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dissertation entitled

An Interlayer Shear Slip Theory for Composite

Laminates with Nonrigid Bonding

presented by

Xianqiang Lu

has been accepted towards fulfillment
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MSU lo An Affirmative ActIoNEquol Opportuntty Insttnrtion
chS—p1

AN INTERLAYER SHEAR SLIP THEORY FOR COMPOSITE
LAMINATES WITH NONRIGID BONDING

By

Xianqiang Lu

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY
in
Engineering Mechanics
Department of Metallurgy, Mechanics, and Materials Science

1991

ABSTRACT

An Interlayer Shear Slip Theory for Composite
Laminates with Nonrigid Bonding
By

Xianqiang Lu

An interlayer shear slip theory (ISST) is presented to study composite laminates
with different kinds of interfacial bonding conditions. It is a displacement-based multiple-
layer technique which satisfies the continuity requirements for the interlaminar shear
stresses across the composite interfaces. By allowing the discontinuity of the displace-
ments on the composite interfaces, the mechanical behavior of nonrigid interface is simu-
lated by a linear shear slip law. The governing equations and associated boundary
conditions for composite laminates with nonrigid bonding are derived from the principle

of virtual displacement.

In assessing the ISST, some numerical results from ISST are compared with those
from elasticity analysis and embedded-layer technique. Excellent agreements are. conclud-
ed. In addition, the comparison between the results from experiments and ISST analysis

are performed. A reasonable agreement is achieved.

A comprehensive application of ISST in the analysis of composite laminates with
delaminations of different kinds of size, location and position is performed. It is concluded
that the transverse shear effect should be considered in the delamination analysis. In addi-
tion, with the use of finite element technique, some cases of static and vibration analysis
are examined. The effect of delamination on the transverse deflection, natural frequencies,
and vibration mode shapes are investigated. The results from these studies further verify

the feasibility of using ISST in studying the composite laminates with nonrigid bonding.

ACKNOWLEDGMENTS

Special tribute goes to my advisor, Dr. Dahsin Liu, for his invaluable guidance,
support and unlimited patience through the course of this research. His enthusiasm and
dedication constitute a very important factor for the conclusion and achievements of this

work.

I am greatly indebted to the members of my thesis committee, Professors N. Altie-
ro, Y. Jasiuk, T. Pence, and D. Yen for the academic support and encouragement provided

through years.

I would also like to express my deep appreciation for the help received from facul-
ties and students in the Department of Metallurgy, Mechanics, and Material Science. In
particular, I like to thank my friends Dr. C. Y. Lee and Dr. C. C. Chiu for many conversa-

tions which helped me a lot in my research.

I am very grateful to my father, mother, and my aunts, Tingsun Lu and Rosa Yu for

their love and support. They showed me how to recognize values of life.

Finally, special thanks are given to my wife, Qingmei Liu. Without her love, en-
couragement and help, this dissertation would never be completed. To her this thesis is

dedicated.

Table of Contents

List of Tables ...................................................................................................................... vii
List of Figures ................................................................................................................... viii
Chapter 1 ~ Introduction ....................................................................................................... 1
1.1 Motivation ................................................................................................................ 1

1.2 Literature Review ..................................................................................................... 3
1.2.1 Perfectly Bonded Laminates ........................................................................... 3

1.2.2 Irnperfectly Bonded Laminates ....................................................................... 6

1.3 Present Research ...................................................................................................... 8
Chapter 2 — Interlayer Shear Slip Theory ............................................................................ 10
2.1 Introduction ............................................................................................................ 10
2.2 Displacement Field ................................................................................................. 11
2.2.1 Rigid Interface ............................................................................................... 14

2.2.2 Nonrigid Interface ......................................................................................... 18

2.3 Equilibrium Equations ........................................................................................... 19
2.3.1 Nonrigid Interface ......................................................................................... 19

2.3.2 Rigid Interface ............................................................................................... 22

Chapter 3 - Assessments of the Interlayer Shear Slip Theory ............................................ 24

3.1 Closed-form Solution ............................................................................................. 24

3.2 Embedded-layer Approach ..................................................................................... 28

3.3 Results and Discussions ......................................................................................... 31
3.3.1 Rigidly Bonded Larrunates ....................... 31

3.3.2 Nonrigidly Bonded Laminates ...................................................................... 41
Chapter 4 - Experimental Verifications .............................................................................. 57
4.1 Test Setup and Specimen Preparation .................................................................... 57
4.1.1 Three-point Bend Test ................................................................................... 57

4.1.2 Free-free Vibration ........................................................................................ 61

4.2 Analytical Solutions ............................................................................................... 61
4.2.1 Static Analysis ............................................................................................... 61

4.2.2 Vibration Analysis................... ...................................................................... 70

4.3 Comparisons and Discussions ................................................................................ 70
4.3.1 Static Test ...................................................................................................... 70

4.3.2 Vibration Test ................................................................................................ 71
Chapter 5 - Applications of the Interlayer Shear Slip Theory ............................................ 73
5.1 Introduction ............................................................................................................ 73

5.2 Static Analysis ........................................................................................................ 73

vi

5.3 Vibration Analysis .................................................................................................. 75
5.3.1 Delamination Length ..................................................................................... 80

5.3.2 Delamination location ................................................................................... 80

5.3.3 Delamination Position ................................................................................... 80

5.4 Conclusions ............................................................................................................ 92
Chapter 6 - Conclusions and Suggestions .......................................................................... 93
6.1 Conclusions ............................................................................................................ 93
6.2 Suggestions ............................................................................................................ 95
Appendix - Analysis of Singular Points Based on Classical Beam 'Iheory ........................ 96
1. Two-layer Laminates ................................................................................................ 96

2. Three-layer Laminates ............................................................................................. 99

List of References ............................................................................................................ 104

vii

List of Tables

Table 3.1 - Comparison of numerical results between elasticity analysis and present

theory ............................................................................................................... 33
Table 4.1 - Central deflections of glass/epoxy beams from three-point bending

test ................................................................................................................... 59
Table 4.2 - Natural frequencies of glass/epoxy beams from free-free vibration

test ................................................................................................................... 64
Table 4.3 - Natural frequencies of glass beams from free-free vibration

test ................................................................................................................... 65

Table 5.1 - Comparisons of natural frequencies calculated from ISST, CLT, and elasticity

analysis ............................................................................................................. 79
Table A1 - Singular points in a [0/0] beam ....................................................................... 102
Table A2 - Singular points in a [90/0] beam ..................................................................... 103

Table A3 — Singular points in a {0/90/IO] beam ................................................................ 103

viii

List of Figures
Figure 2.1 - Nodal variables and the coordinate system ..................................................... 12
Figure 2.2 - Reduced displacement variables and the coordinate system .......................... 17
Figure 3.1- Cylindrical bending of a laminated orthotropic beam ..................................... 25
Figure 3.2 - Pure shear deformation in embedded layer ..................................................... 30

Figure 3.3 ~ Normalized maximum deflection as a function of aspect ratio S in a logarith-

mic scale ........................................................................................................ 32

Figure 3.4 - Comparison of 3 between elasticity analysis and present theory for a [0/90/0]

laminate ......................................................................................................... 35

Figure 3.5 - Comparison of 3,, between elasticity analysis and present theory for a [0/90/0]

laminate ......................................................................................................... 36

Figure 3.6 - Comparison of in between elasticity analysis and present theory for a [0/90/0]

laminate ......................................................................................................... 37

Figure 3.7 - Comparison of 5 between elasticity analysis and present theory for a [90/0]

laminate ......................................................................................................... 38

Figure 3.8 - Comparison of 3,, between elasticity analysis and present theory for a [90/0]

laminate ......................................................................................................... 39

Figure 3.9 - Comparison of in between elasticity analysis and present theory for a [90/0]

laminate ......................................................................................................... 40

Figure 3.10 - Maximum deflections of [0/0] laminates as functions of bonding coeffi-

cients .......................................................................................................... 42

ix

Figure 3.11 - In-plane normal stresses through the thickness in [0/0] laminates ................ 44
Figure 3.12 - Transverse shear stresses through the thickness in [0/0] laminates .............. 45
Figure 3.13 - In-plane displacements through the thickness in [0/0] laminates ................. 46

Figure 3.14 - Maximum deflections of [90/0] laminates as functions of bonding coeffi-

cients .......................................................................................................... 48
Figure 3.15 - In-plane normal stresses through the thickness in [90/0] laminates .............. 49
Figure 3.16 - Transverse shear stresses through the thickness in [90/0] laminates ............ 50
Figure 3.17 - In-plane displacements through the thickness in [90/0] laminates ............... 51

Figure 3.18 - Maximum deflections of [0/90/0] laminates as functions of bonding coeffi-

cients .......................................................................................................... 53

Figure 3.19 - In-plane normal stresses through the thickness in [0/90/0] laminates .......... 54

Figure 3.20 - Transverse shear stresses through the thickness in [0/90/0] laminates ......... 55
Figure 3.21 - In-plane displacements through the thickness in [0/90/0] laminates ............ 56
Figure 4.1 - Three-point bending test ................................................................................. 58

Figure 4.2 — Normalized central deflection of a glass/epoxy beam with a central midplane

delamination of various lengths .................................................................... 60
Figure 4.3 - The dimensions of specimens for free-free vibration test ............................... 62
Figure 4.4 - Block diagram of the apparatus for resonance frequency measurement ......... 63

Figure 4.5 - Normalized natural frequencies of a glass/epoxy beam with central midplane

delamination of various lengths .................................................................... 66

Figure 4.6 - Normalized natural frequencies of a glass beam with a central midplane

delamination of various lengths .................................................................... 67

Figure 5.1 - Normalized central deflection of a glass/epoxy beam with 25.4 mm midplane

X

delamination at various locations .................................................................. 74

Figure 5.2 - Normalized central deflection of a glass/epoxy beam with an end midplane

delamination of different lengths .................................................................. 76

Figure 5.3 - Normalized central deflection of glass/epoxy beams with a central midplane

delamination of various lengths .................................................................... 77

Figure 5.4 - Normalized central deflections of glass/epoxy beams with 25.4 mm and 50.8

mm central delaminations at various positions ............................................. 78

Figure 5.5 - Normalized natural frequencies of graphite/epoxy beams with central mid-

plane delamination as a function of delamination length .............................. 81

Figure 5.6 - Normalized natural frequencies of graphite/epoxy beam with 20% midplane

delamination at different locations ................................................................ 82

Figure 5.7 - Normalized first mode shape of a graphite/epoxy beam with 20% delamina-

tion which has center at x=0.5 ....................................................................... 83

Figure 5.8 - Normalized second mode shape of a graphite/epoxy beam with 20% delami-

nation which has center at x=0.5 .................................................................. 84

Figure 5.9 - Normalized third mode shape of a graphite/epoxy beam with 20% delamina-

tion which has center at x=0.5 ....................................................................... 85

Figure 5.10 - Normalized fourth mode shape of a graphite/epoxy beam with 20% delami-

nation which has center at x=0.5 ............................................................... 86

Figure 5.11 - Normalized first mode shape of a graphite/epoxy beam with 20% delamina-

tion which has center at x=0.2 ................................................................ 87

Figure 5.12 - Normalized second mode shape of a graphite/epoxy beam with 20% delami-

nation which has center at x=0.2 ................................................................ 88

Figure 5.13 - Normalized third mode shape of a graphite/epoxy beam with 20% delamina-

tion which has center at x=0.2 .................................................................... 89

xi
Figure 5.14 - Normalized fourth mode shape of a graphite/epoxy beam with 20% delami-

nation which has center at x=0.2 ................................................................ 90

Figure 5.15 - Normalized natural frequencies of graphite/epoxy beams with 20% central

delamination at different positions ............................................................. 92
Figure A1 - The geometry and coordinate system of a two-layer beam ............................ 97
Figure A2 - The resultant forces and moments at the cross-section C1-C2 ....................... 98
Figure A3 - The geometry and coordinate system of a three-layer beam ........................ 101

Figure A4 - The resultant forces and moments at the cross-section C3-C4 ..................... 101

Chapter 1

INTRODUCTION

1.1 Motivag'on

Fiber-reinforced polymer matrix composite laminates have high in-plane strength
but low density. They are excellent materials for high performance structures such as
space vehicles and aircraft. The pm'pose of stacking layers of dissimilar properties togeth-
er to form a composite laminate is to produce a structural component which is capable of
resisting load in some particular directions. In studying composite laminates, the knowl-
edge of stress and deformation is essentially important to understand the composite behav-
ior. However, because of the heterogeneity of the composite laminates in the thickness
direction and the anisotropy in the individual layers, the design techniques, testing meth-
ods, and analytical approaches developed for conventional isotropic materials cannot be
applied to their composite counterparts directly. Because of the important role of compos-
ite materials in modern engineering design, the investigation of new techniques for study-

ing composite stress and deformation has become an important issue.

The first attempt to study composite laminate is by classical laminate theory
(CLT). Although the stress state in a composite laminate is three-dimensional in nature,
the Kirchhoff assumption used in isotropic plate analysis is borrowed for studying com-
posite laminates. However, it should be noted that CLT is only accurate for thin composite
laminates [1,2]. For thick composite laminates, the transverse shear deformation, which is

neglected in CLT, should be considered.

2

In order to consider the transverse shear efi‘ect, a possible technique is to introduce
a high-order displacement field. The methods which belong to this category are usually
called high-order shear deformation theories (HSDT) [3-7]. Although the fiber orientation
and the material properties of the individual layers are considered, HSDT indeed treats the
composite laminate as a single-layer material. The individual properties are actually
counted in an average sense. In general, HSDT can provide accurate results for displace-
ments and in-plane stresses of moderate thick laminates. However, they are not suitable

for interlaminar stress analysis.

Polymer matrix composite laminates are simply bonded by polymer matrix in the
thickness direction. Sometimes, because of the poor mechanical properties of polymer
matrix and sometimes the weak bonding with fiber materials, delamination can easily hap-
pen on the composite interface. From mechanics viewpoint, delamination is the result of
interlaminar stress concentration, which is caused by the mismatch of material properties
between difl’erent layers. Since delamination can significantly afl'ect the integrity of a
composite laminate, the study of interlaminar stress has become an important issue in the
laminate research. In order to study the interlaminar stress, it is noted that the composite
laminates have to be formulated layer by layer. Such a technique which is usually called a
multiple-layer approach can satisfy the continuity requirements for both displacement and
interlaminar stress across the composite interface. In addition, it can give accurately stress
distribution on the composite interface directly. Several multiple-layer techniques have
been developed by different investigators. A brief review of these techniques will be given

in a latter section.

In addition to poor matrix property and weak bonding between fiber and matrix,
the composite interface may have defects resulting fi'om fabrication and damages due to
service. For a composite laminate with delamination or imperfect bonding on its interface,
displacement discontinuity can take place on the interface. However, the conventional

laminate theories, including CLT, HSDT, and most multiple-layer theories, are based on

3
the assumption of perfect bonding on the composite interface. In this study, it is desired to
develop a laminate theory considering the transverse shear efiect, the continuity require-
ments for interlaminar stresses on the composite interfaces, and the displacement disconti-

nuity on the damaged and imperfect interfaces.

1.2 Literature Review

Many techniques have been developed for composite laminate analysis. Some
comprehensive reviews for these techniques can be found in several articles [8-11]. This
section presents a brief review of the literatures related to develop a theory for composite
laminates with damaged and imperfect interfaces.

1.2.1 Perfectly Bonded Laminates

In conventional composite analysis, the displacements are usually assumed to be
continuous across the bonding surfaces and no slip between layers can take place. Although
the stress state in a composite laminate is three-dimensional in nature. As mentioned in a
previous section, the assumptions used in classical plate theories are first employed in clas-
sical laminate theory (CLT). However, CLT is only accurate for composite laminates with
large dimension-to-thickness ratios, i.e., thin laminates, [1,2]. For thick composite lami-
nates, the transverse shear deformation should be considered. In addition to thickness, the
low shear modulus of polymer matrices also has a significant effect on the transverse shear
deformation [12,13]. Consequently, the transverse shear deformation is an important con-
sideration in the composite analysis.

In order to overcome the shortcoming of CLT, many techniques have been devel-
oped. Among them, the one which receives the most attention in recent years is the so-

called high-order shear deformation theory (HSDF). By introducing different kinds of

4
high-orderdisplaccment fields, many HSDT [3-7] have been presented for both displace-
ment and in-plane stress analysis. Recently, Reddy has indicated that the high-order theo-
ries can be unified by a so-called generalized laminated plate theory (GLPT) [14] which is
developed from a layer-wise displacement field. GLPT can give accurate results for dis-
placements and in-plane stresses for laminate analysis [15]. However, it is not suitable for
interlaminar stresses.

In composite analysis, due to the high interlaminar stresses and weak bonding be-
tween the composite layers, delamination can easily occur on the composite interface. Two
types of delamination, namely edge delamination [16] and central delamination [17], have
been widely investigated. Both of them can be viewed as a result of interlaminar stress con-
centration caused by material property mismatch in the thickness direction. Because delam-
ination can significantly affect the total performance of composite structures, the study of
interlaminar stresses has become an important issue in composite analysis. Since HSDT
and GLPT do not take the interlaminar stress continuity conditions into consideration, the
interlaminar stresses cannot be obtained from the constitutive equations directly. Although
it is possible to recover the interlaminar stresses through the equilibrium equations [1 8], it
is tedious and not suitable for structures with complex configurations.

In order to include the continuity of interlaminar stresses on the composite inter-
face, it is necessary to formulate the composite laminate layer by layer. Ambartsumyan [19]
was among the earliest to present a multiple-layer technique in the composite laminate
analysis. Based on a parabolic distribution for the transverse shear stresses in the composite
layers, he presented a shear deformation theory which satisfied the continuity conditions.
Similar to this work, refined theories for multi-layered symmetric plates were presented by
Whitney [20] and Librescu and Reddy [21]. Another stress-based technique which included
the interlaminar stress continuity was presented by Man, Tong, and Pian [22]. This tech-
nique was named hybrid-stress finite element method. Spilker [23] and many Other inves-

tigators extended this technique for studies with high-order stress assumptions. In addition

5
to the finite element analysis, Pagano [24] assumed a stress distribution in each layer. He
derived the governing equations for laminate analysis with the use of a variational ap-
proach. The continuity of interlaminar stresses was also satisfied in his formulation.

Instead of assuming stresses, DiSciuva [25] presented a displacement-based ap-
proach which had piecewise linear in-plane displacements through the thickness while the
out-of-plane displacement was constant. A variational method was used to formulate the
governing equations. However, due to the low order of assumed displacement field, the
transverse shear stresses were constant through the thickness. Toledano and Murakami [26]
used similar displacement field in their analysis. However, they also assumed quadratic
transverse shear stress distribution for each composite layer independently. Reissner’s
mixed variational principle [27] was used together with the interlaminar shear stress conti-
nuity condition in their formulation. This technique was valid for improving in-plane de-
formation. However, the interlaminar shear stresses were suggested to be recovered by
equilibrium equations because of accuracy reason [26]. In addition, by imposing cubic-
spline function to model the through-the-thickness deformation, Hinrichsen and Palazotto
[28] presented a quasi-three-dimensional nonlinear finite element analysis for thick com-
posite plates. This theory was successfully used to study the transverse deflection and in-
plane stresses and deformation of composite laminates with various thicknesses.

In view of the advantages and disadvantages of the techniques reported, it was
concluded that an accurate theory for interlaminar stress analysis should consider the
transverse shear effect and the continuity requirements for both displacements and inter-
laminar stresses on the composite interface. The interlaminar stresses can then be obtained
directly from the constitutive equations instead of from the equilibrium equations. Be-
sides, it is important that the formulation should be variational consistent [7] and can be
extended to finite element analysis for structures with complex configurations. Based on
these understandings, an interlaminar shear stress continuity theory (ISSCT) was devel-

oped by Lu and Liu [29] with a displacement field refined from the GLPT [14]. Because

6
of its quasi-three-dimension in nature, this technique was suitable for both thick and thin
composite laminates. However, ISSCI' neglected the effect of transverse normal deforma-
. tion which was important under some circumstances [30]. A refined theory accounted for

the transverse normal stress was presented by Lee and Liu [31].

1.2.2 1m B n Lamin tes

All the theories reviewed in the previous sections are based on the assumption of
rigid bonding, i.e., the displacement components on the composite interface remain to be
single-valued. However, in contrast to the strength in the fiber direction, the transversely
tensile and interlaminar shear strengths of composite laminates are relatively low. As a con-
sequence, the composite laminates are susceptible to delamination from a wide variety .of
sources such as fabrication stress, hydrothermal cycling, handling damage, and foreign ob-
ject impact. Under these circumstances, the interlayer connection may become nonrigid. It
has been recognized that the interfacial bonding condition can strongly affect the perfor-
mance of a composite structure. Hence, in order to accurately predict the behavior of a com-
posite structure with imperfect bonding on its interface, it is necessary to account for the
bonding effect in the composite analysis.

Pioneering work in the study of nonrigidly bonded interface for composite struc-
tures was performed by Newmark, Siess, and Viest [32]. Based on Bemoulli-Euler beam
theory, a laminated beam theory with a shear slip on the layer interface was developed. A
linear slip law was employed to model the nonrigid connection between the layers. One of
the applications of their theory was used for studying beams connected by nails. Their the-
ory was later extended for plate analysis by subsequent investigators. In addition, different
slip laws were also introduced for nonrigid interface analysis [33-36]. However, due to the
fundamental assumptions of Kirchhoff hypothesis for both beam theory and plate theory,
the transverse shear effect was not considered in these studies. Barbero and Reddy extended

Reddy’s generalized laminated plate theory (GLPT) to include the discontinuity of dis-

7
placement for damaged interfaces [37]. Their analysis considered the transverse shear ef-
fect. However, the continuity requirements for interlaminar stresses across the composite
interface were not satisfied.

Toledano and Murakami [38], to the author’s best knowledge, was the first to use
a laminate theory which accounted for both transverse shear effect and interlaminar shear
stress continuity in study nonrigid bonding interface. They presented a technique which in-
corporated a slip law in their laminate theory [26]. However, the interlaminar shear stresses
that governed the shear slip at the interface were not accurate because of the low order of
the assumed displacements and stresses. In addition to laminate analysis, elasticity studies
on sandwich beams with nonrigid bonding interface were presented by Rao and Ghosh
[39], and Fazio, Hussein, and Ha [40]. Because of the complexity of analysis, only limited
cases were examined. An alternative way was to use the elasticity-based finite element ap-
proach. However, this technique might not be efiicient because it required a huge number
of elements [41,42].

Inquy the delamination initiation and propagation, it is necessary to study
the interlaminar tensile“ fiagnnejoughness and mterlamrnar shear fracture toughness, i. e. .,
Grc and One, respectively. Many testing procedures have been developed for measuring the
static interlaminar fracture toughness. For example, the technique of double cantilever
beam (DCB) is for Mode I fracture [43] while end notch flexure (ENF) and central notch
flexure (CNF) are for Mode 11 [44,45]. For data reductions, CLT [44,45], HSDT [46], and
two-dimensional elasticity-based finite element approach [45,47] are usually employed for
numerical analysis.

Since delamination can significantly degrade a composite structure, the investiga-
tion of delamination has become an important issue in composite analysis. However,
delamination is usually barely visible. Hence, quite a few destructive and nondestructive
techniques have been developed for delamination detection [48,49,50,51]. One efficient
technique is modal analysis. By studying the degradation of dynamic response of a com-

8

posite laminate, it is possible to detect the size and location of the delamination [51,52].

1.3 Preeent Research

Upon the demanding of an advanced theory for composite analysis, it is the objec-
tive of this study to develop a technique which can be employed to analyze both thin and
thick composite laminates with various types of interfacial bonding condition. It is gig; I
desired-mat the theory satisfies the interlaminar shear stress continuity requirements on the
composite interface. The interlaminar shear stresses can then be directly calculated from
the constitutive equations instead of being recovered from the equilibrium equations. In
addition, if the theory is of variational consistence, it can be easily extended to finite ele-
ment formulation and be used to study composite laminates with complex configurations

and boundary conditions.

In Chapter 2, a so-called interlayer shear slip theory (ISST) is developed to meet
the above demands. It is a displacement-based multiple-layer technique and is extended
from a previous study on perfectly bonded laminates [29]. The nonrigid interface is simu-
lated by allowing the displacement slips, which are complemented by using a linear shear
slip law, on the composite interface. The principle of virtual displacement is used to derive
the governing equations and associated boundary conditions. By adjusting the parameters
for interfacial bonding, the governing equations for cross-ply laminates with different

kinds of interfacial bonding conditions can be presented.

In order to verify the feasibility and accuracy of ISST, the cylindrical bending of
an infinitely long strip examined by Pagano [1] is studied in Chapter 3. me
solutions of [0/0]. [0/90] and {0/90/01 laminates with different kinds of interfacial bonding
conditions are presented. The numerical results from the closed-form solutions for perfect-

ly boned laminates are compared with those from Pagano’s elasticity analysis [1]. In addi-

 

9
tion to ISST, an embedded layer approach is also presented for laminates with imperfect
bondings. The numerical rosults from ISST for imperfectly bonded laminate are compared
with those from embedded layer approach and classical beam theory. The effects of non-
rigid interface on the transverse deflection and stress distribution of the composite lami-

nate are investigated in this chapter.

Chapter 4 can be regarded as an experimental verification for ISST. Two kinds of
tests are performed. One is three-point bend test and the other is free-free vibration mea-
surement for central notched specimens. A finite element scheme based on IS ST is derived
for both static and vibration analysis for composite beams with delamination. Discussion
of the experimental results and finite element analysis are given in this chapter. Comments

on the preparation of specimen are also made.

In Chapter 5, ISST is used to study central notched and end notched specimens un-
der b0th static loading and free vibration. The effects on the transverse deflection, natural
frequencies, and vibration modes in terms of delamination size and location along the

beam length as well as delamination position in the thickness direction are examined.

Chapter 6 is the summary of this thesis. Some important conclusions are drawn

and several suggestions for future studies are given.

10

Chapter 2

INTERLAYER SHEAR SLIP THEORY

2.1 Introduction

In conventional analysis for laminated composite materials, the bonding between
composite layers is assumed to be perfect, namely rigid bonding. In other words, the dis-
placements in the composite laminates are assumed to be continuous across the laminate
interfaces. However, due to the low shear modulus of polymer matrix and imperfect bond-
ing from manufacturing and service, the composite interfaces can be nonrigid. It has been
recognized that the interfacial condition can strongly afl'ect the composite performance.

Hence, it is necessary to account for the bonding condition in the composite analysis.

The objective of this chapter is to present a technique which can be applied to
study composite laminates with different grades of bonding on the interfaces. From the
previous chapter, it has been recognized that a useful theory in the study of composite
laminates, especially those with imperfect bonding interfaces, should consider the trans-
verse shear effect and the continuity requirements for interlaminar stresses on the compos-
ite interface. In addition, in formulating the governing equations, a variationally consistent
approach can be extended to finite element formulation and subsequently used for lami-
nated structures with complex configurations. Based on this understanding, a so-called in-
terlayer shear slip theory (18 ST) is deve10ped to study the effect of interfacial bonding
conditions on the composite behavior. By using the principle of virtual displacement, the

governing equations and associated boundary conditions for cross-ply laminates with both

ll

rigid and nonrigid interfaces are derived in this chapter.

2.2 Diselgement Field

A composite laminate composed of n layers, as shown in Figure 2.1, is considered.
A Cartesian coordinate system is chosen such that the midplane of the laminate occupies a
domain 9 in the x-y plane while the z-axis is normal to this plane. The displacements u, v,

and w at a generic point (x, y, z) in the laminate are assumed as follows.

2 {v.‘P, (x.y)¢,‘° (z) + 0}” mm“) (2) + 52.40:. mg” (2) +32...l (x. y) «31" (z) }

Mam) =
is]
v<x.y.z) = 2 {V.-‘P,(x.y)¢,‘°<z> +V."’(x.y)¢§°(z)+ T2.-,<x.y)¢,“’(z) +7,.-.(x.y>¢§"(z) }.
3'81
(2.1)
W(X.y.2) = w(x.y)
where (bf) are Hermite cubic shape functions and are defined below
¢f° =1-3[(z-z,_,)/h,]2+2[(z-z,_,)/h,]3
¢§° = 3[(z-z,_,)/h,]2—2[(z-z,_,)/h,.]3
2,4525%.
¢3(0 = (z-zi_l) [1— (z-z,_,)/h,]2 (2 2)
chi” = (z-z._,)2(r(z—z._,)/h.-1m.)
¢,“" = I,” = «:9 = ¢,“" = o =<z.-_3 or Dz.-

In the above equations, the superscript (t) represents for the layer number, i.e., the i-th layer
of the composite laminate, and h r the thickness of layer (1'). As shown in Figure 2.1,

0:", v5”, ., 3

(T) (7)
U!" ' Vi.

(a) (a)
vi. ”1‘.
m m
Ur - vi
(a) (a) 1"
Ur . VI
to m
Ui— 1' vi - I

am v.9),

r—l’ r
2k

”1(1),an
Z
U1(B) . V1“)

”(in'vrinvzo

Figure 2.1 - Nodal variables and the coordinate system.

32.-

syfl. 13k,

324* T2:

/,

layer (it)

 

szi-r-th-r /I

NIB-

 

Szr — 2- Tar/£1

layer (1)

 

 

52.772; - 3

W

s2, T2

midplane

layer (k)

 

 

sr'Tr

 

so, To

layer (1)

 

 

>* V MI:-

 

13
U31”) and W”) are the displacements at a point (x,y,z,-) in layer (1') while U5”) and V!“ at the
same point but in layer (i+1). Meanwhile, S and T are the first derivatives of u and v with
respect to z-axis, respectively. More specifically, 32.- and T2, represent for the nodal values

of g; and g"; at the point (x, y, z‘.) in layer (i+1) while S2“ and T2“ at the same point

but in layer (1'). Figure 2.1 depicts the displacement variables. Since the composite interfac-
es are not necessary to be rigidly bonded, displacement slips can exist on the interfaces.
Hence, U!” and VI”) can be different from U!” and VI”, respectively. It is also noted that
the continuity conditions for the interlarrrinar shear stresses must be satisfied for both rigid
and nonrigid bonding interfaces although the continuity conditions are not required for the

interlaminar shear strains. Consequently, 52: and 525.1 are not necessary to be of the same.
neither are T2, and T2“.

The displacement win the thickness direction is assumed to be constant due to the

relatively small value of transverse normal stress, 0,, compared to other stress components

[19,53]. Accordingly. the total number of assumed variables is 8n+l . However, it should
0 1 A ».

{I { (’1'! {I} if 1' +5

at ‘ 11; I 3: ‘1

If the composite laminate of interest is of cross-ply sequence, the constitutive equ'a-

be noted that (32 is important under some circumstances [30].

tions for orthotropic materials can be employed. For layer (1), they are [53]

 

 

 

 

— a (o 1' ~ (I) c - (0
0, 911 912 Qra 0 E,
a) __ Q]; 922 Q23 0 8’
0' 013 923 Qaa e,
1,, o o o 29a; 5.,
b a L - - - ’ (2.3)

 

 

It is assumed in this study that the deformation of composite laminates is within a linear

l4

elastic range. Therefore, the following strain-displacement relations are employed.

g__ a- 5-31-0 e _l(fl 3")
1‘- x , Y‘ay 9 '-3z- r ”-23y 31:
(2.4)
181: 3w lav aw
in = 517*? . in = 5‘5???

In addition, on the composite interfaces, both displacements and interlaminar shear stress-

es have the following relations, i.e.,

AU, = 0}” - 0,") (2.5a)

AV, = vim - vi") (2.5b)
i = I, 2,..., n-I.

W =.hm.m:: <25»

Peri“ " = 32's}? (25d)

A...“
”(I \ _'

where AU, and AV,- are called/interlayer shear ghee, In this study, the interlayer slip in
z-direction is not considered. W- 7 Mi l i

\4 .._‘”-_. a ,1“ J:/
2.2.1 Rigid 1mm

If shear slips vanish, i.e., AU,--AV,=O, the interface between layer (0 and layer
(1' +1 ) becomes a rigidly bonded one, namely rigid interface. And the bonding condition on
the interface is called rigid bonding. Otherwise, it is a nonrigid interface associated with a
nonrigid bonding condition.

For a rigid interface, Equations (2.5a) and (2.5b) result in

(1,") = 0}" = U, and vim .-. vi") = v, (16)

By substituting Equation (2.1) along with Equation (2.4) into Equation (2.3), the stresses

 

15

can be expressed in terms of displacements. More specifically, the interlaminar shear stress-

es have the following expressions.
m B
1;? = Q3) 8;; + Vi-1 (x, y) ¢‘(2 + V: ) (xv Y) ¢§2 + T25-2 (3» Y) $39.), + T2i_1 (x: Y) $.91]

(2.7)

(7) 3 ‘
13‘? = i? a_x +0. I“, Y) ¢1(2+U5( )(KY)¢§2+325_2(I:Y)¢§2+Sz‘°_1(an)¢§;)x]
‘-

With the use of continuity conditions for the interlaminar stresses in Equations (2.5c) and

(2.5d), S2“ and T 2:. r can be expressed as functions of S2,. and T2... respectively.

5 I i
S _ Qs‘s+ ) [952”) 1:le
2i-l " “-7- 25 _.—' ""
9.12 98 3‘
(2.8)
i l r' l
T _ at." {9.2” :law
2r-r - —,— 2.- ——." -
of} (2:2 3’

If both the top and bottom surfaces of the composite laminate are shear-traction

free, the following two equations are valid,

3 = 12 1 = *2 (2.9)

where h is the total thickness of the laminate. With the same fashion as used in obtaining

Equation (2.8), four more variables can be eliminated, i.e.,

3w

80 = 8213-] = "a;
(2.10)

3w

T0 = TZII-l = -$

It then is concluded that it needs only four variables, U i,V‘.l $21., and T2,, to express

16
each nodal point located between the top and bottom surfaces. However, there are only
two independent variables on the top or bottom surface. Hence, including w, the total
number of independent displacement variables is reduced to be 4 n+1 . The reduced vari-
ables are assigned with new notations and shown in Figure 2.2. The displacement field can
then be rewritten as follows.

it . n-l- . n-l 95%.”) aw
“(1’), z) = 2 ”1.0+ ZSj‘Iul'I' [2 (T-Ijej“) _¢§I) —¢§l):|§x_
i=0 ill .31 QSS

(2.11)

 

n . n-l_ . n-l 3‘”) aw
”(MW = 2‘3““ 271‘”? [2 ( 0) '119i(1)‘¢in”«iu)]'a—
i=0 is] '31 Q44 y

W(x.y.2) = WOW)

In the above equations, the shape functions are redefined by the following equations.

 

 

(i) 0-.
. _ 1 j—I-I
4v _ { “a . . layer (i)
2 1:]
of = 0 others
$9 j-i I
I”: = 25‘?” to layer (1')
(o “’4 ) j=i
955
‘1”; = 0 others
(2.12)
ct? . .
\pi = (5+1) 1:1'1
2 Q... (0 layer (1')
<0 “4 ) j=i
QM
‘I’é = 0 Others
9.“) = { “’40) j =1.
1 0 j ¢i

 

Un' Vn' 1.

”fl. vii, 2i.

U... VP 2‘.

'-l-Vl—rh’r-r

H

U], V1, 21

”0h V0 10

Figure 2.2 - Reduced displacement variables and the coordinate system.

 

 

 

 

 

 

 

 

 

[ z
I:
' 2
layer (it)
5.. '3 f' y
- - / 3
5" - 1' Ti-y layer (1)
//
h
/ midplane a
A. _
layer 0:) 'x 1
5&3
layer (1)

 

 

 

18

2.2.2 Nonrigid Interface

Assume that the composite laminate of interest has n , (n15 n- 1) nonrigid bonding
interfaces which are located between layer (1}) and layer (iv-I) as shown in Figure 2.1. A
set II which contains all nonrigid interfaces j), k=1 , 2,..., 11,, is defined. In order to model a

nonrigidly bonded interfacefa linear shear slip law presented by Newmark et al [32] is

employed, i.e.,
A”i. = "if? “i. 2 0
0., 1‘16 II (2.13)
AV. 2 V ‘t ‘ v 20

11 i. y! , j,

The coefficients u. and v are used to represent for interfacial shear bonding conditions. For
rigid bonding, u and v vanish. On the contrary, u and v go to +~ if there is no bonding on
the interface, e. g., the interface is delaminated. Since the shear slips AU and AV should be
finite values, tn and ‘tyz must vanish on the delaminated interface. Hence, the delaminated
interface in this study is defined as an interface which has no ability to transfer shear force.
The values of 11 and v between 0 and .1... are corresponding to imperfect (nonrigid) inter-

faces which have lower ability of transferring shear stresses between layers.

Substituting Equation (2.13) into Equations (2.5a) and (2.5b), the displacements
above and below the nonrigid interface have the following relations.

(a) = . __ ti.) _ (1)
01'. U1. 11.1 , U}. ‘ U}.

I. x:

jke II (2.14)

(B) _ _ U.) _ (1)
Vi. ‘ Vi. ”1.11: . Vi. " V1.

It can be found that Equation (2.14) is also valid for rigid bonding condition as stated by
Equation (2.6) when u and v vanish. Besides, the continuity conditions of interlaminar
shear stresses, Equation (2.8), should remain valid for nonrigid interfaces. Therefore, it
still requires four variables, U,, V,-, S2,, and T2,, to express every nodal point located be-

tween the top and bottom surfaces. Based on the same argument as used in the rigid bond-

19

ing case, it can be concluded that the total number of independent variables remains to be

4n+1. The displacement field for a composite laminate which includes n, nonrigid bond-

 

ing interfaces then becomes
a ' u-l~ . . 1
“(10.2) = 200,4” Elsi‘wr’l‘ng” )ejm)
J- :-
u-l 0+1)
955 (1) 0+1) (2)] (1) (.,}aw
+ ( ‘ -1)6. -u.Q55 6. -¢ -¢ —
{i;1[ 95(5) 1 I I 3 4 ax
1 (2.15)
V(x.y.2) = Zvjdh ZijWg-ujgfimejm)
i-O j-l
u-l 0+1)
9.. . aw
-1911)- _ omega __ (r)_ o.) _
+ {EU Q49 J I “1944 , u m a,
"(1.3.1)= ”(LY)
where
”j = V} = 0 if i¢ 1'1
' ° 2.16
9.”) = { 4’2“) 1:1 ( )
I 0 jaei

2.3 Quilibrigm Egggg'gns
2.3.1 Nonri ' Interf
The principle of virtual displacement is employed to derive the equilibrium equa-

tions and associated boundary conditions. It can be expressed by the following equation.

20

y: )8

15.10““ +0 8: +oH8e +21” Sew-+21%: +21 5+: )dAdz
~ (2.17)

+ Zjou‘J’aw +1mbAV.-)dA In” smu- = o

In the above equation, 11 and v are assumed to be independent of x and y. In other words,
the bonding condition on each interface is uniform. By substituting Equation (2.15) into

Equation (2.4), the strains can be expressed in terms of displacement variables.

13-!

e = 20130-1» 23,,~r’+2— e = Zviyv+zrnvz+12_
i'0 1'1 i-O j-l
l n-l
£,,=§{£Ud +23,-\1",,+(—‘+1)gi;
i'o j-l
"1 (2.18)
Eyfilzvfl”. z+271"”2.z+(?+1)a—y}
i=0 j-l
u-l
a” =.;.{ {2 (u. ,+vj.,)oi+ Z (3,,1/ +r,.~rr’,)+(1t +3.93%”.
1'0 j-l

 

II-l 950*!) l
where 11 2[(” (o -61)’ 9(1) _uiQU+)ej(2)]_ _¢(l)_¢4(n) .

 

i-1 955
H 93“) (1) 0+1) (2) (r) ()

12= .21“ Q}? “)9,- ‘ngu 9,- ]-¢3 ‘°4 ' (2.19)
I.

i“; = (vi-ujes‘émejmn 4’; = (‘Pé-vj93*"9;"’>-

By substituting Equation (2.18) along with Equation (2.3) into Equation (2.17), after inte-

grating by parts and collecting similar terms, the governing equations become as follows.

 

21

5": Q.” ’iux’i 1)!) {2.241.222.2- +fim+P'

+£1w< (vs—31> Wt :21 >

it]

501* N’e..+N’ey.,-Q£ = 0 j = “2 "22" (2.20)
avi: Ngflm -Qi = 0 j: 0,1,...,n
‘0. . . . .+1 2 - aw . . E n
55, o MILZ+MIlI,.’-RIZ-uj(ass ) (Sj+5;)5fi. = O J =1,2,...,n-1 1k
6?" M" +MI’ —Ri-v (Q’*‘)2(i>+a“')5 - o 2-12 n-l j e H
I2 2:»: m r ,- 44 I 33 ii. ' J ' 2 "

5m in the above equations is Kronecker delta. The essential and natural boundary
conditions can also be obtained from the above analysis. They are listed in the following
two columns. It should be noted that one and only one boundary condition, either essential
or natural, in each group needs to be specified.

Essential Boundary Conditions Natural Boundary Conditions

W (Q. +13 4:“ -i.,,% ,ni 15-. (9552613335) )5
all
- 1- u' "+1 2 - 3w
+Q,+(fi ,‘My ,‘25 lm+ git/“91“. ) (Tj.+5;))n,
3‘; Xx’hfi" 21""!
3;; g in": + 23», (2.21)
(1’. Mn +N’x’n’ j = 0.1, ...,n
VJ. N’xynx+N’n j = 0.1, ...,n
§,- M’n +M’,,,n, j= l,2,...,n-1
@- ngynlaeugny j: l.2,...,n-l

The resultant forces and moments shown in Equations (2.20) and (2.21) are defined by the

22
following equations.

It

A
((2,. Q,) = 121.“.9 ”)dz ; (Ni.M;.N{,,) = I: (at, 0, 1,9024:
.5 2

A
(MgbflM’ivM’m )=I5;(o‘I/,.0‘P’2,t’; 1,12” 2)dz

a (2.22)
(in-71,512,) =IE§ (Wailing: llx+2t’12)dz

dd
— )dz

(fir fly): ”I; 2 (1 1341’ an?) dz ; (Q; Q”): -I:; (1’33“!de

h - . _ .
. . - N’ an"
(R1, RI’) = I3! [1882; ‘, 1”; 2)d2

2

2. .2. Ri ' In rf
If all the interfaces are rigidly bonded, i.e., p.=v=0, the governing equations can be
simplified from Equation (2.20).

5w: Qx’x-o-de-i‘u-1,).3,-X,.”+fi‘x+fi,.’+Pt = o

6sz Nix"~i,,,’Qi = 0 j: 0,1,...,n

avj: Ni, ”Ni. -Q’ = 0 j = o, r, ...,n (2.23)
55,-: Mix+M’ix’.’—Ri = 0 j = 1, 2, ...,n—1

5T}: M’k’.x+MI’.,-R’,=O j: l.2,...,n-l

With the same fashion, the associated boundary conditions can be obtained from Equation

23

(2.21).

Essential Boundary Conditions Natural Boundary Conditions

w (Q +fi— u- -;i,..,,)n +(Q, Hi, -i,,- ;X.,.>n,

32—: 22,315,, ,

3‘: it” nx+7tyn

U1. Njn +N’xyn’ j = 0,1, ...,n (2.24)
Vi Niynx+NJn j: 0,1,...,n

s,- Mn ”’11,", j: l,2,...,n-1

i, Manet-+54% j= l.2,...,n-l

Similarly, one and only one of the boundary conditions in each group needs to be specified
in the composite analysis. In Equations (2.23) and (2.24), all the resultant forces and result-

ant moments can also be derived from Equations (2.19) and (2.22) by imposing 11:0 and
v=0.

Chapter 3
ASSESSMENTS OF THE INTERLAYER SHEAR SLIP THEORY

3.1 Closed—form Solution

In order to assess the interlayer shear slip theory for analyzing composite laminates
with rigid and nonrigid interfaces, the cylindrical bending of an infinitely long strip pre-
sented by Pagano [l] is investigated. The configuration of the laminate is given in Figure
3.1. For the beam problem, the displacement components of Equation (2.15) are reduced to _
be functions of x and 2 only. And all the derivatives with respect to y in Equations (2.20)
and (2.21) vanish. By combining Equation (2.22) with Equation (2.20) and using the defi-
nitions of stresses and strains in Equations (2.3) and (2.18), the governing equations can be

expressed in terms of independent displacement variables.

(A55+K55+72+72)w “3+ 2 [(B§5+§§5) U,.,—§§,U,'m]
i-O

n-l
4’ 2 [(05+Cg'5)sn.x- Cllsfim] + ‘2‘”. (Qgs)+12(sh’+'a—7w)—ilw ,xxxx+P: = 0

III 312
" u-l
' ‘ -‘ -' - 3.1
2 (DileIUIS-I-Dgsuj) + 2 (Eilsgn-E§5 “) +Bluw .m ( )
j‘O '31
1= 0.1, ...,n

-(B§5+§§s)w. = o

3

u II-l
Z (Ell‘lUij-E’S‘SUJ') + 2 (Ffisflu-Fg‘sfl) +C‘11W ,m

”o .31 I: l,2,...,n-l

-(C;5+C35)w , -u (Q22,+1 ) (Sm a ) = 126 H

. _. , _. _ _ l' I'
lntheaboveequatrons, B’ll’BJSS’B’SS’CTI’ng’ngs’Dfi’DJS’ 1T, 552F112 5,391455:

24

NIS'NIS’

25

 

 

 

 

 

 

 

 

GB = 1.38 Gpa, and 1112:1213 = ”23 = 0.25.

Figure 3.1 - Cylindrical bending of a laminated orthotropic beam.

26

355» ‘Yge 71 and '72 are the coefficients of laminate properties. They can be expressed as

follows.
a " I ‘2»!

Ass = 2,112.0e‘e’dze Be=1211::-.oe?22..dze Ce = .211.:-.Qe‘?22>.edz
I: I ’3
"so . -_"> . -_"(o

72 = 212:49521'13‘12’ 71 - ELLIQleirdz’ 72 - 212,1,st it‘d}!
i=1 i=1 i=1

5’1: = 2, I::_'Q1(?¢’°lldz; E22,", = 2 1:4wa324"

jsl j-l

Eli] = 21::dge'f9yaxlndz; 5?; = ZI::_'Q§?§‘:le.xdz

is] in]

Dy, = 211:4 {yawn 1517 = 2 12-994932, dz
i=1 1-1

I I
. 8 ‘ . I ‘ a.“
0?, = 2 I‘:-'Q§S7¢l"d>’.xdz; 5;; = 2 Li'dQQdfl‘Plfldz
jal is]

I I
m ’ -'-" . _ 8 “I ”at
Fir = ZI,:_,Q:?V1W1 4" FE? " ELLQQ‘PLfi‘Mdz
i=1 is]

(3.2)

Since the laminate of interest is simply supported at x = O and x = L, the boundary

conditions can be expressed by the following equations.

19(0) = w(L) = o; 1,,(0) = 1,,(L) =0
Ng'w) =N;'(L) =0 m =O,1,2,...,n
mm) = mm = o m =1.2.....n—1

(3.3)

For cylindrical bending, the loading is assumed to be of sinusoidal distribution, i.e.

27

S

P. = Posin ($1!) (3.4)
where B = g . In order to satisfy the boundary conditions listed in Equation (3.3), the fol-

lowing displacement functions are assumed,

w = wsiMBI)
U, = UICOSWI) j = 0,1,2, ...,n ’ (35)
5, = s,cos(Bx) j = 1, 2,...,n-1

where g, [L and ,2,- are coefficients to be determined. It is obvious that w satisfies the
boundary conditions, i.e., w(0)=w(L)=0. The satisfaction of the remaining boundary con- '
ditions are also apparent. By expressing Equation (2.22) in terms of displacements, the

following equations for force and moment resultants can be achieved.

a n-l _
Ax = XBIUUIJ+ 2 d1Sj.x+‘7lW .n

i=0 1-1

3-1
”1' = 2011’er 2813's....wnw (3.6)
i=0 1'1

u-l
M: = 253101.” zmSn,+C—'fiw.

i=0 in]

I!

It is clear that Equations (3.6) are of sine functions and satisfy the boundary conditions
listed in Equation (3.3). Therefore, Equation (3.5) can be a set of solution to the governing
equations, Equation (3.1). By substituting Equation (3.5) into Equation (3.1), it then re-

sults in

28

 

 

 

 

 

 

- ' -' “'ul -0- - - - 1
B§5+B§5+BVHBZ C§+C55+Cnflz+x, (A55+A55+12+12+x1)[3+y,fi3
D§§+Dififlz $95332 (B§,+§§,)B+§§,B3
ewes: Fee+rira2+xe <ce+eee+>e>e+<2ee2
(3.7)
f 1 "P7
U,- T39
. l = 0,1, ...,
0 n
x 5' =
0
. t=1,2,...,n-l
v—w—i .
..-.J
where
M ',+l 2_ ‘,+1 2 .
K1: 2“}.(Q’55 ) ’ K2 = "j.( 55 ) 8,": Jke n

181

The solution of Equation (3.7) gives the coefiicients of the displacement function defined
in Equation (3.5). The closed-form solution for the cylindrical bending of a cross-ply lam-

inate with various interfacial bonding conditions can be achieved.

3.2 Embedded-layer Apmach
Another method to study composite laminates with nonrigidly bonded interfaces is

to introduce embedded layers on the composite interfaces. The controversy of this method

29
is the determination of the thickness and the material properties of the embedded layer. In
this study, the thickness of the embedded layer B is assumed to be 0.0764 mm [53]. Since
only the shear slip is considered, it is possible to simulate the nonrigid bonding by adjusting

the transverse shear modulus 613' The finite element method presented in [54] is used to

study the deformation of the composite laminates with embedded layers. The numerical re-
sults are shown in a following section.

Although the interlayer shear slip theory and the embedded-layer technique are de-
rived from different bases, a relation between them can be found by assuming that an em-
bedded thin layer undergoes a pure shear deformation. As shown in Figure 3.2, ‘ABCD’
becomes ‘abcd’ due to a shear stress 1: acting on the layer. The displacement field of the

layer can be written as follows

x=X+tan(a,.)Z

z=Z i=1> ifZ>Oe i=2. le<0 (3.8)

where x and 2 represent for the coordinates in the deformed configuration while X and Z
the undeformed configuration. Thus, 1: can be expressed as

I = GS’Ianmo = Gfi’tanmg- (3.9)

In addition, the difference between the displacements at z=h72 and z=-T{/2 can also be iden-
tified as

meg) -u(x. -§) = tmn(a,)-mn(a,)1(E/2). (3.10)
If the embedded layer is thin enough to simulate the real interface, the. difference de-

scribed in Equation (3.10) can be regarded the same as the interlayer shear slip defined in
Equation (2.5a). Accordingly, the relation between the embedded-layer approach and the

interlayer shear slip theory is
..E M (3.11)
”‘2 Gene

NI 3'4
£— 39%

NI :4

30

 

 

 

 

Az,Z
D L, C
at d ‘3
Gmra x,X
(2)
a“ A b B G ‘3
(__.
2:

 

Figure 3.2 - Pure shear deformation in embedded layer.

 

31

This equation can help to compare the resultants from the two approaches and provide the

fundamental information for assessing the interlayer shear slip theory.

3.3 R sults Di u ions

In order to assess the accuracy of the interlayer shear slip theory in composite anal-
ysis, the cylindrical bending of [0/0], [0/90]. and [0/90/0] laminates with various interfacial
bonding conditions is investigated. The numerical results are summarized in the following

sections with the nondimensional terms as introduced by Pagano [1].

 

 

 

o (I: z)
6 _ ’ 2’ 121(0’ 2) _ 8214(0, z)
x - P0 1’, = P0 u — hPo
(3.12)
1001.35 L
_ ”(2) - z
w = z = -
L‘Po h

3.3.1 Rigidly Bonded Laminates

A. Transverse Deflection

By setting )3: 0 for Equation (3.7), the closed-form solutions for perfectly bonded
laminates under cylindrical bending can be obtained. The transverse deflection, W, at the
midspan of the laminates is shown in Figure 3.3 as a function of aspect ratio (S=L/h). Ap-
parently, the results from the present theory agree quite well with those from the exact so-
lutions {1] in both large and small aspect ratios. In order to further compare the results
from both techniques, the numerical solutions for all three types of cross-ply laminate are
listed in Table 3.1. In this study, three values of S, i.e., 4, 20, and 100, which represent for

thick, intermediate, and thin laminates, are presented. Besides, three layer numbers are

2.0—1

1.0-A

32

 

[90:0]
-Eufl ”mum
0 Present theory

 

 

 

 

 

[0/90/0]

— Exact solution.
4 Present theory

 

 

 

 

 

[0]
— Exact solution / I

0 Present theory

 

 

 

0.0

r I I

l r ll I T I r
5 7 910 20 40

I

[‘1 I
60 80100

Figure 3.3 - Normalized maximum deflection as a function of aspect ratio S.

 

Table 3.1 - Comparison of numerical results between elasticity

 

 

 

 

 

analysis and present theory.

S Pagano’s present theory
solutions 2-layer 4-layer 6-layer
4 1.9490 1.9672 1.9659 1.9659
[ 0 / 0 ] 20 .5519 .5523 .5523 .5523
100 .4940 .4940 .4940 .4940
4 4.6953 4.7773 4.7812 4.7812
[ 0 / 90 ] 20 2.7027 2.7069 2.7069 2.7069
100 2.6222 2.6230 2.6220 2.6220
4 2.8868 2.9098
[0 / 90 / 0] 20 .6172 .6176
100 .5140 .5140

 

 

34
investigated to study the effect of layer number on the numerical accuracy. From the results
in Table 3.1, it is concluded that the present theory gives excellent results of transverse de-
flection in all three types of lamination although the results in the laminates with lager as-
pect ratio seem to be more accurate. However, the increase in the layer number does not

cause any significant effect on the results.

B. Symmetric Laminate [0/90/0]

Other than [0/0] laminates, [0/90/0] symmetric laminates with S=4 and S=10 are
also investigated. The results of in-plane displacement, in-plane stress, and transverse shear
stress based on a 6-layer model are shown in Figures 3.4, 3.5, and 3.6, respectively. Excel-
lent agreements between the present theory and the elasticity analysis are concluded al-
though a noticeable difference between the two techniques exists around the middle surface
of the in-plane displacement in the case of S=4. It is believed that this is due to the assump-
tion of constant transverse deflection in the present theory. In fact, just because of the con-
stant displacement assumption, the distributions of both in-plane displacement and stress
from the present theory are anti-symmetric to the middle surface while the interlaminar

shear stress is symmetric.

C. Asymmetric Laminate [90/0]
Both S=4 and S=10 for a [90/0] asymmetric laminate are studied. Excellent results
are again concluded. The results are shown in Figures 3.7, 3.8, and 3.9, for in-plane dis-

placement, in-plane stress, and transverse shear stress, respectively.

35

 

 

 

 

 

 

— Exact solution
0 Present theory
S=4
.4 .
firvvflr 'r'r‘21
-1.0 —.5 0.5 1 0
U
.
—.5-
'z'
0.5-
, — Exact solution
0 Present theory
S=10
r r r 1 fi
-10 -5

 

 

Figure 3.4 - Comparison of 5 between elasticity analysis and present

theory for a [0/90/0] laminate.

36

2

0.5
—— Exact solution
0 Present theory

 
 
   
 
 
    

 

NI

0.5
— Exact solution
0 Present theory

S=10

 

Figure 3.5 - Comparison of 3, between elasticity analysis and present

theory for a [0/90/0] laminate.

 

37

 

 

   
  

 

Z

0.5-4

i

..l

V r r l 'r r

.. 1.0 20

“ 7x2

“ S=4-

" 0 Present theory
...,5J — Exact solution

NI

 

S=10

0 Present theory ,
--- Exact solution

 

Figure 3.6 - Comparison of En between elasticity analysis and present

theory for a [0/90/0] laminate.

38

E
0.5

 
   
  

-— Exact solution
0 Present theory

S=4

 

TF7
2 4- 6
G

X

  
 
 
   

4 — Exact solution
, 0 Present theory

4 S=1O

 

‘—

l T
-70 —50 —30 - 0

212*:
50 70

ux

['1
1030

 

Figure 3.7 - Comparison of 3 between elasticity analysis and present

theory for a [90/0] laminate.

39

 

    
  

 

 

   
  

E
0.5-1
..l
r— 2 1 = l
—50 —1o 30
S=4 "x
- 0 Present theory
-— Exact solution
2'
0.5-
..l
..J
F “I— ] r A ]
--2oo -100 200
s=1o ”x
0 Present theory
_ 5_ — Exact solution

 

Figure 3.8 - Comparison of 3,, between elasticity analysis and present

theory for a [90/0] laminate.

40

E

0.5-1
4 -- Exact solution
.1 0 Present theory
" S=4

 

 

 

'5:

0.5—4
" — Exact solution
~ 0 Present theory
‘ S=10

 

 

 

Figure 3.9 - Comparison of in between elasticity analysis and present

theory for a [90/0] laminate.

41

3.3.2; Nonrigidly Bonded Laminates

A. [0/0] Laminates with Nonrigid Interface on Midplane

[0/0] composite laminates with a nonrigid interface on the midplane are studied.
The numerical results from the closed-form solution, Equation (3.7) are summarized be-

low with the use of nondimensional terms defined in Equation (3.12)

a. transverse shear effect

The maximum deflections calculated from different approaches are presented in
Figure 3.10 as a function of interfacial shear bonding coefficient 11 for both S=4 and S=20.
The approach of Bernoulli-Euler beam theory can be referred to Reference [33]. It is con-
cluded that the results from the interlayer shear slip theory (ISST) are close to those from
Bemoulli-Euler beam theory when S=20. However, the difference between them is very
significant for S=4. This difference is apparently due to the transverse shear effect. In ad-
dition, the results from the embedded-layer technique are very close to those from 18 ST for
both S=4 and 20. It verifies that it is feasible to use IS ST to study composite laminates with
nonrigid bonding. Furthermore, it may imply that the analysis used to find the relation be-
tween ISST and embedded-layer approach is acceptable.

b. bonding coefficient
As shown in Figure 3.10, the maximum deflection reaches an upper limit as u ap-
proaches one while lower limit as it goes to zero. The former corresponds to a completely
debonded interface while the latter a rigidly bonded interface. It is verified that the deflec-
tions from ISST are very close to those obtained in the previous section for perfectly bond-
ed laminates. The investigation of complete debonding is performed with 11:101 in this
study. In addition, it is concluded from Figure 3.10 that the deflection changes dramatically

42

 

 

 

 

 

3.0—
2.0—
W
1.0-
H B-E beam theory
a Embedded layer
0 0 o—o Present theory
. I ‘ I ‘ 1
-16 :12 -a -4 o x
,u=10‘ (in3/lb)
(a) S = 20
4.0-
3.0-
v7
2.0-
1.0-
H B-E beam theory
0 Embedded layer
0 O H Present theory
- _ n r l f 2 l
—16 -12 -a -4 o "“
p=1022

(b) s = 4 (in3/lb)

Figure 3.10 - Maximum deflections of [0/0] laminates as functions of bonding

coefficients.

43

within a small range of p. This implies that nonrigid bonding can have a significant effect
on the deflection when the interfacial shear bonding approaches a ‘critical’ range. Howev-
er, beyond the ‘critical’ range, the compositc laminate will quickly lose its integrity and be-
have like two independent layers with a lubricated interface [38]. This phenomenon has
also been reported in Reference [38] though the result beyond the ‘critical’ range are not
presented in details. In addition, it needs to point out that the ‘critical’ range seems to

change as the material property or geometry changes.

c. maximum stresses

Figure 3.11, 3.12, and 3.13 represent for the distributions of in-plane stress, trans-
verse shear stress, and in-plane displacement as functions of interfacial shear bonding co-
efficient u for S=4 and 20. The solid lines represent for the results from Reference [1] for
rigidly bonded laminates while the dash lines from ISST for different shear bonding coef-
ficients. By setting u=10"°, the results from ISST are very close to those from Reference
[1].

From these diagrams, it is also concluded that the in-pane normal stress becomes
larger and larger as the bonding gets less and less rigid. For tt=10°5 and 10", the maximum
normal stresses increase 27% and 97% in 8:20 laminates, respectively, while 25% and
56% in S=4 laminates, respectively. In addition, the maximum transverse shear stress for
8:20 is always smaller than that of rigid bonding. However, the maximum value for S=4
with nonrigid bonding can exceed the value of rigid bonding. This is another evidence in-
dicating the important role of the transverse shear effect in thick laminate analysis. Besides,
the results seem to further verify the conclusion from Reference [38] that the damage in a
composite laminate with nonrigid bonding can take place earlier than the prediction from

the analysis based on rigid bonding assumption.
In addition, it is shown in Figure 3.12 that the delaminated interface, i.e., 0:10",

has no ability to carry any shear stress but the nonrigid interface, e. g. 11:10‘5, can transfer

 

 

 

Z
035—] I, ’II‘
A’ I)”
I I
l ,1
d It
1”
. ’,,;
er /
I” I,
I; I‘
1’ d /
I" I
I” ’4
,-’ x.
’ I
l” I —
”. l" 0!
r—v Y: I’ l I I - 30.9 I I; I I I [:3 I
I
-600 -400 -200 2” 200 , ,400 600
.. ‘1 ll,’
/ ,./’
I u I”
a, ’J’
I
I I,“ o
a” ‘ — Exact solution
”7 o p=10"°
’1’,’// .. k‘A ”=10-5
I’1 ’A B-B 12:10"
I l
r’ t —.5-—
a"
”x
F 2 ,V- > 1
” 30
‘30 ’,,r 20

 

 

Exact solution
12:1 0"°
12:10“
12:10"

Figure 3.11 - In-plane normal stresses through the thickness in [0/0] laminates.

 

45

 

   

 

 

0 “=10“
A [1:104
0 #:10'”
— Exoct solution
$12
(a) 8 =20
D p.=10"‘
A p=10"
o #:10-‘0
— Exoct solution
$22
I , fl
2 3

 

 

Figure 3.12 - Transverse shear stresses through the thickness in [0/0] laminates.

46

 

 

Z
3‘ K\ 0.5“
\ \
‘\\ \a .. Ar-A #:10'5
\
‘1 o p=10“°
\ O
X“. ‘ —— Exoct solution
\ ‘\
\ \
\‘\
\q \x
x x
\ \
‘ \ \x\
\ \\
1‘ '-
\\ \ ux
_ 4 n x \-
fin r W 1 l I 1 rWUtU V "' ' 1 I " ' 7 I j
\\ \
—200 — TQO ‘x 100 200
\\‘\ \Aq
\\ ‘
\\\ A\
x '1 \
\\ ‘\
\I \a
d \\\\
\\
\ x
d ‘\‘\\
\ x
x X.
\ \\
-.5_. x ‘

 

D p: l O"
0 “=10“
v p=10“°

— Excct solution

 

 

 

\\ ux
\-
I 1' I " T 1
\\
\
. -\-\
\\\\
ok“
\ \\
x
__ 54 ‘0 "I

Figure 3.13 - In-plane displacements through the thickness in [0/0] laminates.

47

shear stress up to some extend.

d) singular points
It is interesting to point out that there are four singular (persistent) points in which
the normal stress (or in—plane displacement) and the interlaminar shear stress remain to be
constants even though the bonding condition changes. The locations of these singular

points are near z="_".h/3 for both in-plane normal stress and displacement while around

z=d_‘h/6 for transverse shear stress. A careful discussion about these points is given in Ap-
pendix. It is believed that these singular points have special interest for composite laminates

with embedded sensors.

B. [90/0] Laminates with Nonrigid Interface on Midplane

The numerical results for [90/0] laminates with nonrigidly bonded interface on the
middle plane are shown from Figure 3.14 to Figure 3.17. For maximum deflections, similar
conclusions as obtained in [0/0] analysis can be drawn. By comparing the results from the
present theory with those from Bernoulli-Euler beam approach, it is concluded again that
the shear effect is very important in the study of composite laminates with nonrigidly bond-
ed interface, especially in thick composite laminate analysis. Besides, the results of trans-
verse deflection, in-plane stress and displacement, and transverse shear stress from ISST
for both S=4 and S=20 are very close to those given in the rigid bonding analysis when
u=10"°. It concludes that the results from ISST converge to the results from the perfect
bonding analysis as u approaches to O.

For both S=4 and S=20 laminates, the maximum values of in-plane normal stress
and interlaminar shear stress increase as the bonding gets less and less rigid. For |.t=10‘4
and 11:10“, the increases of the maximum normal stress in S=20 laminates are about 15%

and 34%, respectively, while the increases of the maximum shear stress are around 11%

4.0—

3.0—

5|

2.0-

 

1.0

48

i

H B—E beam theory
0 Embedded layer
o—o Present theory
' 1

 

--16

7.0?
6.0-

5.0-

(I

4.0-

3.0-

2.0-

1.0
-16

 

I ' l T

-12 -8 34 0
p: 1 0" (in3/1b)

X

(a) S = 20

H B-E beam theory
0 Embedded layer
o—o Present theory
I

I ' I f l '

-12 -8 -4 0
p= 1 0" (ian)

X

(b)S=4

Figure 3.14 - Maximum deflections of {90/0} laminates as functions of

bonding coefficient.

49

 

   

— Exact solution
0 p=10“°
Ame ”=10“
G-E} p=10“

 

Exact solution
p.=10“°
p=10‘5
#:10"

 

 

Figure 3.15 - In-plane normal stresses through the thickness in [90/0] laminates.

50

 
  
   
   

,u.=10“
#:10“
#:10-10
—— Exact solution

ODE]

 

 

 

‘ ~~
\ ~.
~\ ~~
\ \

-
’1’ _-‘
-" ,——’
a‘ -—

 

Nl

p=10“
p.=10'°
”=10-‘O
Exact solution

     

|o>a

   

‘1~____——r

r
‘\

~
“-—__..——"‘

 

\

 

 

Figure 3.16 - Transverse shear stresses through the thickness in [90/0] laminates.

51

 

— Exact solution

 

\\ O p: 1 0‘“
1 \\ k—A = 1 0“
\\ "_ -.
\ \ G-B [J—1 O

Exact solution
I": 1 0-10
#:10"
p=10"

 

 

Figure 3.17 — In-plane displacement through the thickness for [90/0] laminates.

52
and 24%, respectively. In addition, it can be verified from Figure 3.16 that the resultant
shear force is independent of bonding coefficient it However, there are only two singular
points which are both located in 0 layers. For both in-plane normal stress and displacement,

it is around z=-O.29 h while near z=-0.09 h for transverse shear stress.

C. [0/90/0] Laminates with Nonrigid Interfaces

[0/90/0] composite laminates with two nonrigid interfaces are also studied. One
nonrigid interface is between 0 and 90 layers and the other 90 and 0 layers. In this analy-
sis, it is assumed that both interfaces have the same nonrigidity. The numerical results of
the maximum deflection and stress distributions are shown from Figure 3.18 to Figure .
3.21. Similar conclusions as obtained for [0/0] and [90/0] laminates about the effects of
nonrigid bonding on the maximum deflection and stress distributions can be drawn. It is
interesting to find that there are two singular points in each 0 layer for the normal stress

and the interlaminar shear stress, respectively. No singular point is found in 90 layer.

 

53

1 H Present theory

 

 

 

 

 

 

5.0-1
W 4.0-<
2.0d
1
G i
00 - r e . - i a .
-16 -12 -a -—4 0 x
p=10" (103/1b)
S=20
9.0-
0-0 Present theory
6.0-4
v?
3.0“ c e
0.0 r r a I r T fl x
~16 -12 -8 -4 0

p=10‘ (in3/lb)

S = 4
Figure 3.18 - Maximum deflections of [0/90/0] laminates as functions of

bonding coefficients.

 

”x
V V T

. a l
650 950 1200

  

 

— ISSCT solution
v ”=10’”
o p- 10" ‘ '
a p: 1 0"

 

-.
”’ ‘o..
I o.
0"
.0

 

 

 

p-10‘"
11-1 0"
pal-10"
ISSCT solution

 

' Figure 3.19 - In-plane normal stresses through the thickness in [0/90/0] laminates.

55

 

lSSCT solution
p- 1 0'"

p- 1 0"

p- 1 O"

 

ao<|

 

 

lSSCT solution
In: 1 0‘”

u- 1 0"

u- 1 O"

 

 

Figure 3.20 - Transverse shear stresses through the thickness in [0/90/0] laminates.

 

 

 

 

 

0".
a“ ‘8
5x
I a r 1
—450 -300 - 150 300 450
Or: """" 8
-— lSSCT solution
v “-10“.
o p=10“
o “-104 '8
as
ta a
6it
fiv I v T r T I ‘ I ‘
—5 —4 -3 —2 3 4 5

Figure 3.21 - In-plane displacements through the thickness in [0/90/0] laminates.

 

lSSCT solution
1" 1 one

up 1 0"

pr- 1 O"

 

Chapter 4

EXPERIMENTAL VERIFICATION S

4.1 Test Setup and Sm'men Mamfion

 

In order to verify the ISST, the following two types of test are performed, one is
three-point bend test and the other is free-free vibration test. Both tests are performed for

laminated beams with a central delarrrination on the midplane.

4.1.1 Ilia-mint Bend Test

The test is performed with an Instmn materials testing machine. The test setup is
shown in Figure 4.1. Specimens of stacking sequence of [06 / 0,] are fabricated from 3M’s
glass/epoxy prepreg tapes. The specimens are cured at 160° C (320° F) and 350 KPa (50
psi) for 45 minutes. The efl'ective dimensions of the specimens are 80 mm x 25.4 mm x 3
mm while the artificial central delamination has length either 25.4 mm or 50.8 mm. The
delamination is achieved by embedding two layers of teflon film on the midplane. The
composite beams are then subjected to three-point bend with the maximum loading of 0.3
kN. The central deflection of the composite beams are measured at the bottom surface of
the beams with a dial gauge. It is observed that the maximum deflections are around 1.8
mm which is within the range of small deflection assumption. The material constants are
given in Figure 4.1. The experimental results are normalized with the deflection of a com-
posite beam without delamination and are shown in the Table 4.1 and Figure 4.2.

57

 

58

loading

delamination

 

 

l/ .t

 

Cb.— dial gauge +

¢ L=80 (m) ———>

 

Eu = 38.6 Gpa, E22 = E33 = 8.27 Gpa, 012 = 013 = 4.14 Gpa

623 = 3.22 Gpa, V12 = V13 = 0.21, V23 = 0.28

Figure 4.1 - Three-point bending test.

59

Table 4.1 - Central deflections of glass/epoxy beams from three-point

 

 

 

 

 

 

 

 

 

 

bending test.
Central Deflection (mm)
Specimen
No. no delamination 25'4 mm 50.8mm
delamination delamination
No. 1 1.792 1.842 1.854
No. 2 1.787 1.812 1.858
No. 3 1.854 1.793 1.908
No. 4 1.761 1.801 1.971
No. 5 1.830 1.762 1.923
Mean 1.8048 1.802 1.9028
S‘W‘Fd 0.0369 0.0291 0.0487
devratron

 

 

 

 

 

normalized deflection

 

 

 

5.0
5 -— central delamination from present theory
1 a testing results
4.0—
3.02
2.04
1 O +
0-0 r r l . l ' 1 ' l r r
0.0 0.2 0.4 0.6 0.8 1.0

normalized delamination length

Figure 4.2 - Normalized central deflections of a glass/epoxy beam with a

central midplane delarrrination of different lengths.

61

4.1; Free-free Vim

Two types of specimen are employed in the vibration test. One is the same glass/
epoxy beams as used in three-point bend test. The other is pure glass beams with artificial-
ly central delamination. More specifically, the beams are made from two pieces of micro—
glass and bonded together by super glue. By carefully adjusting the bonding area, different
lengths of central delamination can be achieved. The dimensions for both types of speci-
mens are given in Figure 4.3. The testing employed to study the effect of delamination on
vibration fiequency is called resonant frequency measurement method. It is an ASTM
standard method designated by C 848-78. Figure 4.4 depicts an overall diagram of the set-
up. The natural frequencies for the first and second bending modes are measured for both
types of specimens. The results are shown in Tables 4.2 and 4.3 and Figures 4.5 and 4.6.

4.2 Analm'cg Solutions

To verify the testing results, a finite element scheme for ISST is developed for both

static and vibration analysis.

4.2.1 Static Analysis

The total potential energy of a composite beam with nonrigid interfaces can be
written as

h n
— _ E 1 1 1
W - J{ -h [idxex-t- ETXZYIZ-PZWsz+ 2 ET‘AU'} dA (4.1)
n 3 i: 1
After considering the continuity requirements on the composite interfaces and assuming 11
to be constant in a section of interest along x-direction, the strain components in layer (1')

can

62

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

delamination
1 3 (m) I I
6 L=125 (mm) +
25.4 (mm)
cross-section
a) glass/epoxy beam specimen
delamination
’ . 2.56 (mm)|
f L=75.7 (mm) b
8.1 (mm)
cross-section

E = 70.7 Gpa, G = 43 Gpa, and v = 0.185

b) glass beam specimen

Figure 4.3 - The dimensions of specimens for free-flee vibration test.

63

 

 

 

 

 

 

 

frequency counter threshold voltage
oscillator controller

 

 

 

 

 

 

 

 

[voltmeter]
oscilloscope

filter &
amplifier

 

 

 

 

 

 

switch

 

 

 

 

 

 

\specimen

 

 

 

 

L/‘ 14%

I
I
x l
I
l vacuum chamber :

Figure 4.4 - Block diagram of the apparatus for resonance frequency

measurement.

 

64

Table 4.2 - Natural frequencies of glass/epoxy beams from free-free

vibration test

 

Specimen

Natural Frequency (Hz)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

N0. mode 1 mode 2 ($603) Comment

0 -l 849 2321 1307 N0 delamination
0 - 2 847 2309 1253 No delamination

0 - 3 822 2210 1281 No delamination

0 - 4 860 2350 1301 No delamination

0 - 5 847 2286 1290 No delamination
Mean 845 2296 1286
33333;: 13.95 52.92 21.18

1 - 1 828 2221 1288 25.4 mm delamination
1 - 2 848 2293 1328 25.4 mm delamination
1 - 3 846 2265 1290 25.4 mm delamination
1 _ 4 852 2290 1314 25.4 mm delamination
1 _ 5 863 2321 1314 25.4 mm delamination
Mean 847 2278 1307

(31:22:31 12.68 37.54 17.24

2 - 1 842 2251 1305 50.8 mm delamination
2 - 2 864 2281 1329 50.8 mm delamination
2 - 3 822 2210 1306 50.8 mm delamination
2 - 4 810 2185 1280 50.8 mm delamination
2 - 5 820 2210 1276 50.8 mm delamination
Mean 832 2227 1299

Standard 21.51 38.19 21.65

deviation

 

 

65

Table 4.3 - Natural frequencies of glass beams from free-free

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

vibration test.
Specimen Natural Frequency (Hz)
Comment
NO- mode 1 mode 2
1 1232 3366 One layer specimen
2 1234 3370 One layer specimen
2466 6376 No delamination
Specimens 1 and 2
1241 3338 75.7 mm delamination
bonded together
2375 3918 38 mm delamination
3 1215 3326 One layer specimen
4 1217 3324 One layer specimen
2432 6650 No delamination
Specimens 3 and 4
1272 3294 75.7 mm delamination
bonded together
2013 3977 57 mm delamination
5 1209 3305 One layer speciemn
6 1210 3303 One layer specimen
2420 6608 No delamination
Specimens 5 and 6
1217 3304 75.7 mm delarrrination
bonded together
2412 7.6 mm delarrrination

 

 

 

 

 

 

 

 

 

 

>\
O
C
Q)
3
O.
Q)
at
“O
Q)
.':‘
E
g 4
0 04—4
C 4 a first mode from testing results

0.3— a second mode from testing results

. —- first mode from present theory
0 2 —- second mode from present theory
. T I I I l’ I I I I 1
0.0 0.2 0.4 0.6 0.8 1.0

normalized delamination length

Figure 4.5 - Normalized natural frequencies of a glass/epoxy beam with a central

midplane delamination of various lengths.

0.6-1

fl
0.5-4
.1

0.4-4

d

0.3-1

d

normalized frequency

67

 

0 first mode from testing results
a second mode from testing results
—— first mode from present theory
-- second mode from present theory

 

 

 

0.2
0.0

f T i f f r I
0.2 0.4 0.6 0.8
normalized delamination length

1
1.0

Figure 4.6 - Normalized natural frequencies of a glass beam with a central

midplane delamination of various lengths.

 

68

be expressed in terms of reduced displacement variables, i.e.,

r dS‘- r —( 1
cf) = {d—x ”410+ 37:111er “'3' ¢§°+Ia‘°¢“’-u 1155.0 ‘1 ’42‘")—}
4’»
+ (31(0¢(0_ uiQ(‘+l)¢(0)_
2 4:2
7‘? _ l 2 (i) —(r'+l)
{U“1d—2+U‘E +S.-.1dz3 +S‘(A1d-§-ut55 E 2)}

(i) (5)
+ (A(D_d_¢4_ HQU+Dd¢2 +l)d_w
1 dz 55 dz dx

[~10] {X9}

 

where
(i+l)
A“) 2”— i=1, 2, ..., n-l
1 ’ 95‘?
1 .
r=n
-(i+l)
Rm 955 - i=1.2....n-1
= —(0
l Q55
0 .
r=n
i (0-(0~(0~(0-(0 l (0-(0-(0-(0-
[~91 = 01.41.411.94 .45.] [~91 = E». 42.413.44.11
aw T
{Xm} = {Ui- 1’ Si- I’Ui’si’d—x =wa

{rope-19116
1 dx ’dx 'dx’dx’de

(4.2)

(4.3)

5”]

 

 

69

In the above equations, the shape functions are defined as follows.

- t '(i) t "(8') (a),
4,1”: ‘9‘"); ¢2 = (I30; ‘93 = 4‘2 .
(4.4)
“(1) I t i r ”(0 I l l i
4». = (A§’¢,"-u,-Q§5*"¢,”); ¢5 =(B"¢”— uQ‘ “M55

Substituting strains and stresses into Equation (4.1) and then integrating through the thick-

ness, it yields

W_“12“AMAWXAD+x<or(3(o+c(o)xm)_Pzw}dA (4,5)
0

ill]

in which A“), B“), and C“) are matrices defined as follows.

4 t (0" (0d 1 , (t') (1)
Ag}: IQIB¢,¢.. 33,2: IQ§5’¢:.2¢md z
(4.6)
0,“): u- (Q‘§*")2 {N }T{NM} 1,»: =1,2,...,s

{N,} = {0. o. o. 1. 1}

Once the assembly in the thickness direction is achieved, the assembly of the finite ele-
ments in the x-direction can be performed. In this study, Hermite cubic shape functions are
also used for in-plane assembly. The details of the finite element formulation can be found
in Reference [54]. Accordingly, the potential energy can be expressed as

W: xTKi‘t—iTF (4.7)

Ni:

where i is the vector of the nodal variables, K is the total stiffness matrix, and F is the
associated external loading vector. By using the principle of minimum potential energy,

the finite element equations in terms of the nodal variables are

Ki - F = 0 (4,3)

 

70

4.2.2 Vibration Analysis

In vibration analysis, the kinetic energy associated with the assumed displacement

field can be written as

It

2
KB = a I (122+v92)dsz (4.9)
-Iu

2

Substituting the time derivatives of u and w into the above equation, it yields

KE = g; 2 «WWW + My)“ (4.10)

is 1
where p, is the density of layer (1') ,

x. (0 3x (0

a ‘ ‘ '(i)
=5; and “9,045.42 «11>

With the same fashion as used in deriving potential energy, the kinetic energy can be ex-

pressed in terms of mass matrix M and the vector of nodal variables i as

[(5 = guru” (4.12)

With the use of the Hamilton’s principle, the finite element equations for free vibration

analysis are

minor = o (4.13)

4.3 m 'son an Dicu ins
4.3.1 Static Test

The normalized transverse deflections obtained from both testing and finite ele-

 

 

7l
ment analysis are shown in Figure 4.2 as functions of normalized delamination length,
which is defined as the ratio of the length of the delamination, a, to the efi'ective length of
the beam, L. In this study, the delamination is located on the midplane of the composite
beams and is symmetric with respect to the beam center, namely central delamination. A
reasonable agreement between experimental results and finite element analysis can be
concluded from this diagram. Figure 4.2 also shows that the central delamination remains
to be insensitive to the delamination length if the delamination does not extend to the
specimen end. However, as the delamination approaches the beam end, a significant in-

crease in the central deflection can take place.

4.3.2 Vibration Test

A. Glass/Epoxy Beams

For a glass/epoxy composite beam with different lengths of central delamination,
the first two natural frequencies of bending type are measured. The measured results are
normalized by that without delamination and plotted as functions of normalized delamina-
tion length as shown in Figure 4.5. There is a pronounced disagreement between the finite
element analysis and the testing results for the second vibration frequency. It is believed
that there are three reasons for the disagreement. One is the effect of inhomogeneity of the
material properties in the composite beams. It is found that the variation of the material
properties is as much as 7% among the different specimens. The inhomogeneity is owing
to the fabrication process or due to the defects in the prepreg tapes. Another reason is the
poor quality control of the specimen thickness. The variation of the thickness in the speci-
mens can be as large as 0.25 mm which is about 8% of the total thickness of the speci-
mens. Unfortunately, it is very hard to control the thickness of the specimen in the
fabrication. The third reason is the effect of the embedded layer. It is reported in Reference
[51] that the embedded layer can strongly afi'ect the results. In addition, it is impossible to

 

72
remove the embedded material after fabrication. However, it is not easy to fabricate a

specimen with natural delamination but without matrix cracking.

B. Glass Beams

In order to eliminate the three undesired causes as stated in the previous section,
pure glass beams are used for vibration test. It is verified that the variations of the modulus
and the thickness of the specimens are less than 1%. In addition, when two pieces of glass
beams are bonded together by super glue, it is confirmed that the composite beam behaves
like a solid beam [56]. Hence, this type of specimen are used for a comprehensive analy-
sis. The delamination is created by carefully adjusting the bonding area. By using the res-

onant frequency measurement method, the effect of delamination length on the natural

frequencies of the beam is studied.

The normalized first two natural frequencies obtained from testing and finite ele-
ment analysis are shown in Figure 4.6 as functions of normalized delamination length.
Reasonable agreements between the experiments and the theoretical results for both
modes are shown in the diagram. It appears that the frequency of the first mode of a glass
beam with a central delamination does not change significantly until the delamination
length extends to the beam end. This conclusion is similar to that from the three-point
bend test. However, both experiment and analysis reveal that there is a considerable
change of frequency for the second mode when the delamination length is moderately
large. By comparing the experimental results with analytical solutions, it can be concluded

that ISST can be employed to study composite laminates with damaged interfaces.

 

 

Chapter 5

APPLICATIONS OF THE INTERLAYER

SHEAR SLIP THEORY

5,1 lugguction

In characterizing the interlaminar shear strength of composite laminates, both end
notch flexure (ENF) and center notch flexure (CNF) tests [44,45] have been widely used.
By measuring the change of compliance of specimens with different delamination lengths,
it is possible to determine the interfacial shear fracture toughness Gm. In section 5.2, the
effect of delamination on transverse deflection is examined by using ISST. In section 5.3,
the ISST is used to study the free vibration of a graphite/epoxy beam with difl'erent kinds
of delamination. The effects of delamination size, location, and position on the vibration

frequencies and mode shapes are examined

5.2 Static Analysis

The finite element scheme presented in 4.2.1 is used to analyze a [0/0] composite
beam with a midplane delamination. The loading, boundary conditions, and material prop-
erties are shown in Figure 4.1. First of all, the effect of delamination location on the trans-
verse deflection is investigated. In this study, the delamination length remains to be 25.4
mm while the center of the delamination is moved along the laminate midplane between
the beam center and end. Figure 5.1 indicates that the change of the central deflection is
moderate unless the delamination approaches the beam end. Secondly, the effect of dela-

73

 

normalized deflection

 

 

 

74

2.01

— midplane delamination

I: end delamination
1.5-4
1.0-
0.5 1 1 . 1 T r . I . .

0.5 0.6 0.7 0.8 0.9 1.0

normalized delamination center

Figure 5.1 - Normalized central deflection of a glass/epoxy beams with a

25.4 mm midplane delamination at various locations.

at

 

75
mination length is studied. The results are shown in Figures 5.2 and 5.3. Based on Figures
5.1, 5.2, and 5.3, it can be concluded that the transverse deflection is very sensitive to the

delamination which covers the beam end.

Delamination is not necessary to be restricted to the composite midplane. It can
take place in any thickness position. The sensitivity of delamination in the thickness direc-
tion is of a major concern in damaged composite analysis. In this study, both 25.4 mm and
50.8 mm delamination are investigated. The delaminations are positioned at the midspan
of the composite beams and can be located between midplane and top surface. Figure 5.4
shows the results from the finite element analysis. Apparently, the delamination of 25.4

mm long only causes very little change in central deflection while that of 50.8 mm shows

noticeable influence when the delamination is close to the midplane. However, both cases ‘

reveal that the effect of delamination on the deflection is getting weaker and weaker as the

delamination moves away from the midplane.

5.3 Vibration Angysis

Similar to static analysis, the finite element scheme from the ISST is employed to
study the effect of delamination on vibration frequency and mode shape of a composite
beam. In this study, a simply supported graphite/epoxy beam with delamination is exam-
ined. The material properties of the beam are given in Figure 3.1. In this analysis, the ef-
fect of transverse shear on vibration frequencies is evaluated. The natural frequencies for
the first four modes calculated from ISST, classical laminate theory, and elasticity analysis
[57] are listed in the Table 5.1. It can be seen that the transverse shear efi‘ect is more signif-

icant to the higher order modes than to the lower order modes.

.-
.5-

 

 

76

ta—a Embedded layer approach
- o—o ISST analysis

Normalized deflection

 

 

O f I r r ' r ' I m n

0.0 0.2 0.4 0.6 0.8 1.0

Normalized delamination length

Figm'e 5.2 - Normalized central deflection of a glass/epoxy beam with an end

midplane delaminations of various lengths.

Normalized deflection

77

 

 

 

 

5..
o—e ISST analysis

‘ a—a Embedded layer approach
4-
3-
2-l
1L 4 k +
O l l l T l j l
0.0 0 2 0.4 0 6 0 8 1 0

Normalized delamination length

Figure 5.3 - Normalized central deflection of a glass/epoxy beam with a central

midplane delamination of various lengths.

 

 

normalized deflection

78

 

1.30-

d

1.25-1

d

1.20-

-l

1.15-
1.10" ‘\‘
1.05"I “\‘

1.00-

d

0.95 T I V r
0.5 0.6 0.7

— 25.4 mm center delamination
-- 50.8 mm center delamination

 

 

 

T

I . r .
0.8 0.9 1.0

 

normalized delamination position

Figure 5.4 - Normalized central deflections of glass/epoxy beams with 25.4 mm

and 50.8 mm central delaminations at various positions.

79

Table 5.1 - Comparisons of natural frequencies calculated from ISST, CLT,

and elasticity analysis.

 

Natural Frequency (Hz)

Natural Frequency (Hz)

 

 

 

 

 

S = 20 S = 100
model mode2 mode3 mode4 model mode2 mode3 mode4
ISST 5.3498 18.550 35.171 52.850 .2264 .8991 1.9987 3.4962
CLT 5.6681 22.672 51.103 90.690 .2267 .9096 2.0405 3.6276
Elasticity 5.3444 18.5394 35.161 52.838 .2262 .8980 1.9961 3.4907

 

 

 

 

 

 

 

 

 

. dint".
A

 

 

80

5.3.1 Delamination Lengt_h

Figure 5.5 reveals the efiect of delamination length on the natural frequencies for
the first four mode. The normalized frequencies and delamination lengths shown in this di-
agram have the same definitions as those given in Chapter 4. It is noted that the effect of
the central delamination on the vibration frequencies is more sensitive to the even modes

than to the odd modes.

5.3.2 Delamination Locagq' it

Figure 5.6 shows that the normalized frequencies are functions of delamination lo-
cation. The delamination length in this study remains to be 20% of the beam length while
its center is moved along the midplane from the beam center to the end. For all four
modes, the frequencies experience significant reductions as the delaminations move from
the beam center to the beam end. In addition, the patterns of the change seem to match
with the vibration mode shapes. The largest changes in the individual mode seem to coin-
cide with the nodal points. This implies that the vibration modes may have significant
change if the delamination is located at the nodal point. In order to further verify this re-
sult, the change of the first four vibration mode shapes are studied with the center of a
delamination located at x=0.5 or x=0.2. For the case of x=0.5, it covers a nodal point for
both the second and the fourth modes while for the case of x=0.2 only the fourth mode.
Numerical results are shown fi'om Figure 5.7 to Figure 5.14. When the delamination cen-
ter is located at x=0.5, the second and the fourth modes experience noticeable changes in
vibration mode shapes. However, only the fourth mode shape shows a significant change

when the delamination center is located at x=0.2.

5.3.3 Delamination Position

The change of vibration frequency due to the position of a 20% central delamina-

 

normalized frequency

81

 

 

 

 

 

1.0 -
0.9-<
0.8—l
\‘ ' \\~. ‘v. - , - . d.
O. 7 A \“\ “O—-0-——0-—-‘\
“\
4 \'~\
0.6-4 o—e first mode “mm-u.“
B--EJ second mode ' --..------..--
‘ one third mode
v -v fourth mode
0.5 T l I I I I I T l’ 1
0.0 0.2 0.4 0.6 0.8 1.0

normalized delamination length

Figure 5.5 - Normalized natural frequencies of graphite/epoxy beams with central

midplane delamination of lengths.

 

 

normalized frequency

82

1.00

 

 

4 9—0 first mode
i B--B second mode
. <>--<> third mode

0.85 v -v fourth mode

r
0.5 0.6
center

edge

 

l'

r T
0.8 0.9 1.0

r
0.7

normalized delamination center

Figure 5.6 - Normalized natural frequencies of graphite/epoxy beams with

20% midplane delamination at different locations.

normalized vibration amplitude

83

 

 

O I I I I r l T I U T

0.2 0.4 0.6 0.8 1.0

normalized length

I
....
.Ll

ant-3 with 20% delamination
-4... — no delamination

 

Figure 5.7 - Normalized first mode shape of a graphite/epoxy beam with 20%

delamination which has center at x=0.5.

normalized vibration amplitude

 

 

 

4-
3.4
24
J
1 ..
O ‘ l ' I T ‘ I T j
. 0.2 0.4- 0.6 0.8 1.0
-1- ."l normalized length
_2_ \\ ’III
-3- “ ...-
. lam-E.) with 20% delamination
_4_ — no delamination

Figure 5.8 - Normalized second mode shape of a graphite/epoxy beam with 20%

delamination which has center at x=0.5.

normalized vibration amplitude

85

 

 

 

4n
:5—
.l
2—l
1 _
O F l ‘ f I f ‘7
l 0.2 0.8 1.0
71" normalized length
.l
-2..
-34
. Elna with 20% delamination
_4_ — no delamination

Figure 5.9 - Normalized third mode shape of a graphite/epoxy beam with 20%

delamination which has center at x=0.5.

normalized vibration amplitude

86

 

 

 

 

4-
3-
1
2-l
J
1" .
O ' l "l, j I I vT
4 0.2 {v' 0.4 1.0
-1“ I; normalized length
_2_ I“ I.
_3_ “K“
. h" B--El with 20% delamination
__4_d — no delamination

Figure 5.10 - Normalized fourth mode shape of a graphite/epoxy beam with 20%

delamination which has center at x=0.5.

 

,. MEI

normalized vibration amplitude

87

 

 

4.-
3-
2-1
1 ..
O V I r I I I U I I j
. 0.2 0.4 0.6 0.8 1.0
'4: normalized length
-22
-3-
. Elna with 20% delamination
-44 -— no delamination

Figure 5.11 - Normalized first mode shape of a graphite/epoxy beam with 20%

delamination which has center at x=0.2.

88

 

 

4-
g .
3 3‘
a d
E 2‘
O ...
c 1 -« '
o x
E O I",
L- r I F I I . I . 7
g . 0.2 0.4 x' 0.6 0.8 1.0
8 _1_ ‘ " normalized length
N d “
z: -2..
E
a -3— ,
c i 5...; WIth 20% delamination
_4J — no delamination

Figure 5.12 - Normalized second mode shape of a graphite/epoxy beam with 20%

delamination which has center at x=0.2.

normalized vibration amplitude

 

89

  
 

normalized length

13--El with 20% delamination
—— no delamination

Figure 5.13 - Normalized third mode shape of a graphite/epoxy beam with 20%

delamination which has center at x=0.2.

90

  
  

 

T
1.0

normalized length

normalized vibration amplitude

~, / B--EJ with 20% delamination
—— no delamination

 

Figure 5.14 - Normalized fourth mode shape of a graphite/epoxy beam with 20%

delamination which has center at x=0.2.

normalized frequency

91

 

 

 

 

 

1 .00 c e i 3 ---—
<>- 4 4 * ...... I" '
oo,"" ' ’ ’
.l ''''' . . '
1" '
o""" - o ' ’
---‘ ........ " I
II """"""" , ’
0.95— . . '
I 1‘ '
- , zv- ' ‘ .
I» - ' ' "
e—e first mode
I:-'I--EI second mode
one third mode
v -v fourth mode
0.90 I I I I l I I j 1
0.5 0.6 0.7 0.8 0.9 1.0

normalized delamination position

Figure 5.15 - Normalized natural frequencies of graphite/epoxy beams with

20% central delamination at different positions.

 

 

92
tion in the thickness direction, as shown in Figure 5.15, is not significant although the even
modes seem to be more sensitive than the odd ones. However, they seem to reveal that the
effect of delamination on the natural frequencies becomes smaller and smaller as the

delamination moves away from the midplane.

5.4 on In ions

For a simply supported composite beam, the efl‘ects of delamination on the vibra-

tion frequencies can be summarized as follows.

(1) A delamination covering the beam end has more significant effect on the trans-

verse deflection and first vibration mode than those not extending to the beam end.

(2) The effect of delamination on the central deflection and all vibration modes
(first to fourth) becomes weaker and weaker as the delamination moves away from the

midplane to the surface of the beam.

(3) In vibration analysis, if a delamination covers a nodal point of a vibration

mode, the efl'ect of the delamination on the mode shape is very significant.

Based on the above conclusions, it has been found that delamination has strong ef-
fect on the composite response wherever the transverse shear stress reaches a maximum,
e.g., nodal point, midplane, and beam end. This conclusion implies that the transverse

shear effect should be considered in the study of composite laminates with delamination.

Chapter 6

CONCLUSIONS AND SUGGESTIONS

6.1 ncl ions

By allowing displacement discontinuity on the composite interface, an interlayer
shear slip theory (ISST) is developed to study composite laminates with both rigid (per-
feet) and nonrigid (imperfect) bonding conditions. This study is based on multiple-layer
technique. It is derived from an interlaminar shear stress continuity theory [29] and an as-
sumed linear shear slip law on the nonrigid interface. A comprehensive assessment of us-
ing the ISST to study composite laminates with difl‘erent degrees of interfacial bonding is
given while limited cases for experimental verifications and delamination analysis are also

presented. As a summary, the following conclusions are drawn:

1. IS ST can be employed to study composite laminates with both rigid and nonrigid
bonding interfaces. It has been verified that ISST can be used to calculate displacements,
in-plane stresses, and interlarrrinar shear stresses in composite laminates accurately. In ad-
dition, the interlaminar shear stresses, which play a critical role in interfacial bonding, can
be obtained directly from the constitutive equations instead of being recovered from the
equilibrium equations because of the consideration of the continuity of interlarrrinar shear

stresses on the composite interfaces.

2. By comparing the results from ISST with those from other approaches, e.g., elasticity
analysis and embedded-layer technique, excellent agreements are concluded. In addition,
a reasonable agreement between the results from ISST and the experiments is also con-

cluded. These comparison provide evidences for the conclusion that it is feasible to use

93

94

ISST to study composite laminates with nonrigid bonding.

3. It is shown that ISST can be used to study the effects of delamination on both static

and vibration performances of laminated beams.

4. Some closed-form solutions for cross-ply laminates with different kinds of interfacial
bonding are presented. These results are valuable for more studies in this area since very

limited elasticity solutions exist.

5. Because of the variational consistence in the development of IS ST, this theory can be
extended to finite element formulation and applied to the study of imperfectly bonded

laminated structures with complex configuration and various boundary conditions.

6. From the results of stress distributions of composite laminates with nonrigid bonding
interfaces, it is concluded that the maximum in-plane normal stress and the maximum
transverse shear stress are underestimated by using theories based on perfect bonding as-

sumption.

7. It is interesting to find that at some special locations, namely singular points, the
transverse shear stress or in-plane normal stress and displacement remain insensitive to
the condition of interfacial bonding. These results have important application in composite

laminates with embedded sensors.

8. Nonrigid bonding can have a significant effect on transverse deflection when the in-
terfacial shear bonding approaches a ‘critical’ range. However, beyond the ‘critical’ range,
the composite laminate will quickly lose its integrity and behave like two independent lay-

ers with a lubricated interface.

9. For a composite beam with a delamination, the effect of delamination on the deflec-
tion is strongly dependent on the location and position of the delamination. In addition, the
effects of delamination on vibration frequencies and mode shapes also depend on the loca-
tion and position of the delamination. It is found that the effects are very significant if

delamination is located in an area where the transverse shear stress is very high. This con-

 

 

 

95
clusion is helpful in developing nondestructive testing method for delamination investiga-

tion.

6.2 u stions

Based on the work performed in this thesis, the following studies are suggested for

future investigation.

1. In this study, the continuity requirements for transverse displacement and trans-
verse normal stress are neglected. The eflect of the transverse normal stress is considered
in the Reference [31]. It is believed that it will be a very useful work to extend interlayer
shear slip theory to include the displacement slip in the thickness direction for studying

mode-I interface fracture and delamination buckling.

2. In order to study the initiation and propagation of delamination in composite
laminates, it is suggested to combine the stresses analysis provided by ISST with numeri-

cal schemes to calculate the strain energy release rates [37].

3. The present study is mainly for laminated beams. The study for delamination in
composite laminates is recommended. This study will have more significant impact in the
study of composite structures. In addition, it can be extended to include nonlinear strain-

displacement relations and dynamic analysis.

4. More experimental studies in the effect of delamination on the static, vibration,

and buckling performances of composite laminates are suggested.

 

 

Appendix

ANALYSIS OF SINGULAR POINTS BASED ON

CLASSICAL BEAM THEORY

1. Two-layer Laminates

Consider a composite beam made of two layers shown in Figure A1. The interface
between the layers is of nonrigid bonding.

 

 

 

 

 

 

 

C2

Figure A1 - The geometry and coordinate system of a two-layer beam.

X ,0,Y,- is the local coordinate system of layer (0 which has the thickness of h,-. For simplic-
ity, it is assumed that h 1 is equal to hz. In addition, only beams with large length-to-thick-
tress ratio are considered in this analysis. Therefore, the classical beam theory can be used

in the following derivation.

Assume that the beam is subjected to a transverse load. The normal stress in each

layer at a generic cross-section Cl-C2 can be expressed as follows,

96

97

———— (A-l)

(1).]?1
.-,.—l

where A, and I,- are the cross-sectional area and moment of inertia of layer (i). F ,- and M,- are

the corresponding resultant force and moment and are shown in Figure A2.

 

 

 

C2

Figure A2 - The resultant forces and moments at the cross-section Cl-C2.

Because of the equilibrium requirements, the following relations can be obtained.

F, = F2 = F; M = M, +M2—Fh, (A-2)

M is the total moment applied at the cross-section Cl-C2. Moreover, it is assumed that
both layers have the same curvature. Therefore,

M, M2 M+Fh,

m = E = W. (A—3)
In the above equation, E I and E; are Young’s Modulus of layer (1) and (2), respectively.
Combining Equation (A-2) and Equation (A-3) together, and substituting the result into
Equation (A-l), it yields

.Lfl“ ham... (.-.,
x A, I,E,+12E2 l 1’ 2 A2 I,E,+1252 2 2'

Equation (A-4) can be rewritten as

98

Pk E Y
(i) = Jim—1L -__"L__. _
0" (Ar 1151+1252) 1151+125251Y1 (A 5a)
Fh E Y
(2) _ ( F r 2 2 J M
o _ .._-___ -————EY (A-Sb)
1 A2 I,E,+1252 I,E,+1252 2 2

Equations (A-5a) and (A-5b) are expressions for normal stress distribution in the individu-
al layers. It is important to note that they are true for any kind of interfacial bonding as
long as the two layers have the same curvature. In addition, the moment M, the moduli E,
and E2, and the geometrical parameters 1,- and A,- in Equations (A-Sa) and (A-5b) are inde-
pendent of bonding condition. In fact, only F is affected by the interfacing status. Define

the following two variables.

g = (11%). g =(.£-L"1§3’;) (M)
l A, I,I£,+lzli’2 ’ 2 A2 1,153,4-12152

If the composite beam has no interlaminar shear bonding on its interface, F will vanish.
Consequently, 5,, and Q are zeros. In Chapter 3, it was defined that the singular point is a
point at which the normal stress, transverse shear stress, or in-plane displacement is inde-
pendent of bonding condition. Since the second terms in Equations (A-Sa) and (A-Sb) are
not affected by the interfacial bonding, the singular points for the normal stress in each

layer can be determined by the following equations.

F FhIEIYl
g = _ _ __ =
1 ( A 1151+,252) for layer (1) (A-7a)
F ”,1521'2
g2 = (.3; _ W) = 0 for layer (2) (A-7b)

Furthermore, if it is assumed that layer (1) and layer (2) have the same cross-section, i.e.,
A, equals to A2 and I , equals to 12, the associated singular points for normal stress are lo-

cated at

 

(E +E )II ,
(n) _ l 2 I _
l — 1251 m 135’“ (I) 9 (A 83)
(E +E )hz .
(n) _ _ 1 2 -
Y2 _ 1 2 rn layer (2) . (A 8b)

By considering the equilibrium of a free body in the beam, the transverse shear
stress in each layer can be expressed as
II, M
<0 - 7 ii- I...‘
t" — IY1(A,dX, ’ldXIJdY,
(A-9)

1(2) :4”: 3.91-3112 4y
n -'-2 Izdx2 lzdxz ‘

Similar to the normal stress, the singular points for the transverse shear stress are

k
Y,m = 2Y,(") - -2—1 in layer (1) , (A-10a)
xi“ = 523+ 25"” in layer (2) . (A-lOb)

Since there is no X , variables explicitly involved in the derivation of the singular points for
the normal stress, the singular points for the in-plane displacements are the same as those

given in Equations (A-8a) and (A-8b).

2. Three-layer Laminates

For a three-layer beam as shown in Figure A3, only the case in which layer (1) and

layer (3) have the same geometry and material constants is presented.

100

 

 

C4

Figure A3 - The geometry and coordinate system of a three-layer beam.

Hence, it results in
(A-ll)
M1 = M33 15, = 53

The resultant force F and moments M, and M2 at a generic cross-section C3-C4 are

shown in Figure A4.

loading

    

Y:

01
02

03

Figure A4 - The resultant forces and moments at the cross—section C3-C4.

101

The normal stress in each layer can be expressed by the following equations.

,0) = .Lfly,
’ Ar ’1
M
(2) _. 2
6(3) _ _f__‘_‘gy
x - Al 11 3

 

_ = — = —- _ (A'13)
1,13, 1252 13153 2E,I,+l:’212

where M is the total moment exerted on the cross-section. After combining Equations

(A-12) and (A- 13) together, Equation (A-12) can be rewritten as

(2) F (h, + 1:2)52 M52

“x = '_Yz‘——72

a“) = (£_F(h,+h2)8, ,)_ ME, Y! - (A-14)
r A, 2£,I,+5212 215,1,“5212

25,1, + £212Y3

A3 21'3,1,+15212 3

If it is further assumed that the geometry of the layer (2) is the same as that of layer (1),

the singular points for the normal stress in each layer are

n h (213 +15) .
y,” = __‘ 2415, 2 rnlayer(1), (A-15a)
25 E
,m - -33_‘__3_‘_‘_2_’_ in layer (3), «NW

3 ‘ 245,

Y2“) = 0 in layer (2). (A-lSc)

 

102
It can be found that the singular point in layer (2) is a trivial solution and will not be con-'
sidered in the following analysis.
Following a similar fashion as used for two-layer beams, the singular points for the
in-plane displacements are the same as those given in Equation (A-15) while the singular

points for the transverse shear stress are as follows.

It
y,m = 21,90- 31 in layer (1), (A-16a)
ygn = $.24") in layer (3). (A-16b)

Assume that the material of interest is given in Figure 3.1. The singular points in
the [0/0] composite beam are given in Table A1.

The singular points in the [90/0] beam are listed in Table A2. Since the coordinate
of the singular point cannot be larger than half of the thickness of the associated layer,

there is no singular point in 90 layer.

The singular points in the [0/90/0] beam are shown in Table A3.

Table A1 - Singular points in a [0/0] beam.

 

 

 

 

 

 

normal stress and in-plane displacement yf") = %
laycrU)

transverse shear stress ylm _._ '96—!

normal stress and in-plane displacement Y 2(n) = £63
layer(2)

transverse shear stress Y-In = {’63

 

 

 

 

 

103

Table A2 - Singular points in a [90/0] beam.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

normal stress and in-plane displacement none
layerU)
transverse shear stress none
normal stress and in-plane displacement 1,20!) = "13,2
150
laycr(2)
transverse shear stress Yén = 29:2
Table A3 - Singular points in a [0/90/0] beam.
. . 51h2
normal stress and rn-plane displacement YO!) = _
2 600
laycrU)
33h2
transverse shear stress Y") = __
2 150
normal stress and in-plane displacement Yé") ___ _ §1_"2
600
layer(3)
33h2

 

transverse shear stress ym =

 

 

 

 

 

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