysis is

carried out by using the extension of the short-fiber com-

posite model and part of the results obtained in

Chapter 3. An averaging procedure is first applied to

the six off-axis 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 in-plane 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 elastic-viscoelastic 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 In-Plane Randomly Oriented
Short-Fiber Composites

For in-plane randomly oriented short-fiber 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 off-axis short-fiber 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 off-axis 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 4.2-4.4 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 in-plane randomly oriented
short-fiber composites behave as planar isotropic mate-
rials with two independent material constants. The
subscript r represents randomly oriented short-fiber

For isotropic materials, the following relations


SEr 4.9

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

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)


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 short-fiber com-

posites, according to the elastic-viscoelastic 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 3.32-3.35. After separation of the real and

imaginary parts, the material damping constants nr and nGr

of in-plane randomly oriented short-fiber composites can

be obtained.



An alternative way of determining Er and Gr is given in

Appendix C.


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

Figure 5.1: Sketch of Laminated Fiber Composite Plate







All A12



























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 in-plane stiffness, coupling stiffness and

reduced flexure stiffness, respectively. They are expressed

in terms of Qij and total thickness of the plate, h, as


Aj =
Aij -h/2

J h/2
Bij =

J h/2
Dij =

Qij dz =

Z Qij dz

Qij dz

k (Qij)k (hk1hk-1)


n 1 2 2
E 7(Qij~k (hk-hk1)-
k=1 5.2b

n1 _
k 3(Qij)k (1k k-1


where Qij can be expressed in vector form

1 1

cos 2 6

cos 4 e

Ui (i = 1, 2 5)

Ul =

U2 =-

U3 =
U3 -8

U4 -

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)















Qij (i, j = 1, 2, 6) are known to be, by references [38,

S = EL


12 I-vLT VTL

Q66 = GLT


= LT ET 5.6

where EL, ET, GLT, and vLT are defined in Equations 3.13,

3.15-3.17. Interesting relations should be noted that

Qij (i, j = 1, 2, 6) of Equation 5.3 will be equal to Qij
of Equations 4.2-4.5, 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



A12 A22

A16 A26

= [A] {E }

A16 Ex

A26 I E

A66 0 xy


rD1I D12

D22 D26

D26 D66




{M} = [D] {k}

From Equation 5.8a and the definition of Aij, one can

express in-plane moduli Eij (i, j = 1, 2, 6) as

E = 1 1
-1 h
ij Aij






< N



M y





where h is the total thickness of laminated composite and

A-i 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


According to elastic-viscoelastic 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, 3.15-3.17)

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.


{M*} = [D ] {k) 5.13b

The corresponding complex form of in-plane complex moduli

Eij can be expressed as

S1 + 5.14
E (= -1 Eij + iEij

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 in-plane material damping inij and flexure

material damping Fnij are defined as follows:

= _Ei i, j = 1, 2, 6 5.15
Iij Ej

SDi i, j = 1, 2, 6 5.16
F ij Dj

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

off-axis 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



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 in-plane 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 three-dimensional finite-element 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 two-dimensional,

classical, laminated plate theory to a three-dimensional

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

q (q D)cyc
n= q= 6.1
Z 27 qUs
2 U

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


(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

short-fiber 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 elastic-viscoelastic 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,

C = TL L6.4

changes into

V '
S(TL + i TL) (EL + i L)
12 1-(L + iv" ) (VT + )6.5


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



1 1
qUs =


(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




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 three-dimensional finite element


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(U-We) = 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


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 displacement-strain


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 in-plane 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.


7.1 Introduction

Free vibration decay [9], band-width method [9],

resonant-dwell method [60], forced-vibration 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


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


7.2 Apparatus

The composite plates or beams are fixed by two alumi-

num blocks at one end. A non-contact probe (KD-2310-3U,

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 set-up 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

graphite-epoxy 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


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 amplitude-to-thickness

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 short-fiber composite


through impulse hammer technique. Some of those experi-

mental results for unidirectional short-fiber composites

presented in this chapter are compared with the analytical




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-4 *44

u O


2 I1

t- $4


u to




4-) CO
I -r
to Cd




rd i
(*2.. E

U 0u2

0 r.n

/,- /


U~ .,-4


0 to

41 rz:
po 0


p 0H

N Z 4-j

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P4 o

t4-4 i


ZO 0


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Figure 7.3:



fb fr f

Enlarged Schematic Drawing of Real Part of
Frequency Response Function


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 so-called 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., two-design variables), one

should choose an equilateral triangle; and in three dimen-

sions (i.e., three-design variables), one should choose a


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 two-dimensional case as


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 t-hen

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


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


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.





Figure 8.1: Rule 1 of Sequential Simplex method





Figure 8.2: Rule 2 of Sequential Simplex method


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 graphite-epoxy,

Kevlar-epoxy and the hybrid of those two composite

materials, so that only two sets of fiber moduli are


Two different cases are considered in optimization


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


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

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


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

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

1 = -T (D11 + 2D12 + 4D66 + D22) 8.9

The natural circular frequency of the first mode, wl, is

defined as

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,


S 2 1 h
S= -2 (DI + 2D2 + 4D6 + D
2 p 11 12 66 22

After separating the real and imaginary parts and

neglecting the higher-order terms of the binomial

expansion of the quantity on the right-hand side in

Equation 8.12, the following expression is obtained.

(1 = W' + i ;"




w = -


- (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



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-)

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


f, (P/d, s/d)


of two variables p/d and s/d, where fl is assumed in the


fl (p/d, s/d) = T P- + T2 -
D' D'
o 0
Po P- tlm


fl (p/d, s/d) = (T1 + T- ) 8.25
P ( 1 2 Tm

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


F(x) = 81 cos Xlx 81 cosh X1x + sin X1x sinh Xlx


H(y) = 1 cos AXy B1 cosh Xly + sin Xly sinh Aly


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]


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


The mathematical formulation of optimization for this

case is to seek the maximum value of the objective


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





f2 (Pl, s/d) = T1 + T2-
0 0
0P- T IM




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





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 practically-used 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 in-plane randomly oriented short-fiber


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,


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