Table Of ContentCOMPOSITE MATERIALS SERIES
Series Editor: R. Byron Pipes, Center for Composite Materials, University of
Delaware, Newark, Delaware, USA
Vol. 1 Friction and Wear of Polymer Composites (K. Friedrich, Editor)
Vol. 2 Fibre Reinforcements for Composite Materials (A.R. Bunsell, Editor)
Vol. 3 Textile Structural Composites (T.-W. Chou, Editor)
Vol. 4 Fatigue of Composite Materials (K.L. Reifsnider, Editor)
The figures on the cover show a typical free-edge laminate under axial tension
and the nomenclature used throughout the text. In general, all six stress components
are present, as shown in the small inset (bottom, right). Central-plane delamination
cracks may be seen in the laminate. The source of the delamination is the interlaminar
normal stress distribution shown by the curve on the left.
C o m p o s i te M a t e r i a ls S e r i e s, 5
I N T E R L A M I N AR
R E S P O N SE OF
C O M P O S I TE
M A T E R I A LS
edited by
N.J. Pagano
Materials Laboratory, Air Force Wright Aeronautical Laboratories, Wright Patterson
Air Force Base, OH 45433, USA
ELSEVIER
Amsterdam — Oxford New York — Tokyo 1989
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Library of Congress Cataloging-in-Publication Data
Interlaminar response of composite materials/edited by N.J. Pagano.
(Composite materials series; 5)
p. cm. —
Includes bibliographies and indexes.
ISBN 0-444-87285-X (U.S.)
1. Composite materials—Delamination. I. Pagano, Nicholas J.
II. Series.
TA418.9.C6I56 1989 89-1543
620.1' 18—dc 19 CIP
ISBN 0-444-87285-X (vol. 5)
ISBN 0-444-42525-X (Series)
© Elsevier Science Publishers B.V., 1989
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iv
INTRODUCTION
In recent times, delamination has become the most feared failure mode in lami
nated composite structures. It can exhibit unstable crack growth, and while delamina
tion failure in itself is not usually a catastrophic event, it can perpetrate such a
condition due to its weakening influence on a component in its resistance to
subsequent failure modes. In graphite-epoxy structural components, delamination
resistance has become synonymous with "toughness".
What had begun in 1970 as somewhat of an academic curiosity blossomed in
recent years into a beehive of research activity. Studies of delamination became the
most prominent topic in composite mechanics research. In this book we attempt to
summarize the essence of this activity by presenting chapters by some of the major
contributors.
In chapter 1, N.J. Pagano and S.R. Soni give a historical and technical review of
the evolutionary development of response models that ultimately led to the so-called
global-local model along with some of the experimental findings which aided the
growth of the model. The global-local model is an approximate formulation based
on Reissner's variational theorem to predict the stress field in composite laminates
and is applied here to study the nature of the stresses in the neighborhood of a
laminate free edge.
Chapter 2 reviews some of the modeling efforts of A.S.D. Wang. At the heart of
this research is the use of finite-element modeling and concepts of classical fracture
mechanics to study the onset of free-edge delamination in composite laminates.
R.Y. Kim examines the response of laminates before and after onset of delamina
tion by experimental methods in chapter 3. Methods of experimental observation,
monitoring the onset of delamination, and failure theories, including mode-I and
mode-II interactions, are featured in this chapter.
J.M. Whitney discusses the topic of test methods to be used in the characterization
of delamination in chapter 4. Although the subject is experimental in nature,
significant analytical modeling is presented to justify the approaches taken and to
provide procedures for data reduction.
We have tried to adopt consistent terminology and notation as much as possible
within this book, however, one should consider each chapter as a self-contained unit.
For the convenience of the readers, one will observe that, throughout the entire
book, the following consistent notation is employed with regard to the important
free-edge problem in laminate elasticity:
loading direction = x,
width direction = y,
thickness direction = z,
laminate thickness = 2ft,
V
vi Introduction
while:
ply thickness = h in chapters 1, 3 and 4,
0
2t in chapter 2.
Parentheses ( ) and brackets [ ] are used interchangeably to denote laminate
stacking sequence. The two symbols have identical meaning. Finally, two of the
chapters (chs. 1 and 4) contain numerous mathematical equations and symbols. For
each of these two, a detailed listing of symbols is provided at the end of the chapter.
April, 1989 NJ. Pagano
Interlaminar Response of Composite Materials
edited by Ν J. Pagano
© Elsevier Science Publishers B.V., 1989
Chapter 1
Models for Studying Free-edge Effects
NJ. PAGANO
Materials Laboratory, Air Force Wright Aeronautical Laboratories, Wright Patterson Air Force Base,
Ohio 45433, USA
and
S.R. SONI
AdTech Systems Research Inc., 1342 North Fairfield Road, Dayton, Ohio 45432, USA
Contents
Abstract 1
1. Introduction 2
2. Effective-modulus theory 3
3. Free-edge boundary-value problem 4
4. Finite-difference solution 5
5. Analysis of stacking sequence effect 11
6. Primitive delamination model 16
7. Elastic plate model for σ 17
ζ
7.1. Model formulation 18
7.2. A look at an incorrect use of the present model 22
8. Extended Reissner variational principle for composite laminates (local model) 25
8.1. Laminate variational principle 27
8.2. Model development 29
8.3. Summary of field equations and boundary conditions 34
8.4. Solution for the free-edge problem 36
8.5. Comparative results 43
8.6. Model summary 48
9. The global-local laminate variational model 49
9.1. Global-local model results 55
9.2. Final comments on the model 61
10. Recent work in free-edge models 61
List of symbols 62
References 66
Abstract
A historical/technical review is given, depicting the development of analytical
models of the free-edge delamination phenomenon. Emphasis is placed on the role
ι
2 Ν J. Pagano and S.R. Soni
and importance of the free-edge problem in laminate elasticity in fostering an
understanding of interlaminar stresses and their influence on composite response.
From the early modeling work of Hayashi in 1967, and the definitive experiments
of Foye and Baker in 1970, we trace analytical developments over the past two
decades. The concept of ply elasticity or effective-modulus representation is dis
cussed, as well as its consequences in laminate modeling. The first elasticity solution
of the free-edge problem by Pipes and Pagano using finite differences is then
described. This solution has been very instrumental in defining the general character
of the interlaminar stress field in the neighborhood of a free edge. We then provide
results for the elementary modeling of the effect of stacking sequence on laminate
response and derivation of simplified equations to either optimize or minimize this
effect in test specimens. Thence, a description of a model based upon the concept
of a plate on a smooth foundation is given to capture the essence of the distribution
of interlaminar normal stress, its boundary layer zone, and its implication on
initiation of delamination. This model leads naturally to the formulation of a
variational theorem for laminates that provides an accurate way to compute com
posite stresses. The chapter culminates with a derivation of the global-local model,
which provides a practical way to describe the stress field in a multi-layered
composite laminate, and a review of some recent modeling activities that have their
basis in the concepts given earlier here. Numerous results are shown for stress fields
within laminates which have both practical and theoretical importance.
1. Introduction
The study of delamination phenomena in structural composite laminates began
with analytical and experimental observations of the response of such bodies in the
vicinity of a free edge. In 1967, Hayashi [1] presented the first analytical model
treating interlaminar stresses in what has come to be known as the "free-edge
problem". Characteristically, this work focused on the computation of interlaminar
shear stress, as in the early stages of composite research interlaminar and delamina
tion effects were viewed as being synonymous with interlaminar shear. The presence
of interlaminar normal stress, being of a more subtle origin and also seemingly
defying common intuition, was not appreciated until many years after the pioneering
work of Hayashi.
The development of the Hayashi model was based upon the implicit assumption
that the in-plane stresses within a given layer did not depend upon the thickness
coordinate. The magnitude of the maximum interlaminar shear stress was calculated
to be a relatively large value in a glass epoxy [0°/90°] laminate. Unfortunately,
however, owing to the omission of the interlaminar normal stress, the computed
stress field within each layer does not satisfy moment equilibrium.
The first reported experimental observations involving free-edge delamination
were made by Foye and Baker [2], In that work, tremendous differences in fatigue
life of boron-epoxy composite laminates as a function of layer stacking sequence
were reported. Severe delaminations were witnessed in that work and were identified
as the primary source of strength degradation in fatigue.
Models for studying free-edge effects 3
From this early work to the present time, the free-edge laminate problem has
been the most prominent device utilized in the study of composite delamination.
Hence, in this chapter we shall examine the various models that have been developed
through the years to predict the stress field in such a body. Our emphasis shall be
placed on the work which led to the development of the global-local model, a
model which attempts to circumvent the overwhelming difficulties and complexities
associated with stress analysis of multi-layered composite laminates. In this model,
three-dimensional elasticity problems are transformed into two-dimensional prob
lems in a self-consistent approach that features realistic satisfaction of boundary
and interface conditions. Other methods of analysis, including finite-element model
ing, are described in subsequent chapters in this book.
2. Effective-modulus theory
At this point we shall provide a discussion regarding the level of abstraction of
the theoretical models treated in this chapter, as well as in the remaining chapters.
This level of abstraction of modeling detail is called effective-modulus theory, or
"ply elasticity", an accurate descriptive term coined by A.S.D. Wang (see ch. 2). At
the heart of this concept is the representation of each layer within a laminate as a
homogeneous, anisotropic, usually elastic, body. The laminate itself is viewed as a
collection of such layers which are, in most cases, bonded together at their interfaces.
Thus, the ensuing models lead to a piecewise constant representation of the stiffness
matrix C in the (z) thickness direction, i.e., discontinuities in Cy occur at the
i}
various interfaces. This form of representation is, admittedly, artificial, however, it
is widely used, almost exclusively so, in practice and research, and has proven to
be a valuable tool in composite mechanics.
In a physical composite laminate, each layer is reinforced by hundreds or
thousands of fibers per square inch of cross-sectional area. Obviously, modeling
such details in a precise manner is an impossible task. It is equally clear that the
effective-modulus predictions cannot be interpreted as providing a precise point-by-
point description of the stress field. Only very few studies, such as refs. [20,21],
have attempted to study the relationship between "microstresses" (constituent
material level stresses) and the stresses given by effective-modulus theory in the
presence of steep stress gradients. These studies were accomplished approximately
a decade ago and were susceptible to errors caused by computer limitations. The
studies [20,21] were also necessarily of narrow scope, however, they do tend to
support the use of effective-modulus theory provided the results are averaged over
distances comparable to unit-cell dimensions. Also note that no geometric sin
gularities were treated in these studies - only the (artificial) free-edge singularities
of effective-modulus theory were considered. Much could be learned by repeating
or extending these studies with the aid of modern computer facilities.
There is considerable evidence to suggest the utility, if not precision, of effective-
modulus theory, even in the presence of macroscopic stress gradients. The solution
of the free-edge problem and the successful predictions of problems involving local
damage, such as layer flaws, testify to this success. It should also be observed,
4 Ν J, Pagano and S.R. Soni
however, that the effective-modulus models have been correlated with failure by
such empirical failure laws as those of classical elastic fracture mechanics and the
average-stress criterion. This does not prove that these models are fundamentally
sound as direct experimental evidence regarding the detailed localized influence of
these geometric features simply does not exist. Thus the physics of the failure process
in composites remains as much a mystery as in homogeneous materials and our
understanding of failure is strongly dependent on experimental information. Hence,
the use of effective-modulus modeling should be viewed as an approximation, rather
than a fundamental truth, and as such requires experimental confirmation of the
phenomenon being studied.
3. Free-edge boundary-value problem
Consider the class of boundary-value problems in which a prismatic symmetric
laminate having traction-free edges at y = ±b and surfaces z = ±h is loaded by
tractions applied only on its ends χ = const., such that all stress components are
functions of y and ζ only. Each layer is composed of unidirectional fiber-reinforced
material such that the fiber direction is defined by its angle θ with the x-axis, where
0(z) = 0(-z). We treat each layer as a homogeneous anisotropic body represented
by its effective moduli.
The elastic properties of various composite materials considered in the examples
in this chapter are shown in table 1. These are not necessarily exact properties
measured in laboratory tests, but values that have been used in conducting analytical
studies throughout the course of the work reported herein.
TABLE ι
Properties of composite materials used. Ε symbolizes
Young's modulus and G the engineering shear modulus.
The subscripts denote the L, Τ, ζ directions, where L is the
fiber direction, Τ the in-plane transverse direction, and ζ
the thickness direction, v is the Poisson ratio measuring
tj
strain in the j direction caused by uniaxial stress σ,.
Property Material
I II III
(GPa) 137.9 137.9 206.85
Er (GPa) 14.48 9.69 14.82
E (GPa) 14.48 9.69 —
z
(GPa) 5.86 5.52 4.69
G (GPa) 5.86 5.52 —
Lz
G (GPa) 5.86 4.14 —
Tz
0.21 0.3 0.19
"LT
0.21 0.3 —
0.21 0.6 —
Models for studying free-edge effects 5
Fig. 1. Laminate geometry.
4. Finite-difference solution
The first attempt to solve the free-edge problem by use of the theory of anisotropic
elasticity without any simplifying assumptions was given by Pipes and Pagano [3].
In that presentation, a laminate consisting of four unidirectional fibrous composite
layers, two with their axes of elastic symmetry (fiber direction) at +Θ and two at
-0 to the longitudinal laminate axis was considered. Figure 1 shows the laminate
geometry and coordinate system. Each layer is of thickness h. The constitutive
0
relations for each layer with respect to the laminate coordinate axes are given by
"C C, c 0 0 ~ε~
n 2 13 χ
°~y C22 C23 0 0 c2t By
c C23 C33 0 0 ε
13 ζ (1)
0 0 0 C44 C45 0
Jyz Ifyz
Τχζ 0 0 0 C45 C55 0 Ύχζ
_c16 C26 c36 0 0 Ifxy
The thirteen anisotropic material constants are related to the nine constants with
respect to the material symmetry axes through the well-known stiffness transforma
tion law. The layer-strain-displacement relations are
(2)
7yz = W,y + V,z> 7xz = W,x + Uz, 7xy = V, + U.
x y
