DYNAMIC STABILITY OF SHEAR DEFORMABLE VISCOELASTIC · DYNAMIC STABILITY OF SHEAR DEFORMABLE...

DYNAMIC STABILITY OF SHEAR DEFORMABLE VISCOELASTIC

COMPOSITE PLATES

by

NARESH K. CHANDIRAMANI

Thesis submitted to the Faculty of the

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

in

ENGINEERING SCIENCE AND MECHANICS

V v DVL. LIBRES'e"U , .. Cliainnan

I w. E. R. JOHNSON

APPROVED:

7 DECEMBER 1987

Blacksburg, Virginia

Dr. R. H. PLAUT

Dr. H. F. BRINSON

DYNAMIC STABILITY OF SHEAR DEFORMABLE VISCOELASTIC

COMPOSITE PLATES

by

NARESH K. CHANDIRAMANI

Dr. L. LIBRESCU , Chairman

ENGINEERING SCIENCE AND MECHANICS

(ABSTRACT)

Linear viscoelasticity theory is used to analyze the dynamic stability of composite,

viscoelastic flat plates subjected to in-plane, biaxial edge loads. In deriving the associated

governing equations, a hereditary constitutive law is assumed. In addition, having in view that

composite-type structures exhibit weak rigidity in transverse shear, the associated governing

equations account for the transverse shear deformations, as well as the transverse normal

stress effect. The integro-differential equations governing the stability are solved for simply-

supported boundary conditions by using the Laplace transform technique, thus yielding the

characteristic equation of the system.

In order to predict the effective time-dependent properties of the orthotropic plate, an

elastic behavior is assumed for tile fiber, whereas the matrix is considered as linearly

viscoelastic.

In order to evaluate the nine independent properties of the orthotropic viscoelastic ma-

terial in terms of its isotropic constituents, the micromechanical relations developed by

Aboudi [24] are considered in conjunction with the correspondence principle for linear

viscoelasticity. The stability behavior analyzed here concerns the determination of the critical

in-plane normal edge loads yielding asymptotic stability of the plate. The problem is studied

as an eigenvalue problem.

The general dynamic stabili~y solutions are compared with their quasi-static counter-

parts. Comparisons of the various solutions obtained in the framework of the Third Order

Transverse Shear Deformation Theory (TTSD) are made with its first order counterpart. Se-

veral special cases are considered and pertinent numerical results are compared with the

very few ones available in the field literature.

Acknowledgements

I owe many thanks to my principal advisor, Dr. L. Librescu, without whose guidance and

encouragement this work would have never been possible. The training he has provided will

stand me in good stead in my future pursuits.

I would also like to express my sincerest gratitude to Dr. R. H. Plaut and Dr. E. R.

Johnson for their useful suggestions throughout the period of my research work. Thanks also

to Dr. H. F. Brinson for reviewing the manuscript and providing useful suggestions.

My special thanks to Dr. J. Aboudi without whom this work would never have gotten

started.

My thanks are also due to all my friends at Virginia Tech who have helped me through-

out rny stay here.

Thanks to for her patience and excellent typing of the manuscript.

Last but not least, I would like to thank my parents whose moral support and encour-

agement shall remain unparalleled.

Acknowledgements iv

Table of Contents

CHAPTER 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Background ......................................... ·. . . . . . . . . . . . . . . . 1

1.2 Scope ............................................................. 3

CHAPTER 2. PROBLEM FORMULATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Introduction and Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Viscoelasticity Preliminaries ........................................... , 6

2.2.1 Differential Form of Stress-Strain Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.2 Creep Compliance and Relaxation Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.3 Integral Form of Stress-Strain Constitutive Relations---Boltzmann Constitutive

Law and Stieltjes Convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Equations Governing the Stability of Viscoelastic Flat Plates Using a Third Order Re-

fined Theory (TTSD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.1 Strain-Displacement Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.2 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.3 Equations of Motion . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.4 The Elastic-Viscoelastic Correspondence Principle ....................... 36

Table of Contents v

2.3.5 Derivation of the governing stability equations using the correspondence principle

(TTSD) ............................................................ 37

2.4 Equations Governing the Stability of Viscoelastic Flat Plates Using a First Order Trans-

verse Shear Deformation Theory (FSDT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.5 Alternative Representation of the Governing Equations (2.86) for the FSDT Theory . . 43

2.6 Determination of the Transverse .Shear Correction Factor K2 • • • • • • • • • • • • • • • • • • 50

CHAPTER 3. DETERMINATION OF MATERIAL PROPERTIES OF A FIBER-REINFORCED

COMPOSITE WITH VISCOELASTIC MATERIAL BEHAVIOR .•.•...•...••........•. 51

3.1 Introduction ........................................................ 51

3.2 Use of Micromechanical Model to Determine' Material Properties . . . . . . . . . . . . . . . 52

3.2.1 Use of the 3-parameter solid (or standard linear solid) to model viscoelastic mate-

rial behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.3 Numerical Inversion of the Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.3.1 Use of Bellman's Technique to Obtain E.. (t) . . . . . . . . . . . . . . . . . . . . . . . . . . 64 11mn

CHAPTER 4. SOLUTION OF THE STABILITY PROBLEM . . . . . . . . . . . • • . • . . • . • . • . . • 67

4.1 Definition of Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.2 Stability Analysis Using the Third Order Transverse Shear Deformation Theory (TSDT) 68

4.3 Stability Analysis Using a First Order Transverse Shear Deformation Theory (FSDT) 75

4.4 Stability Analysis Using the Equations Representing the Interior Solution in the

Framework of the FSDT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.5 Stability of a Transversely Isotropic Viscoelastic Plate Undergoing Cylindrical Bending 78

CHAPTER 5. NUMERICAL RESULTS AND CONCLUSIONS .......... , . • • . . . . . . • • • . 86

5.1 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Table of Contents vi

Appendix A. THE LAPLACE TRANSFORM AND ASSOCIATED THEOREMS .•.••..••.. 135

Appendix B. MICROMECHANICAL MODEL FOR UNIDIRECTIONAL FIBER-REINFORCED

COMPOSITES . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . • . . . . . . . . . . • . . . • . . 137

Appendix C. NUMERICAL INVERSION OF THE LAPLACE TRANSFORM USING BELLMAN'S

TECHNIQUE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . 142

Appendix D. EXPLICIT FORM OF THE CHARACTERISTIC EQUATION. . ....•.•• , . , . , 145

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . 155

Vita .................................•....................•.•...... 158

Table of Contents vii

List of Illustrations

Figure 1. Variation of Poisson's ratio for the epoxy matrix. . .................... 94

Figure 2. Layup of plate and coordinate system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Figure 3. 3-Parameter solid element. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Figure 4. Variation of Young's modulus for the epoxy matrix. . . . . . . . . . . . . . . . . . . . . 97

Figure 5.

Figure 6.

Figure 7.

Figure 8.

Figure 9.

Figure 10.

Figure 11.

Figure 12.

Figure 13.

Figure 14.

Figure 15.

Figure 16.

Figure 17.

Figure 18.

Material property for the orthotropic plate - ......................... 98

Material property for the orthotropic plate - ......................... 99

Material property for the orthotropic plate - .................. ' ..... 100

Material property for the orthotropic plate - . .......... -· ............ 101

Material property for the orthotropic plate - ........................ 102

Plate in cylindrical bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Arrangement of fibers in matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Representative volume element for a fiber-matrix composite. . . . . . . . . . . . 105

Stability boundary for orthotropic viscoelastic plate; Uh = 4.8; biaxial com-pression; c5A = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Stability boundary for orthotropic viscoelastic plate; Uh = 4.8; biaxial com-pression; c5A = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Stability boundary for orthotropic viscoelastic plate; L/h = 4.8; uniaxial com-pression; c5A = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Stability boundary for orthotropic viscoelastic plate; L/h = 4.8; uniaxial com-pression; OA = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Stability boundary for orthotropic elastic plate; L/h = 4.8; biaxial compression; c5A = 1 or c5A =O .............................................. 110

Stability boundary for orthotropic elastic plate; L/h = 4.8; uniaxial com-pression; c5A = 1 or c5A = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

List of Illustrations viii

Figure 19. Stability boundary for isotropic viscoelastic plate; L/h = 4.8; biaxial com-pression; OA = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Figure 20.

Figure 21.

Figure 22.

Figure 23.

Figure 24.

Figure 25.

Figure 26.

Figure 27.

Figure 28.

Figure 29.

Figure 30.

Figure 31.

Figure 32.

Figure 33.

Figure 34.

Figure 35.

Figure 36.

Figure 37.

Stability boundary for isotropic viscoelastic plate; L/h = 4.8; biaxial com-pression; oA = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Stability boundary for isotropic viscoelastic plate; L/h = 4.8; uniaxial com-pression; OA = 1 or OA = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Stability boundary for isotropic elastic plate; L/h = 4.8; biaxial compression; oA = 1 or OA = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Stability boundary for isotropic elastic plate; L/h = 4.8; uniaxial compression; OA = 1 or OA = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Comparison of stability boundaries for orthotropic viscoelastic plate; L/h = 4.8; biaxial compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Comparison of stability boundaries for orthotropic viscoelastic plate; L/h = 4.8; uniaxial compression ......................................... 118

Comparison of stability boundaries for orthotropic elastic plate; L/h = 4.8; biaxial compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Comparison of stability boundaries for isotropic viscoelastic plate; L/h = 4.8; biaxial compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Comparison of stability boundaries for isotropic viscoelastic plate; L/h = 4.8; uniaxial compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Comparison of stability boundaries for isotropic elastic plate; L/h = 4.8; biaxial compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Comparison of stability boundaries for isotropic elastic plate; L/h = 4.8; uniaxial compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Stability boundary for orthotropic viscoelastic plate; L/h = 24; uniaxial com-pression; OA = 1 or c5A = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Stability boundary for orthotropic elastic plate; L/h = 24; biaxial compression; OA = 1 or oA = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Stability boundary for orthotropic elastic plate; L/h = 24; uniaxial compression; c5A = 1 or oA = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

Stability boundary for isotropic viscoelastic plate; L/h = 24; biaxial com-pression; oA = 1 or c5A = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Stability boundary for isotropic viscoelastic plate; L/h = 24; uniaxial com-pression; oA = 1 or <5A = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Stability boundary for isotropic elastic plate; L/h = 24; biaxial compression; c5A = 1 or c5A = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Stability boundary for isotropic elastic plate; L/h = 24; uniaxial compression; oA = 1 or oA = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

List of Illustrations ix

Figure 38. Comparison of stability boundaries for orthotropic elastic plate; L/h = 4.8; biaxial compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Figure 39. Comparison of stability boundaries for orthotropic elastic plate; L/h = 4.8; uniaxial compression ........................................ . 132

Figure 40. Comparison of stability boundaries for orthotropic elastic plate; L/h = 24; biaxial compression ......................................... . 133

Figure 41. Comparison of stability boundaries for orthotropic elastic plate; L/h = 24; uniaxial compression ........................................ . 134

List of Illustrations x

List of Tables

Table 1. Material properties of boron fibers and epoxy matrix. . . . . . . . . . . . . . . . . . . . 63

List of Tables xi

CHAPTER 1. INTRODUCTION

Advanced fiber-reinforced composites have gained increasing attention due to their

widespread use in the design of primary and secondary load-bearing members where the

strength/stiffness to weight ratio is of paramount importance. Such applications include, e.g.,

the aircraft and aerospace structures in which elevated temperature gradients arise due to

high-speed flight; rocket engines; nuclear reactors where thermal insulation is the governing

criterion during design, etc. Under the influence of such high temperatures, these composite

materials exhibit time-dependent properties which could be modelled by linear (or nonlinear)

viscoelastic constitutive laws.

For such advanced composite structures exhibiting viscoelastic properties, it is essential

to determine their stability behavior under compressive load systems.

1.1 Background

1

The analysis of viscoelastic composite plates requires a complete knowledge of their

time-dependent material behavior. The relaxation moduli (and creep compliances) for a

CHAPTER 1. INTRODUCTION

transversely isotropic laminate are given by McQuillen [1]. However, in this analysis which

uses micromechanical relations developed by Hashin [2], a quasi-elastic (also referred to as

quasi-static) approximation is implied. In this manner, the micromechanical relations for the

viscoelastic body in a 2-D state of stress were derived from their elastic counterparts by

merely replacing elastic constants with time-dependent material properties. Similar quasi-

elastic analyses were also performed by Wilson [3] in which the Halpin-Tsai relations were

employed, and by Sims [4] who used the method developed by McQuillan [1]. Both Sims [4]

and Wilson [3] assume an elastic behavior in dilatation for the viscoelastic matrix.

Stability of viscoelastic composite panels undergoing cylindrical bending was studied

by Malmeister et al. [5] for the case when the composite exhibits viscoelastic properties in

transverse shear only. However, it is evident that this is a restriction being imposed on the

material behavior since the extensional moduli in the direction perpendicular to the fibers are

also time-dependent due to the viscoelastic behavior of the 0matrix.

Wilson [3] analyzes the stability of rectangular, viscoelastic, orthotropic plates subjected

to biaxial compression. In deriving the equations governing the stability, he uses the principle

of stationarity of the potential energy functional. The foregoing variational principle may be

valid for a quasi-elastic approximation of the problem. Strictly speaking, however, the use of

the aforementioned variational principle is questionable due to the presence of internal

damping in viscoelastic systems which results in a non-conservative problem. Both Sims [4]

and Wilson [3], by using the quasi-elastic approximation, analyze the system as an instanta-

neous time-dependent elastic system, thereby not implying the hereditary material behavior.

In all these methods, linearized sytems were considered and stability was analyzed over

an infinitely long period of time (i.e., the asymptotic behavior of motion was studied). How-

ever, another interesting problem concerning viscoelastic systems is the determination of a

critical time defined by that instant when the deflection or its rate of change become infinite.

Grigoliuk and Lipovtsev [6] have studied the snap-through buckling of a viscoelastic shell and

have pointed out that introduction of either geometric or physical non-linearities may result in

the existence of such a critical time, t = tc, (the time at which snap-through buckling occurs).

CHAPTER 1. INTRODUCTION 2

Szyszkowski and Glockner [7] analyze the stability of a viscoelastic column using a geomet-

rically non-linear formulation of the problem, thus allowing them to determine a critical time.

They point out that the problem of critical time evaluation becomes important for loads lying

between the safe-load-limit, Pv, (defined by asymptotic instability of the viscoelastic structure)

and the instantaneous Euler buckling load, PE. Thus, when Pv < P <PE, the system "may" be-

come unstable at a finite time defined as the "critical time". However, such a study goes be-

yond the scope of the present work.

1.2 Scope

In this study, a method of analyzing the linearized stability of viscoelastic composite

plates has been developed. To this end, an "exact" dynamic approach has been used in the

formulation and solution of the problem. The composite material was modelled through a 3-D

linearly viscoelastic, hereditary, constitutive law. Effects of transverse shear deformations

(which are highly pronounced for materials exhibiting high degrees of anisotropy, and/or the

plate thickness-to-length aspect ratio is high) have also been incorporated in the analysis.

Emphasis has also been given to the effect of transverse normal stress, o-33 , which was neg-

lected by Wilson [3).

The analysis is done for an orthotropic, viscoelastic plate in the framework of a third-

order transverse shear deformation theory (TTSD) and its first-order counterpart (FSDT).

Then, making use of the single equation representing the interior solution (discussed in Sec.

2.5), it has been ~hown that for an isotropic plate, the results obtained by solving the exact

system of three coupled equations agree very well with the approximate solutions obtained

via the "single equation'. This result constitutes an extension for the viscoelastic case of the

elastic counterpart obtained by Librescu (8, 9].


Comparison studies between TTSD, FSDT and the classical Kirchhoff theory of plates are

also presented.


CHAPTER 2. PROBLEM FORMULATION

2.1 Introduction and Problem Statement

The increased interest of the utilization in aeronautical and aerospace structures of

composite material systems requires the development of adequate analytical methods for

their rational design. This demand becomes more stringent in the case of materials exhibiting

time-dependent properties as is the case for high-speed aircraft structures operating at ele~

vated temperatures. The behavior of such time-dependent materials may be described by a

linear viscoelastic model.

Another aspect of equal importance is the effect of transverse shear deformations which

is highly prevalent in composite structures. The importance of this becomes more prominent

when the material exhibits high degrees of anisotropy or when the plate (or shell) is thick

enough. The classical theory does not account for this effect, thereby implying infinite rigidity

in transverse shear. Moreover, the assumption involved in the classical theory of plates does

not allow the fulfillment of the boundary conditions on the external bounding planes of the

panel. In addition, the simultaneous consideration of zero transverse normal stress and zero

transverse normal strain implies a contradiction which is to be eliminated in the framework

CHAPTER 2. PROBLEM FORMULATION 5

of the refined theories of plates. This calls for the use of refined theories when analyzing

beams, plates, etc., in which the contradictory assumptions as well as the shortcomings of the

classical theory, stated earlier, are removed.

In this study an anisotropic rectangular plate is analyzed for dynamic stability. The

composite material considered herein consists of a linearly viscoelastic matrix reinforced by

elastic fibers. This, however, is not a restriction on the behaviour of the fibers which may also

be treated as linearly viscoelastic if desired, as will be seen in Chapter 3. The plate is subject

to constant inplane edge loads and assumed to be simply supported along its edges. The

plate is considered as a moderately thick one with h/L ratio of about 5.

2.2 Viscoelasticity Preliminaries

The time-dependent behavior of a typical epoxy resin material is shown in Figs. 1 and 4

and was used by Schapery [10) in his analysis. This requires the use of a constitutive law

different from the elastic Hooke's law. In the following there are two methods of describing

the constitutive law, wh.ich are entirely equivalent.

2.2.1 Differential Form of Stress-Strain Relations

The differential equation relating stresses to strains for a linearly viscoelastic material

(ref. Christensen [ 11)) may be expressed as:

or in a compact form as,


P[D] au= Q[D] eij (2.1)

where,

M

P[D] = LPk Dk k=O

M

Q[D] =I qk Dk k=O

Pk and qk being the viscoelastic characteristics of the material. Relation (2.1) corresponds to

a Voigt model represented as a combination of springs and dash pots (in series and parallel).

The resulting coefficients Pk and qk are algebraic functions of the spring and dash pot constants

(see, e.g., Flugge (12)).

2.2.2 Creep Compliance and Relaxation Modulus

The Laplace transform of Eq. (2.1) yields

N k N k

P[s] au[s] - LPk I sr-1 a~tr)[O] = Q[s] eu[s] - I qi I sr-1 e~k-r)[O] (2.2) k=1 r=1 k=1 r=1

where,


and

N

P[s] = LPk[s]k k=O

N

Q[s] = L qk[st k=O

d(k-r) G.

u(k-r)[OJ = - 11 I t = 0 I} dt(k-r)

with a similar definition for eif-'>[O]. The pverbars denote the Laplace transform (LT.) with s

as the LT. variable. Now the coefficients of various powers of s must be equal on either side

of (2.2). For arbitrary initial conditions, this requires that the coefficients of powers of s be

equal in the terms containing these initial conditions on either side of (2.2). Equating the co-

efficients of like powers in the terms involving the initial conditions results in,

N N "\'1 (r-k) _ "\'1 (r-k) t__lr au [OJ - 6 q, eij [O], k = 1,2, ... ,N r=k- r=k

Hence (2.2) reduces to

P[s] au[s] = Q[s] eu[s]

from which we obtain

a[s] = Q[s] e[s] '1 P[s] 'J

or


where,

e[s] ~ Q[s] Si> P[s]

Let the input be a constant strain represented as,

(2.3)

where H[t] is the Heaviside distribution. Thus after inverting into the time domain, (2.3) yields

(2.4)

where,

E[t] = 2-1(~) = 2-1(J... Q[s]) · s s P[s] '

2-1 denoting the inverse Laplace transform.

Thus E[t], the relaxation modulus, is the resulting stress for a unit input strain. The inverse

of (2.3) would yield a strain-stress law involving D[t], the creep compliance, which is the re-

suiting strain for unit input stress.


2.2.3 Integral Form of Stress .. Strain Constitutive Relations---Boltzmann

Constitutive Law and Stieltjes Convolutions

Using the superposition principle for a linear viscoelastic material results in the

constitutive law involving the Stieltjes convolution (ref. Pipkin (13]). This relates the time-

dependent stresses to the time-varying input strains (or vice-versa). The derivation considers

the input strains (or stresses) to be a superposition of Heaviside functions with various step

sizes. The output stresses (or strains) are then related to the input quantity by using (2.4) (or

its inverse) and summing over all the step intervals.

The resulting constitutive law is,

and the inverse law is,

Generalizing to an anisotropic material for a 3-dimensional state of stress we obtain, (ref. [12])

(2.Sa)

(2.Sb)

Here EifmnUJ and F,JmnUJ are the tensors of relaxation moduli and creep compliances

which exhibit the following symmetry properties:


and

Eijmn[t] = Emnij[t]

the former arising due to the symmetry of the stress and strain tensors while the latter can

be proved by using Onsager's principle in irreversible thermodynamcis (see Biot (14)). The

same symmetry properties result for the creep compliance tensor (F;jmn)·

Now for the sake of generality let the initial condition for the input strains be represented

by

Substituting this in (2.5a) yields,

where we have used the fact that the derivative of the Heaviside function is the Dirac delta

function. The above equation thus becomes,

(2.6)

where the second term in (2.6) represents the response due to initial strain emn[O].

Integration of (2.6) by parts yields,


-it d Eijmn[t- T] d[t - T]

o+

where we have used the fact that

t-> 0

d[t- T] [ ] d d-r emn T T

On replacing [t - T] by s we obtain, (see Malmeister et al., [5])

(2.7)

(2.8)

where the first term in (2.8) represents the elastic part of the response. Again making use of

(2.7) we can change the lower limit Q+ in (2.6) to 0. Equations (2.6), (2.8) are equivalent forms

of (2.5) which is the integral constitutive law for an anisotropic viscoelastic body in a

3-dimensional state of stress, and is also referred to as the Boltzmann hereditary constitutive

law.

Taking the Laplace transform (2.6) (which is the form of the constitutive law used henceforth)

we obtain,

Introducing (2.7) in the above yields


(2.9a)

and

~J = s F;jmn[s] <Tmn[s] (2.9b)

which is similar to (2.3), obtained by the differential form of the constitutive law.

As is readily seen, Eqs. (2.9) take into account the initial conditions of the input variable

(emn[t]) or (amn[t]).

2.3 Equations Governing the Stability of Viscoelastic Flat

Plates Using a Third Order Refined Theory (TTSD)

As noted earlier for anisotropic composite plates, the assumptions involved in the clas-

sical theory disregarding the influence of transverse shear deformations and transverse

normal stresses are to be eliminated by use of a refined theory of plates. In the following

development we extend a method developed by Librescu [8] (see also Librescu and Reddy

[15], and Librescu, Khdeir and Reddy [16]) to the viscoelastic case. A geometrically nonlinear

theory for a viscoelastic flat plate is developed and the governing equations are derived in the

Laplace transformed space. These are then inverted to the time domain.

We shall consider the case of a flat plate of uniform thickness h. By S± we denote the

upper and lower bounding planes of the plate, symmetrically disposed with respect to its

mid-plane a.

We denote the edge boundary surface (see Fig. 2) by n. It is assumed that S± and a

(defined by X3 = ± h/2 and x3 = 0, respectively) are sufficiently smooth without singularities.

The points of the 3-D space of the plate will .be referred to a rectangular Cartesian system of


coordinates X;. where x.(a = 1,2) denote the in-plane coordinates, x3 being the coordinate

normal to the plane x3 = 0 .

2.3.1 Strain-Displacement Equations

The higher-order theory of plates is developed by using the following representation of

the displacement field across the thickness of the plate:

N (n) V<X[xw, x3, t] = I(x3)n V<X [xW' t] (2.10a)

n=O

R (n) V3 [x<X, x3, t] = I (x3)n V3 [xw, t] (2.10b)

n=O

where the Greek indices range from 1 to 2, while the Latin indices range from 1 to 3, with t

being the time variable. The numbers N and R denote two natural numbers identifying the

truncation levels in the displacement expansion. At this point it is worthwhile to note that the (2r) (2r+1)

terms corresponding to the stretching state of stress are v., V3 and those corresponding to (2r+1) (2r)

the bending state are v. and V3.

For a third-order bending theory which retains the assumption of the inextensiblity of the

transverse normal elements, we may write the following expansions for the displacement

components:

(2.11a)

(2.11b)

where


(n) (n) V; = V; [xcc, t] (2.11c)

We now introduce the nonlinear Lagrangian strain tensor as,

where a comma denotes differentiation with respect to the index following it. Using the Von-

Karman approximation which neglects the nonlinearities involving the inplane displacement

components or their gradients (i.e., v •. , VP."' etc.), we obtain,

(2.12a)

1 2 e22 = V2 2 + -(V3 2) ' 2 '

(2.12b)

(2.12c)

(2.12d)

(2.12e)

(2.12/)

Introducing the displacement expansions (2.11) into (2.12) and neglecting the resulting non-<n>

linearities associated with v. and their derivatives, we obtain

(2.13a)

(2.13b)


(2.13c)

(2.13d)

(2.13e)

(2.131)

Examining (2.13) we see that only e11 ,e22 and e12 are affected by the Von Karman type nonline-

arities. Taking the L.T. of equation (2.13) yields

(2.14a)

(2.14b)

(2.14c)

(2.14d)

(2.14e)

where the overbars denote the Laplace transform (l.T.) of the quantity free of overbars,

(n) (n) vi= Vi[xix, s]

and .ft! is the L.T. operator defined a~.


with s being the L.T. variable.

2.3.2 Constitutive Equations

Reproducing the constitutive law in the form as expressed in (2.6), we have,

Taking the L.T. of the above in conjunction with (2.8) yields (2.9), which is reproduced as fol-

lows:

(2.15)

Writing (2.15) in its full form for an orthotropic body (see, e.g., Librescu [8]), for which the

components of ~imn[s] involving indices repeated once or thrice are to be considered zero,

we obtain

(2.16a)

(2.16b)

(2.16c)

(2.16d)

(2.16e)


(2.161)

Now we introduce the following notation which is helpful in further developments:

- _. S Eljmn = E;1mn

where the star(*) along with the overbar (-) identifies the Carson transform which is the L.T.

multiplied by s. Thus Eijmn is the Carson transform of the tensor of relaxation moduli. Thus,

introducing e33 from (2.16c) into (2.16a) and (2.16b), we obtain,

(2.17a)

~ -. 7\ .. u22 = E2211 e11 + E2222 e22 + t5 A E2233 U33 (2.17b)

(2.17c)

(2.17d)

(2.17e)

where,

_. _. E:x.{133 E33anr

(2.1 Ba) E3333

and

(2.18b)


- -In (2.18a) E:11.,,, represents the Carson transform of the reduced stiffness E,11.,,, (analogous to its

elastic counterpart) (see Librescu [8]). Also the tracer c5A identifying the presence of a33 will

take the value 0 or 1, according to whether this influence is ignored or included. Thus we can

write (2.17) in compact form as,

-. A+

<icr.p = Ecr.pwn eW'ff: + 0A Ecr.p33 U33 (2.19a)

(2.19b)

Inverting (2.19) back to the time domain, we get by using Borel's Theorem (see Appendix [A))

and Eqns. (2.18)

it . A A

+ c5A (Erip33[t- -r] + Ecr.p33[0] o[t- -r]) <133[-r]d-r

0

~-

:. • 0 p[t] ~ I"~.p-[t- T] •w.[TJ dT + E•Pw•[O] •w•[t] + i'&,µ33[1 - T] o33[T] dT

o Jo

A

+ Er.xp33[0] a33[t] (2.20a)

and


(2.20b)

where: (as shown in Appendix [A])

(2.20c)

(2.20d)

Eqs. (2.20c,d) allow one to infer that all these time-dependent moduli attain their correspond-

ing elastic values at t = 0. We also note that in (2.20a) and (2.20b)

E [ ] ~ 2'-1 {J_( E - s Erx{J33 s E33w-rr )} = 2'-1 {E - Erx{J33 E33w-rr } rx{Jw-rr t s S rx{Jw-rr . E rx{Jw-rr E

s 3333 3333 (2.20e)

and

(2.201)

and the overdots (.) denote time derivatives.

2.3.3 Equations of Motion

The equations of motion in Lagrangian description for a 3-D continuum undergoing finite

deformations have been derived by Green and Adkins (17] (see also Amenzade (18) and Fung

(19]). They are:


[<r·k(b·k + V· k)] ·+pH·= p ii J I I, , J I I (2.21)

where aik is the second Piola-Kirchhoff stress tensor (symmetric), b;k is the Kronecker delta,

p is the mass density of the medium and H; are the body forces per unit mass. We now define

the unsymmetric tensor S;1 (referred to as the first Piol.a-Kirchhoff stress tensor) as follows:

Introduction of the above definition into (2.21) yields,

The above equation is another form of the equation of motion (in Lagrangian description) of

a continuum undergoing finite deformations.

Writing (2.21) in full form, we obtain (see Librescu [20] for the FSDT counterpart),

(2.22a)

(2.22b)

(2.22c)

Neglecting the nonlinearities associated with gradients of in plane displacements and stresses,

we may write (2.22) as follows:

<r13,1 + <r23,2 + <r33,3 + (<r11 V3,1),1 + (<r12 V3,2),1 + (<r21 V3,1),2 + (<r22 V3,2),2

+ (<r31 V3,1),3 + (a32 V3,2),3 + P H3 = P V3


(2.23a)

(2.23b)

(2.23c)

21

where in (2.23) the fact that V3,3 = 0 was also used (see equation (2.11) concerning displace-

ment expansions).

The stability problem when formulated in terms of the displacement field may be re-<1> (3) (0)

duced to a system of equations in 5 unknown quantities, Vs, Vs and V3• In order to obtain the

governing equations in terms of these displacement quantities we need five equations of

equilibrium. To this end we consider the moment of order zero and one of the first two

equations of motion (2.23(a) and (b)) and the moment of order zero of the third equation

(2.23c). Also we note that henceforth we neglect .all body forces.

Taking the moment of order one of the first two equations of motion (2.23(a) and (b)),

we obtain:

where, (in a general form)

and

(1) (1) f +h/2 (1) L11,1 + L21,2 + 0"31,3 X3 dx3 = c5c f1

-h/2

(1) (1) I +h/2 (1) L12,1 + L22,2 + 0"32,3 X3 dX3 = c5c f2

-h/2


(2.24a)

(2.24b)

(2.25a)

(2.25b)

22

(n)

and c5c is a tracer which identifies the presence or absence of rotary inertia terms (i.e., r. ) by

taking values 1 and 0, respectively. Now consider the term f +hf2a3• 3 x3 dx3 appearing in (2.24). -h/2 •

An integration by parts yields,

where (in a general form),

and

Introducing (2.26) into (2.24), we obtain,

(1) (1) (1) (0) (1) L11,1 + L21,2 + P1 - L13 =Jc f1

(1) (1) (1) (0) (1) L12,2 + L22,2 + P2 - L23 = Oc f2

The above when written in compact form yield,

(1) (1) (0) (1) Lap, fl+ Pa - La3 = r5c fa

(2.26)

(2.27a)

(2.27b)

(2.28)

Equation (2.28) represents the first two equations governing the motion of a flat plate. The

Laplace transforms of (2.28), (2.25), (2.27) yield


where,

(1) (1) (0) (1)

L(tp, fJ +Pa. - La.3 = Jc fa. (2.29)

(2.30a)

(2.30b)

(2.30c)

(2.30d)

In deriving (2.29) and (2.30) we have used the fact that the order of integration performed over

x3 and t can be interchanged. (1) (0) (1) (0)

Now we represent L.p,p and L.3 in terms of v. and V3 . To this end we introduce (2.14d)

and (2.14e) into (2.19b) to obtain,

(2.31)

Writing (2.14) in compact form yields,

(2.32a)

(2.32b)


where,

(2.33a)

(n) (n) V1 = Vi[x<X. s] (2.33b)

as defined previously.

The introduction of (2.32b) into (2.19a) considered in conjunction with (2.33) yields,

(2.34)

- -where E:.p.,,, and £:p33 have been defined in (2.18).

Determination of <133(x •• x3,s)

The third equation of motion represented by (2.23c) when written in compact form yields,

(2.35)

The Laplace transform of (2.35) yields,

(2.36)

We shall first deal with the two nonlinear terms in the above equation. Consider the term

2 {(a u6 v3, 6), u}·

Introducing (2.2Da) for a u6[x •. x3, t] , (2.13) for e.,,,[x •• x3, t] in conjunction with (2.11), we

obtain for the above term,


Consider the term !i' {(u311 V3, 11),3} • Similarly introducing (2.20b) for u311 , (2.13) for eA3 in con-

junction with (2.11), we obtain for this term,

(2.38)

Introducing (2.37), (2.38) into (2.36) and then integrating (2.36) through the segment [O, x3) in

conjunction with (2.11), (2.33), we obtain for <733,


_ (x )2 (1) (x )4 (3) (x )2 (1) + J... E [OJ[-· -3 - V [t] + - 3 - V [t] + - 3 - V [t] 2 µow11: 2 w, n 4 w, 11: 2 ir, w

(2.39)

Introducing a33 from (2.39) into (2.34), we obtain,


_ (x )2 (1) (x )4 (3) (x )2 (1) +-1..E [OJ[-3-V [t]+-3-V [t]+-3 -V [t] 2 µown 2 w, .n 4 w, n 2 n, w

(1)

Introducing (2.40) in (2.30a) in conjunction with (2.11 c), (2.33b) and (2.40), we obtain, for L.p.

(2.41)

(1)

We see that the last term in the expression of L.p , i.e., the one under the Laplace transform (0) (0) (0)

operator (2) contains the cubic nonlinearity of V3,., V3," Vu . This represents the product of

small disturbances and can thus be neglected for a linear analysis of the problem. Thus on (1)

linearizing L.p we obtain,


(2.42)

(3)

We now solve for V" in order to eliminate it from the governing equations of stability. Towards

this end we recall the constitutive equations (2.8) which are as follows:

(2.43)

with the inverse relation being,

(2.44)

Introduction of ek1[t'] in a;1[t] above yields:

"'I[ I] - r ( [ E,jkl[ I - Tl + E11k/[O] O[t - Tl] f J" klmh - T'] + F klmo[O] 6[ T - T'J] "mol T'] d<') <h 0

(2.45)

Interchanging the order of integration in the expression above yields:

"11[ I] - r (I) Eijk/[ I - Tl + E11kl[OJ J[ I - Tl] [F klmol T - "l + F klmo[O] 6[ T - T'J] dT) "mol T'] dT' 0

(2.46)

Due to the symmetry property of the stress tensor a;Jt] , we may write the following ex-

pression (see !Ilyushin and Pobedria [21 )):


aij[t] = ~ [Jim Jjn +Jin Jjm] O'mn[t] = r ~ J[t- T'J[ Jim Jjn +Jin Jjm] O'mn[T'] dT' (2.47) 0

By virtue of (2.46) and (2.47), we obtain,

Setting T' = 0 in (2.48) and then taking the LT. of the resulting expression yields,

(2.49)

From (2.49) we thus obtain the following result for an orthotropic body:

(2.50)

In writing (2.50), we have also assumed that the relaxation moduli E;imn[tJ and creep compli-

ances F;imn[tJ are continuous functions and thus possess unique Laplace transforms E;imn[s]

and F;imn(s) exhibiting the same symmetry properties as their originals. Making use of (2.50)

in (2.31) yields,

(2.51)

In order to satisfy the boundary conditions on the boundary planes, we get by introducing

(2.51) in (2.30c) the result:

(2.52)


(3)

Solving for V" from (2.52), we get,

(2.53)

Introducing (2.53) into (2.42), in conjunction with (2.50) and the symmetry property

= :: (1)

C..p.," = C..p""' , we obtain for L.11 the following:

(2.54)

Now introducing (2.31) into (2.30d), and then making use of (2.53) in conjunction with (2.50), (0)

yields for L.,3 the following:

(O) 2h - [(i) (O) J 1 (i) Lw3 =-3-s Ew3;.3 V;. + V3,;. + 3Pw (2.55)

Upon introducing (2.54), (2.55) into the Laplace transformed equations of motion (i.e., (2.29))

we obtain, after multiplying throughout by ~~ ,

(2.56)

(1) (1)

In (2.56), r. is obtained by first evaluating f. and then taking its Laplace transform. Introducing (1)

(2.11a) into (2.25b), we .obtain for r.,


(2.57)

Inverting (2.53) into the time domain, we obtain,

(2.58)

Differentiating twice w.r.t. T we obtain (using Leibnitz's rule),

(2.59)

Introducing (2.59) in (2.57) yields, for zero !;Urf;ice tractions,

.. .. (1) - (.K_ (1) - h3 (0) ) frr. - P 15 Va 60 V 3, a (2.60)

Its L.T. yields,

(2.61)

Introduce the notation

(2.62a)

and

" -1 (- Eap33) Err.pwA. = fe Ew3A.3 ----E3333

(2.62b)


- • Ewm[O] E.pn[O] Also, E.pwi[O] = ] (see Appendix [A]). Substituting (2.61), (2,62a) into (2.56), we

E3333[0 obtain for zero surface tractions,

- - (- - ) ~ (0) ~ (1) i\. (1) (0)

Ert.{Jwn V 3, nw{J - 4 Ert.{Jwn V w, n{J + 4 <5 A E'1.{3wA VA, w{J + V 3, ).w{J (2.63)

i\. ( 2 (0) (0) (0) ) 40 ~ (1) 40 ~. (0) - So A Os p Ert.{333 s \13, {3 - s V3, p[O] - V3, p[O] + h2 Ert.3A.3 VA+ h2 Eci3A.3 V3, A

Inverting the above in the time domain, we have, using Beret's theorem for convolution inte-

grals,

(2.64)

Equations (2.64) are the first two (of three) governing equations for a geometrically nonlinear

theory of an orthotropic, linearly viscoelastic fiat plate subject to zero in plane surface tractions

(.6: = o), using a third order transverse shear deformation theory.

Now the Laplace transform of the third equation of motion (2.23c) yields,


(2.65)

Here the tracer b0 identifies the transverse inertia term. Taking the moment of order zero of

equation (2.65), we obtain,

[ (0) ((0) )] - G +h/2 La.3 + 2 La.p V3; p . , a.+ [0'33 + 2(0'31 V3,1) + 2(0'32 V3,2)Lh/2

2(0) (0) (0) - b0 p h (s V3 - s V3[0] - V3[0]) = 0

From the above we obtain,

(0) (0) (0) (0) (0) (0) (0) (0) (0) (0) (0) - - ( L13,1 + L23,2 + 2 L11 V3,11 + L22 V3,22 + 2 L12 V3,12 + [L11,1 + L21,J V3,1

[ (O) (0) J(O) ) + L12,1 + L22,2 V3,2

(2.66)

[ (0) (0) ] +h /2 ( (0) (0) (0) ) + U33 + 2(0'31 V3,1) + 2(0'32 V3,2) -h/2 - bop h s2 V3 - s V3[0J - V3[0J = 0

The zeroth order moments of (2.23b) and (2.23c) yield,

(0) (0) (0) L11,1 + L12,2 + P1 = 0 (2.67a)

(0) (0) (0) L12,1 + L22,2 + P2 = 0 (2.67b)

(0) (ll) Substituting (2.67) in (2.66) and neglecting surface tractions (which implies p1 = p2 = 0 and (0) p3 = 0 ), we obtain,

(2.68)

For constant edge loads, equation (2.68) converts to a linear form which reads,

CHAPTER 2. PROB.LEM FORMULATION 34

Introducing (2.55) into (2.69) yields,

(2.70)

Inverting (2.70) into the time domain, we obtain,

2 t [(1) (0) J 2 [(1) (0) J 3 h L Ew3A3[t- T] VA, co[T] + V3, Aco[T] dT + 3 h Eco3A3[0] VA, co[tJ + V3, Aco[t] (2.71)

(0) (0) (0) (0) (0) (0) (0) + L11 V3,11 [t] + L22 V3,22[t] + 2 L12 V3,12[t] - <50 Ph V3 = 0

Equation (2.71) represents the third equation governing the stability of an orthotropic, linearly

viscoelastic, shear deformable flat plate.

In the next section we introduce the elastic-viscoelastic correspondence principle.

Boundary conditions

Equations (2.64) and (2.71) represent a sixth-order governing equation system. Its sol-

ution must be determined in conjunction with the prescribed boundary conditions (which are

to be three at each edge). For a simply supported plate, i.e., hinged-free in the normal di-

rection, we have the following boundary conditions:

(1) (0) (1) V2 =V3 =L11 =0 atx1 =0,/1

(1) (0) (1) V1 = V3 = L22 = 0 at x2 = 0, 12

The above equations represent the conditions of zero tangential displacement, zero trans-

verse displacement and zero moment resultant along the eqges of the plate.


2.3.4 The Elastic-Viscoelastic Correspondence Principle

Consider the equations of motion for infinitesimal strain theory which write as:

(2.72a)

The linearized strain-displacement relations are:

(2.72b)

Also the constitutive equations for an elastic continuum write as:

(2.72c)

and the boundary conditions are,

(2.72d)

(2.72e)

where Sq denotes that part of the boundary S on which the components of the stress vector

are prescribed, while S, is that part of S over which the components of the displacement

vector are prescribed. Sq and S, are required to remain invariant with time. For a viscoelastic

continuum the equations of motion, strain-displacement equations and boundary conditions

are identical to (2. 72a, b, d and e), respectively, the only change being in the constitutive

equations. For a linearly viscoelastic anisotropic material, these were given in (2.5) and are

reproduced here as,

(2.72d)


Taking the L.T. of (2,72c) yields (for the elastic body):

(2.73a)

The corresponding L.T. of the viscoelastic constitutive equation (2.72d) yields,

(2.73b)

Now the L.T. of (2.72a, b, d, e) would be identical for the elastic and viscoelastic continua. This

fact when considered in conjunction with (2.73a) and (2.73b) gives rise to the elastic-

viscoelastic correspondence principle (see Christensen [11]). The principle states that the L.T.

of the governing equations for a viscoelastic continuum can be obtained by taking the Laplace

transform of the corresponding governing equations of an elastic continuum and then replac-

ing the moduli and compliances by their Carson transforms. This means that E;imn (or F;imn)

which are constant for the elastic body (and are thus unaffected by taking the Laplace trans-

form of the governing equations pertaining to the elastic body) should be replaced by their

Carson transforms in order to obtain the corresponding transformed governing equations for

the viscoelastic body.

We note that the equations of motion and strain-displacement relations used in estab-

lishing the correspondence principle (C. P. hereafter) are those pertaining to the infinitesimal

displacement theory. However we shall see in the following section that the C.P. can be used

for deriving the governing equations of the linear stability problem.

2.3.5 Derivation of the governing stability equations using the

correspondence principle (TTSD)

Making use of the general procedure followed in Sec. 2.3.3 for an orthotropic viscoelastic

flat plate, we may derive the equations governing the stability of an orthotropic, elastic, flat


plate. Neglecting the inplane tractions and the transverse loads on the bounding surfaces

(i.e., ~: = ~~ = 0), we obtain the following system of equations governing the stability of an

orthotropic, elastic, flat plate:

(2.74)

where,

(2. 75)

and,

2 ((1) (0) ) (0) (0) (0) (0) (0) (0) (0) 3 Ew3A3 h VA, w + V3, Aw + L11 V3,11 + L22 V3,12 + 2 L12 V3,12 - Po h V3 = 0 (2.76)

At this stage we note that (2.74), which represents the first two governing equations when fi-

nite displacements are considered, remains unaltered when compared with its counterpart

obtained for infinitesimal displacement theory (see Librescu and Reddy [15]). The third gov-<1> (0)

erning. equation (2.76) may be obtained, for the case when P. = 0, by replacing p3 by ~ ~ ~ ~ ~ ~ p3 + L11 V3,11 + L22 V3,22 + 2 L12 V3,12 in the third governing equation derived in conjunction with the

infinitesimal displacement theory (see Librescu and Reddy [15]). In equations (2.74) and (2.76)

the quantities E.11w"' E.1133 , Ewm• E3333 and E.1133 are constants defined by (2.20c), (2.20d). They £3333

are the elastic constants which coincide with their viscoelastic counterparts at time t = 0 (see

equations (2.20) and (2.62)).

Furthermore we also observe that setting the time derivatives of material properties to

zero in equations (2.64) and (2.71) in conjunction with (2.20), (2.62) yields equations (2.74) and

(2.76), respectively. Also note that in (2.71) tMe edge loads are assumed constant and hence


(0) (Q) (0)

are equal to L11 , L22 and L12, whereas in (2.76) this restriction is not necessarily preser.t so that (0) (0) (0)

L11 , L22, L12 are functions of [x •• x3; t] and don't take on the meaning of edge loads. However for

the following analysis we assume constant edge loads.

Now using the correspondence principle (as stated in Sec. 2.3.4) in equations (2.74),

(2.75), (2.76) we obtain,

(2.77)

where,

(2.78)

(2.79)

(0) (0) (0)

where in writing (2.79) the stress resultants L11 , L22 , L12 are constant and could be interpreted

as the edge loads as noted above.

Now inverting (2.77), (2.78), (2.79) into the time domain in conjunction with (2.18), (2.20),

(2.62) and Borel's theorem, we obtain:


(2.80)

and,

(2.81)

(0) (0) (0) (0) (0) (0) + L11 V3,11UJ + L22V3,22UJ + 2 L12 V3,12UJ - '50 Po h V3 = 0

(1)

In eqn. (2.80) r. is as defined in (2. 75). Equations (2.80), (2.81) are the governing equations of

stability for the orthotropic, linearly viscoelastic, flat plate subject to constant inplane edge

loads and obtained by using the correspondence principle. The comparison of (2.80) and

(2.81) with (2.64) and (2.71) shows that the two sets of governing equations coincide. This

proves again that the C.P. is a powerful tool for analyzing problems in linear viscoelasticity,

and will be used in developing the stability problem in the framework of the first order trans-

verse shear deformation theory (FSDT).


2.4 Equations Governing the Stability of Viscoelastic Flat

Plates Using a First Order Transverse Shear Deformation

Theory (FSDT)

In the framework of this theory, the following representation of the displacement field is

postulated:

(2.82a)

(2.82b)

Disregarding the influence of the transverse normal stress a33 in the constitutive equations

(2.17) and making use of the general procedure developed in Sec. 2.3.1 - 2.3.3, the governing

equations of stability may be derived for a viscoelastic flat plate. However, employment of the

procedure developed in Sec. 2.3.5 yields an identical system of governing equations, as was

shown in Sec. 2.3.5. Thus, using the latter method, we proceed in the following manner. The

governing equations for the elastic plate using the FSDT theory and the infinitesimal strain

assumption were obtained by Librescu [8] (see also Librescu and Reddy [15]) and are written

as,

h3 - (1) ·. 2 . ((1) (0) ) (1) (1) 12 Erx.flµp V11 , pa - K h Ep3;.3 V;. + V3,;. +Pp - c5c m1 V fl= 0 (2.83a)

2 ((1) (0) ) (0) (0) K h Ep3;.3 V;., fl+ V3, ;.p + p3 - c5 0 m0 V3 = 0 (2.83b)

where K 2 is the transverse shear correction factor associated with the transverse shear . p h3

moduli Ep313 ; m0 = p h and m1 = --. 12


(1) For the case when the in plane surface tractions are neglected (i.e., Pp= 0), we obtain the

(0)

equations corresponding to (2.83) for finite displacements by replacing p3 by a a a a a a a p3 + L11 V3,11 + L22 V3,22 + 2 L12 V3•12• Furthermore, neglecting transverse normal loads (i.e.,

z~ = 0), we obtain the following equations governing the stability of the orthotropic, elastic

plate:

(2.84a)

and

((1) (0) ) {O) {O) (0) {O) (0) {O) {O)

K 2 h Ep3;,3 V;,, fl+ V3, ).{J + L11 V3,11 + L22 V3,12 + 2L12 V3,12 - bo mo V3 = 0 (2.84b)

Using the correspondence principle in (2.84) for the case of constant edge loads, we have:

(2.85a)

and

(2.85b)

Equations (2.85) represent the Laplace transformed equations governing the stability of an

orthotropic, viscoelastic, flat plate subject to constant inplane edge loads and no surface loads

present. As noted earlier, in (2.85b) the stress resultants represent the edge loads as the

latter are assumed constant.

Inverting (2.85) into the time domain in conjunction with (2.18), (2.20) and Beret's thee-

rem, we obtain:


and

2 J,t. [(1) (0) J 2 [(1) (0) J K h 0Ep3,t3(f-T] V,t,p(T]+V3,,tp(T] dT+K hEp3,t3[0] V,t,p[t]+V3,,tp[t]

(0) (0) (0) (0) (0) (0) (0)

+ L11 V3,11 + L22 v3,22 + 2 L12 V3,12 - c5o mo V3 = 0 (2.86b)

Equations (2.86a and b) are the equations governing the stability of an orthotropic viscoelastic

flat plate subject to constant in plane edge loads . The problem of determining the transverse

shear correction factor K2 is discussed later in Sec. 2.6.

2.5 Alternative Representation of the Governing Equations

(2.86) for the FSDT Theory

We now make use of the procedure developed by Librescu (see Librescu [8, 22, 9]) al-

lowing one to recast the stability problem governed by (2.86a and b) in terms of two inde-

pendent equations, for a transversely isotropic body. For a transversely isotropic, elastic

material we have (see Librescu [8). pp. 402-403),

(2.87a)


(2.87b)

where E, v denote the Young's modulus and Poisson's ratio corresponding to the plane of

isotropy and E', v', G' are the Young's modulus, Poisson's ratio and transverse shear modulus

for the plane normal to the plane of isotropy.

Using the correspondence principle in (2.87), we obtain:

(2.88a)

(2.88b)

Introducing (2.88) for a transversely isotropic body in (2.85), we obtain,

h3 r [(1> <1> ] h3 r <1> 2 -(<1> <o> ) --- V -V +- V -K hG' V +V 24 1 + -v· . fl,µµ µ,µfl 12 1 _ (v·)2 µ,µp fl 3,fl

(2.89a)

( (1) (1) (i) ) - <Sc m1 s2 V fl - s V fl[O] - V fl[O] = 0

(2.89b) - ) -(0) (0) (0)

- 2 L12 V3,12 - V3, ,u = 0

Introducing the 2-D permutation symbol r.,p (where r.,2 = - r.21 = 1, r.11 = e22 = 0), we may write t

the following result:

(1) (1) (1) V p, µµ - V µ,µfl = &fly &µA VJt, A.y (2.90)

Introducing (2.89b) in (2.89a) in conjunction with (2.90), we obtain:


(0) (0) (0) (0) (0) (0) (0) - - - ) -- L11 V 3, 11 p - L22 V 3, 22p - 2 L12 V 3, 12p . - V 3, . .up J

(2.91)

Equation (2.91) represents the L.T. of the first two equations governing the stability of a (1)

transversely isotropic plate. Solving for Vp from (2.91}, we obtain,

(2.92)

[ - . J h3 E 1 (O) (o) (O) +- . [ _. (<5.omo s2 V3 p - s V3 p[O] - V3 p[O]

12 1 - ("V)2 K2hG' ' ' '

Now we define the potential function </> = </> [x1, x2, t] as fol.lows:

(2.93)

Inverting (2.92) into the time domain, we obtain:

(2.94a)

where we define

(2.94b)


Introducing {2.93) into {2.92), we obtain the following result:

(2.95)

Introducing (2.95) in (2.93) and using the properties of the permutation symbol s«P results in:

Multiplying throughout by (K 2hG'' + c\m1s 2), we obtain,

. . h3 £ <t> <1> <1> <1> _ 24 -1 - • [c5cm1 (sV p, yy [OJ - svy, py [OJ + v p, yy [OJ - Vy, py [OJ) + Spy 4>, cocoy]

+v

(2.96)

(2.97)

Inverting (2.97) to the time domain and assuming zero initial conditions, i.e., (1) (1)

Vp [OJ= Vp[OJ = 0 in conjunction with (2.94a) and (2.94b), we obtain:

(2.98)


Using the property eyp = - ep, in (2.98), we obtain

h3 f,t . h3 . 2 J,t . - 24 /G[t-T]4>,mrc.o[?']dT-242G[OJ4>,nnc.o[t]+K h 0G'[t--T]4>,c.o[T]dT (2.99)

Integrating (2.99) w.r.t x., yields:

h3 it . h3 2 it . - 24 0 2G(t- T]4>, 11'11'[T]dT -242G[OJ<P, 11'11' [t] + K h 0 G'[t - 1']4>[T]dT (2.100)

Equation (2.100) represents the first equation governing the stability of the transversely

isotropic plate. We observe that it is an equation expressed in terms of the potential function

4>[x1. x2, t] only.

Introducing (2.95) in (2.89b) and assuming zero initial conditions for the displacement

quantities, we obtain (also note that e.,p ~,.,p = 0):


\

Multiplying throughout by (K 2hG'• + 00m1s2), we obtain:

(2.101)

Inverting (2.101) into the time domain for zero initial conditions, we obtain:

ft (0) (0) (0) (0) (0) (0) (0) (0)

0 D[t- T]V3, ,i,ipp[T]dT + D[OJV3, upµ [t] - (L11 V3, 11 [t] + L22 V3, 22 [t] + 2L12V3, 12 [t]) (2.102a)

2 (0) (0) (0) (0) (0) +--2 C3[0](L11 V3 11 [t] + L22V3 22 [t] + 2L12V3 12 [t]) aa 3K ' ' ' ',,,.,


(0) (0) (0) (0) (0) (0) + C2(O)(l11V3,11[t] + l22V3, 22[t] + 2l12V3, 12 [t])]

where we have defined,

D[O] = .K_( E(O) ) 12 1 - v2[0]

(2.102b)

C [t] = 2-1 .lL.1. E = .lL2 ..1. G { 2 ( -- )} 2 { ( - )} 3 . 8 s G'°(1 - (V°)2) 4 s G'.(1 - -v» (2.102c)

with h2 G[O]

C3 [O] = 4 G'[0](1 - v(O]) (see Eqn. 2.94b)

(2.102d)

Equation (2.102) represents the second equation governing the stability of the transversely

isotropic plate subject to constant inplane edge loads. We observe that (2.102) is expressed (0)

in terms of the transverse displacement quantity (i.e., V3) only. Thus for zero initial conditions,

the coupled form of the equations (2.86) governing the stability of a transversely isotropic plate

(transversal isotropy being a special case of orthotropy) can be recast into two independent

equations, (a) one governing the basic state of stress, i.e., the interior solution, (2.100), and (b)

the other one governing the boundary layer solution, (2.102). We also observe that setting the

time derivatives of material properties to zero, we obtain the corresponding elasticity


equations from (2.100) and (2.102) which coincide with those obtained by Librescu [8, 9) (see

also Librescu and Reddy (15]).

2.6 Determination of the Transverse Shear Correction

Factor K2

Following a procedure similar to that described in Sec. 2.5, we can derive the counter-

parts of (2.100) and (2.102) for a transversely isotropic, viscoelastic plate in the framework of

the TTSD. These equations were obtained by Librescu and Reddy (15) for the elastic case. (1)

Here, they point out that in the absence of inplane surface tractions (i.e., P. = 0) the de-coupled

system of equations for the FSDT coincide with those for the TTSD when the transverse shear

correction factor K2 --+ · ~ in the FSDT and the transverse normal stress is neglected in the

TTSD (i.e., c5A = '58 = 0). The value of K2 does not change for the viscoelastic case as it is a

constant.


CHAPTER 3. DETERMINATION OF MA TE RIAL

PROPERTIES OF A FIBER-REINFORCED

COMPOSITE WITH VISCOELASTIC MATERIAL

BEHAVIOR

3.1 Introduction

In order to render explicitly the constitutive law for an anisotropic viscoelastic body in

a 3-D state of stress {see equation {2.5)), we require the relaxation moduli (E;imn) as functions

of time. Towards this end we seek a suitable micromechanical model that predicts the overall

behavior {i.e., effective properties E;imnl of the unidirectional fiber-reinforced composite in

terms of the properties of its constituents (i.e., fiber and matrix).

Wilson [3] determines the effective properties by using the Halpin-Tsai equations, them-

selves formulated by a strength of materials approach. Other, more rigorous methods have

CHAPTER 3. DETERMINATION OF MATERIAL PROPERTIES OF A FIBER-REINFORCED COMPOSITE WITH VISCOELASTIC MATERIAL BEHAVIOR 51

been developed by Aboudi (see [23]. [24], [25]) and Hashin (see Hashin and Rosen [26]). The

model of Aboudi [24] was adopted in the treatment of the problem. In the following section,

a brief description of this model is presented (see also Appendix[B]).

3.2 Use of Micromechanical Model to Determine Material

Properties

By using the results presented in Appendix [BJ along with the elastic-viscoelastic cor-

respondence principle (hereafter C.P.), we obtain the relevant micromechanical equations

pertaining to a viscoelastic fiber-reinforced composite. Using Eqn. (B-10) and the C.P. for a

viscoelastic transversely isotropic matrix yields

E-'(m) _ E-'(m) + 4K-(m)(-*(m))2 1111- A VA

E__.(m) _ E__.(m) _ 2K-(m)-*(m) 1122 - . 1133 - v A

-'(m) --'(m) -(m) O.SEr E2222 = K + _ *(m)

--'(m) _ ___.(m) E1212 - GA

1 + vr

-'(m) -'(m) --'(m) E2222 - E2233 E2323 = 2

(3.1a)

(3.1b)

(3.1c)

(3.1d)

(3.1e)

(3.1 t)


The relevant expressions for viscoelastic, transversely isotropic fibers are obtained from (3.1)

by replacing the superscript (m) with (f). Now the resulting constitutive law for an orthotropic

body relating average stresses (;,ii) to average strains (e;1) are from [24]. After using the C.P.

for an orthotropic viscoelastic body, we obtain the relevant constitutive law in the L.T. domain.

These relations write as,

(3.2a)

(3.2b)

(3.2c)

(3.2d)

(3.2e)

(3.21)

where (see [26] for the elastic counterpart)

(3.3a)

(3.3b)

(3.3c)

(3.3d)

(3.3e)


(3.3f)

(3.3g)

(3.3h)

(3.3i)

We have the following relations for the terms appearing in (3.3):

(3.4a)

(3.4b)

-,,• _.(f) _,,, _.(f) _,,, . h2 _.(m) _,,, 11 _.(m) _,,, Q 1 = V11(E2233 T1 + E2222Tg) - vdh E2233 Ts+ /E2222 Tg)

1 2

h1 _.(m) _,,, 12 _,,(m) _,,, --(m) _,,, -'(m) _,,, - V21(~E2233 T1 + f;E2222 T13) + vdE2233 Ts+ E2222 Td

(3.4c)

(3.4d)

(3.4e)

(3.4f)

CHAPTER 3. DETERMINATION OF MATE.RIAL PROPERTIES OF A FIBER-REINFORCED COMPOSITE WITH VISCOELASTIC MATERIAL BEHAVIOR 54

In addition Q";, o;. Q; can be obtained from o; (given by 3.3a) by replacing T; in the latter by

T;+1• Ti+2• Ti+3• respectively. Following the same procedure, we obtain O';, o';, Q'~. from Q'~.

(given by 3.3b) and O";, O";, Q"~. from Q"~. (given by 3.3c).

We have the following relations for the terms appearing in (3.4):

(3.Sa)

(3.Sb)

(3.Sc)

(3.Sd)

(3.Se)

(3.5/)

(3.Sg)

_.. 1 _... _... _... ._...... _... --- _.... ---Ta= -:::;-(A2 A6 A7 + A3 [A4 A9 - A5 A7 ]) (3.Sh)

D

(3.Si)

(3.Sj)


(3.Sk)

(3.5/)

(3.Sm)

(3.Sn)

(3.So)

(3.Sp)

where (see [24)),

....... ...... ...... 0 A1 A2 A3 ...... ....... ......

....... A4 0 As A5 D = det ....... ...... -A? Aa Ag 0

(3.6)

...... ...... ....... A10 A11 0 A12

and

(3.7a)

(3.?b)

(3.7c)


- -(m) h1 -(t) (3.7d) A4 = E2222(h) + E2222 2

- -"(t) As= E2233 (3.7e)

- -'(m) 12 (3. 7f) A5 = E2233(/) 1

- -'(t) A1 = E2233 (3.7g)

- _.(m) h2 Aa = E2233(f1;) (3.7h)

- -"(t) -'(m) 11 (3.7i) Ag= E2222 + E2222(/) 2

- -"(m) h1 A10 = E2233(f1;) (3.7j)

- -'(m) 12 A12 = E2222(J;" + 1) (3.7k)

3.2.1 Use of the 3-parameter solid (or standard linear solid) to model

viscoelastic material behavior

The 3-parameter solid (see Flugge [12]) can be viewed as a Kelvin element and a spring

in series (see Fig. 3). The differential equation of the material behavior (e.g., in a state of

uniaxial tension) writes as:

(3.8a)


where

(3.Bb)

By taking the L.T. of (3.8) and introducing (2.3) and (2.4), we obtain the relaxation modulus,

E(t), as:

Re-writing (3.9), we have,

(0) (1) <2> E(t) = E +Ee-Et

where we have defined,

<2> 6 1 E =-

P1

(3.9)

(3.10a)

(3.10b)

(3.10c)

(3.10d)

Henceforth, we shall consider the form of the relaxation modulus given by (3.10a). Repres-

enting the material properties of the transversely isotropic matrix (i.e., the 5 independent

constants E},,m>, v)f>, GAm>, E~m>, v'f'» ) as a 3-parameter solid, we write,

(3.11a)

(3.11b)


(3.11c)

(3.11d)

(3.11e)

Upon taking the Carson transform of (3.11), we obtain,

(3.12a)

(3.12b)

(3.12c)

(3.12d)

_. (0) (1tm) G (m) = G (m) + GA s

A A (2) s + G1m)

(3.12e)

Now using the C.P. in (B.11) for the transversely isotropic matrix yields

(3.13)

For an isotropic matrix, we make use of the following simplifications:


-"(m) _ _.(m) _ -"(m) EA -Er -E

-*(m) -*(m) -*(m) vA = vr = v

(3.14a)

(3.14b)

(3.14c)

The relevant equations pertaining to the transversely isotropic or isotropic fibers can be ob-

tained by replacing the superscript (m) with (f) in (3.11), (3.12), (3.13) and (3.14).

Now introducing (3.12), (3.13) [and (3.14) if the constituents are isotropic] into (3.1), we

obtain the relevant Carson transformed relaxation moduli of the constituents, i.e., E';j<;;~ ahd

E';j~n· Then upon introducing these along with (3.4), (3.5), (3.6), (3.7) into (3.3), we obtain the

Carson transformed relaxation moduli of the orthotropic fiber-matrix composite, i.e., E';jmn[s]. Th · · E";imn[s] · h t' d · bt · th I t' d I' . en upon inverting s into t e 1me omam, we o · am e re axa ion mo u 1, 1.e.,

E;'.Jmn[t] for the orthotropic composite. At this stage, it is worth noting that for square fibers (I.e.,

h1 =11 ) arranged at equal spacing (i.e., h2 = 12 ) we have the following result [after comparing

3.3(d) with 3.3(f), 3.3(g) with 3.3(h) , 3.3(b) with 3.3(c) and using the fact that continuous func-

tions possess unique inverse Laplace transforms]:

(3.15a)

(3.15b)

(3.15c)

Thus the 9 independent constants of the orthotropic composite reduce to 6 independent con-

stants in the case of equally-spaced square fibers.

The material properties for the isotropic, elastic, boron fibers were considered from

Aboudi et al. [27] while the properties of the isotropic, visoelastic, epoxy were taken from

Mohlenpah et al. (28] (see fig. 4) and Schapery [10] (see fig. 1).


Fiber properties:

The properties of the isotropic, elastic, boron fibers considered from Aboudi et al. [27]

are as follows:

K(f) = 33.2 x 106p.s.i.

Using the above properties, we obtain the following:

(3.16a)

(f)( ;.<f) 2G (f)) E(f) - G 3 + = 6 426 107 . - . X p.S.I.

A.(f) + G(f) (3.16b)

(3.16c)

Matrix properties

Using the properties for the isotropic, visoelastic matrix in Figures 4 and 1, we can rep-

resent them as a 3-parameter solid in the following manner:

-3 E(m)[t] = 0.8 x 105 + 0.18 X 106e-0.4115x 10 t for 0 :s; t :s; 2000 hrs

-2 v(m)[t] = 0.372 - 0.007e-0.2403x 10 t for 0 :s; t :s; 2000 hrs

where the time t is in minutes.

(3.17a)

(3.17b)


The material properties given by (3.16) and (3.17) are shown in Table 1, which is used in

Eqn. (3.12) (and its counterpart for the fibers) in order to obtain the relaxation moduli E;jmn(t)

for the orthotropic plate.

3.3 Numerical Inversion of the Laplace Transform

In order to obtain the time-dependent relaxation moduli, E;jmn[t] , for the viscoelastic

orthotropic composite plate, we must invert the corresponding Laplace transformed moduli

A number of approximate techniques for Laplace transform inversions are discussed by

Cost [29] and Swanson [30]. However, a lot of these methods are applicable only for specific

problems which impose restrictions on the nature of the desired inverse L.T. For example,

Schapery's direct method assumes that the function f(t) (which is the desired inverse L.T. of

f[s] ) has a linear variation with log t. This means that i is proportional to+· This is an un-

necessary restriction on the nature of f[t] and, moreover, since we desire an exponential var-

iation of f[t] with time (t), it is inconsistent to use this method of L.T. inversion for our problem.

A second example is the general inversion formula developed by Post and Widder (see Cost

[29] and Bellman [31]). This represents the general case of a set of inversion methods that

have been considered, namely, the methods of Alfrey, ter Haar and Schapery (see [29]).

Widder's general inversion formula is written as (see {29)),

( -1)n sn+1 dn -t[t] = lim [ (f[s])J I -n s=nlt

n-+oo n. ds (3.18)


Table 1. Material properties of boron fibers and epoxy matrix.

(0) (1) (2) (O) (1) (2) E E E y y y

p. s. i. p.s.i. (mins- 1 ) (mins- 1 )

Boron fibers 0.6426 0 0 0.1990 0 0 (elastic) x 10 1

Epoxy matrix 0.8000 0.1800 0. 4115 0. 3720 0.0070 0.2403 at temp = x 10 5 x 10' x 10- 3 x 10- 2

25° c (viscoelastic)

V,= 0.45

As shown by Bellman [31). this method relies on the fact that the derivatives of f(s] are eval-

uated accurately since small inaccuracies in the derivatives produce large errors in f [t], es-

pecially for t - 0.

A rigorous treatment of the problem of LT. inversion is given by Bellman and Kalaba [31)

and this method, which was chosen for our problem, has been effectively used by Swanson

[30] for dynamic viscoelastic problems. In this method, known as Bellman's technique, the

definition of the LT. is used to invert the LT. by means of a Gaussian quadrature using

orthogonal polynomials. Due to their excellent convergence properties, Legendre

polynomials were mainly used by Bellman [31). It was observed that for the order of the

polynominal N = 10, the convergence had been attained to provide accurate results. For N

= 15, the results were almost the same as for N = 10. A brief description of Bellman's

technique is provided in Appendix C.

3.3.1 Use of Bellman's Technique to Obtain E.. (t) 11mn

Using (C.7) with g[x] = E;jmn[ - In x], we can write,

(3.19)

Thus, we can evaluate E;imn[t] for values oft given by,

(3.20)

where X1r are the roots of the Nth order shifted Legendre polynominal P~[r] and the coefficients

of MP" are tabulated in [31] for N ranging from 3 to 15.

Using N = 15, we can obtain values of E;imn[t] for 15 discrete values oft. Using (3.20) and

the tabulated values of X; from [31). we see that we can obtain values of E;jmn[t] in the interval

CHAPTER 3. DETERMINATION OF MATERIAL PROPERTIES OF A FIBER-REINFORCED. COMPOSITE WITH VISCOELASTIC MATERIAL BEHAVIOR 64

0 < t < 5 minutes. In order to obtain E;imn[t] for larger values of time, we make use of the

following property of the LT. (known as the change of time scale property), i.e.,

If

i[s] = .P{t[t]}

then

i[~a] = .P{t[at]} (3.21)

Using (3.21) in conjunction with (3.19), we can write,

p, k = 1,2, ... N

Thus by changing the value of a, we obtain values of f[t] for larger times [t].

The results obtained for E;imn[t] versus time [t] are shown in Figures 5-9 for the case of

equally-spaced square fibers. When these plots are fitted by an exponential series of the form

(3.10(a)) (which represents a 3-parameter solid}, we obtain the following results:

E1111 = 0.2903 x 108 + 0.2500 x 106e- <0·3746x 10- 3)t

--3 E2222 = E3333 = 0.3212 x 106 + 0.6769 x 106 e - <0·3966x 10 )t

-3 E1122 = E1133 = 0.1294 x 106 +0.2633x106e-(o.37s5x 10 )t

E2233 = 0.1304 x 106 + 0.2609 x 106 e - (0.3687x 1 o-")t

E1212 = E1313 = 0.6921 x 105 + 0.1548 x 106 e- (0.4356x 1o-3)t

E2323 = 0.5321 x 105 + 0.1194 x 106 e- (0.4365x 10-3)t

(3.22a)

(3.22b)

(3.22c)

(3.22d)

(3.22e)

(3.22/)


Equations (3.22) represent the desired time-dependent relaxation moduli for the orthotropic

plate with equally-spaced, square elastic fibers (boron) and a viscoelastic matrix (epoxy).


CHAPTER 4. SOLUTION OF THE ST ABILITY

PROBLEM

4.1 Definition of Stability

The equations governing the stability of viscoelastic composite plates in biaxial com-

pression, as derived in Chapter 2 (see eq. (2.64 and 2.71)), represent a system of linear

integro-differential equations which must be solved for the displacement field represented by (1) (0)

V~ (x"';t) and V3 (x"';t) in order to yield the stress solution of the problem. However, a stability

analysis does not require the explicit solution of the governing equations. Therefore, instead

of determining the response of the system to a given input (i.e., edge loads in this case), we

merely seek the conditions on the edge loads that would lead to instability of the system (re-

presented in the present case by the plate).

We define instability as the phenomenon characterized by displacements increasing

unboundedly as time unfolds. In doing so, we consider the stability (or instability) of a certain

equilibrium configuration of the system referred to as the undisturbed equilibrium state. In

addition, we also consider disturbed forms of motion, close to the undisturbed equilibrium

CHAPTER 4. SOLUTION OF THE STABILITY PROBLEM 67

state. We now define stable equilibrium (or stability) as that state in which small disturbances

yield small deviations from the undisturbed equilibrium state. If these deviations tend

asymptotically to zero as time unfolds, then the equilibrium configuration is known as

asymptotically stable. However, if these disturbances, no matter how small, cause a finite

deviation from the undisturbed state of equilibrium, then this undisturbed state of equilibrium

of the system is unstable (in the small, according to the Liapunov definition of stability for

dynamical systems). Thus, we consider the plate to be deformed by in plane edge loads which

are small enough such that the flat configuration of the plate is the only possible equilibrium

state, and this is a stable equilibrium state. If these edge loads are increased, the flat con-

figuration of the plate may become unstable, i.e., the plate may pass, under the effect of

negligibly small perturbations, to a new equilibrium state with a curved configuration.

Therefore, in order to determine the stability of the plate, we need to analyze the behavior of

the disturbed configuration, i.e., the stability equations (ref. Ambartsumian (32). Also ref.

Meirovitch [33) and Porter [34) for other equivalent definitions of stability for linear dynamical

systems).

4.2 Stability Analysis Using the Third Order Transverse

Shear Deformation Theory (TSDT)

It was noted in Chapter 2 that the solution of the equation governing the stability of the

plate requires the fulfillment of the boundary conditions given in sec. 2.3.3. To this end, the (1) (0)

following representation of the displacement field V.(xw;t) and V3(xw;t) which satisfies the

boundary conditions of the plate is postulated:


(1) 00 00

V1 = ,L ,L Amn cos[).mx1] sin[2nx2JfmnUJ m=1 n=1

(1) 00 00

V2 = I I Bmn sin[A.mx1] cos[lnx2JfmnUJ m=1 n=1

(0) 00 00

V3 = I L Cmn sin[A.mx1] sin[A.nx2Jfmn[tJ m=1 n=1

(4.1a)

(4.1b)

(4.1c)

where Am= ~~ , A.n = ~: and Amn• Bmn• Cmn are constants representing the amplitudes of the

displacement quantities. Now the L.T. of equations (2.64) and (2.71) are (2.63) and (2.70), re-

spectively. Thus, introducing (4.1) into (2.64) for the free index a= 1 in conjunction with Eqn.

(2.16) yields the following equation:

00 00 I I (Ymn[s]fmn + lmn[s]) cos[A.mx1] sin[..l.nx2] = 0 (4.2a) m=1 n=1

where,

and

(4.2c)


The equation corresponding to the free index rJ. = 2 can be obtained from equation (4.2) by

replacing the index 1 with 2 (and vice versa). We also note that repeated indices in equation

(4.2) do not imply a summation over their range. Examining (4.2) (and its counterpart for the

index rJ. = 2), we see that due to the orthogonality of the sine and cosine functions, we have

the following result:

(4.3)

where Ymn[s] and i;,,n[s] are polynomials with s as the variable. From the above, we observe

that theoretically it is possible to obtain fmn[t] as a sum of exponentials e•t by inverting (4.3)

into the time domain. Now for the initially undisturbed system, i.e., i;,,n[s] = 0, we obtain,

(4.4)

Thus, for non-trivial solutions of fmn[t] (i.e., imn[s] =!= 0) we have,

Ymn[s] = 0 (and its counterpart for the index rJ. = 2) (4.5)

Now introducing (4.1) into (2.71) for the case of uniform biaxial compression (i.e., (0) (0) (0)

L11 = .z1 1 h, L22 = .z22 h, L12 = 0) we obtain,

00 00 L L (Wmn[s]fmn[s] + Jmn[s]) sin[Am x1] cos[An x2] = 0 (4.6a) m=1 n=1

where

- 6 2_.. 2-- 2-- 2 2-- 2 Wmn[s] = Amn(3 h E1313,.lm) + Bmn(3 h E2323An) + Cmn(3 h E1313,.lm + 3 h E2323An)

+ Cmn(h[211A~ + £22A~]) + Cmn(c5o ph)s2 (4.6b)

and

(4.6c)


Now for an initially undisturbed body, i.e., J,,,n[s] = 0 we obtain (by using the orthogonality

property of the sine and cosine functions),

(4.7)

and for nontrivial solutions of fmn[t], we have,

(4.8)

Equations (4.5) (and its counterpart for index IX = 2) represent the first two equations governing

the stability of the plate, whereas equation (4.8) represents the third equation governing the

stability. This set of three equations represents a homogeneous system of equations in terms

of the unknown amplitudes Amn• Bmn• Cmn (which represent the eigenvector). Thus, using (4.5)

and (4.8) in conjunction with (4.2) and (4.6), we can write this system of homogeneous

equations in the following form:

where,

Z12 Z13

Z22 Z23

Z32 Z33 l [Amn] ,

~:: =0

CHAPTER 4. SOLUTION OF THE STABILllY PROBLEM

(4.9a)

(4.9b)

(4.9c)

(4.9d)

71

(4.9e)

(4.91)

(4.9h)

(4.9i)

(4.9J)

Now from {4.9a) we see that for non-trivial solutions of Amn• Bmno Cmn , we must have the fol-

lowing identity:

Equation (4.10) yields a characteristic equation of the form,

P~n[s] ---=0 Omn[s]

(4.10)

(4.11)

where P mn[s] and Omn[s] are polynomials in s. Thus, the zeros of equation (4.11) are deter-

mined by writing,

Pmn[s] = 0 (4.12)


Equation (4.12) is the characteristic equation of the system (i.e., the plate subjected to uniform

biaxial compression). The zeros of this equation, i.e., the roots s; of P mn[s] , are the

eigenvalues of the system, which are complex quantities in general.

Now we recall that the time dependency of the displacement field is given by fmn[t] which

has the general solution of the form e•1 • Thus, we see that the eigenvalues (s] of the system

decide the nature of fmn[t] and hence the stability of the system. Representing the eigenvalues

in the form s = a + ib, a and b being real numbers, we see that when a(= Re[s]) is greater

than zero .. fmn[t] becomes unbounded as time unfolds and hence the system becomes unsta-

ble. In general, we may have the following cases arising as a result of the nature of s:

1. (Re[s] = a) > O; (lm[s] = b) = 0

Thus, we have fmn[t] = e•1. Therefore, fmnUJ grows exponentially with time, and we have

instability by divergence.

2. (Re[s) = a) > 0 ; (lm[s] = b) =I= 0

Thus, we have fmn[t] = e•1eibt. Therefore, fmn[t] has an oscillatory growth with time and the

amplitude of oscillations is given by e•1• This leads to instability by flutter.

When a ~ 0, we have asymptotic stability or marginal stability according to whether a <

0 or a = 0, respectively. However, when we have a double root lying on the imaginary

axis, i.e., a = 0, the following cases will arise:

3. (Re[s] = a) = O; (I m [s] = b) = O; double root

Thus, we have fmn[t] = t. Therefore, f,,m[t] grows linearly with time, and we have diver-

gence instability.

4. {Re[s] = a) = O; (lm(s] = b) =I= O; double root

Thus, we have fmn[t] = feibt. Therefore, fmn[t] has an oscillatory growth with time and the

amplitude of oscillations is given by t. This leads to instability by flutter.

Instabilities of type (3) and (4) are also referred to as t-type instability and correspond to

the situation when a double root lies on the imaginary axis (ref. Porter (34]).


Thus, we see that the stability problem is reduced to one in which we examine the nature

of the zeros of the characteristic equation of the system given by (4.12). The coefficients

of the characteristic polynominal, P mn[s] , in equation (4.12) can be varied by suitably

varying the in plane edge loads Q'11 and Q'22 in order to yield the stability boundaries of the

system.

Initial conditions

In the preceding analysis, we have assumed, in Eqs. (4.2a) and (4.6a), zero initial con-

ditions (i;,,r;[s] = ]mn[s] = 0) for an initially undisturbed system. For non-zero initial conditions,

the counterparts of Eqs. (4.4) and (4.7) would write as,

In the above equations, the terms 1/Ymn and 1/Wmn represent the LT. of the response fmn[tJ of

the system (i.e., fmn[s]) for a unit impulse input (i.e., the Dirac Delta function as defined in

Appendix [A]). We also note that the response of a linear system (of the type considered

above) to an arbitrary input may be represented as a superposition of impulse responses (see

Meirovitch [35]). In view of this fact, we may conclude that the stability of the system is gov-

erned by whether its impulse response (i.e., .2'-1[1/YmnJ and .2'-1[1/WmnJ in the present case)

remains bounded as t--+ oo (ref. Porter (34]). Therefore, the initial conditions, represented by

- -lmn and Jmn• are not required in a stability analysis of a linear dynamical system and may be

assumed to be zero without any loss of generality. However, it is obvious that the initial con-

ditions will play an important role in a response analysis of the problem.


4.3 Stability Analysis Using a First Order Transverse

Shear Deformation Theory (FSDT)

Examining the equations governing the stability of the plate in the framework of the FSDT

(i.e., equation (2.85)), we see that this represents a sixth-order governing equation system, the

solution of which requires that three boundary conditions be prescribed at each edge of the

plate. Thus, for a hinged-free (in the normal direction) set .of boundary conditions, we may

represent the boundary conditions in exactly the same form as done for the TTSD in Sec. 2.3.3.

Following a procedure analogous to that in Sec. 4.2 and considering the LT. of equation

(2.85), we obtain the characteristic equation of the system in exactly the same form as given

by (4.12) but with different coefficients. These coefficients are determined by eqn. (4.10) in

which, for the FSDT, we have,

(4.13a)

(4.13b)

(4.13c)

(4.13d)

(4.13e)

(4.131)

(4.13g)

(4.13h)


(4.13i)

Proceeding in exactly the same manner as described in Sec. 4.2, we obtain the stability

boundaries of the orthotropic plate subjected to uniform in plane edge loads.

4.4 Stability Analysis Using the Equations Representing

the Interior Solution in the Framework of the FSDT

As was shown in Sec. 2.5 (see Librescu [8, 22, 9)), the coupled equations governing the

stability of a transversely isotropic plate (i.e., (2.86)) can be recast into two independent

equations, (2.100) and (2.102). We now solve (2.102) independent of (2.100) with the aim of

showing, through numerical comparison with the solution obtained in Sec. 4.3, that (2.102)

represents the interior solution of stability and is thus by itself sufficient to analyze the stability

of transversely isotropic plates. To this end, we consider the following representation for the (0)

transverse displacement V3 given as:

(0) 00 00

V3 [t] = L L sin[..l.m x1] sin[A.nx2] fmnUJ (4.14) m=1 n=1

In equation (4.14) A.m and A.n are as defined in Sec. 4.2, while fmn[tJ represents the time de-<O> (0)

pendent amplitude of V3. The representation of V3 given by (4.14) satisfies the two boundary

conditions, given below, for each edge of the plate:

(0) (0) V3=0; V3, 11=0, at x2 =0,L2


(0) (0) V3 = 0 ; V3, 22 = 0, at x1 = 0, L1

By introducing (4.14) into the L.T. of equation (2.102) [i.e., (2.101)], we obtain:

00 00 L L (Rmn[s] f mn[s] + Omn[s]) sin[Am x1] sin[An x2] = 0 (4.15a) m=1 n=1

where

(4.15b)

and,

(4.15c)

The terms O', C' and c· appearing in equation (4.15) are defined in equation (2.102). Using the

orthogonality property of the sine and cosine functions and following a procedure similar to

that in Sec. 4.2, we obtain the characteristic equation of the initially undisturbed system as,

(4.16)

Now proceeding in the same manner as described in Sec. 4.2, we can establish the stability

boundaries of the transversely isotropic plate subject to uniform in plane edge loads.


4.5 Stability of a Transversely Isotropic Viscoelastic Plate

Undergoing Cylindrical Bending

Consider a transversely isotropic plate with an infinitely large aspect ratio (i.e.,

L2/t.1 -+ oo ) and simply supported along its edges x1 = 0, L1• The plate is subjected to a uni-

form compressive force applied along its edges. Thus, the plate undergoes cylindrical bend-

ing and in this case the operator -88 -+ 0 . In the following developments, we analyze the X2

stability of the plate in the framework of the FSDT by making use of the correspondence

principle. To this end, we consider the elastic counterpart of equation (2.102) in conjunction

with the condition that 88 -+ 0 , and we obtain, X2

.. .. (0) (0) (0) 2 (0) (0) (0) 2 (0)

OV31111 [t] - L11 V3 11 [t] +--2 C3L11 V3 1111 [t] + c5omo V3 [OJ - - 2-C3 V3 11 [t] ' ' 3k ' 3k '

(4.17)

In equation (4.17), D, C2, C3 are the elastic counterparts (defined at t = 0) of the quantities D[t].

C2[t], C3[t] defined in equation (2.102).

Now we represent the engineering material constants given in (4.17) in terms of the

tensorial constants by making use of equation (2.87). Thus, we obtain the following results for

a transversely isotropic body:

- E E1111 = 2 1-v (4.18a)

G' = E1313 (4.18b)


G= E 2(1 + v)

(4.18c)

Introducing Eqs. (4.18) into eqs. (2.102b,c,d), evaluated at t = 0, we obtain,

(4.19a)

(4.19b)

Now introducing (4.19) into (4.17), neglecting the effect of rotary and transverse inertias (i.e.,

<5 0 = J0 = 0), and postulating that K2 = 5/6, we obtain:

_ I 3 (O) (O) (O) I 2 E- (O) (O) 1 1 1111

E111112 V3,1111 - L11 V3,11 + ffi E1313 L11V3,1111=0 (4.20)

Now we consider the constitutive equations for a transversely isotropic plate undergoing cy-

lindrical bending and having viscoelastic properties in shear only. These equations write as,

-a11[t] = E1111e11[t] (4.21a)

(4.21b)

Making use of the C.P. in equation (4.20) in conjunction with (4.21) yields

E'1111 2<o> (0) _ h3 (0) <o> <o> 10 sE1313 h L11 v3,1111 - E111112V3,1111 + L11V3,1111=0 (4.22)

Introducing (2.50) into (4.22), we can write,

Inverting (4,23) into the time domain, we obtain,


h2 (0) - f,t. (0) h2 (0) - (0) -10 L11E1111 0 F1313[t- -r]V3, 1111[r]d-r -10 L11E1111F1313[0]V3,1111

- h3 (0) (0) (0) - E111112 V3, 1111 + L11V3,11=0

(4.24)

Equation (4.24) is the equation governing the stability of a viscoelastic transversely isotropic

plate of infinite aspect ratio, exhibiting viscoelastic properties in transverse shear only.

Solution of stability equation:

Now we consider the following representation of the creep compliance in transverse

shear which corresponds to a 3-Parameter solid:

(1) (2) <3> F [t] F F - F1313I

1313 = 1313 - 1313e (4.25)

We also assume the following representation of the transverse displacement (valid for the

simply supported boundary conditions at x1 = 0, L1 ):

(0) 00

V3 = I fm[tJ sin[A.m x1] , (4.26) m=1

Introducing (4.25) and (4.26) into the LT. of (4.24), we obtain,

(4.27)

Invoking the orthogonality property of the sine function along with the argument of non-trivial

solutions (i.e., fm "I= 0) in (4.27) yields the following characteristic equation of the system:


(4.28)

Equation (4.28) is a linear equation in the unknown s (i.e., the eigenvalue), therefore pas-

sessing only real solutions (since the coefficients are real). Thus, we conclude that instability

could occur by divergence only. Having in view the fact that for divergence instability, s = 0

in (4.28), we obtain,

(4.29)

Equation (4.29) gives us the value of the applied inplane edge load when divergence occurs.

It is easily seen that the lowest value of this load corresponds to m = 1 for which we obtain,

(4.30)

(3)

Considering (4.28) with F1313 = 0 (i.e., the elastic case), it is easy to verify that the static

buckling load for the elastic transversely isotropic plate would be,

( 4.31)

(1) (2) (3)

Since F1313 , F1313 and F1313 in (4.25) are all greater than zero, we see, upon comparing (4.30) and

(4.31), that the instability (divergence) load for the viscoelastic case is lower than that for the

elastic case. Equations (4.30) and (4.31) coincide with the solutions obtained by Malmeister

et al. [5] (where the stability problem is solved as a special case of the response problem


using a time domain analysis). We notice that for the classical Kirchhoff theory F1313 --. 0 and

hence equations (4.30) and (4.31) yield the same divergence loads for the elastic and

viscoelastic case, which is the Euler load for a wide column given by,

(4.32)

This evident result is a direct consequence of the assumption implied in eqs. (4.21) which

considers the viscoelasticity in the transverse shear direction only.

Alternative method of deriving the equation governing the stability of viscoelastic transversely

isotropic plate undergoing cylindrical bending

In the absence of in plane surface loads and rotary inertia effects, the first order moments

of the first two equations of motion (i.e., (2.29)) when specialized for the case of cylindrical

bending (yielding _!}_--. 0 ) results in: UX2

(0) (0) L11 - L13 = 0 (4.33a)

(1) (0) L21,1 - L23 = 0 (4.33b)

In the absence of transverse inertia effects (i.e., '50 = 0). the LT. of the third equatjon of motion

(2.62) when specialized for cylindrical bending writes as,

(1) (0) (0) L13,1 + L11V3,11=0 (4.34)

Introducing (4.33a) into (4.34), we obtain,

(1) (0) (0) L11,11 + L11V3,11=0 (4.35)


Equation (4.35) is the desired equation of motion of the plate undergoing cylindrical bending. (1) (0)

We shall now solve for L11 ,11 in terms of V3 in order to obtain the governing equation expressed

in terms of the displacement field. To this end, we postulate the following representation for

the transverse shear stress u 13 in the framework of the first order transverse shear stress

theory (S.T.) [which yields identical results as the FSDT (see Librescu [8, 9])]:

(4.36a)

1 2 h 2 t[z] == -[(x3) - -]

2 4 (4.36b)

The constitutive equations for the 2-D elastic body (i.e., the elastic counterpart of (4.21)) write

as follows:

"'11 e11 =-~-.-

E1111

(4.37a)

(4.37b)

We now see that in this case we must consider the following functional dependence for the

displacement field:

(4.38a)

(4.38b)

The substitution of the strain-displacement relation (4.12e) considered in conjunction with

(4.38b) and (4.36) into (4.37b) yields,

(4.39)

Integrating (4.39) across the interval [O, x3) in conjunction with (4.38), we obtain:


(4.40)

Introducing (4.40) into (4.37a) in conjunction with (2.12a) results in,

Replacement of (4.41) into (2.25a) in conjunction with (4.38) yields,

(4.42)

Now in order to obtain the equation governing the stability in terms of the displacement.field,

we need to express t/J.1 in terms of the displacement field. To this end, by introducing (4.36)

into (2.27b), we obtain

which yields,

(4.43)

Upon introducing (4.33) and (4.35) into (4.43), we obtain,

(4.44)

Substitution of (4.44) into (4.42) results in

(4.45)

CHAPTER 4. SOLUTION OF THE STABILITY PROBLEM 84 '

Upon introducing (4.45) into (4.35), we obtain the desired governing equation of an elastic

transversely isotropic plate undergoing cylindrical bending. This equation coincides with Eqn.

(4.20), i.e., the one obtained by considering the single equation which represents the interior

solution (i.e., (2.102)). From here onwards, the procedure for obtaining the viscoelastic

counterpart of the equations governing the stability is the same as that used at the beginning

of this section.


CHAPTER 5. NUMERICAL RESULTS AND

CONCLUSIONS

5.1 Numerical Results

The stability boundary was obtained by solving the characteristic polynomial (e.g., Eqn.

(4.12)) using the IMSL subroutine ZPOLR. The following problems were considered in this

context:

Problem (1). The TTSD (also referred to as the higher order shear deformation theory -

HSDT) represented by Eqn. (4.12).

Problem (2). The FSDT counterpart of Eqn. (4.12).

Problem (3). The "single equation" representing the interior solution of stability in the

framework of the FSDT, represented by (4.15b).

CHAPTER 5. NUMERICAL RESULTS AND CONCLUSIONS 86

Problems (1) and (2) above were considered for an orthotropic viscoelastic plate and were

specialized for the orthotropic elastic, isotropic viscoelastic, and isotropic elastic cases.

Problem (3) was considered for an isotropic viscoelastic plate and was specialized for

its elastic counterpart.

All cases above were considered so as to obtain an "exact" dynamic solution, i.e., for

c5A = c58 = c5c = c50 = 1 where c58, c5c, c50 are tracers identifying the dynamic effect of a33 , rotary

inertia and transverse inertia, respectively, and c5A is a tracer identifying the overall (i.e., static

and dynamic) effect of a33 . It was observed that the inclusion or exclusion of the inertia terms

does not affect the results.

All elastic counterparts were solved as special cases of the corresponding viscoelastic

problem by considering the initial value theorem (ref. Appendix [A]) for the Laplace trans-

formed material properties appearing in Eqns. (4.9), (4.13) and (4.15b). Comparisons were

made for the orthotropic elastic plate with Ambartsumian [32] and Ashton and Whitney [36]

(ref. Figures 38-41) and the results show an excellent agreement.

The results associated with the classical Kirchhoff theory were obtained as a special

case of the FSDT by considering K 2 ~ = . The results obtained in this study are not universal

since a non-dimensional analysis was not possible due to the inherent complexity of the

problem.

Discussion of Figs. 13-41

The stability boundaries for the following cases are shown in Figs.13-41.

(i) Orthotropic, viscoelastic, flat plate

(ii) Orthotropic, elastic, flat plate

(iii) Isotropic, viscoelastic, flat plate

(iv) Isotropic, elastic, flat plate.


Cases (i)-(iv) are considered for thick plates (L/H = 4.8) as well as thin plates (L/H = 24). In

addition, the following sub-cases were considered:

(1) Biaxial compression: For this case, the aspect ratio (A.R. = L1/L2 ) of the plate was

taken as unity. The values of the in plane, normal edge loads !!11

versus !Zzz are plotted to obtain the stability boundaries.

(2) Uniaxial compression: In this case, the aspect ratio, A.R., was varied and the corre-

sponding value of !!11 was plotted in order to obtain the stability

Fig. 13:

boundaries. The inplane normal edge load, !Z11 , is applied in the

direction of the fibers, at the edges X1 = 0, L1.

For all plots shown, M and N denote the mode numbers in the

X1 and x2 direction, respectively (see Eqn. (4.1)). It was observed

that for biaxial compression, the stability boundaries corre-

sponding to M = 1 were the lowest ones, whereas for uniaxial

compression, those corresponding to N = 1 were the lowest ones.

Therefore, in each of these two sub-cases, only the lowest sta-

bility boundaries were displayed. For all the cases, unless oth-

erwise indicated, instability occurs by divergence only. Flutter

boundaries are indicated on the figures. For uniaxial com-

pression cases, flutter instability occurs to the right of the arrow

indicated in the figures.

This plot depicts the stability boundaries for the case of biaxial com-

pression of an orthotropic, viscoelastic, thick plate (L/H = 4.8). The

solid line corresponds to the result obtained in the framework of the

HSDT with <5A = 1, whereas the dotted line corresponds to its FSDT

counterpart. It is observed that the inclusion of the transverse normal

stress ( o-33) in the HSDT has a beneficial effect on the stability by

yielding higher stability boundaries as compared to the FSDT counter-


Fig. 14:

Fig. 15:

Fig. 16:

Figs. 17, 18:

Figs. 19, 20:

parts. Moreover, the inclusion of o-33 could yield a change in the char-

acter of the instability boundaries, i.e., the conversion of flutter (for

FSDT) into divergence (for HSDT) instability boundaries.

The results shown here are for a similar case as those shown in Fig.

13, except that t5A = 0. Thus, we observe that when the effect of o-33 is

neglected in the HSDT, the stability boundaries are lowered and coin-

cide with those obtained in the framework of the FSDT.

Here the results obtained for uniaxial compression of an orthotropic,

viscoelastic, thick plate are displayed. On comparing the results ob-

tained for the HSOT with t5A = 1 (solid line) with those obtained for the

FSDT (dotted line), we observe once again that the former yields

higher stability boundaries. The undulating nature of these plots could

not be interpreted.

This plot shows that for the case considered in Fig. 15, when o-33 is

neglected in the HSOT (i.e., bA = 0), the stability boundaries for the

HSDT are lowered and coincide with those obtained in the framework

of the FSDT.

These are results for the elastic counterparts of those displayed in

Figs. 13 and 14. From these plots, we infer that o-33 has only a minimal

but beneficial effect on the stability boundaries as compared to its ef-

fect for tile viscoelastic case considered in Figs. 13 and 14.

Fig. 19 depicts the results obtained for biaxial compression of an

isotropic, elastic, thick plate in the framework of the HSDT with t5A = 1

(solid line), the FSDT (dotted line) and the FSDT "single equation"

(broken line). The results for the two FSDT cases coincide, whereas

those for the HSDT represent higher stability boundaries when com-

pared to their FSOT counterparts. However, for the critical stability

boundary (corresponding to M = N = 1), the three cases yield identical


Fig. 21:

Figs. 22, 23:

Fig. 24:

Fig. 25:

results. In Fig. 20, the only difference is that u33 = 0 for the HSDT, and

we observe that for all modes, the three cases yield identical stability

boundaries.

This plot shows the results for uniaxial loading of an isotropic,

viscoelastic, thick plate. The stability boundaries for the HSDT with

c5A = 1 are slightly higher than those for the FSDT and FSDT "single

equation". The latter two cases coincide.

Fig. 22 displays the results for the biaxial compression of an isotropic,

elastic, thick plate, and Fig. 23 shows the results for the corresponding

uniaxial compression case. In both these plots, we obs.erve that the

HSDT {with or without the effect of u 33), the FSDT, and the FSDT "single

equation" yield identical stability boundaries.

This plot shows the comparison between results obtained by using the

FSDT and the classical theory of plates for the biaxial compression of

an orthotropic, viscoelastic, thick plate. It is evident from this figure

that the classical theory of plates yields much higher stability bounda-

ries than the FSDT and, hence, we conclude that the transverse shear

deformation effects play a very important role for the case considered

in this plot.

Here the results for the uniaxial compression of an orthotropic,

viscoelastic, thick plate obtained in the framework of the FSDT are

compared with those obtained by using the classical theory.

It is evident from this plot that the classical theory over-predicts the

stability boundaries. However, we observe that for large aspect ratios

{l1/l2) the two theories tend to yield similar results. This is because

for large aspect ratios, the quantity, L1/H, increases which means that

in fact the plate tends to become a thin plate.


Fig. 26:

Figs. 27, 28:

Figs. 29, 30:

Figs. 31, 32, 33:

Figs. 34, 35, 36, 37:

Figs. 38, 39:

This plot compares the results obtained by the FSDT and the classical

theory for the biaxial compression of an orthotropic, elastic, thick plate.

It is evident that the classical theory over-predicts the stability bound-

aries.

In Fig. 27, comparison between the FSDT, FSDT "single equation", and

classical theory are shown for the biaxial compression of an isotropic,

viscoelastic, thick plate. Here it is observed that the two FSDT results

coincide with each other. We observe that the classical theory over-

predicts the stability boundaries. Fig. 28 shows the corresponding re-

sults for the case of uniaxial compression. It can be observed that for

low aspect ratios, the classical theory over-predicts the stability

boundary, but as the A.R. increases, the classical theory results tend

to coincide with the FSDT ones.

These results correspond to the elastic counterpart of the results ob-

tained in Figs. 27, 28, in which similar trends are observed. However,

for the biaxial compression case, the classical theory yields the same

critical boundary (i.e., for M = N = 1) as its FSDT counterpart.

These plots correspond to those obtained for the cases considered in

Figs. 15, 17 and 18. Here a thin plate (UH = 24) was considered. it

is observed that for the thin plate (orthotropic, elastic or viscoelastic),

the effect of 0'33 is minimal.

These results correspond to those obtained for the cases displayed in

Figs. 19, 21, 22, 23, but restricted here for a thin plate. The effect of

0'33 when included in the HSDT is negligible, as can be seen from these

figures.

In these plots, comparisons are made between the results for the

HSDT (where the effect of 0'33 is included) and the results obtained by

Ambartsumian [32] (for which the effect of 0'33 is neglected). However,


Figs. 40, 41:

for the case when the effect of 0'33 is included, the stability boundaries

are slightly higher. The cases considered in these figures pertain to

orthotropic, elastic, thick plates.

In these figures, the results obtained by the classical theory are com-

pared with those obtained by Ashton et al. [36). The cases considered

in these figures are those for an orthotropic, elastic, thin plate. It may

be seen that these results compare extremely well.

5.2 Conclusions

In this study, a stability analysis of orthotropic and transversely isotropic, viscoelastic

rectangular plates has been done. The equations governing the stability were derived by us•

ing a direct approach as well as the correspondence principle technique. The material prop-

erties were obtained by considering the micromechanical relations developed by Aboudi [24).

In the modeling of the problem, the Boltzmann hereditary constitutive law has been used for

a 3-D viscoelastic medium. The stability problem was analyzed in the Laplace transformed

space in order to determine the asymptotic stability behavior.

The special cases considered in the numerical applications allow one to conclude the

following:

1. The stability boundary determined for a viscoelastic plate is lower than that for its elastic

counterpart.

2. Incorporation of transverse shear deformation effects yields a stability boundary which is

much lower than in the case of its transversely rigid (classical) counterpart. This effect

is more pronounced in the case of an orthotropic viscoelastic plate than in its isotropic

counterpart, for which the effect is minimal.

CHAPTER 5. NUMERICAL RESULTS ANO CONCLUSIONS 92

3. The above conclusion remains valid in the case of their elastic counterparts.

4. The results show that a33 may influence the viscoelastic stability boundary in a strong and

beneficial way, especially in the case of an orthotropic viscoelastic plate.

5. In addition, we may conclude that transverse shear deformation effects are more pro-

nounced in viscoelastic plates than in their elastic counterparts. In this regard, we ob-

serve that for the special case when viscoelasticity is considered in the transverse shear

direction only, the exclusion of transverse shear deformation effects results in identical

solutions for the viscoelastic and elastic plates (see Sec. 4.5).

6. The analysis performed here allows one to obtain the nature of loss of stability, i.e., either

by divergence or by flutter. It was observed that for an isotropic, viscoelastic plate the

instability occurs by divergence only. However, for an orthotropic viscoelastic plate, the

instability may result both by divergence and by flutter.

7. It was observed that the boundary layer effect considered for an isotropic viscoelastic

plate has no effect on the stability boundary.

8. It is observed that for large aspect ratios (L1/L2) the stability boundary tends to coincide

with the classical Kirchhoff theory of plates.


n :c )>

~ m :u ~

z c 15: m :u n )> .-::u m Cll c ~ )> z 0 n 0 z n .-c Cll 0 z Cll

0 50 r----------------------------- 1.5

(i 4 ll lfj 20 I vg f/Q I ( 1Tlo'1)

Ve. - Poi.sson'.s ..-~ti.o for c.re.e.p test ( c.onsta.nt str1.ss.)

Vy - Poi.sson's Y"Qt L.o for ,.-e.l~ XQ.t Lon test ( c.onst~nt !It r"Cl..i.n)

Q.I -+ 5 hi-ft fo..c.t Or'

Figure 1. Variation of Poisson's ratio for the epoxy matrix.

e QI

- <II >-<II QI

- I'll c :s .. 0 0 CJ "ti c I'll QI

- I'll Q. - 0 a. :I >-I'll ~

c-i QI .. :I C

l Li:

CH

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CLU

SION

S 95

... w

0 lJJ

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96

!! ..-~~~~~~~~~~~~--;;:--~~~~~~--,s

·r Sd ·(1) 3

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~

"' i: -i:. ·--

E ......

:aC ·i:: - «I E

>o ~

Q

a. Ill Ill .c - ... Q

-Ill :I :; "O

Q

E

Ill bi c :I

~ - Q

c Q

; «I ·i::

~

..; Ill ... :I C> ~

97

0 0 ....-~~~~~~~~~~~~~~~~~~~~~~~~.......... 0 IO

.... 0 0 0 0 ....

I

/ 0 0 0

/ IO

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

QI

G)

.... CD

IO """

Cl)

.... (~sd)~

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-; ~ .. .. iJ II N

N

N

uj

GI -:; Q

. u Q

. 0 .. - 0 .c - .. 0 GI .c - .. 0 -~

GI Q

. 0 .. Q

.

ii ·;: QI

- IU == .,; QI .. ::i Cll

u:::

98

0 x )> 400000 ~ m ;u !-" z c 360000 s: m ;u n )> r ;u

300000 m (/) ...... c .~

~ <II ~

)> z ~ 260000 c

0 0 z 0 r c (/)

0 200000 :z (/)

160000

100000-+-~~~~~~~~~~~~~~~--~~~~~-.-~~.,--..,..--.--1

0 6000 10000 16000

Figure 6. Material property for the orthotropic plate • E1122 = E1133

I I I )

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 IO

0

IO

0 IO

..

"' "'

"' w

-

(15d) ~

CH

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0 0 0 0 0 -

0 0 0 0 -0 0 0 IO

0 -~ ::; N

u.i' 41 1;j Q

. (J

·c. Q

... 0 .c: t: Q

aJ .c: - ... Q

->-t: aJ Q

. Q

... Q

.

-;; ·;: aJ 1;j :i

...: 41 ... ::I C

l ii:

100

0 0

0 0

0 0

0 0

0 0

0 ·o

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

• ('I

0 CD

G ..

('I 0

CD G

('I

('I ('I -

--

-....

c1~dJ ·cma

CH

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0 0 0 IO

-0 0 0 0 -0 0 0 IO

0 -j ... ...

uj

II N

;;; u

j

QI

- ctl ii I.I

'Q.

0 .. 0 .c - .. 0 QI .c ... .. 0 -. >-t: Q

I Q

. 0 .. Q

.

'; ·;: QI 'la :s cci QI .. :: C

l Ii:

101

0 0 ....-~~~~~~~~~~~~~~~~~~~~~~....-~.....,.-

0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

CD (Q

•

('I 0

.... ....

.... -

.... (15d) ~

CH

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0 0

0 0

0 0

0 CID

(Q

0 0 0 0 "It IO

-0 0 0 0 -0 0 0 IO

0 -~ ... N

i:r QI -;; Q

. y

'Q.

0 .. 0 ; .. 0 QI .&:.

- .. 0 -~

... QI Q

, 0 .. Q

,

c;; ·;:: QI -;; == o; QI .. ::::J C

) u:::

102

E'

- _µ

2 -=

0-..J .....

l 1

1 1

x N

)(

t I

I I

T - ..,µ z -()_J

-.J

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.I --1

I C

l c :s c QI .C

l

ii u ·;: "'C .5 "& .5 QI

- Ill a: 0 ....

103

CJ) a:: LL.I co LL

N

><

D

D ·D

D

D D

D

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,,, )(

>< >< ·c ~

E

.5 rn ... QI ..a :;:::

- 0 - c QI E

QI C

l c <11 ... ... <C

..: .... QI ... :I C

l ii:

104

n :c )lo

~ ::0 ~

:z c s: m ::0 n )lo r-::0 m (JJ c ~ )lo :z c n 0 :z n r c (/)

0 :z (/)

..... 0 UI

X2

1.-- J, -•~ ~.~ 12 •I

h /3 X2 L-<I) X2 L i(2) I -(I) -(I)

I =I X3 /3= I 3 _ Y=I Y=2

-(2) - - - - - - - - -_(2) x2

h2

x,

X2

/3=2 r =I

I L _(I) I

X3 : /3=2 I y =2 I

Figure 12. Representative volume element for a fiber-matrix composite .

L -<2> X3

X3

z c s:: m ;u n )> r-;u m (/) c ~ (/)

)> z c 0 0 z 0 r-e (/)

0 z (/)

..... 0 OI

60000

,...... 40000 ·-' tJ) ll..

20000

0 0

' ', ·-~--- '-...

--- ...... :: .... -::.-... -- ....... ' ...... ' ,._ ---.:-::~

~,

'

n N .. 1 e N .. 2 B N .. 3 .. N .. 4 * ..

FLUTTER

/ ~FLUTTER

126000 SIG11 lpsi.)

Figure 13. Stability boundary for orthotropic viscoelastic plate; Uh = 4.8; biaxial compression; tJA = 1

M •1

HSDT FSDT

---260000

() :c )>

~ ::a ~

z c 3: m ::a 0 )> r ::a m (/) c ~ )> z CJ () 0 z () r c (/)

0 z (/)

,--.. ...... <[) Cl ..

~ ~

eoooo.....-Fi-~~~~~~~~~~~~~~-.-~~~~-.-~~~~~

40000

20000

0

FLUTTER

0 125000 SIG11 (p.i:,l)

n N ... 1 • N ... 2 a N .,,3 6 N .. 4 * "'

fLUTT ER

HSDT FSDT

260000

Figure 14. Stability boundary for orthotropic viscoelastic plate; Uh = 4.8; biaxial compression; DA = 0

(") :c )>

~ ::a ~

z c s: m ::a r; )> r-::0 m (I> c ~ )> z c (") 0 z (") r-e (I>

0 z (I>

..... 0 CD

0 ~ Q -;: i

200000 ~q;. ~

~" '<,~ «."' /

~ '<.."

160000

100000

60000

n M .. 1 .. M ·2 a M .. 3 • M .. 4 * Ill

.. ·-------· - ........ ~ ... _ -----·-~::~ ___ .. ..,. ___ ,.

--r- .=:~;:~::--~ FLVTTER

FLUTTER

HSDT FSDT

Q-i-,r-r-.--r-ir-r-r--r-ir-r-r--r-.--.-......-T-.--.--.--r--r-..--r-r-r-r--r-r-r-.--r-.--.-.....-...-..--.--.-..--1

0 2 4

A.A. 6 8

Figure 15. Stability boundary for orthotropic viscoelastic plate; Uh = 4.8; uniaxial compression; .5A = 1

0 :r )> "ti -I m ::a !-" z c s: m ::a ii )> r ::a m (/J c r -I (/J

)> z c 0 0 z 0 r c (/J

0 z (/J

... 0 co

.-----. .-• ·,/) LI..

;: ~

"'

250000 1:1 M •1 HSDT

M •2 " • M .. 3 FSDT • M •4

N ·2 200000 * .. «,,~

,~

,v\J '(

<c,~ "\"\

150000 ~\,\)

100000

FLUTTER

50000

12500 O-r-i,,..-r-.,-,,,..-.-.,-,-r--,-..--r-r--,-..--.-..--.-'1--r--.-..,-r-r-r-~.-.--.--.-.--.-~...-.-...---..--1

0 2 4

AA

6 8

Figure 16. Stability boundary for orthotropic viscoelastic plate; Uh = 4.8; uniaxial compression; DA = 0

(") ::c )>

~ ;o ~

z c s: m ;o n )> r ;o m en c ~ )> z C1 (") 0 z (") r-e (I)

0 z en

..... ..... 0

-----i 100000

0

- ... .,.._----- ...... __ _

n N •1 • N ·2 a N •3 • N •4 * 1!15

------------ ..

200000 SIG11 l psi)

Figure 17. Stability boundary for orthotropic elastic plate; Uh = 4.8; biaxial compression; OA = 1 or OA = 0

HSDT FSDT

400000

n :z: )>

~ m :0 ~

z c s: m ~ n )> ,... :0 m (/) c ,... -I (/)

)> z c n 0 z n ,... c (/)

0 z (/)

.... .... ....

400000 t:1 M .. 1 HSDT .. M .,,2

J;I .. M .. 3 FSDT I I • M .. 4 I

I

* .,,5 N .. 1 350000

,,--~

.-.J

£1L 300000 -:i= i

, 250000 , , ,

,,A

---200000

1600001-r--,-r-r-,--,--.-r-r-,--r-.-r--r-.,--,--.-,,....-r-.,--,---,-,,...-.,--,--,-,;--r--.-.--,-.,.--,-.-.--r-.-~

0 2 4

A.A. 6

Figure 18. Stability boundary for orthotropic elastic plate; Uh = 4.8; uniaxial compression; DA = 1 or DA = 0

8

0 0 0

I 0 0

...... ......

,... U

) (0

......

nff nff d' w

I

C')

C') ,...

I .....

f::: .....

b ....

I c

c r.n

en en

::c ilL

ILL. I

I I

I I

II

I " "°

I c 0

I .,...,,

-~

.... Cll

I '")

QI

.a.. ...

...__. c.

I 0

E

I 0

~ 0 u

0 ni

I 0

fl) ·;c

I ID

Ill

/ :a

I =

I /

..; II

f /

.c

I - _. Q

I

/ -

I I

Ill ii

/ u ;

I I

Cll Ill

/ Q

j 0 u

I )

/ Cll

·:;; u

/ ·c.

I I

0 ... 0

I /

-~ ...

" .2

0 ~

Ill

0 0

0 "O

c

0 0

:I 0

0 0

..Q

0 0

>-C

'I ,...

:!: :a

C1sd) ~ets

Ill

- (IJ ai .... Q

I ... :I C

'I u::

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112

n :c )>

~ m :u ~

z c s: m :u 0 )> r-:u m (/) c !::j (/)

)> z c n 0 z n r-c (/)

0 z (/)

.... .... (,)

,.--.,. ·,fi Ll..

~ i

20000 nN ... 1 HSDTC3 Eqsl • N ... 2 FSDTC3 Eqsl "'N ..,3 • N .. 4 FSDTC1 Eq.l

*

10000

o+--.--.-tt-.-----.--.~r---.--.---.~~-.-----.-~.....--.-~~..--~~~__J

0 60000 SIG11 (psl)

100000

Figure 20. Stability boundary for isotropic viscoelastic plate; Uh = 4.8; biaxial compression; c5A = 0

(') 120000 :c HSDTC3 Eqsl > uM •1 ~ "M .. 2 FSDTC3 Eqsl m

a M .. 3 :ti ~ .. M .. 4 FSDTC1 Eq.) z *M .. 5 "' c :5: m :ti n > 80000 r-:ti ::7-m JI [I) c il.

~ .........

(I) ;:: > (!J z m c (') 0 z n r- 40000 c (I) -5 z (I)

0-t-r-i~-r-.---.--r--r-r-i--r-r--,.-.--r--r--ir-r-r-.--,.--.-...-r-r--.--.--.--r--t

0 2 4 6

Figure 21. Stability boundary for isotropic viscoelastic plate; L/h = 4.8; uniaxial compression; OA = 1 or OA = 0

0 ::i::: )> "1J -I m :;u !-" z c :ii: m :;u n )> r :;u m

"' c !:; "' )> z c 0 0 z 0 r c "' 0 z "'

-::;-If) fl_

~ i

100000 HSDTC3 Eqs) n N .. 1

• N .. 2 FSDTC3 Eqsl "N .. 3

• N .. 4 FSDTC1 Eq.>

* ..

60000

o+---.---!S---..~--.----.:---.---,.---Til.--.----.-~...---r-~-.--...~-.----..~~~~~~

0 200000 SIG11 ( psL)

400000

Figure 22. Stability boundary for isotropic elastic plate; Uh = 4.8; biaxial compression; bA = 1 or OA = 0

z c s: m :a ~ .-:a m en c ~ )> z c n 0 z n r-e en 0 z en

..... QI

400000

:i= i 200000

HSDTC3 Eqsl

FSDTC3 Eqsl

FSOTC1 Eq.)

o+-.,.-.-~.,.......,.---.-..,._,,_,..-.-........ ---.--.-.-...--.-...--.---.--.-.......... -r--.-.---.-~ .......... -..--.--..-.---r-i.-...--.--.-1

0 2 4

AA e

Figure 23. Stability boundary for isotropic elastic plate; Uh = 4.8; uniaxial compression; OA = 1 or OA = 0

8

2 c: :s: m ::a

~ ,... ::a m (/) c:

~ )> 2 c 0 0 2 0 ,... c: (/)

0 2 (/)

.... .... ....,

200000

I •------·--------L fcUTTER

------------- ..... ----------~-----------. 100000

·-------·-------z------------·--------------------· FLUTTER

~ J FLUTfE.R

- -0 ~ -. . ~ I . -.., T T

0 100000 200000 SIG11 ( p'.:.l.)

FSDT

CLASSICAL

. 300000

Figure 24. Comparison of stability boundaries for orthotropic viscoelastic plate; Uh = 4.8; biaxial compression

n ::c )>

~ ::.u

z c s: m ::.u n )> r ::.u m en c ~ )> z c n 0 z n r c en 0 z en

...... ...... Q)

200000

,, ... , -~ 160000

100000

50000

I \

' \ \

' I ' I I\

\

it.

FLUTTER

'

fl UTTER

' ' ' '.. ' ' ', • ' ' •

f SDT CLASSICAL

o-+-.....-.-.-....--r-1-.--r--r--ir-r-.--.-.--.-.-...,-..-..,_,..-..-..,.--,,.........-.--.-...---.-.--.-.--.-.-~~~~~

0 2 4 AA

6

Figure 25. Comparison of stability boundaries for orthotropic viscoelastic plate; Uh = 4,8; uniaxial compression

8

n :c )> .,, -4 m ;;u ~

z c s: m ;;u 0 )> r ;;u m (/) c r -4 (/)

)> z 0 n 0 z n r c (/)

0 z (/)

..... .... '°

,.-, ... ·.Jl {l.

8t ~

450000 n N ... 1 FSDT e N .. 2 .. N ,..3 CLASSICAL AN .. 4 ... __ --·-- M .. 1

300000

150000

0 ;--.--.--.---.--.---.--r-.-~---r---.r--ir--r-....--r--~...--.--.---,--.--.--.--r--.--....--.---.--,~ 0 160000 300000

SIGU (psL)

450000

Figure 26. Comparison of stability boundaries for orthotropic elastic plate; L/h = 4.8; biaxial compression

z c: s: m ::u 0 > r ::u m en c:

~ > z c 0 0 z 0 r c: en 0 z en

... ~ 0

eoooo..--~~~~--.,-~~~~~~~~~--r~~--..-_-_-F_s_o_T_C3~Eq_s_>~-.-.

nN ... 1

~ 40000

0

" N .. 2 - - - -CLASSICALC3 Eqs> s N .,.3 - - -FSDTC1 Eq.) • N .,.4 - - CLASSICALC1 Eq.l *N .,.5 M ... 1

-----------~---

-..........._

-----------......._-----------

------------------------* -------=:...:::::::-~ .. ------.. ~ •

-==---....:::::·:::-------------- ------50000

SIG11 ( p51)

------100000

Figure 27. Comparison of stability boundaries for isotropic viscoelastic plate; Uh = 4.8; biaxial compression

z c 3: m :u n )> r-:u m Cll c ~ )> :z c 0 0 z 0 r-e Cll 0 z Cll

.... I\.) -....

,,-· ..... .-• (>) (L

1soooo.--~~~~~~~~~~~~~--.-~---.~-=-----~~~----. - FSDTC3 Eqs)

100000

50000

nM .. 1 • M .. 2 a M ,,.3 • M .. 4 •M .. 5

- - - -CLASSICALC3 Eqs) --- FSDTC1 Eq.) - - CLASSIC.ALU Eq.) N .. 1

----------------~ -------------r

0-t-r--r--r--i--r-,-,r-r-.---,--.-r--r--.--.--r-,-,r-r-.---,--.-r-,--,--,--r-..--.-I

0 2 4 e AA

Figure 28. Comparison of stability boundaries for isotropic viscoelastic plate; Uh = 4.8; uniaxial compression

2 c ~ m ;o n ,.. I ;o m "' c ~ ,.. :z c n 0 :z n I c "' 0 :z "'

..... N N

r, -~ ~) .a..

~ i

250000.,.--~~~~~~~~~~~~~-.~~.---::====-=-~~--. nN •1 - FSDTC3 Eqsl

200000

150000

100000

50000

·----.,....__ --

.. N •2 - - - - CLASSICALC3 Eqsl a N •3 - - - FSDTC1 Eq.> • N •4 - - CLASSICALC1 Eq.) •N .. 5 M .. 1

------..

--------------------------. *

*

o-+--.---l~-r~-r--r~...---.-_..;;;~--.-~..---r:"'9-..----.--~..----,-----..---..---.~--.---I

0 200000 SIG11 ( ~~t)

400000

Figure 29. Comparison of stability boundaries for isotropic elastic plate; Uh = 4.8; biaxial compression

z c s: m :;o

~ r-:;o m (/I c ~ )> z 0 n 0 z n r-e (/I

0 z (/J

.... N w

sooooo~~~~~~~~~~~~~~~-.-~---.~~~~~->~---. n M ,.1 - FSDTC3 Eqs

;: ~ 250000

0 0 2 4

AA

• M ... 2 - -- · CLASSICALC3 Eqs) a M ,.3 - - · FSDT<1 Eq.l " M .,,4 - - CLASSICALC1 Eq.l •M ... 5 N .. 1

6

Figure 30. Comparison of stability boundaries for isotropic elastic plate; L/h = 4.8; uniaxial compression

8

0 x )> 'V -f m ;:u ~

z c: s: m', :u n )> r :u m en c: !::j en )> z c 0 0 z 0 r c: en 0 z en

60000

40000 C'

<I) .a. :i:: i

20000

0 2 4 A.A.

I:l • B

•

M .. 1 M .. 2 M "'3 M "'4

N •1

6

Figure 31. Stability boundary for orthotropic viscoelastic plate; Uh = 24; uniaxial compression; .SA = 1 or .SA = 0

HSDT FSDT

8

z c ;: m :0 n )> r :0 m en c ~ ~ c 0 0 z 0 r c en 0 z en

..... N UI

40000 -.------------------.--N-1--..,-----HS-DT-, l::t .. .. N .. 2 B N .. 3 FSDT .. N .. 4 * .. 5 M 1111

~ 20000 ----- --- - --- -- -- ... _

o~-.----~.--r---r~..--.---,-----,ra--.--r---.~.--.-~~..--.----.----,~

0 40000 SIG11 (psi)

Figure 32. Stability boundary for orthotropic elastic plate; Uh = 24; biaxial compression; OA = 1 or oA = 0

80000

z c s: m :a n > r ;u m en c ~ > z c n 0 z n r c en 0 z en

aoooo~~~~~~~~~~~~~~~.--n-M~ ... -1~-,~~~H-s_o_t'. • M ... 2 B M ..,3 FSDT • M ..,4

* "'

;: ~ 40000

o+-.--r-...--.-,-....---r-,--r-T"-r--T"-r-r-,.,...~.-..-,--,..--.--.-,--r-r-,--r-r-r--,.,...~.-..-,--,,--r-1

0 2 4

A.A.

e

Figure 33. Stability boundary for orthotropic elastic plate; Uh = 24; uniaxial compression; OA = 1 or OA = 0

8

() :c )> "D -I m :u !" 2 c :s: m :u n )> r :u m (/I c ~ )> 2 0 () 0 2 () r c (/I

0 2 (/I

... N ......

2500 HSDTC3 Eqsl nN .. 1

• N ... 2 FSDTC3 Eqsl

a N .. 3 FSDTC1 Eq.l

2000 ... 1

,........ ·-' <.fl 1600 .Q_

81 i

1000

600

0+--r--rf!-r--r--r--r--r--r--.--.--.--.--..--.---.---.--.--.--.----.----.---r--r---r--r--r--r--.--.--I

0 2600 6000 7600 SIG11 ( psL)

Figure 34. Stability boundary for isotropic viscoelastic plate; Uh = 24; biaxial compression; fJA = 1 or fJA = 0

z c 31:: m :n

~ r-:n m en c ~ )> z c 0 0 z 0 r-e Cl)

0 z en

.... N Q)

~ (/) a.. ;::

8000-.-~~~~~~~-,-~~~~~~~-,.-n-M-~-1-,-~~-H-S-DT_C_3_E_q_s_)__,

e M .. 2 a M ... 3 FSDTC3 Eqsl

FSOTC1 Eq.l * ...

~ 4000

o-+-.--..-.--r-,.....,r-r-r--r-i~--.-....-r-r-.--.--r--r-.--r-..---.r-r-r-....-r-r---.---,-r--r-.--.-.--..-.--.-,--t

0 2 4

AA 6

Figure 35. Stabilit~ boundary for isotropic viscoelastic plate; L/h = 24; uniaxial compression; OA = 1 or OA = 0

8

0 :c )>

~ m ~

!-" z c s: m ~

0 )> r-~ m rn c r-(if )> z 0 0 0 z 0 r-c rn 0 z rn

.... N (Q•

7500

5000 ~---";;) D-. __,

~ SI

2500

0 0 10000

SIG11 {p:;,i.)

uN .. 1 "N .. 2 a N .,3

* ..

Figure 36. Stability boundary for isotropic elastic plate; Uh = 24; biaxial compression; .;A = 1 or .;A = 0

HSDTC3 Eqsl

FSDTC3 Eqsl

FSDTC1 Eq.l

20000

(') :c )>

~ ;u ~

z c s: m ;u n )> r ;u m (/I c ~ )> z 0 (') 0 z (') r c (/I

0 z (/I

_., w 0

:i::

20000-r-~~~~~~~~~~~~~~--rn-M_m_l....,-~~H~SD~T~C3~Eq~s7l--i

• M m2 11 M .,3 FSDTC3 Eqsl .. M ,.4 FSDTCl Eq.) * Nm

Ii 10000

o+-.-.--r-T-r-...-.--.-T"-"T"-.-r-r-,--r-r--r-;r-r-,-,r-r-.,.-,-.-.-r-r--r--r-.-r-r-.-r-r-.,.-,r-r-1

0 2 4 A.A.

8

Figure 37. Stability boundary for isotropic elastic plate; Uh = 24; uniaxial compression; 6A = 1 or 6A = 0

8

()

0 0 0 0 'lilt

.i 5 ,,,, I-

i c

53 .c

0

~ 'iii

"'i Cll

:c 41 ...

I ::E

Cl.

E

0 u

,... C\I t") 'V

U')

ii ..

·;; II

II II

II II

I ca

I :a

:t?:if :;;: ~

I I a:;

I "-"

' ..; ...... .,.,)

II 0

a_

---.::

I 0

I ;::

:J I

0 ~

Qi I

0 I

-I

0 ca

I a.

I <'I

u I

; I

Cll ca

I Gi

4 u ·a. 0 ... ... 0 .:: t: 0 ... 0 -Cll 41 ·;: ca "ti c :I 0

.Q

>-

= :a ca - Cll - 0

0 c

0 0

0 0

Cll ·;:

0 0

ca 0

0 C

l. E

0

0 0

0 0

(J

<'I ,....

a:i (1s-d)~

M 41

... :I C

l ii:

CH

APTER 5.

NU

ME

RIC

AL R

ESULTS A

ND

CO

NC

LUSIO

NS

131

z c s: m ::u n ):> r-::u m (I) c ~ ):> z 0 0 0 z 0 r-e (/I

i5 z (I)

.... w N

........

400000-.-~~~~~~~~~~~~~---irtM~.-1-r-_-_-H-so~T~~~--,

•M ·2

360000

•M ·3 - - - · Ambartsumian

N •1

I I

~

..... 300000 ~

, ,• ,

i 260000

200000 -· ,.,

___ _,..,.,, ----=::::11:=::: ______ _ g;;;; ••• .-.a:a:s----

.it' ,•'

,• , ,

,• I .

, , I ,

•' ,/

, , , , I

, I

160000+-,.-,--,~.,-,....-.-.-r-r--r-T-.-i.-r-,.-,-,...,.-r-.-.--r-r-.-i-r--r-,--r-r-,--,~,.-,--r--Tj

0 2 4 AA

e

Figure 39. Comparison of stability boundaries for orthotropic elastic plate; Uh = 4.8; uniaxial compression

8

z c s: m :.!:! 0 )> r ;u m UI c ~ )> z 0 0 0 z 0 r c UI 0 z UI

.... w w

60000

~ 25000

0 0 26000

SIG11 ( fSL)

i:N .. 1 -CLASSICAL &N ·2 sN .. 3 ---·Ashton AN .. 4

60000

Figure 40. Comparison of stability boundaries for orthotropic elastic plate; Uh = 24; biaxial compression

76000

z c 3: m :u n > r-:u m en c ~ > z c 0 0 z 0 r-e UI 0 z UI

eoooo-.-~~~~~~~~~~~~~---irz:M~ .. -1--r-_-_-C-L-AS_S_~_A_L~~-, ·M .. 2

i:: s 40000

aM .. 3 ·M .. 4

---·Ashton

o+-..-.-..-.--,.---...,.....--r--,-,r-r-.,-,...-..-.--,.---...,..,..--r---r-ir-r-"T-r..-r-r.,-,--r-..-.-,...,..,....,. 0 2 4

A.A.

8

Figure 41. Comparison of stability boundaries for orthotropic elastic plate; Uh = 24; uniaxial compression

8

Appendix A. THE LAPLACE TRANSFORM AND

ASSOCIATED THEOREMS

The Laplace Transform (L.T.) of f(t) is defined as follows:

2'[t[t]] = f[s] =Loo e-st t[t]dt

The L.T. of f[t] is said to exist if the integral in Eqn. (A.1) converges for some value of s.

Borel's Theorem and the convolution integral

If 2-1[t[s]] = f(t) and 2-1[g[s]] = g[t], then

and

frg = g*f

Appendix A. THE LAPLACE TRANSFORM AND ASSOCIATED THEOREMS

(A.1)

(A.2)

(A.3)

135

Initial value theorem

If the limit indicated below exists, then

lim t[t] = Lim si[s] t->0 S--+oo

L.T. of derivatives

If 2[t[t]] = f[s], then

The inverse L.T.

If 2[t[t]] = f[s], then t[t] = ..<.e-1[f[s]] where 2-1 is the inverse L.T. operator.

Now if f[O] = 0, then

However, if f [OJ * 0, then

..c.e-1[sf[s]] =ht]+ f[0]15[t]

where c5[t] is the Dirac delta or unit impulse function.

Use of initial value theorem in determining elastic values of material properties

(A.4)

(A.6)

(A.7)

Introducing Eqn. (A.4) into Eqns. (2.18a), (2.18b) and (2.62b) and noting that for a phys-

ically real material, the limit of sE;Jmn[s] as s--+ oo must exist, we obtain the required values

of E.pµw[t], E.1133[t] and E.pw.1UJ as t--+ 0 .

Appendix A. THE LAPLACE TRANSFORM AND ASSOCIATED THEOREMS 136

Appendix 8. MICROMECHANICAL MODEL FOR

UNIDIRECTIONAL FIBER-REINFORCED

COMPOSITES

The continuum model for fiber-reinforced composites developed by Aboudi [23, 24, 25)

assumes that continuous fibers extend in the X1 direction and are arranged in a doubly peri-

odic array in the x2 and X3 direction (see Fig. 11). The cross-section of the rectangular fibers

is h1 11 , and h2, 12 represent their spacing in the matrix. Due to this periodic arrangement, we

need to analyze only a representative element as shown in Fig. 12. This representative cell

contains four sub-cells identified by {3, y = 1, 2. Four local coordinate systems defined by X1,

xf>, x1v>, and having their origins at the center of each sub-cell, are shown in Fig. 12. The fol-

lowing first-order displacement expansion in each sub-cell is considered:

v<fiy) - wlfJY) + #),1..(/JY) + jf-Y),,,<fiy) I - I 2 '!'/ 3 'I'/ (B.1)

where wyiv> are the displacement components of the center of each subcell with <f>fv>, ifi'f1v>

characterizing the linear dependerce of the displacements on the local coordinates xf>, x~v>.

Appendix B. MICROMECHANICAL MODEL FOR UNIDIRECTIONAL FIBER-REINFORCED COMPOSITES 137

We note that in (B.1) and for the remainder of this appendix repeated greek indices do not

imply a summation. The infinitesimal strain tensor writes as:

e<flY> = ..1.[v(/JY) + v<f3Y>] I) 2 j,I I,) (B.2)

where in (B.2) for indices 2, 3 the differential is w.r.t. the respective local coordinates. At this

point, we note that for the field variables (or microvariables) w1, </; 1, l/1 1 we have,

(B.3a)

(B.3b)

(B.3c)

with wpv>, <f;'fv>, ijl'fv> being the values of w1, </> 1, l/1 1 evaluated at the center of each subcell.

The displacement continuity at the interfaces between sub-cells requires that the fol-

lowing relations be satisfied (see [23] for a complete derivation):

w(11) = w(12) = w(21) = w(22) = w-' I I I I

I ,,,(JJ1) +I ,,,(JJ2) = (/ +I ) aw, 1 'I' I 2'1' I 1 2 OX3

Now the average strain in the composite is written as,

where

Appendix B. MICROMECHANICAL MODEL FOR UNIDIRECTIONAL FIBER-REINFORCED COMPOSITES

(S.4a)

(B.4b)

(B.4c)

(S.Sa)

138

(B.5b)

and

(B.5c)

Introducing (8.2) into (8.5) in conjunction with (8.1) and (8.4), we may obtain (see [23] for a

detailed derivation) the following:

(B.6)

Now the average stress in the composite is written as,

2 v 1 ~ (jJy) a/j=V L_.i VpySij (B.7)

p, y=1

where sg1Y> is the average stress in each sub-cell, given by,

s<f3Y) = - 1- P Y a(f3r>dtfl2 >di-3Y) f h 12f112 I} V I}

py -hp/2 -i/2 (B.8)

For transversely isotropic, elastic constituents, the constitutive law for each subcell writes as,

where omitting the superindices (fly) for simplicity,

E1122 = E1133 = 2kv A


(B.9)

(B.10a)

(B.10b)

139

E2222 = E3333 = k + 0.5Er/(1 + vr) (8.10c)

E2233 = k - 0.5Er/(1 + vr) (8.10d)

(8.10e)

(8.101)

In (8.10) EA, vA, GA are the axial Young's modulus, Poisson's ratio and shear modulus and

Er, vr are the transverse Young's modulus and Poisson's ratio of the material identified by the

subcell (p-y). In (8.10), k is given by,

(8.11)

Now upon introducing (8.9) for a transversely isotropic body into (8.8) in conjunction with

(8.5), (8.6) and (8.1), we may obtain (see [23] for details),

5 <PY) _ E(py) ~ + E(fly) (,1.. (py) + .1.<PY)) 11 - 1111 11 1122 '1'2 '1'3

S(py) - E(py) ~ + E(py) ,1..(/ly) + E{py) .1,<PY) 22 - 1122 11 2222'1'2 2233'1' 3

s<PY) - E(py) ~ + E(py) ,1..<PY) + E<PY) .1,<PY) 33 - 1122 11 2233'1'2 2222'1'3

s<PY) - E(fly) ( 8w2 + ,/.. (py)) 12 - 1212 ax- '1'1

1

s<PY) - E(py) ( 8w3 + .1,<PY>) 13 - 1212 ax- . '1'1 1

S(py) _ E(py) (,1.. {py) + .1.<PY)) 23 - 2323 '1'3 '1'2


(8.12a)

(8.12b)

(8.12c)

(8.12d) \

(8.12e)

(8.12/)

140

Upon imposing the conditions for the continuity of tractions along the interfaces of the

subcells, we can obtain (see [23] for a complete derivation) the following:

(B.13a)

s!ff 1) = s!/f2) (B.13b)

Now using the above expressions, we can derive the constitutive equations for a

unidirectional composite in explicit form (see [23] for details). The brief outline of the proce-

dure is as follows:

1. Using the displacement continuity equations (B.4) and the traction continuity relations

(B.13), we can obtain a set of equations in terms of the microvariables <f/fly), ljJ<fv>.

2. Solving for the microvariables above, we can obtain the explicit constitutive law relating

average StreSSeS (~;) tO average Strains eij by intrOdUCihg the miCrOVariableS </J'fYl, i/J'fYJ

into (B.7).

The detailed expressions for the constitutive law of the elastic body can be found in [23].

In Chapter 3, the viscoelastic counterparts of these equations are obtained using the corre-

spondence principle. In order to obtain the corresponding equations pertaining to the elastic

body, we only need to replace Carson transformed quantities by their elastic constants.

Appendix B. MICROMECHANICAL MODEL FOR UNIDIRECTIONAL FIBER-REINFORCED COMPOSITES 141

Appendix C. NUMERICAL INVERSION OF THE

LAPLACE TRANSFORM USING BELLMAN'S

TECHNIQUE

Consider the LT. of a function f(t) defined by,

f[s] =Loo e-stt[t]dt

Now we introduce the change of variable x = e-t in (C.1), yielding

- r1 s 1 f[s] =Jo x - f[ - In x]dx

Writing g[x] = f[ - In x] we have from (C.2),

- r1 s 1 f[s] = J, x - g[x]dx.

0

Appendix C. NUMERICAL INVERSION OF THE LAPLACE TRANSFORM USING BELLMAN'S TECHNIQUE

(C.1)

(C.2)

(C.3)

142

Now consider the following representation of the integral defined by I.

(C.4)

where r, are the points at which f is evaluated and w1 are weighting functions. Using a stand-

ard Gaussian quadrature to evaluate the integral in (C.4), it can be shown that (see Bellman

(31]) r1, w1 are the roots and weights, respectively, of the shifted Legendre polynomial P~[r]

defined by,

. PN[r] = PN[1 - 2r] (C.Sa)

where PN(r) is the nth order Legendre polynomial defined by,

m (2n - 2k)!rn-2k P [r] = ~ ( -1/------

N f::o 2nk!(n - k)!(n - 2k)! (C.Sb)

and w1 are shown to be,

(C.Sc)

Introducing (C.4) into (C.3) we obtain,

N - ~ s-1 f[s] = ~ w;x1 g[x;] (C.6)

i=1

Now letting s in (C.6) assume N different values, say s = 1, 2, ... ,N, yields a linear system of

N equations in the N unknowns, g[x1], i = 1, 2, ... ,N, given by,

Appendix C. NUMERICAL INVERSION OF THE LAPLACE TRANSFORM USING BELLMAN'S TECHNIQUE 143

N - '\"1 k1 f[k] = f....J W;X; - g(x;] ,k= 1, ... ,N (C.7)

i=1

where w,, x1 are the weights and roots of P~[r] given by (C.5). Equation (C.7) can be inverted

to obtain the solution of g[x1J and we can write,

,i,k=1, ... ,N (C.8)

where i; = i[s] I.=, and M,1 is the inverse of the NxN matrix given by (w,xf-1). Bellman [31] has

tabulated the weights (w1) and roots (r,) of P~[r] and also the coefficients of the matrix M,k for

values of N ranging from 3 to 15.

Appendix C. NUMERICAL INVERSION OF THE LAPLACE TRANSFORM USING BELLMAN'S TECHNIQUE 144

Appendix D. EXPLICIT FORM OF THE

CHARACTERISTIC EQUATION.

The characteristic equation given by Eqn. (4.11) may be written as:

n Pmn[s] '°' [ ] = L};E;[s] = o Omn s .

1=1

For the TSDT, n = 92 and we have the following relations for T; and E;[s]:

T.2 =~hA.6 3 m

- - --. -. -. E1 = E1111E2222E1212

- --. -. 2 E2 = E1111(E1212)

~ -. /\•

E3 = E1111E1212E2222

--. .- -. E4 = E1111E2323E1212

- --. -. 2 Es= E2222(E1212)

Appendix D. EXPLICIT FORM OF THE CHARACTERISTIC EQUATION. 145

-Ta T1s=--2

-. -. --Ea= E1111E2222E2323

-. -. --Eg = E1212E2222E2323

/\• -. _,,,,, E10 = E1111E2222E2323

"• -.... _. E11 = E1111E1212E2323

--. /\• -.. E14 = E1122E2222E1212

--. -- -. E1s = E1122E2323E1212

-.. /\• ....... E15 = E1212E1122E2222

~ ....... "• E1s = E1212E1122E1122


2 /\+ /\+ -.

E21 = s E1122E2233E1212

........ /\+ ......

E23 = E1122E1111E2323

...... "• ........ E2s = E2323E2211 E1111

....... -.. I\+

E21 = E2323E1122E2211

/\+ ........ --

E2a = E2211E1212E2323

2-.... "* -... E30 = s E1212E1133E2222

2 -- /\• -. E33 = s E2323E2233E1111


-. -. E3s = E1111E1212

-. -. E3s = E1212E2222

I\+ -.

E42 = E2222E1212

T = 1600 q + 160 q J 2 44 3 mn h mn n

h

-. ..... E44 = E1212E2323

-. ..... E4s = E1111E2323


T6o = -c5 A T54

i\.. _.. E49 = E1111E2323

...... ...... Es1 = E1111E2222

~ A•

Es3 = E1111E2222

11. ......

Ess = E1111E2222

2 Es6 = s Ess

i\.. -... E57 = E1111E1212

2 Esa = s Es1

/\; I\~

Esg = E1111E2222

2 E6o = s Ese

-. II+

E61 = E1212E1122


Tea= -Ts1 ~ 2

Eaa = (E1122)

T57 = -Ts2 2 Ea1 = s Eaa

..... ..... Tee= -2Ts1 Eaa = E1212E1122

Tag= -2Ts2 2 Eag = s Eaa

-. A+ T1o=<>ATs1 E70 = E1122E2211

T11 = oATs2 2

E11 = s E10

-. /\+

T12 = oATs1 E12 = E1122E1122

T13=0ATs2 2 E13 = s E12

-. ii. T14 = T12 E14 = E1212E2211

T1s = T13 2 E1s = s E14

/\+ /\+

T1a = -o A T12 E1a = E1122E2211

T71 = -oAT73 2 E77 = s E76


where,

4 ..... Eao = s E2323

2 Ea2 = s Ee1

2 Eas = s Eas

2 Eee = s Ea7

2 E90 = s E69

4 E91 = s

For the FSDT, n = 38, and we have the following relations for T; and E,[s]:


-. -. ...... Ee= E2222E1212E2323

-. - ...... Ea= E1122E1212E2323

- --. 2-E10 = (E1122) E1212

- -. E12 = E1111E1212

Appendix 0. EXPLICIT FORM OF THE CHARACTERISTIC EQUATION. 152

-. --E15 = E1111E2323

-. -. E11 = E2222E1212

::::::. --E21 ::;:: E1212E2323

-. -. E23 = E1111E2222

- --. -. E2s = E1122E1212


-2-E29 = s E2222


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3. Wilson, D.W., "Viscoelastic Buckling Behavior of Laminated Plates", M.MAE thesis, Univ. of Delaware, Newark, DE (Aug. 1982).

4. Sims, D.F .. "Viscoelastic Creep and Relaxation Behavior of Laminated Composite Plates", Ph.D. Dissertation, Southern Methodist University (May 1972).

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14. Biot, M.A., "Theory of Stress-Strain Relations in Anisotropic Viscoelasticity and Relaxa-tion Phenomena", Journal of Applied Physics, Vol. 25, No. 11 (Nov. 1954).

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---~------- - ---------

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BIBLIOGRAPHY 156

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BIBLIOGRAPHY 157

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