24
It may be noted that the above equations are nonlinear
and lead to multiple solutions. Solving these two equations
we obtain the coefficients c and d as follows:
One possible solution set is:
P
(2.10)
c = 
2 AE
d = 0
(2.11)
This solution describes the behavior of the cracked
portion of the beam where the compressive load is less than
the buckling load Pcr. Since d=0, there is no deformation in
z direction. The relation between compressive load P and
axial displacement Uo takes a simple form:
Px
(2.12)
The end shortening Sc at x=2a is given by:
(2.13)
The other possible solution set is:
3 P 8n2I
(2.14)
c = +
2 AE A(2a)
107
At x=0
w = w,, y/ = y/x, V = VX, MM]
(A. 9)
At x=a
w=w2, if/ = iÂ¡/2, V = V2, M=M2 (y. 10)
Solving Equations A.7 and A.8 and using the boundary
conditions in Equations A.9 and A.10, we obtain relations
between Vir Mi and wir 0X. Then FZi and C can be expressed in
terms of wir Gi as:
l a a PA<*\
FZ1 =tK'i ~w2  V2 TTTT^)
A, a
2 24Â£,/1
1 i a Poh\a\
FZ2 = + V', +w2 + ^2 +T77Tr)
4 a
2 24Â£,7,
_ w, A,/, a w,
C, = pr + ^) ( L_L + t ) + '
24
a 4/4, 2/4,
+ PoV3 PoV
2 4Ax a 48Â£,/,/4,' 4
2/4, a 4/4, 2/4,
a /?0/2,a3 PoKa
+ ( r + LJ) + 1 1
4/4, a 48 Â£,/,/4,
(411)
(412)
(413)
4
(414)
78
denoted as <1>, <2> and <3>. Free body diagrams of the
three elements are shown in Figure 4.4. At each end of the
element there are three displacements, u, w, and y, and
three corresponding forces, Fx, Fzr and C. The u and w are
the displacements in the x and z directions and y is the
rotation of the cross section. In the list of forces C
represents the couple corresponding to the rotation y. We
use the Timoshenko beam theory in modeling the beams. The
equations of equilibrium of a general element are given in
the Appendix. These equations can be easily solved as shown
in the Appendix to obtain a relation between the six forces
acting at the two ends of the element and the six
corresponding displacements. In order to relate the forces
in the three different elements we use force and moment
equilibrium equations, and compatibility equations at the
junctions (nodes) where the elements meet. Further, the
boundary conditions at the ends of the beam can also be
implemented. This procedure is similar to assembling
elements in the Finite Element Method except the stiffness
matrix of each element is obtained exactly by solving the
differential equations of equilibrium. The forces and
displacements in each element are shown in Figure 4.4. It
t
should be noted that F is the contact force acting on the
beam, and because of symmetry F/2 is assumed to act at Node
1 in Element 1. The compatibility relation for the axial
method to provide this reinforcement is to use throughthe
thickness stitching.
When the delaminated composite is under compression,
it may undergo postbuckling. When the energy release rate
is greater than critical energy release rate, the
delamination will propagate in an unstable manner and
finally cause catastrophic failure of the composite. In
this dissertation analytical and finite element methods are
proposed to investigate the effect of throughthethickness
stitching on the postbuckling of a delaminated composite.
In particular the studies focus on the effect of stitching
on buckling load, loaddisplacement behavior, and energy
release rate.
When a delaminated composite is subject to a low
velocity impact, the crack may propagate and finally cause
catastrophic failure. In this dissertation an analytical
model is proposed to understand the effect of stitching on
improving impact damage resistance. In particular the
studies focus on the effects of stitching when the impact
energy, delamination length and location, and the fracture
toughness are varied.
vi
Energy release rate G(lb/in.
68
Figure 3.12 Energy release rate vs. end shortening
(a=l" h/H=l/4)
83
The expressions for Blf B2 and B3 used in Equation 4.14a are
as follows:
Gj A\ 12EX /,
G2A2 12 E2I2
B 3 = +
G3A3 12 E3I3
Note that Ei and Gi are the equivalent Young's modulus and
shear modulus, respectively, and and Ii are area of cross
section and moment of inertia of the corresponding beam
element.
In Equation 4.14 the forces f2 f5 are known. The
force fi that is related to the contact force F is the
unknown. On the other hand the deflection w: (same as the
variable q used in impact equations) can be treated as
known, and the other five displacements are unknowns. Thus
we have five equations for five unknowns. The equations can
be solved for a given wL to determine fx or the contact
forcedisplacement Fq relations can be developed.
The flowchart in Figuation 4.6 describes the procedure
used in developing the Fq relations. Further at each
displacement increment the energy release rate G is computed
Stitch strains
69
Figure 3.13 Stitch strains vs. end shortening
(a=l" h/H1/4)
Stitch: Glass 1 (16 ssi)
109
c,=
2 AL
Q 1^2^2 \ ^4
+ V'^ + ^^) +
4A2 a 2 A2
+ ,<* E2)  Poh2a3 Pohia
4 4A2 a ME212A2 4
(4.19)
r ~^w3
U4
/ a Â£2A\ w4
r + ^3( r ) + 17
2^ 44 a 2^
a Â£2/2 P0/22a3 p0h2a
+ ( r + ) + r
442 48Â£2/242 4
(A.20)
A2 is area of cross section, E2 is Young's modulus and G2 is
shear modulus in part <2> and A2 is given by:
4, =
a
G2 A2
12 E2I2
The free body diagram of part <3> is shown in Figure A.5:
Figure A.5 Free body diagram in part <3>
77
length is denoted by a and L is uncracked length in Figure
4.3.
Figure 4.2 A delaminated beam with stitches
subject to impact load F
F/2
Figure 4.3 Right half side of
specimen in Figure 4.2
The right half beam can be divided into three elements
Element 1, Element 2 and Element 3. In Figure 4.3 they are
17
subjected to compressive loads or to lowvelocity impact.
The purpose of the present research is to use analytical
and finite element methods to study the effect of stitching
on postbuckling and lowvelocity impact response of
composite structures. In particular the study will focus on
the effect of stitching on buckling load, energy release
rate, and impact damage resistance.
The organization of this dissertation is as follows:
In Chapter 2 an analytical solution for postbuckling of a
delaminated composite beam with stitching will be derived.
In Chapter 3 a finite element analysis will be given
related to postbuckling of a delaminated composite beam
with stitching. In Chapter 4 an analytical model for impact
response of stitched composite beam is presented.
Conclusions drawn from this study and discussions for
future research are presented in Chapter 5.
APPENDIX
DERIVATION OF FORCEDISPLACEMENT RELATIONS
IN THE BEAM ELEMENT
In this appendix the derivations for external forces
FXi, Fzi and moments Ci described in Chapter 4 are given:
h H
a
Figure A.l Free body diagram in part <1>
Consider Part <1> of
the beam shown
in
Figure A.l
. In
Figure A.l
Pif VJf and
Mi (i
=1,2) are
internal force
and
moment. p0
is shear
force
provided
by
stitches.
The
relations
between external
forces
and
internal force
resultants
at left side
of a
beam and
right
side of a
beam
are given as follows:
F = P
1 x\ 1 1
fa = K
C, = M, (A1)
104
3
thickness of the laminate. Only small amounts of outof
plane reinforcement are needed in order to significantly
change the mechanical properties of the structure. A
stitched laminate can be formed by using a sewing machine
to stitch fiber yarns into a composite preform before
curing. Having a stitch in a laminate does not greatly
alter the original laminated structure. Further, stitching
can be done very fast in automated machines, and is
suitable for large structures such as aircraft wings.
Under compression the delaminated part of composite
laminate may undergo postbuckling behavior. A delamination
may propagate when a foreign body drops on a delaminated
composite. Many researchers have studied these subjects by
using analytical and numerical methods. Some studies have
investigated the effect of stitching on mechanical and
impact properties, such as interlaminar strength and
fracture toughness.
A literature review is provided in the next section.
1.2 Literature Review
The literature review is divided into three parts: (a)
postbuckling behavior of delaminated composites; (b)
impact response of composite structures; and (c) through
thethickness stitching of composite laminates.
102
1. From the analytical models a single nondimensional
parameter K' was identified:
A,E,NB
K
It may be noted that K' is actually the ratio of the
stitch stiffness and the equivalent Young's modulus of
the beam. Thus one needs to use high density stitches
with stiffer yarn material if the flexural modulus of
the laminate is also higher. It is found that stitching
increases the buckling load, and reduces the energy
release rate at the cracktip.
2. The FE models also support the conclusions of the
analytical models. The strains in the stitches depend on
the axial stiffness of the stitches, and they determine
if the stitch will break during the postbuckling
process and thus reduce the effectiveness of stitching.
3. From the static simulations of a delaminated beam it is
found that the percentage increase in apparent fracture
toughness is more for laminates with lower inherent
fracture toughness. The impact force at which the
delamination begins to grow is not dependent on
stitching. However the extent of delamination growth
depends on the stitching parameters. Stitching is
Load P (lb)
63
Figure 3.7 Compressive load vs. end shortening
at crack length (a=l in.)
EFFECT OF THROUGHTHETHICKNESS STITCHING ON
POSTBUCKLING AND IMPACT RESPONSE OF
COMPOSITE LAMINATES
By
HUASHENG ZHU
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1999
To my parents Deqian Zhu and Guirong Liu
and my beloved YingFeng Liu
ACKNOWLEDGMENTS
The author would like to express her sincere gratitude
to her advisor, Dr. Bhavani V. Sankar, for his patient
guidance, constant encouragement, and endless support of
her study.
The author is also grateful to Dr. Nicolae D.
Cristescu, Dr. Peter G. Ifju, Dr. Edward K. Walsh, and
Dr. Reynaldo Roque for serving as her committee members.
Finally the author would like to express her gratitude
to her parents, brothers, and her love, Mr. YingFeng Liu,
who share the happiness, sadness, difficulty with the
author, for their encouragement, support, and patience.
11
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ii
ABSTRACT v
CHAPTERS
1 INTRODUCTION AND LITERATURE REVIEW 1
1.1 Background 1
1.2 Literature Review 3
1.2.1 Research on Postbuckling of Delaminate
Composite 4
1.2.2 Research on Low Velocity Impact Response and
Damage of Delaminated Composite Materials...9
1.2.3 Research Related to Delaminated Composite
Materials with Stitches 11
1.3 Objectives and Scope of the Present Study 16
2 AN ANALYTICAL SOLUTION FOR POSTBUCKLING OF
A DELAMINATED COMPOSITE BEAM WITH STITCHING 18
2.1 Basic Assumptions 19
2.2 Analytical Model 20
2.2.1 Delaminated Composite Beam
without Stitching 20
2.2.2 Delaminated Beam with ThroughtheThickness
Stitches 28
2.3 Results and Discussion 34
3 FINITE ELEMENT ANALYSIS OF A DELAMINATED COMPOSITE
WITH STITCHING 44
3.1 Specimen 45
3.2 Finite Element Model 46
3.3 PostBuckling Analysis 50
3.4 Energy Release Rate 52
3.5 Results and Discussion 55
iii
4
EFFECT OF STITCHING ON LOWVELOCITY IMPACT RESPONSE...71
4.1 Basic Assumptions 73
4.2 Analytical Model 75
4.2.1 Relation between Contact Force and
Beam Deflection 76
4.2.2 Impact Response 85
4.3 Numerical Examples, Results, and Discussion 88
5 CONCLUSIONS AND FUTURE WORK 101
5.1 Summary 101
5.2 Suggestions for Future Work 103
APPENDIX DERIVATION OF FORCEDISPLACEMENT RELATIONS
IN THE BEAM ELEMENT 104
REFERENCES 112
BIOGRAPHICAL SKETCH 118
IV
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
EFFECT OF THROUGHTHETHICKNESS STITCHING ON
POSTBUCKLING AND IMPACT RESPONSE OF
COMPOSITE LAMINATES
By
Huasheng Zhu
December 1999
Chairman: Bhavani V. Sankar
Major Department: Aerospace Engineering, Mechanics, and
Engineering Science
Fiber composite materials have very high specific
strength and stiffness. This structural efficiency enables
composite materials to be used in a wide variety of
applications. However, laminated composites have also some
disadvantages such as poor interlaminar strength, low
impact resistance, and poor delamination resistance. During
the manufacture of fiberreinforced composite laminates,
imperfections can cause initial delamination, and the
impact on laminated composites by foreign objects during
service can cause delamination propagation. Delamination is
one of the most dominant forms of damage due to lack of
reinforcement in the thickness direction. An effective
v
method to provide this reinforcement is to use throughthe
thickness stitching.
When the delaminated composite is under compression,
it may undergo postbuckling. When the energy release rate
is greater than critical energy release rate, the
delamination will propagate in an unstable manner and
finally cause catastrophic failure of the composite. In
this dissertation analytical and finite element methods are
proposed to investigate the effect of throughthethickness
stitching on the postbuckling of a delaminated composite.
In particular the studies focus on the effect of stitching
on buckling load, loaddisplacement behavior, and energy
release rate.
When a delaminated composite is subject to a low
velocity impact, the crack may propagate and finally cause
catastrophic failure. In this dissertation an analytical
model is proposed to understand the effect of stitching on
improving impact damage resistance. In particular the
studies focus on the effects of stitching when the impact
energy, delamination length and location, and the fracture
toughness are varied.
vi
CHAPTER 1
INTRODUCTION AND LITERATURE REVIEW
1.1 Background
Fiber composite materials have very high strength and
stiffness but low density. This structural efficiency
enables these composite materials to be used in a wide
variety of applications, especially in aerospace
structures. However, advanced composites also have some
disadvantages such as poor interlaminar strength, low
damage resistance, low damage tolerance, and poor
delamination resistance. Since most of the commonly used
advanced composites are made by laying up fibrous
reinforcement in a surrounding matrix and the mechanical
properties between the plies are dominated by a relatively
weak matrix, this weakness causes low damage resistance and
low damage tolerance and finally results in low
compressionafterimpact strength.
A delamination is an interface crack or a debond
between two adjacent plies. It is one of the critical
failure modes in composite materials. During the
manufacture of fiberreinforced composite laminates,
1
2
imperfections such as air entrapment or resin starvation
may cause initial delamination in the composite. Impact on
laminated composites by a foreign body during service may
also cause delamination while the surface of the composite
structure remains undamaged to visual inspection. This kind
of delamination can randomly occur somewhere in the
structure. No matter where it occurs it may decrease the
overall stiffness and loadcarrying capacity of the
composite structure. The growth of delamination cracks
under the subsequent application of external loads leads to
the rapid deterioration of mechanical properties and may
cause catastrophic failure of the composite structure. One
of the most dominant forms of damage is delamination due to
lack of reinforcement in the thickness direction. There are
some approaches that can be adopted in order to improve the
strength in the thickness direction. Using fully integrated
3D composites such as weaves and braids is one of the
methods; however, due to their complexity, limited shape
ability, and processability, their applications have been
limited. Inserting a pin in the thickness direction can
also improve reinforcement in the thickness direction, but
currently it is difficult to use this technique in large
structures. Another method for providing a reinforcement in
the thickness direction is to use stitching through the
3
thickness of the laminate. Only small amounts of outof
plane reinforcement are needed in order to significantly
change the mechanical properties of the structure. A
stitched laminate can be formed by using a sewing machine
to stitch fiber yarns into a composite preform before
curing. Having a stitch in a laminate does not greatly
alter the original laminated structure. Further, stitching
can be done very fast in automated machines, and is
suitable for large structures such as aircraft wings.
Under compression the delaminated part of composite
laminate may undergo postbuckling behavior. A delamination
may propagate when a foreign body drops on a delaminated
composite. Many researchers have studied these subjects by
using analytical and numerical methods. Some studies have
investigated the effect of stitching on mechanical and
impact properties, such as interlaminar strength and
fracture toughness.
A literature review is provided in the next section.
1.2 Literature Review
The literature review is divided into three parts: (a)
postbuckling behavior of delaminated composites; (b)
impact response of composite structures; and (c) through
thethickness stitching of composite laminates.
4
1.2.1 Research on Postbuckling of Delaminated Composites
In many cases, when delaminated composites are under
external compressive loads the delaminated structure will
have the capability of carrying loads beyond their buckling
loads and may fail in the postbuckling regime. Therefore
bifurcation analysis may not be adequate in describing the
whole damage process of delaminated composites up to final
failure; a postbuckling analysis may be required.
The initial development and growth of delamination in
compressed composites are affected by various geometrical
parameters, loading conditions, material properties and so
on. The different combinations of these parameters can
result in different types of buckling and delamination
growth behavior. Most studies were based on the static
postbuckling solution of homogeneous or laminated plate
theory. The energy release rate (G) is considered as the
critical parameter and different methods were used to
calculate the energy release rate. An assumption that uses
Griffithtype fracture criterion with a specific constant
critical fracture energy release rate (Gc) was adopted in
these studies. When the energy release rate G is greater
than critical fracture energy release rate Gc the
delamination in the composite may propagate.
5
Simple analytical and finite element models seem to be
popular in analyzing onedimensional delaminations. Chai et
al. [1] used an analytical method to obtain energy release
rate by differentiating the strain energy with respect to
delamination length. Yin and Wang [2] used the Jintegral
method to obtain an expression for energy release rate in
the postbuckled delaminated plate. Whitcomb [3] adopted
the crack closure technique and analyzed the effects of
various parameters on energy release rate in a beamlike
structure using geometrically nonlinear finite elements.
Sheinman and Soffer [4] investigated the effects of
bendingextension coupling and initial imperfection on the
postbuckling behavior of a composite beam with various
delamination geometries. It was found that the coupling
effect significantly reduced the buckling load and
increased the postbuckling deformations, and the global
postbuckling deformation was shown to be very sensitive to
initial imperfections.
Kardomateas and Schmueser [5] proposed a one
dimensional beamplate model that accounts for the
transverse shear effect. Analytical solutions for the
critical instability load and the postbuckling deflections
were obtained with aid of the perturbation technique.
Kardomateas [6] also modeled the postbuckling behavior of
6
a laminate with a thin delaminated sublaminate through a
procedure that is based on large deflection theory of the
delaminated layer.
Chen [78] proposed a shear deformation theory and
obtained energy release rate by a variational energy
principle. He found that the energy release rate is larger
with transverse shear effect included for the same applied
load. Therefore the initiation of delamination growth will
occur at a lower applied load than that evaluated with the
classical lamination theory. He also found that the
magnitude of the transverse shear effects depends on the
delamination location and size.
Davidson and Krafchak [9], Davidson [10], Davidson and
Krafchak [11] used crack tip element analysis to determine
total energy release rate and individual mode I and II
energy release rate in different loading conditions or
different delamination cases.
Kyoung and Kim [12] presented an energy method to
obtain an analytical solution for determining the buckling
load and the growth of delaminated beamplate structures.
In their study the delamination was arbitrary located. They
also found that the shear effects decrease the buckling
loads and increase energy release rate. The results from
their analysis show that the buckling load of a center
7
located delamination is higher than an offcenter
delamination.
Lee et al. [13] proposed a finite element method based
on a layerwise laminated composite plate theory for
solving the postbuckling problem. In their study a contact
algorithm, which overcame the physically unallowable
overlapping between delaminated surfaces, was used.
There are also several works concerning two dimension
postbuckling analysis. Yin and Fei [14] presented an
elastic postbuckling analysis of a delaminated circular
plate under axissymmetric compression along its clamped
boundary. They found that certain features of the post
buckling behavior are qualitatively similar to the buckling
of an axially loaded beam plate containing a one
dimensional delamination. Whitcomb and Shivakumar [15] used
virtual crack closure technique to calculate the total
energy release rate in a locally postbuckled laminate with
embedded delamination. They found that there is a large
variation of energy release rate along the delamination
front for square and rectangular plates, and delamination
growth depends on current delamination aspect ratio, the
strain level, and the size of the delamination.
Suemasu [16] investigated an effect of multiple
delaminations of composite panels on the buckling loads by
8
using an analytical approach and verified the model using
experiments and finite element methods. He pointed out that
the buckling load reduces significantly due to the
existence of multiple delaminations.
Naganarayana and Atluri [1718] used a multiplate
model, in conjunction with a 3noded quasicomforming shell
element, to model the delaminated plates. The Jintegral
technique was used for delamination growth prediction in
term of pointwise energy release rate distribution along
the delamination edge. The effects of structural parameters
such as delamination thickness, size and shape on the post
buckling behavior and on the delamination growth were
examined.
Shen and Williams [19] studied a postbuckling
analysis of laminated plates subjected to combined axial
compress and uniform temperature loading. The analysis used
a perturbation technique to determine buckling loads and
postbuckling behaviors in antisymmetrically angleply and
symmetrically crossply laminated plates. Klug et al [20]
adopted Mindlin plate finite element to perform the post
buckling analysis and to compute energy release rate at the
delamination front with aid of the crack closure method in
composite laminates which have elliptic delamination shape.
The results show that this method is efficient and accurate
9
compared to threedimensional finite method for calculation
of the energy release rate.
1.2.2 Research on LowVelocity Impact Response and Damage
of Delaminated Composite Materials
Lowvelocity impact response and damage caused by low
velocity projectile impact has received attention from some
researchers. When the lowvelocity foreign body drops on a
structure such as the surface of an airplane wing, it may
cause damage such as a delamination which is invisible on
the surface and difficult to detect. Therefore
understanding the lowvelocity impact response of a
delaminated composite is very important, and a method is
needed to predict the damage due to lowvelocity impact.
Graves and Kantz [21] used the maximum shear stress failure
criterion to estimate the damage area. Clark [22] provided
a model only for qualitative prediction of the delamination
size. Grady and Sun [23], and Grady and Depaola [24]
provided an estimation of the delamination growth due to
impact. Choi et al.[25] adopted a dynamic finite element
method associated with failure analysis to predict
threshold of impact of damage and initiation of
delamination. Sankar and Hu [26] and Hu [27] used a finite
element method to analyze dynamic delamination growth in a
10
composite beam. The delaminated beams were modeled as two
offset beams and spring elements were used to connect the
beams in the uncracked portion. The dynamic energy release
rate was computed from the strain energy in the cracktip
element. Razi and Kobayashi [28] performed finite element
analysis and experiments to study damage growth and
distribution in crossply laminate beams and plates. A
quasistatic finite element analysis coupled with a Gu> Guc
criterion was proposed for the delamination growth, and Gu <
Giia criterion was used to arrest crack propagation. Sun and
Yih [29] investigated the quasistatic characteristics of
impact responses and the impactinduced delamination growth
of composite laminates subjected to lowvelocity impact.
They used a springmass model to predict contact force
history and the peak force was associated with the energy
release rate. Thermal loads were included in computing
energy release rate. Wang and VuKhanh [30] also used the
finite element method to model carbon fiber/PEEK crossply
damaged laminates. They found that a delamination occurred
in the form of a Mode IIdominated unstable crack growth
and subsequent arrest.
Kwon and Sankar [3132] performed experiments to study
the lowvelocity impact damage of quasiisotropic and
crossply graphite/epoxy composite laminates. They found
11
that the impact force history and delamination radius could
be predicted using the results from a static indentation
flexure test. Sankar [33] reviewed the necessary concepts
for understanding lowvelocity impact response and damage
in fiber composite materials and gave several algorithms
for predicting the impact force history to different
degrees of approximations. Sankar [33] presented a semi
empirical method for predicting impact damage in
composites.
1.2.3 Research Related to Delaminated Composite Materials
with Stitches
In transversely stitched composites the delamination
is not completely separated but is held together by the
stitches. Therefore the propagation of cracks can be
prevented in stitched composite materials. Several studies
have investigated the effect of stitching on mechanical and
impact properties, failure loads, interlaminar strength,
and fracture toughness.
In general there are three types of stitches as shown
in Figure 1.1: modified lock stitch, chain stitch, and
standard lock stitch. Some parameters of stitches are:
Stitch density, stitch yarn, and yarn material density. We
define the stitch density in a composite laminate by the
12
number of stitches per square inch and represent this
density by the stitching pattern as: (number of stitches
per inch) X (spacing between two stitch lines), for
example, 4X1/4 means a stitch density of 16 where there are
4 stitches per inch and distance between two adjacent
stitch rows is 1/4 in. Stitch yarn is the bobbin yarn used
in the stitching process. The material density of the
stitch yarn is expressed as denier. One denier is the mass
in grams of 9000 meters of yarn.
Modified Lock Sliteh
m
Needle
thread
Bobbin
thread
Standard Lack Stitch
nrnmr
JUUL
Needle
thread
Sohain
thread
Chain Stitch
Figure 1.1 Types of lock stitches
(Courtesy, Sharma and Sankar [46])
Sawyer [34] used an experimental method to determine
the effect of stitching on the static load of bonded
composite single lap joint. Up to 38% improvement in static
failure load compared with unstitched results was obtained
by a single row of stitches near the end of the overlap.
13
Additional row of stitching or different stitch spacing has
little effect on static joint failure load.
Su [35] improved the delamination resistance in
composite laminates by the use of thermoplastic matrix
resin and stitching. There was a 2030% further improvement
of delamination resistance by using stitches. Dexter and
Funk [36] conducted an experimental investigation on impact
resistance and interlaminar fracture toughness of stitched
guasiisotropic graphite/epoxy composites. The stitch
, TM
materials were polyester or Kevlar yarns with several
parameters. They found that a significant drop in damage
areas of stitched laminates compared to unstitched
laminates for the same impact energy. The results shown
that interlaminar fracture toughness Glc of stitched
laminated with the most effective Kevlar yarn was 30 times
higher than unstitched laminates.
Pelstring and Madan [37] proposed a semiempirical
formula to relate a damage tolerance of a composite
laminate to stitching parameters. The results showed that
Mode I critical energy release rate of a stitched laminate
was 15 times higher than an unstitched laminated.
Dransfield et al. [38] reviewed the effect of stitching on
improving some mechanical properties such as compress
afterimpact strength and delamination resistance of
14
composite materials. The advantages and disadvantages of
adopting a stitch in a composite structure and the
necessary technigues of manufacture were examined. Shu and
Mai [3940] used analytical approaches to investigate the
effect of stitching on buckling load and energy release
rate of thin film structure. They assumed that the stitches
followed a Winkler elastic foundation type of stress
separation relation. The results shown that the strength of
composite laminates under edgewise compression can be
significantly increased. Jain and Mai [41] proposed two
analytical models to study the effect of stitch parameters
such as stitch density, stitch thread diameter, and
matrix/stitch interfacial shear stress etc. on improving
Mode I delamination toughness. They found that a large
matrix/stitch interfacial shear stress, high stitch density
and a small stitch thread diameter were desirable in order
to maximize the delamination growth resistance. Jain and
Mai [42] also used an analytical method to study the effect
of stitching parameters on Mode II energy release rate in a
laminated composite and found that throughthickness
stitching can improve Mode II fracture toughness also. Chen
and Sun [43] used a finite element method to propose an
effective energy release rate for measuring Mode I fracture
toughness of a stitched composite laminate. Kang and Lee
15
[44] did an experiment to study the effect of stitch on
mechanical and impact properties of woven laminated
composites. The mechanical properties were improved at an
optimum stitch density compared with unstitched woven
laminated composites. The damage area caused by repeated
impact was far smaller in stitched woven composites than
that in unstitched composites. Wu and Liau [45] also used
an experimental method to investigate the behavior of
stitched laminates under low velocity impact. They also
found that stitching could significantly reduce the
delamination area. With increasing the stitch density the
damage mechanism changed from delamination in unstitched
laminates to the plastichinge type of local deformation at
the impact location in stitched laminates. Sharma and
Sankar [46] performed experiments to study the effects of
stitching on CAI (CompressionAfterImpact) strength,
fracture toughness in Mode I and Mode II. They found that
CAI strength, Fracture toughness of Mode I and Mode II can
be improved significantly by stitching. Sankar and Sonik
[47] used Finite Element and analytical models to study the
effect of stitching on Mode II fracture toughness.
Dickinson [48] adopted a micromechanics approach to
characterize the effects of stitching on the elastic
constants, inplane strength and interlaminar mechanical
16
response of undamaged composite laminates. Sankar and
Dharmapuri [49] presented an analytical mode to describe
delamination growth in laminated DCB (Double Cantilever
Beam) specimen with stitches and proposed a simple method
to compute the energy release rate due to delamination in
the stitched specimen. Jain and Mai [50] studied the effect
of reinforced tabs on the Mode I fracture toughness of
stitched composites. They indicated that reinforcing
aluminum tabs along the length of stitched DCB specimens
can alter the failure mechanism of the stitch threads and
hence erroneous values for Mode I fracture toughness.
Glaessgen et al [51] used finite element method in
conjunction with crack closure technique to study the
effect of stitching on Mode I and Mode II energy release
rate for various debond configurations.
1.3 Objectives and Scope of the Present Study
From the above sections we can realize that one of the
most dominant damage in composites is delamination growth
due to lack of reinforcement in the thickness direction.
Stitching is found to be one of the effective reinforcing
methods in the thickness direction of laminated composites.
However it is not clearly understood how stitching can
delay the delamination propagation of composite laminates
17
subjected to compressive loads or to lowvelocity impact.
The purpose of the present research is to use analytical
and finite element methods to study the effect of stitching
on postbuckling and lowvelocity impact response of
composite structures. In particular the study will focus on
the effect of stitching on buckling load, energy release
rate, and impact damage resistance.
The organization of this dissertation is as follows:
In Chapter 2 an analytical solution for postbuckling of a
delaminated composite beam with stitching will be derived.
In Chapter 3 a finite element analysis will be given
related to postbuckling of a delaminated composite beam
with stitching. In Chapter 4 an analytical model for impact
response of stitched composite beam is presented.
Conclusions drawn from this study and discussions for
future research are presented in Chapter 5.
CHAPTER 2
AN ANALYTICAL SOLUTION FOR POSTBUCKLING OF A DELAMINATED
COMPOSITE BEAM WITH STITCHING
In this chapter an analytical solution for post
buckling behavior of a delaminated composite beam with
stitching is derived based on RayleighRitz method. The
effect of stitching on buckling load and energy release
rate is discussed. Nondimensional parameters for buckling
load, energy release rate [G) crack length (a), and stitch
stiffness (K) are derived. Although finite element methods
can model a stitched composite more realistically,
analytical methods have several advantages. Analytical
methods provide a closedform solution to all the desired
quantities such as energy release rate, and critical
buckling loads. Further, analytical methods allow
introduction of nondimensional parameters, and the results
expressed in terms of these dimensionless parameters are
applicable over a wide range of the actual problem
variables such as beam thickness, delamination length, and
stitch density.
18
19
2.1 Basic Assumptions
When we model the postbuckling behavior of a
delaminated composite beam with stitching the following
assumptions are made:
1.A stitch is modeled as a linear cohesive spring. The
stiffness of the spring is a function of stitch
density, Young's modulus E of the stitch, diameter
of the stitch yarn, and thickness of the stitched
composite beam. Stitches are smeared averaged on the
whole length of the beam, and thus the smeared
stitch stiffness should account for the stitch
spacing also.
2. A delamination is assumed to be preexisting before
the compressive loading is applied and is located at
the center of the beam both in length direction and
in thickness direction. Therefore the beamlike
structure is symmetric (see Figure 2.1).
3. The deformation shape of the stitched composite beam
under postbuckling behavior is assumed to be the
same as in an unstitched beam under postbuckling.
4. The laminated beam is approximated as a homogenous
beam with an equivalent Young's modulus.
20
2.2 Analytical Model
In this section we first derive the buckling load Pcr
and energy release rate (G) in an unstitched composite beam
using the RayleighRitz method and then obtain the buckling
load Pcr and energy release rate (G) for a stitched
composite beam. A single crack is located at the center of
the structure.
2.2.1 Delaminated Composite Beam Without Stitching
A composite beamlike structure without considering
throughthethickness reinforcement is shown in Fig. 2.1.
4
2L
Figure 2.1 A delaminated beam
The length of the beam is 2L, B is the width, 2h is the
thickness of the beam, 2a is the delamination length and P
is the compressive load.
21
The delaminated composite beam is composed of two
parts: One part is the uncracked portion and the other is
the cracked portion. The cracked portion is modeled as two
sublaminate beams, called top sublaminated and bottom
sublaminate. When the compressive load P reaches the
buckling load Pcr/ the cracked portion of the beamlike
structure will deform. Therefore let us consider the
deformation of the delaminated portion first. After the
compressive load P exceeds the buckling load Pcr, the
cracked portion of the beam will deform as shown in
Figure 2.2.
Z
/'
/'
o A
o A
P
X
2a
\4
Figure 2.2 Deformed shape of sublaminates
The Strain energy density of cracked portion is
(2.1)
22
here
Â£ x ~I
dx 2\dx )
du 1 (chv\2
+
(2.2)
dx 2\dx,
Here Uo is the midplane displacement in the x direction, w
is the displacement of top sublamated beam in the z
direction. sx is the normal strain of top sublaminated beam
in x direction. The crosssectional area of the
sublaminates is represented by A and E is the Young's
modulus. Assume the forms of u0 and w as: u0=cx,
w=dsin"7ix/2a, where c and d are unknown coefficients to be
determined.
The strain energy of the delaminated portion of the
beam can be derived as:
2 a
U=Â¡Uldx
0
+
3 7T2d4 2ElnAd2
32(2 af (2a)2
+
(2.3)
23
where I is the moment of inertia of each sublaminate.
The potential energy V of the external loads is given by
V = ~{P)c(2d)
Total energy FI:
n = u + v
(2.4)
(2.5)
Using the RayleighRitz method:
an
dc
an
dd
= o
(2.6)
(2.7)
Substituting for the energy terms U and V we can obtain the
following two equations:
Ea\^ + 2c(2a) 1 + P(2a) = 0
(2.8)
+ 37r4d2 
+ 8(2a)3J
4 Eln*
+ (2)3
= 0
(2.9)
24
It may be noted that the above equations are nonlinear
and lead to multiple solutions. Solving these two equations
we obtain the coefficients c and d as follows:
One possible solution set is:
P
(2.10)
c = 
2 AE
d = 0
(2.11)
This solution describes the behavior of the cracked
portion of the beam where the compressive load is less than
the buckling load Pcr. Since d=0, there is no deformation in
z direction. The relation between compressive load P and
axial displacement Uo takes a simple form:
Px
(2.12)
The end shortening Sc at x=2a is given by:
(2.13)
The other possible solution set is:
3 P 8n2I
(2.14)
c = +
2 AE A(2a)
25
2 4P(2a)2 32/
EAn2 A
(2.15)
In order for d2 > 0, we have:
P > P.
8 n2El
(2a)2
(2.16)
here Pcr is the critical buckling load.
This solution set describes the behavior of the
cracked portion of the beam where the compressive load is
greater than the buckling load Pcr. There is a deformation
in the z direction. The relation among compressive load P
and axial displacement u0 and deformation shape w are:
3 Px 8 n2Ix
2AE A(2a)2
(2.17)
w = d sin2
TJX
2 a
(2.18)
here d is obtained from Equation 2.15 for a given P. End
shortening 8C at x=2a can be expressed as shown as:
u
0 \x=2a
' 3 P
k2AE
8n2l >
A(2a)2 J
(2a)
(2.19)
26
Based on Equation 2.13 and 2.19 the relation between the
compressive load P and the end shortening Sc will be as
shown in Figure 2.3.
Figure 2.3 The relation between compressive
load P and end shortening 80
In Figure 2.3 Scr is end shortening at critical buckling load
Pcr and dd is end shortening at an arbitrary compressive
load Pj.
Based on Figure 2.3, the strain energy Ui of the
cracked portion can be written as:
V>=\P',S',+\(.PtP'r)(.8aS',)
_ 3Px2a 4nAEl1
~ 2EA Aa3
(2.20)
27
We can extend the results to the entire composite
beam. After the compressive loads P reaches the buckling
load Pen the beamlike structure will deform as shown in
Figure 2.4.
H H
*
2L
*
Figure 2.4 Deformed shape of the entire delaminated beam
The relation between the compressive load (P) and end
shortening (S) is:
EAS
L
(P
(2.21)
P =T~j(8 +T) (P>Pcr)
2a + L Aa
Aa
(2.22)
28
The strain energy U of the entire beam is the sum of
strain energy UÂ¡ in the cracked portion of the beam and
strain energy U2 in the uncracked portion of the beam, and
it has the following form:
U=Ul+U2
_ "3P2a 4EI27T4} P\La)
v 2EA Aa3 j 2EA
(2.23)
Strain energy release rate (G) at the crack tip is obtained
by differentiating total strain energy U with respect to
half of crack length a [53]. Then the expression for G can
be derived as:
G = 
2 B da s
1 f 4AES2 SEI;r2S(Sa + 2L)
~ 2B [(4a + 2L)1 {Aa + 2aLf
\6EI27v4(\2a2 +4 aL) 12Â£/2;r4
+ A(4a3 + 2a2 L)2 Aa4 J (2'24^
2.2.2 Delaminated Beam with ThroughtheThickness Stitches
A delaminated beam with stitching is depicted in
Figure 2.5.
29
Stitch
k
2a
M N
2L

Figure 2.5 A stitched delaminated beam
First let us consider the delaminated portion of the beam
with stitching. We assume that the buckled shape of the
sublaminates is similar to that without stitches as
described in the previous section. Thus we assume u0=cx and
w=dsin2 7rx/2a where unknown coefficients c and d will be
different for the unstitched case. The deformed shape of
the cracked portion is depicted in Figure 2.6.
30
Z
2a
H M
Figure 2.6 Deformed shape of crack portion with stitch
Strain energy U of the delamated portion of the beam
with stitching includes two parts: strain energy Dnostitch
from the delaminated portion of the beam without stitching
and the energy Ustitch contributed by the stitches. The total
strain energy U is of the following form:
^ ^ nostitch ^ stitch
= Unostitch + \ \K (2w)2 dx (2.25)
here K is foundation constant that represents the stiffness
of the smeared stitches. The foundation constant K is
related to the actual stitch parameters as:
31
NB
l 2 h J
(2.26)
here As is the area of cross section of a stitch (2 bobbin
yarns); Es is Young's modulus of stitch material (bobbin
yarn); 2h is the total of thickness of uncracked beam; N is
the stitch density (number of stitches/unit area), and B is
the width of the beam in the ydirection.
After using a procedure similar to that for an
unstitched beam, we derive the buckling load Pcr of a
delaminated beam with stitches as follows:
%Eln2  3K(2a)
(2.27)
From Equation 2.27 we can see that the first part of Pcr is
the same as the buckling load of an unstitched beam, and
the second part of the equation is contributed by the
stitches. Therefore stitching indeed can improve the
buckling load.
Now we can extend the result to the entire stitched
beam. The deformation of the beam undergoing postbuckling
is shown in Figure 2.7.
32
The relation between compressive loads P and end
shortening 8 is:
P =
AE
~T
8
(P
AE 41 tv2 24 Ka\
(8 + + r)
2 a + L Aa EAn2
iP>P.)
(2.29)
2a
\4 N
K
2L
Figure 2.7 Deformed shape of the delaminated
beam with stitches
When compressive loads P is greater than the buckling
load Pcr of the stitched beam total strain energy U of the
entire delaminated beam with stitching is:
U =
P\La) 
3 P2a
2EA
a ,2EIn2 12Ka\2
+ )
EA a 7T
2EA
(2.30)
33
Strain energy release rate G (when P>Pcr) is
j_du_
2 B 8a
1 [576Â£2(20a6 +12a5I) \92KI(4a2 +4 aL)
2B { (4a + 2L)2 EAk4 + A(4a + 2L)2
48K8(8a3 + 6 a2L)
n2 (4a + 2L)2
48K1
720K2a4}
A
EAn4 J
(2.31)
Let us define the nondimensional parameters as:
Pcr=Pcr /EA, a=G/Eh, a'=a/L, B'=B/L, h'=h/L,8 '=8/L, K'=K/E. Pcr'
is nondimensional buckling load; G' is the nondimensional
energy release rate; a' is the nondimensional half crack
length of the structure; B' is the ratio of the width of the
structure to the half length of the beam; h' is the ratio of
the thickness to the length of the structure; 8' is the
nondimensional end shortening of the beam along the axial
direction; K' is the nondimensional stiffness of the
stitches. After tedious algebraic manipulations we can
derive the following relations:
Per
n2h \2K a
Â¡T
6 a
n~Bh
(2.32)
34
2 S'2 7T4h\6a2+ 2a) 288Â£'2(20a'6 + 12a5)
(4a +2)2+ 9(4a3 + 2a'2 )2 ;r4Â£ 2h'2 (4a + 2)2
n2hi28(%a + 2) SKh (4a2 + 4a ) 2Kh
+ 3(4a2 + 2a )2 5'(4a+2)2 + B
24K8 (&a3 + 6a2) n4U4 360K'2a4
7r2B hi (4a +2)2 24a4 + ;t45'2/j2
In the following section the results of the analytical
model are discussed in term of the nondimensional
parameters.
2.3 Results and Discussion
The results presented in this section are based on
Equation 2.32 and 2.33. In all numerical examples h'=0.02
and B' =0.08.
Figure 2.8 shows the variation of buckling load Pcr' as
a function of semicrack length a' for various values of
stitch parameter K'. From Figure 2.8 we can observe that the
nondimensional buckling loads Pcr' is not affected by
stitching when the range of the nondimensional crack length
a' is from 0.05 to 0.1. When a' is greater than 0.1, Pc/ for
35
an unstitched beam decreases as the nondimensional crack
length a' increases. This is understandable because a
structure buckles more easily with increasing crack length.
When we look at stitched beams in the range 0.1
Fig. 2.9) the buckling load is always higher than that of
unstitched beam, and further the buckling load Pcr' increases
as the stiffness of stitching K! increases. In a stitched
beam an interesting phenomenon occurs: the buckling load Pcr'
decreases as the crack length a' increases in the range
0.1
not decrease due to increasing crack length a', but in
contrast Pcr' increases to some extent depending on the
stiffness of stitching. The reason is that the longer
crack, the more stitches involved in the structure and the
stitches hold the structure together more tightly,
therefore the buckling load Pcr can be improved in the
stitched beam even with longer delamination. A similar
behavior was obtained by Shu and Mai [39] for the case of
thin film delamination with stitches. Figure 2.9 shows this
phenomenon in detail.
In Figure 2.10 the relation between nondimensional
energy release rate G' and nondimensional crack length a' and
the effect of K' on energy release rate G' for a
36
nondimensional end shortening 5'=0.1 are presented. When
crack length a' is less than 0.1 the stitching has no effect
on nondimensional energy release rate G'. As the
nondimensional delamination size a' grows beyond 0.1 the
effect of stitching becomes significant. The energy release
rate G' decreases in a stitched beam compared to that in an
unstitched beam. The larger the stiffness of the stitching,
smaller is the nondimensional energy release rate in
stitched beams. Therefore using a stitching can decrease
the energy release rate at the crack tip, and the crack
propagation can be delayed. The crack will propagate when
the energy release rate is greater than the critical energy
release rate (fracture toughness Gc) The variations of G'
have similar tendency in both stitched beam and unstitched
beam. Nondimensional energy release rate G' dramatically
increases with crack length for smaller delaminations, and
reaches maximum Gmax at certain crack length a' and then G'
goes down as the delamination size increases. Whether a
crack will propagate or not depends on fracture toughness
Gc. Whether it will be a stable crack propagation or an
unstatble propagation depends on the rate of change of
energy release rate G with delamination size a. Referring
to Figure 2.10, we can note that in the beginning the crack
37
will propagate rapidly in an unstable manner. For a'>0.1,
dG'/da' is negative and hence the crack propagation will be
stable.
In Figure 2.11 the relation between nondimensional
energy release rate G' and nondimensional stitch stiffness
at a given crack length (a'=0.2) is examined and the effect
of nondimensional end shortening 5' is shown. It
demonstrates that energy release rate G' decreases with
increasing the stiffness K' of stitching. The more non
dimensional end shortening 8' is, the more nondimensional
energy release rate G' has.
The variation of G' with S' for various stitch
stiffnesses is shown in Figure 2.12. The results correspond
to a0.2. The relations were very similar for other crack
lengths also. From Figure 2.12 one may note that the G'
remains close to zero in the beginning, but increases after
certain critical S'. The variation of G' is approximately
quadratic in S'. Further, it may be noted that for a given
S', G' decreases with the K'. This is brought out clearly in
Figure 2.12, where G' is plotted as a function of K' for
various S'. One can see that the dramatic decrease in G' with
increasing K'.
38
In summary stitching always increases the critical
load for buckling in delaminated beams. It reduces the
energy release rate at the cracktip thus preventing or
delaying the onset of delamination propagation. Both these
effects are very much dependent on the stitch parameter K'
which is given by (see Equation 2.26):
K' =
AES NB
2 h E
(2.34)
From Equation 2.34 one can see that K is proportional to
the stitch axial stiffness and stitching density, but
inversely proportional to the equivalent Young's modulus of
the parent laminate. Thus one needs to use high density,
high stiffness stitches for stiffer laminates.
cr
39
0.0 0.1 0.2 0.3 0.4
a'
Figure 2.8 Variation of Pcr' with a' for different K'
in the range 0
cr
40
a'
Figure 2.9 Variation of Pcr' with a'
for different K'
in the range 0.1
41
a'
Figure 2.10 Variation of G' with a' for a given (8'=0.1)
42
K'
Figure 2.11 Variation of G' with K' for a given a'
(a=0.2)
43
8'
Figure 2.12 Variation of G with 5 for a given a (a =0.2)
CHAPTER 3
FINITE ELEMENT ANALYSIS OF A DELAMINATED COMPOSITE WITH
STITCHING
As discussed in Chapter 2 delaminations can greatly
reduce the compressive load carrying capacity of laminated
composite structures. Throughthethickness stitching seems
to be helpful in increasing the critical load at which a
delaminated beam becomes unstable under compressive
loading. In Chapter 2 an analytical solution for post
buckling of a stitched composite with a delamination was
given. Several assumptions were made in order to obtain a
closed form solution. In this chapter finite element
analysis is used to investigate the effect of stitching on
the postbuckling behavior of a stitched composite.
Although analytical models provide insight into the effects
of stitching, more detailed behavior of a structure can be
obtained using finite element models. Several restrictions
of the analytical approach can be easily removed in FE
models. For example, the analytical model assumes the beam
is homogeneous and orthotropic, Whereas FE models can
handle laminated composites. Unlike in the analytical
44
45
Method no restriction is placed on the location of the
crack in the thickness direction. Further the stitches do
not have to be smeared as continuous springs; each stitch
can be modeled individually, if necessary. In this chapter
the FE model is used to understand the effects of stitch
parameters, especially stitch density and stitch yarn
material properties, on the loadend shortening relation
and the energy release rate of a delaminated composite
under axial compression.
3.1 Specimen
The specimen considered for finite element analysis is
made up of 48 unidirectional AS4 graphite/epoxy plies. The
material properties for each lamina are: Â£=19.4X106 psi,
Â£=1.3X106 psi, Gi=0.7X106 psi, vir=0.28. The length L of
the specimen is 5 inches (127mm). The width B is 1 inch
(25.4mm) and the thickness H is 0.27 inch (6.86mm). A
single delamination (length of 2a) is located at a distance
h from the top surface of the beam. The specimen is
depicted in Figure 3.1. The specimen is under axial
compression. The FE analysis is used to simulate the
compression test. In particular we are interested in the
loadend shortening relationship and energy release rate as
a function of the load.
46
Three stitch yarn materials were considered: Glass 1
(3570 denier), Kevlar (1600 denier) and Glass 2 (5952
denier) bobbin yarn. The stitch densities are 4X1/4" and
8X1/8". For example 4X1/4" stitching pattern means a
number of 16 stitches per square inch where the pitch is
1/4 inch and the distance between two adjacent stitch lines
is 1/4 inch.
Stitch
L
h H
Figure 3.1 A delaminated beamlike structure with stitche
3.2 Finite Element Model
r
The commercial finite element software ABAQUS [54] was
used to simulate the stitched composite with through the
width delamination. This specimen is modeled by two
47
separate beams, top beam and bottom beam. The nodes of each
beam are distributed uniformly along the axial direction.
Type B21 element, which considers transverse shear effect
is adopted. The stiffness matrix of this type is given in
ABAQUS Theory Manual [55]. The element size is chosen as
0.0125 inch. In an undelaminated portion the connection
between top and bottoms is achieved by using the EQUATION
command. The boundary conditions that were implemented are
depicted in Figure 3.2.
Stitch
Lx
Figure 3.2 Boundary condition in a delaminated beam
with throughthethickness stitches
The relations between the top and bottom beam
displacements for the uncracked portion of the composite
are as follows:
48
h, h.
, u =yY', + y^
(3.1)
wt wb = 0
(3.2)
Â¥, = n
(3.3)
In the above three equations t and b denote the top and
bottom beam respectively, u is the nodal displacement in
the x direction, w is the nodal displacement in the z
direction and i// is the rotation.
Throughthethickness stitches are modeled as linear
springs using SPRING2 elements in ABAQUS. Two spring
elements are used to model each stitch. One spring
represents the resistance offered by the stitch to relative
transverse displacements of the top and bottom beams. The
second spring element represents the relative displacement
in the axial (longitudinal) direction. These spring
elements connect the two corresponding nodes on the top and
bottom beams. The relative displacement across a SPRING2
element is the difference between the ith component of
displacement of the spring's first node and the ith
component of displacement of the spring's second node. If
49
we denote the axial and transverse displacements by u and
w, then relative displacements are (see Figure 3.3):
A u = uxu2, A w = wxw2 (34)
AW2 t W1
 /W
u2 2
Figure 3.3 Node specification
for SPRING2 element
The procedure for determining the spring constants of
the stitches is as follows. Figure 3.4 shows a stitch where
the beam surfaces have relative displacements in both x and
z directions. The initial length of the stitch is the total
thickness of the beam, H. The relative displacements in the
x and z directions are Au and Aw, respectively. The
elongation of the stitch AL is related to the displacement
components by
o
1
Au = ALcos?
Aw = AL sin d
(3.5)
50
here 9 is the inclination of the stitch as shown in Fig.
3.4. The tensile force in the stitch is given by
F = ^AL (3.6)
H
here As and Es are the area of cross section and Young's
modulus of the stitch yarn. It should be remembered that
there are two yarns in each stitch (see Figure 1.1). The x
and z components of the force F are: Fx=Fcos6 and FzFsin6.
Substituting for F from Eq. 3.6 we obtain
A F A F
Fx = ^ALcosGAu
x H H
A F A E
Fz =^LJLALsm6=Aw (3.7)
H H
From Equation 3.7 it may be seen that the stiffness of the
spring in the FE model should be equal to ASES/H. In fact
the stiffness of both the springs, horizontal as well as
vertical, happens to be the same.
3.3 PostBuckling Analysis
The postbuckling analysis requires that an initial
imperfection be specified for the beam. Usually the
imperfection shape is a certain linear combination of
51
several buckling mode shapes. Thus a linear bifurcation
analysis has to be performed to determine the buckling mode
shapes. In the present analysis only the first mode shape
was considered in the initial imperfection. The
imperfection amplitude was 5% of the eigenmode calculated
by ABAQUS.
Top beam
1
Aw AL
u
A u
Bottom beam
Figure 3.4 Stitch deformation
The nonlinear analysis used the Riks algorithm in
order to determine the successive equilibrium positions. To
analyze a postbuckling problem, it must be turned into a
problem with continuous response instead of bifurcation.
This effect can be accomplished by introducing an initial
imperfection into a "perfect" geometry so that there is
some response in the buckling mode before the critical load
is reached. The postbuckling problem is a geometrically
nonlinear static problem. Static equilibrium states during
52
the unstable phase of the response can be found by using
the modified Riks method. This method is used for the case
where the load magnitudes are governed by a single scale
parameter. The method can provide solutions even in the
case of complex, unstable response. The Riks method uses
the load magnitude as an additional unknown; it solves
simultaneously for load and displacements. Therefore,
another quantity must be used to measure the progress of
the solution; ABAQUS uses the arc length, 1, along the
static equilibrium path in loaddisplacement space. This
approach provides solutions regardless of whether the
response is stable or unstable. The modified Riks algorithm
is described in detail in [55].
3.4 Energy Release Rate
The purpose of stitching or any other forms of
throughthethickness reinforcement is to increase the
buckling load and also to prevent the cracks from
propagating further. Energy release rate at the crack tip
has been found to be a useful parameter in predicting crack
propagation in laminated composites. The delamination is
expected to propagate further when the energy release rate
exceeds the corresponding fracture toughness value of the
composite material system. Delaminations have a higher
53
propensity for propagation along the plane of delamination
as the interlaminar fracture toughness is much lower for
most laminated composites. In particular Mode I (opening
mode) interlaminar fracture toughness is about onehalf of
Mode II (shearing mode) fracture toughness. For example,
the values for a typical AS4/35016 graphite/epoxy
composite are [46] : Gjc = 300 N/m and Guc = 670 N/m. Thus
the effectiveness of stitching can be judged by the
reduction in energy release rate, G, at the crack tip.
There are three different methods to compute G: (i)
computing stress intensity factor at the cracktip; (ii)
Strain Energy Density Method; (iii) Virtual crack growth
method. The first method requires detailed three
dimensional stress field near the crack tip, which can only
be obtained by using threedimensional finite elements with
a fine mesh. Since we are using structural elements in the
present analysis, the stress intensity factor approach
cannot be used. The strain energy density method developed
by Sankar [52] requires computation of energy densities
just behind and ahead of the crack tip. However in the case
of stitched composites, the stitch forces affect the energy
densities significantly, and hence a very fine mesh is
needed near the crack tip. Further ABAQUS does not provide
the energy density values readily, and they need to be
54
computed from the force and moment resultant just ahead and
behind the crack tip. Hence the third method is used in the
present study, and it is explained below.
Energy released rate G is computed by using the
following formula [53]:
1 dU
2 B da
(3.8)
here U is the strain energy in the beam for a given
displacement, a is the crack length, and B is the width of
the beam. In this method the loadend shortening relation
PS is developed using the aforementioned nonlinear FE
analysis. For a given displacement (end shortening) the
area under the PS diagram up to the displacement represents
the strain energy stored in the beam. Let it be equal to
U(S) The change in strain energy is computed numerically by
repeating the nonlinear FE analysis for a crack length
a+Aa, where Aa is the incremental crack length. The area
under the new PS curve represents the strain energy for the
new crack length. Then G is obtained by numerically
differentiating U{S) (see Fig. 3.5):
55
1 MJ
2 B Aa
>
5
Figure 3.5 Strain energy change
due to crack growth
3.5 Results and Discussion
In this section the results of the nonlinear FE
analysis are presented. The effects of stitching are
understood by analyzing the loadend shortening diagrams,
P8 curves, for various types of stitches. The second
quantity of interest is the energy release rate G at the
crack tip. The G is also plotted as a function of 8 for
various stitches. Another important and interesting
information is the amount of axial strain in the stitches,
56
especially during the postbuckling process. Although
stitches prevent delamination propagation by reducing the
G, they are subjected to strain by the beam segments that
tend to buckle. Thus the strains in the stitches are also
plotted as a function of 8.
Three types of stitch yarns were used in the numerical
example: Kevlar (1600 denier), Glass 1 (3570 denier), and
Glass 2 (5952 denier). Their properties are listed in Table
3.1. Two types of stitch patterns were considered: 4x1/4"
(16 stitches per square inch) and 8x1/8"(64 stitches per
square inch). The properties of the graphite/epoxy and the
specimen dimensions were described in section 3.1. The
delamination was assumed to be at a distance equal to l/4th
of the total thickness from the top surface of the beam,
i.e., h=H/4.
Table 3.1 Stitch parameters
STITCH
YARN
LINEAR
DENSITY IN
DENIERS
(gm/9,000m)
MASS
DENSITY
(gm/cm3)
AREA OF
CROSS
SECTION
As (cm2)
YOUNG'S
MODULUS
Es
(GPA)
AXIAL
RIGIDITY
ASES (N)
Kevlar
1600
1.50
0.1185x
10~2
130
15.405*
103
Glass 1
3570
2.49
0.1593*
10'2
85.5
13.620*
103
Gl3.ss 2
5952
2.49
0.2656*
102
85.5
22.709*
103
57
The P8 diagrams for 16 ssi (stitches per square inch)
and 64 ssi are given in Figure 3.6 and 3.7, respectively.
It is clearly seen that stitching can increase the peak
load by 20%30% depending on the stitch material and stitch
density. It is also seen that the difference in maximum
loads due to different stitch materials is not significant.
The effects of stitch density on the maximum load can be
inferred from Figure 3.8. It may be seen that there is a
slight increase in the maximum load due to an increase in
stitch density from 16 ssi to 64 ssi. The results presented
in Figure 3.6 through 3.8 are for a crack length of 1 inch.
Figure 3.9 shows the effects of various stitches on
the maximum load as a function of the crack length. It
should be noted that the beam length is 5 inches, however
the effective length is only 3 inches (see Figure 3.2).
From Figure 3.9 it is seen that stitching has no effect for
very short delaminations (a
in. the 64 ssi specimens carry almost the same maximum load
as the undelaminated specimens. That is, the high density
stitching is able to completely suppress delamination
buckling and restore the original load carrying capacity.
The 16 ssi specimens also show marked improvement in the
maximum load. However for a>2.0 inches the effect of
stitching seems to diminish. This is because the
58
delamination length approaches the total length of the
beam, and global buckling takes over the subdelamination
buckling. During global buckling the delamination is under
more shearing mode than opening mode, and the stitches are
not that effective in suppressing the relative sliding
motion of the delaminated surfaces.
The effects of stitching on energy release rate G are
shown in Figure 3.10 through 3.12. Figure 3.10 corresponds
to a stitch density of 16 ssi. It is seen that G is almost
negligible until the sublaminate buckling begins.
Thereafter the G rises rapidly with the end shortening of
the beam. The stitches are able to delay the point when G
begins to rise. For example, in Figure 3.10, G begins to
rise at 50.03 in. in an unstitched beam, whereas it is
delayed until 50.55 in. in stitched beams. Further, the G
is greatly reduced compared to the unstitched beam. For
5=0.1 in. the value of G is reduced from about 800 lb./in.
in an unstitched beam to about 100 200 lb./in. in
stitched beams. Further it can be seen that stitches with
higher axial rigidity (see Table 3.1) cause the greatest
reduction in G.
Similar results for 64 ssi stitching are presented in
Figure 3.11. Both stitch densities are compared with
59
unstitched beam in Figure 3.12. It may be seen that 64 ssi
stitching offers slightly higher reduction in G.
As mentioned earlier monitoring the strains in the
stitches is an important consideration in analyzing
stitched specimens. Because stitch yarns could break if the
strains exceed the allowable limit and this could trigger a
domino effect of successive stitch failures and dynamic
delamination propagation, and lead to catastrophic failure.
The strains in various stitches as a function of end
shortening are shown in Figure 3.13. It should be noted
that the crack length is 1 inch and the stitch pattern is
4x1/4". Stitch 3 (Figure 3.13) is at the center of the beam
and thus undergoes large strains. Stitch 1 is actually at
the crack tip and hence there is no strain at all. Stitch 2
is in between stitches 1 and 3. The variation of stitch
strain with end shortening is strikingly similar to that of
G. discussed earlier.
The strains of stitches located at the center of beam
in various stitch materials and stitch patters are shown in
Figure 3.14. It may be seen that the strains are sensitive
to the stitch material as well as the stitch density. In
general the strains are significantly less for higher
stitch density. Further, stitches with lower axial
stiffness (ASES) undergo larger strains.
60
From the above discussion we can arrive at the
following conclusions:
1. Stitching can improve the buckling loads of
delaminated beams. The increase is not sensitive
to the three types of yarns used in this study.
The yarns had axial rigidities in the range of
13,000 N 223,000 N and they seem to perform
equally well. However, the maximum load increases
with the stitch density. This is consistent with
the experimental observation of Sharma and Sankar
[46] .
2. Throughthethickness stitches significantly
reduce the energy release rate at the crack tip.
They also increase the end shortening required to
cause sublaminate buckling when the energy release
rate rises rapidly. By reducing the energy release
rate stitching prevents the crack propagation
which may lead to catastrophic failure of the
specimen. The axial rigidity of the stitches also
affects the energy release rate. The Higher axial
rigidity the greater is the reduction in G.
3. The maximum strain in the stitches depends
strongly on the axial rigidity of the stitches and
the stitch pattern. The stitches with small axial
61
rigidity undergo large axial strains during
sublaminate buckling.
Load P (lb)
62
End shortening 5 (in.)
Figure 3.6 Compressive load vs. end shortening
at crack length (a=l in.)
Load P (lb)
63
Figure 3.7 Compressive load vs. end shortening
at crack length (a=l in.)
Load P (lb)
64
Figure 3.8 Compressive load vs. end shortening
at crack length (a=l in.)
10
Maximum load P (lb)
65
Crack length a (in.)
Figure 3.9 Maximum load vs. crack length
Energy release rate G (lb/in.
66
Figure 3.10 Energy release rate vs. end shortening
(a=l" h/H=l/4)
Energy release rate G (lb/in.
67
End shortening 5 (in.)
Figure 3.11 Energy release rate
vs.end shortening (a=l" h/H=l/4)
Energy release rate G(lb/in.
68
Figure 3.12 Energy release rate vs. end shortening
(a=l" h/H=l/4)
Stitch strains
69
Figure 3.13 Stitch strains vs. end shortening
(a=l" h/H1/4)
Stitch: Glass 1 (16 ssi)
Maximum Strain of stitch
70
Figure 3.14 Maximum strain in a stitch
vs. end shortening (a=l" h/H=l/4)
CHAPTER 4
EFFECT OF STITCHING ON LOWVELOCITY IMPACT RESPONSE
In the previous two chapters we noted that a
delaminated beam is susceptible to buckling failure under
axial compression and that stitching can greatly improve its
compressive loadcarrying capacity. During sublaminate
buckling the stitches are predominantly under tension, and
the fracture mode is close to Mode I or the opening mode.
Another type of loading that can be detrimental to
delaminated beam is lowvelocity foreign object impact.
Composite structures are prone to low velocity impact damage
due to dropped hand tools, runway debris, hail stones etc.
If the composite structure has prior delaminations, then the
impact force can cause them to propagate, leading to
catastrophic failure. Translaminar reinforcements such as
stitching are expected to prevent delamination propagation
during foreign object impact.
In this chapter we study the effects of stitching on
the impact response of delaminated beams. Several
assumptions are made to simplify the problem so that the
focus can be on the effects of various stitching parameters.
71
72
The impact problem is depicted in Figure 4.1. The force
acting on the beam is not known a priori, and has to be
calculated as part of the solution of the problem. A major
difference between the impact loading and the compressive
loading discussed in the earlier chapters is that the
delamination is under pure Mode II (shearing mode)
conditions. Previous experimental studies by Sharma and
Sankar (46] have found that under pure Mode II the stitches
try to plough through the matrix, and the resistance offered
by the matrix is responsible for the increase in apparent
fracture toughness. Thus the stitch model has to be modified
to account for this phenomenon.
M0, V,
F(t)
Impactor
V7
+
F(t)
M0: Impactor mass
V0: Impact velocity
t: Time Variable
Figure 4.1 A structure under impact
73
In the following sections we discuss the assumptions
made in simplifying the impact problem, the stitch model and
the impact simulation procedure. The results focus on the
extent of delamination propagation due to impact in
specimens with different stitch densities.
4.1 Basic Assumptions
The following assumptions are made to simplify the
impact simulation:
1. The velocity of impact is low compared to the
velocities of wave propagation in the composite
beam.
2. The projectile is assumed to be rigid compared to
the target. Therefore the impactor can be treated as
a rigid body and its equation of motion is greatly
simplified.
3. The target, laminated beam in the present case, is
highly flexible. The deflection of the beam is
expected to be much higher than the local
indentation, and hence the Hertzian indentation
effects can be neglected.
4. The impactor mass is much greater that that of the
beam, and hence the impact duration will be very
long compared to the fundamental period of vibration
of the beam. Therefore the target can be represented
74
by a simple springmass system. The stiffness of the
spring can be approximated by the static beam
stiffness k at the impact location.
5. An embedded central delamination is assumed to be
preexisting in the laminated beam. The delamination
is symmetrically located in the simply supported
beam.
Because of Assumption 4 above we need to compute only
the static stiffness of the delaminated beam due to a
central transverse force. We will assume that the
delamination will propagate along the same plane when the G
exceeds Gllc, Mode II fracture toughness of the parent
laminate.
The resistance offered by the matrix under shearing
mode is computed as follows. The ploughing resistance of the
stitches can be represented as a distributed shear traction
(force/unit length) at the interface of the two sublaminates
that are stitched together. The traction p0 is estimated as:
B Dha
Po =
s n
here B is the beam width, D is the diameter of stitch yarn,
h is the greater of the two sublaminate thicknesses,
the yield stress of the surrounding matrix, s is the stitch
spacing in the width direction, and n is the number of
75
stitches per inch. In terms of stitch density the
distributed traction can be written as:
p0 = NBDh
where N=l/(nxs) is the number of stitches per unit area. As
the delamination propagates new stitches come into action in
the freshly created delamination areas, and they offer
additional shear resistance. This assumption is consistent
with the experimental observations of Sharma and Sankar
[46] .
4.2 Analytical Model
The equation of motion of the impactor along with the
initial conditions can be written as:
M,^r = F(q)
at
(4.1 a)
<7(0) = 0
(4.1 b)
% =v
(4.1 c)
here M0 is the impactor mass, q is the impactor displacement
which is same as the transverse deflection of the target
beam at the point of impact, V0 is the initial velocity of
the impactor or the impact velocity, and F (q) is the contact
76
force. The contact force F is a function of the beam
deflection. The contact force F will be a linear function of
g if there were no stitches or delamination propagation.
However in the present case it will be a nonlinear function.
Once F(q) is determined, then the equation of motion
Equation 4.1a4.1c can be numerically integrated to obtain
q(t). From q(t) one can compute the impact force history
F (t) using the Fq relation. In the following section we
discuss the procedure for determining the Fg relation in a
stitched delaminated beam.
4,2.1 Relation between Contact Force and Beam Deflection
The problem to be solved in this section is depicted in
Figure 4.2. Because of symmetry only one half of the beam
will be analyzed. The problem is to find the relation
between transverse force F and deflection g at the center of
the beam in the delaminated stitched beam. Further the
energy release rate G at the crack tip needs to be computed
also. In the numerical simulation the crack will be
propagated by a small distance (symmetrically on both sides)
if the G exceeds the Mode II fracture toughness GIIC of the
parent laminated material system.
First we will provide an overview of the procedures to
be followed. Since the structure is symmetric we can analyze
the right half of the beam as shown in Figure 4.3. The crack
77
length is denoted by a and L is uncracked length in Figure
4.3.
Figure 4.2 A delaminated beam with stitches
subject to impact load F
F/2
Figure 4.3 Right half side of
specimen in Figure 4.2
The right half beam can be divided into three elements
Element 1, Element 2 and Element 3. In Figure 4.3 they are
78
denoted as <1>, <2> and <3>. Free body diagrams of the
three elements are shown in Figure 4.4. At each end of the
element there are three displacements, u, w, and y, and
three corresponding forces, Fx, Fzr and C. The u and w are
the displacements in the x and z directions and y is the
rotation of the cross section. In the list of forces C
represents the couple corresponding to the rotation y. We
use the Timoshenko beam theory in modeling the beams. The
equations of equilibrium of a general element are given in
the Appendix. These equations can be easily solved as shown
in the Appendix to obtain a relation between the six forces
acting at the two ends of the element and the six
corresponding displacements. In order to relate the forces
in the three different elements we use force and moment
equilibrium equations, and compatibility equations at the
junctions (nodes) where the elements meet. Further, the
boundary conditions at the ends of the beam can also be
implemented. This procedure is similar to assembling
elements in the Finite Element Method except the stiffness
matrix of each element is obtained exactly by solving the
differential equations of equilibrium. The forces and
displacements in each element are shown in Figure 4.4. It
t
should be noted that F is the contact force acting on the
beam, and because of symmetry F/2 is assumed to act at Node
1 in Element 1. The compatibility relation for the axial
79
displacements at the junction of all three elements is
depicted in Figure 4.5.
Figure 4.4 Free body diagrams of part <1>, <2>, <3>
Figure 4.5 Compatibility of displacement
in axial direction at joint position
The force and moment eguilibrium eguations, the
compatibility eguation and boundary conditions are as
follows:
80
Force and moment equilibrium equations:
Fx2 +Fx4 + FxS
(4.2)
+ F* f
(4.3)
+^z4 +FzS =
(4.4)
C2+C4 + C!+F.JFIt = 0
(4.5)
Compatibility equations at the joints:
h2
U2 =5 +y^5
(4.6)
K
4 = w5y^5
(4.7)
to
ii
ta.
n
(4.8)
to
II
II
O
(4.9)
Boundary conditions:
u] = u3 =0
(4.10)
o
II
ro
II
(4.11)
w] =w3
(4.12)
W6 =Fx6=C6= 0
(4.13)
81
Fxi and Fzi are external forces in the axial direction and
transverse direction respectively. C is moment of a beam.
Odd indices i (i=l,3,5) denote the left end node of each
element and even indices (i=2,4,6) correspond to the right
end nodes. Expressions of Fxi/ Fzi and C are derived in the
Appendix.
After implementing the aforementioned element
equilibrium conditions and displacement compatibility
conditions at the nodes, a compact set of 5 equations are
obtained for the five displacements wlf u5, w5, y/5f and as
shown in Equation 4.14:
"*n
*12
*13
*14
*15'
7,'
*21
*22
*23
*24
*25
5
7
*31
*32
*33
*34
*35
<:
W5
> = 
*4.
*42
*43
*44
*45
^5
7
.*51
*52
*53
*54
*55.
/5.
(4.14)
The stiffness coefficients kj (i=1,5, j=l,5) and the
generalized forces F2...F5 on the right hand side of Equation
4.14 are given in Equation 4.14a:
k]2 k15 k21 k23 k25 k32 k5! k52 0
1 1
=
B,a B2a
k k 
/v13 n 31
1 1
Bx a B2a
82
1 4 = 41 =
1 1
25, 2 B2
'22
A,E,
_ .li^i 2^2
rv >)
a
a
h h
*24 *42
AxExh2 A2E2hx
2a
2 a
, 1 1 1
&33 H H
5, a B2a B2L
34 =43 = + ~
25, 2B2 253
35 53
25,
5,7, a 5,7, a EJ, L A.EM A,E,h2
44= +
+ 2 2 +
+ +
a 45, a 452 7, 453 4a
+ 112 +
4a
J FI
h 1. ^ 'tL373
"45 ~ "54
453 I
A = EiLl + _L
Z 45,
/i ~~ +
7r 5oV2 Poh2a
+ 
2 2AEX1XBX 24 E2I2B2
/3=
5
24 E2I2B2 245,7,5,
_ 5oi3 5o23
J4
48 5,7,5, 48527,5,
/2=/5= o
(4.14a)
83
The expressions for Blf B2 and B3 used in Equation 4.14a are
as follows:
Gj A\ 12EX /,
G2A2 12 E2I2
B 3 = +
G3A3 12 E3I3
Note that Ei and Gi are the equivalent Young's modulus and
shear modulus, respectively, and and Ii are area of cross
section and moment of inertia of the corresponding beam
element.
In Equation 4.14 the forces f2 f5 are known. The
force fi that is related to the contact force F is the
unknown. On the other hand the deflection w: (same as the
variable q used in impact equations) can be treated as
known, and the other five displacements are unknowns. Thus
we have five equations for five unknowns. The equations can
be solved for a given wL to determine fx or the contact
forcedisplacement Fq relations can be developed.
The flowchart in Figuation 4.6 describes the procedure
used in developing the Fq relations. Further at each
displacement increment the energy release rate G is computed
84
using the procedure explained in the next paragraph. If the
value of G exceeds the Mode II fracture toughness Gllc, the
length of the delamination is increased by a small amount.
The numerical value of the extension is arrived by a trial
and error method until G equals Gllc for that deflection
increment.
In order to compute the G we used the strain energy
density method derived by Sankar [52] This method is very
much suitable for the present model as the force and moment
resultant ahead and behind the crack tip can be obtained in
closedform from the solution of the differential equations
of equilibrium and the energy density values thus calculated
does not have any numerical errors. Consider the three
elements surrounding the crack tip as shown in Figure 4.3.
There are two (Elements 1 and 2) behind the crack tip and
one ahead of the crack tip (Element 3). The G is derived as:
(4.15)
here UL represents the strain energy density (strain energy
per unit length of the beam), the superscripts (1) and (2)
denote the cross sections immediately behind the crack tip
and (3) denotes the cross section immediately ahead of the
crack tip and B is the width of the beam in the ydirection
85
The strain energy density in terms of force and moment
resultants is given by:
U
L
P2
~EA
M2
El
~GAy
(4.16)
where P, M and V are the axial force, bending moment and
shear force resultants; EA, El and GA are the equivalent
axial, flexural and shear rigidities of the beam cross
section.
4.2.2 Impact Response
After computing the Fq relation for a beam, the impact
equations (Equation 4.1c) can be solved numerically. The F
q relations were stored in a spread sheet program (Excel) .
The expression for F(q) on the RHS of Equation 4.1a can be
approximated by a linear interpolation in each small
integration step as:
F{q) = Frf Fa F\ (??,) (4.17)
(02
where qÂ¡ and q2 are impactor initial and final displacements
and Fj and F2 are initial and final contact force in each
increment.
86
From equation 4.1a the relation between contact force
F and velocity V can be derived as follows:
M^ = F{q) (4.18a)
at at
where
dq = v (4.18A)
dt
Substituting Equation 4.18b into Equation 4.18a and a new
equation Equation 4.18c relating the contact force F(q) and
the impact velocity V can be obtained:
M^
dq dt
= ~F{q)
(4.18c)
Substituting Equation 4.18b into Equation 4.18c and
integrating both sides of the equation, the relation between
V and F(q) can be derived as:
M(V2 V2) = f F(q)dq (4 18 d)
9i
Substituting Equation 4.17 into Equation 4.18d the velocity
can be expressed in following form
v, =
(4.19)
87
where V2 and V2 are initial and final velocities in each
step.
The corresponding impact time can be derived using
Equation 4.18b by assuming that the impact velocity in each
step varies linearly:
V(q) = Vl +Vl ~V' (qq,) (4.20a)
?2 "?i
here g2 and q2 are impactor initial and final displacements
in each step. Integrating Equation 4.18b and then
substituting Equation 4.20a we derive the following
integral equation:
,y,tx
(?2 )
Integrating Equation 4.20b an expression for time t2 can be
obtained as:
t2 /,
+ iiin
(Vivj V,
(4.20 c)
here tj and t2 are initial and final time in each step.
Equation 4.20c provides the qt relation for the impact
problem. From that using Fq relations, we can obtain the F
t relation or the impact force history. Since we know the
delamination length at each displacement in the static
88
problem, i.e., aq relation, we can translate that into at
relation, and thus the propagation of delamination can be
followed.
4.3 Numerical Example, Results, and Discussion
The specimen dimension and material properties used in
the impact simulation are as follows: initial uncracked
length L0=74.2 mm, initial crack length a0=27.4 mm, beam
width B=25.4 mm, equivalent Young's modulus Â£eg=90.375 GPa,
equivalent shear modulus Gxy=6.8GPa. Stitching material is
3570 denier glass yarn. Nine different examples were
studied. In these examples, the impactor mass, impact
velocity, position of the delamination in the thickness
direction and the Mode II fracture toughness Gllc were
varied. The parameters used in these examples are listed in
Table 4.1.
In Table 4.1 H is the total thickness of the beam
and hj is the distance of delamination from the top surface
(impact surface). For each example three different cases 
beam without stitching, 16 ssi stitches, and 64 ssi stitches
 were considered. Thus a total of 27 impact simulations
were performed. The simulations were stopped when the
contact force becomes equal to zero denoting the contact
between the impactor and the beam has ceased. In two cases
(Example 2, no stitches and 16 ssi stitches) the simulation
89
has to be stopped when the delamination propagated all the
way to the ends of the beam.
Table 4
.1 Various parameters used
in numerical examples
EXAMPLE
Mo
Vo
T
POSITION OF
Guc
NO.
(kg)
(m/s)
(J)
DELAMINATION
(hi/H)
(N/m)
1
5
1.5
5.625
0.5
530
2
5
1.5
5.625
0.5
300
3
1.25
3
5.625
0.5
530
4
2.5
1.5
2.813
0.5
530
5
2.5
1.5
2.813
0.5
300
6
5
1.5
5.625
0.25
530
7
5
1.5
5.625
0.25
300
8
2.5
1.5
2.813
0.25
530
9
2.5
1.5
2.813
0.25
300
Note: M0
energy).
(Impactor
Mass),
V0 (Impac
t velocity) T
(Kinetic
From the static analysis we obtain the loaddeflection
relation (Fq) and the delaminationdeflection relation (a
q). Sample Fq relations for Example 1 are shown in Figure
4.7. The Fq relation is linear until the crack begins to
propagate. It may be noted that the load at which the crack
begin to propagate is almost the same for all three cases
(no stitches, 16 ssi, and 64 ssi stitches). After that the
curves take different shapes depending on the stitch
density. The maximum load that the beam can carry very much
depends on the stitch density. The 64 ssi beam carries about
50% more load than the unstitched beam. The unloading was
assumed to be linear and hence the unloading curve was a
90
straight line joining the point of unloading and the origin.
This assumption is validated by the Mode II experiments
conducted by Sharma and Sankar [46].
Another interesting result that can be deduced from the
static loaddeflection curve is the apparent fracture
toughness of stitched laminates. The area enclosed by the
loaddeflection diagram (Fig.4.7) denotes the work of
fracture. Since we know the extent of delamination
propagation we can compute the apparent fracture toughness
from:
AW
here AW is the work done and Aa is the new delamination
surface created. The apparent fracture toughness for various
cases is presented in Table 4.2 along with that for
unstitched laminates. The numbers in parentheses are the
percentage increase in apparent fracture toughness. It may
be seen that the percentage increase in Gllc is higher for
laminates with lower fracture toughness.
The results for each impact analysis include the
complete impact force history (Ft) and the delamination
propagation history (at). A sample impact force history is
shown in Figure 4.8. In general, stitched beams carry more
impact force, provided the impact energy is sufficient to
cause delamination propagation. For low impact energies, the
91
impact force history will be identical in stitched and
unstitched beams, because the stitches come into effect only
when there is sufficient energy to propagate the
delamination.
Table 4.2 Comparisons of Gllc
CASE
Guc(N/m)
Gnc for 16ssi
Guc for
64ssi
1
530
595 (12%)
707
(33%)
2
300
344 (15%)
430
(43%)
The results presented in Table 4.3 show for each case
impact energy, Gllc of the parent material system, the
contact force F at which the delamination began to
propagate, the final crack length amax, and the maximum
contact force Fmax during the impact event. The extent of
delamination propagation is also shown in the bar charts in
Figures 4.9 and 4.10. Figure 4.9 considers the examples
wherein the Gllc 530 J/m2' and Figure 4.10 corresponds to
Giic= 300 J/m2.
There are many interesting observations that can be
made from the results presented in Table 4.3. The contact
force at which the delamination propagates is almost the
same in unstitched and stitched beams. This initiation force
depends only on the Gllc and the position of the crack
92
(hl/H). Thus it does not depend on impact parameters such as
impact energy.
In general the extent of crack propagation at the end
of the impact event is the least in the 64 ssi beam and the
highest in the unstitched beams. The result for 16 ssi beams
are somewhere in between. This can also be observed readily
from the bar charts in Figures 4.9 and 4.10. However the
amount of delamination propagation depends also on the
impact energy, Gllc and hl/H. Stitching is very effective in
the beam with lower inherent fracture toughness. In
Example 2 (Gllc = 300 N/m) the crack propagates all the way
to the ends of the beam in the unstitched and 16 ssi
specimens, whereas the crack propagated up to a=59.4 mm in
the 64 ssi beam. In Example 1 {Gllc = 530 N/m) the stitches
were able to reduce the delamination extension by about 6 
12 mm.
Stitches are also more effective when the crack is in
the middle plane of the laminate (hl/H= 0.5) compared to the
cases wherein the crack is near the top surface of the beam
(hl/H=0.25). This can be explained as follows. In an
undelaminated beam the shear stresses are higher at the
midplane (1.5 times the average shear stress) compared to
the plane at 1/4 distance from the top. Thus when the
delamination is at the midplane the tendency for propagation
is much higher. In fact the energy release rate G is higher
93
for midplane delaminations. Thus the stitches play a very
useful role in preventing crack propagation. This situation
is similar to the effectiveness of stitches for various
Guc s.
The stitches become more effective at higher impact
energies when the propensity for crack propagation is also
higher. Comparing Examples 1 and 4 in Figure 4.9 one can see
this phenomenon. In Example 4 the impact energy was very low
so that stitching was not necessary. However in Example 1
the effectiveness of stitch density could be inferred.
From the above discussion we can arrive at the
following conclusions:
1. Static simulations of delaminated stitched beam
provide an estimate of the apparent fracture
toughness of the stitched laminates. The stitch
density significantly affects the increase in
apparent fracture toughness. The percentage
increase is more for laminates with lower inherent
fracture toughness.
2. The impact force at which the delamination begins
to grow is not dependent on stitching. However
after the delamination growth is initiated stitches
come into play, and the extent of delamination
growth depends on the stitching parameters. In
general the extent of crack propagation at the end
94
of an impact event is small for higher stitch
densities.
3. Stitching is effective when the impact energies are
higher and the propensity for delamination growth
is also higher.
4. Stitching is more effective when the delamination
is at the center.
95
Table 4.3 Contact force vs.crack length in various examples
EXAMPLE
hi/H
CASE
Fi (N)
3max (nun)
Fmax (N)
Example 1
T=5.6125J
Gnc=530N/m
0.5
No stitch
553
56.4
645
16 ssi
562
50.4
678
64 ssi
577
44.4
713
Example 2
T=5.6125J
Gnc=300N/m
0.5
No stitch
416
101.4
431
16 ssi
423
101.4
465
64 ssi
438
59.4
679
Example 3
T=5.6125J
GIIC=530N/m
0.5
No stitch
553
56.4
645
16 ssi
562
50.4
678
64 ssi
577
44.4
713
Example 4
T=2.813J
Gnc=530N/m
0.5
No stitch
545
27.4
545
16 ssi
546
27.4
546
64 ssi
548
27.4
546
Example 5
T=2.813J
Gnc=30 ON/m
0.5
No stitch
416
49.4
478
16 ssi
425
44.4
496
64 ssi
438
39.4
517
Example 6
T=5.6125J
Gnc=53ON/m
0.25
No stitch
568
47.4
708
16 ssi
571
46.4
711
64 ssi
578
44.4
721
Example 7
T=5.6125J
Gnc=300N/m
0.25
No stitch
428
74.4
646
16 ssi
431
67.4
667
64 ssi
437
60.4
692
Example 8
T=2.813J
Gnc=530N/m
0.25
No stitch
547
27.4
547
16 ssi
547
27.4
547
64 ssi
549
27.4
549
Example 9
T=2.813J
Gnc=30 ON/m
0.25
No stitch
428
43.4
507
16 ssi
431
41.4
511
64 ssi
437
40.4
518
96
a=crack length
L=uncrack length
F=impact force
q=deflect at center point
G=energy release rate
Figure 4.6 Flow chart for computing the Fq relation
Contact force F(N)
97
Center deflection q (m)
Figure 4.7 Contact force F vs. center deflection q
(GIIC=530J/m2)
Impact load F(N)
98
Figure 4.8 Impact load F vs. time t
99
60
Example 1 Example 4 Example 6 Example 8
Initial crack length of a specimen
Final crack length of an unstitched specimen
Final crack length of a specimen with stitching (16 ssi)
Final crack length of a specimen with stitching (64 ssi)
Figure 4.9 Crack growth in different examples
(Gllc=530 J/m2)
100
120
100
80
&
ji
o 60
M
O'
u
<0
20
0
Initial crack length of a specimen
Final crack length of an unstitched specimen
Final crack length of a specimen with stitching (16 ssi)
Final crack lenth of a specimen with stitching (64 ssi)
Figure 4.10 Crack growth in different examples
(GiIC=300J/m2)
CHAPTER 5
CONCLUSIONS AND FUTURE WORK
5.1 Summary
In this dissertation analytical and finite element
methods are used to investigate the effects of throughthe
thickness stitching on the mechanical behavior of
delaminated composite laminates. In particular the buckling
and postbuckling behavior and lowvelocity impact response
of delaminated composites are studied. These two loading
modes were selected as they are very detrimental
to effective functioning of composites containing
delaminations.
The analytical modelsuse the RaleighRitz method in
which stitches are modeled as smeared foundation springs.
The finite element models have the capability of modeling
I
individual stitches. The lowvelocity impact response
calculations use the static loaddeflection behavior of the
stitched composite beam.
From the results presented in the previous chapters
the following conclusions can be reached:
101
102
1. From the analytical models a single nondimensional
parameter K' was identified:
A,E,NB
K
It may be noted that K' is actually the ratio of the
stitch stiffness and the equivalent Young's modulus of
the beam. Thus one needs to use high density stitches
with stiffer yarn material if the flexural modulus of
the laminate is also higher. It is found that stitching
increases the buckling load, and reduces the energy
release rate at the cracktip.
2. The FE models also support the conclusions of the
analytical models. The strains in the stitches depend on
the axial stiffness of the stitches, and they determine
if the stitch will break during the postbuckling
process and thus reduce the effectiveness of stitching.
3. From the static simulations of a delaminated beam it is
found that the percentage increase in apparent fracture
toughness is more for laminates with lower inherent
fracture toughness. The impact force at which the
delamination begins to grow is not dependent on
stitching. However the extent of delamination growth
depends on the stitching parameters. Stitching is
103
effective when the impact energies are higher and when
the delamination is at the center.
5.2 Suggestions for Future Work
The present research assumed a single central
delamination. The models, especially the FE models, can be
extended to beams with multiple delaminations. Further, FE
models can handle laminated plates with stitching and other
forms of translaminar reinforcements. Since it will not be
possible to model individual stitches in larger plates,
eguivalent stitch parameters such as foundation spring
constants may have to be used. More complex constitutive
relations based on experimental observations can be used to
model the stitch yarn or other translaminar reinforcements.
Since stitching results in increased cost and weight,
optimization studies can be undertaken to determine the
effective stitch materials and stitch pattern in the design
of laminated composite structures.
APPENDIX
DERIVATION OF FORCEDISPLACEMENT RELATIONS
IN THE BEAM ELEMENT
In this appendix the derivations for external forces
FXi, Fzi and moments Ci described in Chapter 4 are given:
h H
a
Figure A.l Free body diagram in part <1>
Consider Part <1> of
the beam shown
in
Figure A.l
. In
Figure A.l
Pif VJf and
Mi (i
=1,2) are
internal force
and
moment. p0
is shear
force
provided
by
stitches.
The
relations
between external
forces
and
internal force
resultants
at left side
of a
beam and
right
side of a
beam
are given as follows:
F = P
1 x\ 1 1
fa = K
C, = M, (A1)
104
105
F P
1 X2 *2
F V
1 Z 2 v 2
C, = M\
Now we consider the internal force P in the axial direction
shown in Figure A. 2.
dx
l 4
Po
>P+dP
Figure A.2 Free body diagram
in axial direction
The differential equations of equilibrium are:
A,E.
d2u dP
dx dx
= Po
(A3)
M4)
Here Aj is area of cross section of part <1>, E: is young's
modulus of part <1>. Substituting boundary condition at
x=0, u=Ui, at x=a, u=u2 into Equations A. 3 and A. 4 and
consider the equations in Equations A.l and A.2, we have
following expressions of Fxl and Fx2:
106
rr A\E\ .. P0a
Ex\ ~ (U\ Ul) +
a
(A. 5)
r AEi _;y
ri2 (m2 mi)+ _
a 2
(6)
We consider the segment shown in Figure A.3 for deriving
the expressions for Fzi and Ci (i=l,2):
Vx v
A
n nr
4 4 4
Po
X
Figure A.3 Free body diagram
in an arbitrary section
The moment and force resultants M and V in an
arbitrary cross section can be expressed as:
Â£,7,^ = A7 = A7, +V]xp0x^ (A7)
1 1 dx 1 1 y 2
(V/ + ^r) = V = V] (A. 8)
ax
Here TÂ¡ is moment of inertia and Gj is shear modulus of part
<1>.
107
At x=0
w = w,, y/ = y/x, V = VX, MM]
(A. 9)
At x=a
w=w2, if/ = iÂ¡/2, V = V2, M=M2 (y. 10)
Solving Equations A.7 and A.8 and using the boundary
conditions in Equations A.9 and A.10, we obtain relations
between Vir Mi and wir 0X. Then FZi and C can be expressed in
terms of wir Gi as:
l a a PA<*\
FZ1 =tK'i ~w2  V2 TTTT^)
A, a
2 24Â£,/1
1 i a Poh\a\
FZ2 = + V', +w2 + ^2 +T77Tr)
4 a
2 24Â£,7,
_ w, A,/, a w,
C, = pr + ^) ( L_L + t ) + '
24
a 4/4, 2/4,
+ PoV3 PoV
2 4Ax a 48Â£,/,/4,' 4
2/4, a 4/4, 2/4,
a /?0/2,a3 PoKa
+ ( r + LJ) + 1 1
4/4, a 48 Â£,/,/4,
(411)
(412)
(413)
4
(414)
108
where
_ 1 a2
A, 1
G,4 \2EJX
Free body diagram of portion <2> is shown in Figure
A. 4:
Po
Figure A.4 Free body diagram in part <2>
The procedure for deriving the expressions for Fxi, Fzi,
and Ci (i=3,4) is similar to that one in part <1>, therefore
the derivation is omitted here.
Poa A2E2
f*3 = ^ + ^~Mw3 ua)
2 a
Poa a2e2
= + u3)
2 a
1 p0h2a3 y/3a \Â¡/xa
Fz4 = (Â£JLi w3+1* + w.+ ll)
A2a 24E2I2 3 2 4 2 '
F = L (ME + Wj Vt _
A2a 24 E,1
21 2
(15)
(A. 16)
(17)
2
2
(18)
109
c,=
2 AL
Q 1^2^2 \ ^4
+ V'^ + ^^) +
4A2 a 2 A2
+ ,<* E2)  Poh2a3 Pohia
4 4A2 a ME212A2 4
(4.19)
r ~^w3
U4
/ a Â£2A\ w4
r + ^3( r ) + 17
2^ 44 a 2^
a Â£2/2 P0/22a3 p0h2a
+ ( r + ) + r
442 48Â£2/242 4
(A.20)
A2 is area of cross section, E2 is Young's modulus and G2 is
shear modulus in part <2> and A2 is given by:
4, =
a
G2 A2
12 E2I2
The free body diagram of part <3> is shown in Figure A.5:
Figure A.5 Free body diagram in part <3>
110
We can derive expressions for Fxi, Fzi, and C (i=5,6) as
shown below. The procedure for deriving these expressions
is similar to that one in part <1>.
^=^(3.) <^2i)
^.=^(".5) (4.22)
4'z 2v43' /43Z 2/43'
w. If/ w, iy,
F7, = r + ^liT + +
4j L 2^3 /43 Z, 24
C5 =
^5 ,Â£373 L
w
EJ, L
2 A.
r + (rL + ~)Â¥s +TTr + ( 7^ +
4/4
2/4,
4/4,
C6 =
W5
2/4,
/ Â£3/3
+ ( +
w, ,Z7Z,
Vs + T~r + (7a + TT^Vs
4/4, 2/4, Z, 4/4,
(/4.23)
(24)
(25)
(/4.26)
A3 is area of cross section, E3 is Young's modulus and G3 is
shear modulus in part <3>, and A3 is given by:
1  Z2
G3A3 12Â£3Z3
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Stitched Laminates by Line Loading," Journal of Composite
Materials 28(17), 1994: 16401658.
[46] Sharma, S.K., and Sankar, B.V., "Effect of Through
theThickness Stitching on Impact and Interlaminar
Fracture Properties of Textile Graphite/Epoxy Laminates,"
NASA Contractor Report 195042, 1995.
[47] Sankar, B.V., and Sonik, V., "Modeling EndNotched
Flexure Tests of Stitched Laminates," Proceedings of the 8th
annual technical meeting of American Society of Composites,
Technomic Publishing Co., Lancaster, Pennsylvania, 1995:
172181.
[48] Dickinson, L.C., "TransLaminarReinforced (TLR)
Composites," Department of Applied Science, PhD
dissertation, College of William and Mary, Williamsburg,
Virginia, 1997.
[49] Sankar, B.V., and Dharmapuri, S.M., "Analysis of a
Stitched Double Cantilever Beam," Journal of Composite
Materials 32(24), 1998: 22032225.
[50] Jain, L.K., Dransfield, K.A., and Mai, Y., "Effect of
Reinforcing Tabs on the Mode I Delamination Toughness of
Stitched CFRPs," Journal of Composite Materials 32(22),
1998: 20162041.
116
[51] Glaessgen, E.H., Raju, I.S., and Poe, C.C., "Fracture
Mechanics Analysis of Stitched StiffenerDebonding," AIAA
Journal 20(22), 1998: 26332647.
[52] Sankar, B.V., and Sonik, V., "Pointwise Energy Release
Rate in Delaminated Plates", AIAA Journal 33(7), 1995:
13121318.
[53] Anderson,T.L., Fracture Mechanics Fundamentals and
Applications, 2nd Ed., CRC Press, Boca Raton, Florida, 1995.
[54] ABAQUS /Standard User's Manual (Version 5.7).
[55] ABAQUS Theory Manual (Version 5.7).
BIOGRAPHICAL SKETCH
Huasheng Zhu was born on July 17, 1966, in Shenyang,
People's Republic of China. She received a B.E. degree from
Northwestern Polytechnical University in the Department of
Aircraft in China in July 1987. Then she worked at Shenyang
Institute of Aeronautical Engineering four years. She came
to the United State for her graduate study in August 1991.
She obtained her M.S. degree in Aerospace Engineering from
the Department of Aerospace Engineering, Mechanics, and
Engineering Science, University of Florida, in May 1994 and
continued her Ph.D. study at the same department. She will
receive her Ph.D. in December 1999.
117
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of the Doctor of
Philosophy.
Bhavani V. Sankar, Chairman
Professor of Aerospace
Engineering, Mechanics, and
Engineering Science
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of the Doctor of
Philosophy.
Nicolae D. Cristescu
Graduate Research Professor
of Aerospace Engineering,
Mechanics, and Engineering
Science
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of the Doctor of
Philosophy.
Aerospace Engineering,
Mechanics, and Engineering
Science
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of the Doctor of
Philosophy.
Edward K. Walsh
Professor of Aerospace
Engineering, Mechanics, and
Engineering Science
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of the Doctor of
Philosophy.
Professor of Civil Engineering
This dissertation was submitted to the Graduate
Faculty of the College of Engineering and the Graduate
School and was accepted as partial fulfillment of the
requirements for the degree of Doctor of Philosophy.
December 1999
M. J. Ohanian
Dean, College of Engineering
Winfred M. Phillips
Dean, Graduate School
Load P (lb)
62
End shortening 5 (in.)
Figure 3.6 Compressive load vs. end shortening
at crack length (a=l in.)
ACKNOWLEDGMENTS
The author would like to express her sincere gratitude
to her advisor, Dr. Bhavani V. Sankar, for his patient
guidance, constant encouragement, and endless support of
her study.
The author is also grateful to Dr. Nicolae D.
Cristescu, Dr. Peter G. Ifju, Dr. Edward K. Walsh, and
Dr. Reynaldo Roque for serving as her committee members.
Finally the author would like to express her gratitude
to her parents, brothers, and her love, Mr. YingFeng Liu,
who share the happiness, sadness, difficulty with the
author, for their encouragement, support, and patience.
11
96
a=crack length
L=uncrack length
F=impact force
q=deflect at center point
G=energy release rate
Figure 4.6 Flow chart for computing the Fq relation
29
Stitch
k
2a
M N
2L

Figure 2.5 A stitched delaminated beam
First let us consider the delaminated portion of the beam
with stitching. We assume that the buckled shape of the
sublaminates is similar to that without stitches as
described in the previous section. Thus we assume u0=cx and
w=dsin2 7rx/2a where unknown coefficients c and d will be
different for the unstitched case. The deformed shape of
the cracked portion is depicted in Figure 2.6.
12
number of stitches per square inch and represent this
density by the stitching pattern as: (number of stitches
per inch) X (spacing between two stitch lines), for
example, 4X1/4 means a stitch density of 16 where there are
4 stitches per inch and distance between two adjacent
stitch rows is 1/4 in. Stitch yarn is the bobbin yarn used
in the stitching process. The material density of the
stitch yarn is expressed as denier. One denier is the mass
in grams of 9000 meters of yarn.
Modified Lock Sliteh
m
Needle
thread
Bobbin
thread
Standard Lack Stitch
nrnmr
JUUL
Needle
thread
Sohain
thread
Chain Stitch
Figure 1.1 Types of lock stitches
(Courtesy, Sharma and Sankar [46])
Sawyer [34] used an experimental method to determine
the effect of stitching on the static load of bonded
composite single lap joint. Up to 38% improvement in static
failure load compared with unstitched results was obtained
by a single row of stitches near the end of the overlap.
82
1 4 = 41 =
1 1
25, 2 B2
'22
A,E,
_ .li^i 2^2
rv >)
a
a
h h
*24 *42
AxExh2 A2E2hx
2a
2 a
, 1 1 1
&33 H H
5, a B2a B2L
34 =43 = + ~
25, 2B2 253
35 53
25,
5,7, a 5,7, a EJ, L A.EM A,E,h2
44= +
+ 2 2 +
+ +
a 45, a 452 7, 453 4a
+ 112 +
4a
J FI
h 1. ^ 'tL373
"45 ~ "54
453 I
A = EiLl + _L
Z 45,
/i ~~ +
7r 5oV2 Poh2a
+ 
2 2AEX1XBX 24 E2I2B2
/3=
5
24 E2I2B2 245,7,5,
_ 5oi3 5o23
J4
48 5,7,5, 48527,5,
/2=/5= o
(4.14a)
EFFECT OF THROUGHTHETHICKNESS STITCHING ON
POSTBUCKLING AND IMPACT RESPONSE OF
COMPOSITE LAMINATES
By
HUASHENG ZHU
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1999
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of the Doctor of
Philosophy.
Bhavani V. Sankar, Chairman
Professor of Aerospace
Engineering, Mechanics, and
Engineering Science
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of the Doctor of
Philosophy.
Nicolae D. Cristescu
Graduate Research Professor
of Aerospace Engineering,
Mechanics, and Engineering
Science
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of the Doctor of
Philosophy.
Aerospace Engineering,
Mechanics, and Engineering
Science
52
the unstable phase of the response can be found by using
the modified Riks method. This method is used for the case
where the load magnitudes are governed by a single scale
parameter. The method can provide solutions even in the
case of complex, unstable response. The Riks method uses
the load magnitude as an additional unknown; it solves
simultaneously for load and displacements. Therefore,
another quantity must be used to measure the progress of
the solution; ABAQUS uses the arc length, 1, along the
static equilibrium path in loaddisplacement space. This
approach provides solutions regardless of whether the
response is stable or unstable. The modified Riks algorithm
is described in detail in [55].
3.4 Energy Release Rate
The purpose of stitching or any other forms of
throughthethickness reinforcement is to increase the
buckling load and also to prevent the cracks from
propagating further. Energy release rate at the crack tip
has been found to be a useful parameter in predicting crack
propagation in laminated composites. The delamination is
expected to propagate further when the energy release rate
exceeds the corresponding fracture toughness value of the
composite material system. Delaminations have a higher
To my parents Deqian Zhu and Guirong Liu
and my beloved YingFeng Liu
56
especially during the postbuckling process. Although
stitches prevent delamination propagation by reducing the
G, they are subjected to strain by the beam segments that
tend to buckle. Thus the strains in the stitches are also
plotted as a function of 8.
Three types of stitch yarns were used in the numerical
example: Kevlar (1600 denier), Glass 1 (3570 denier), and
Glass 2 (5952 denier). Their properties are listed in Table
3.1. Two types of stitch patterns were considered: 4x1/4"
(16 stitches per square inch) and 8x1/8"(64 stitches per
square inch). The properties of the graphite/epoxy and the
specimen dimensions were described in section 3.1. The
delamination was assumed to be at a distance equal to l/4th
of the total thickness from the top surface of the beam,
i.e., h=H/4.
Table 3.1 Stitch parameters
STITCH
YARN
LINEAR
DENSITY IN
DENIERS
(gm/9,000m)
MASS
DENSITY
(gm/cm3)
AREA OF
CROSS
SECTION
As (cm2)
YOUNG'S
MODULUS
Es
(GPA)
AXIAL
RIGIDITY
ASES (N)
Kevlar
1600
1.50
0.1185x
10~2
130
15.405*
103
Glass 1
3570
2.49
0.1593*
10'2
85.5
13.620*
103
Gl3.ss 2
5952
2.49
0.2656*
102
85.5
22.709*
103
13
Additional row of stitching or different stitch spacing has
little effect on static joint failure load.
Su [35] improved the delamination resistance in
composite laminates by the use of thermoplastic matrix
resin and stitching. There was a 2030% further improvement
of delamination resistance by using stitches. Dexter and
Funk [36] conducted an experimental investigation on impact
resistance and interlaminar fracture toughness of stitched
guasiisotropic graphite/epoxy composites. The stitch
, TM
materials were polyester or Kevlar yarns with several
parameters. They found that a significant drop in damage
areas of stitched laminates compared to unstitched
laminates for the same impact energy. The results shown
that interlaminar fracture toughness Glc of stitched
laminated with the most effective Kevlar yarn was 30 times
higher than unstitched laminates.
Pelstring and Madan [37] proposed a semiempirical
formula to relate a damage tolerance of a composite
laminate to stitching parameters. The results showed that
Mode I critical energy release rate of a stitched laminate
was 15 times higher than an unstitched laminated.
Dransfield et al. [38] reviewed the effect of stitching on
improving some mechanical properties such as compress
afterimpact strength and delamination resistance of
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of the Doctor of
Philosophy.
Edward K. Walsh
Professor of Aerospace
Engineering, Mechanics, and
Engineering Science
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of the Doctor of
Philosophy.
Professor of Civil Engineering
This dissertation was submitted to the Graduate
Faculty of the College of Engineering and the Graduate
School and was accepted as partial fulfillment of the
requirements for the degree of Doctor of Philosophy.
December 1999
M. J. Ohanian
Dean, College of Engineering
Winfred M. Phillips
Dean, Graduate School
74
by a simple springmass system. The stiffness of the
spring can be approximated by the static beam
stiffness k at the impact location.
5. An embedded central delamination is assumed to be
preexisting in the laminated beam. The delamination
is symmetrically located in the simply supported
beam.
Because of Assumption 4 above we need to compute only
the static stiffness of the delaminated beam due to a
central transverse force. We will assume that the
delamination will propagate along the same plane when the G
exceeds Gllc, Mode II fracture toughness of the parent
laminate.
The resistance offered by the matrix under shearing
mode is computed as follows. The ploughing resistance of the
stitches can be represented as a distributed shear traction
(force/unit length) at the interface of the two sublaminates
that are stitched together. The traction p0 is estimated as:
B Dha
Po =
s n
here B is the beam width, D is the diameter of stitch yarn,
h is the greater of the two sublaminate thicknesses,
the yield stress of the surrounding matrix, s is the stitch
spacing in the width direction, and n is the number of
23
where I is the moment of inertia of each sublaminate.
The potential energy V of the external loads is given by
V = ~{P)c(2d)
Total energy FI:
n = u + v
(2.4)
(2.5)
Using the RayleighRitz method:
an
dc
an
dd
= o
(2.6)
(2.7)
Substituting for the energy terms U and V we can obtain the
following two equations:
Ea\^ + 2c(2a) 1 + P(2a) = 0
(2.8)
+ 37r4d2 
+ 8(2a)3J
4 Eln*
+ (2)3
= 0
(2.9)
57
The P8 diagrams for 16 ssi (stitches per square inch)
and 64 ssi are given in Figure 3.6 and 3.7, respectively.
It is clearly seen that stitching can increase the peak
load by 20%30% depending on the stitch material and stitch
density. It is also seen that the difference in maximum
loads due to different stitch materials is not significant.
The effects of stitch density on the maximum load can be
inferred from Figure 3.8. It may be seen that there is a
slight increase in the maximum load due to an increase in
stitch density from 16 ssi to 64 ssi. The results presented
in Figure 3.6 through 3.8 are for a crack length of 1 inch.
Figure 3.9 shows the effects of various stitches on
the maximum load as a function of the crack length. It
should be noted that the beam length is 5 inches, however
the effective length is only 3 inches (see Figure 3.2).
From Figure 3.9 it is seen that stitching has no effect for
very short delaminations (a
in. the 64 ssi specimens carry almost the same maximum load
as the undelaminated specimens. That is, the high density
stitching is able to completely suppress delamination
buckling and restore the original load carrying capacity.
The 16 ssi specimens also show marked improvement in the
maximum load. However for a>2.0 inches the effect of
stitching seems to diminish. This is because the
54
computed from the force and moment resultant just ahead and
behind the crack tip. Hence the third method is used in the
present study, and it is explained below.
Energy released rate G is computed by using the
following formula [53]:
1 dU
2 B da
(3.8)
here U is the strain energy in the beam for a given
displacement, a is the crack length, and B is the width of
the beam. In this method the loadend shortening relation
PS is developed using the aforementioned nonlinear FE
analysis. For a given displacement (end shortening) the
area under the PS diagram up to the displacement represents
the strain energy stored in the beam. Let it be equal to
U(S) The change in strain energy is computed numerically by
repeating the nonlinear FE analysis for a crack length
a+Aa, where Aa is the incremental crack length. The area
under the new PS curve represents the strain energy for the
new crack length. Then G is obtained by numerically
differentiating U{S) (see Fig. 3.5):
36
nondimensional end shortening 5'=0.1 are presented. When
crack length a' is less than 0.1 the stitching has no effect
on nondimensional energy release rate G'. As the
nondimensional delamination size a' grows beyond 0.1 the
effect of stitching becomes significant. The energy release
rate G' decreases in a stitched beam compared to that in an
unstitched beam. The larger the stiffness of the stitching,
smaller is the nondimensional energy release rate in
stitched beams. Therefore using a stitching can decrease
the energy release rate at the crack tip, and the crack
propagation can be delayed. The crack will propagate when
the energy release rate is greater than the critical energy
release rate (fracture toughness Gc) The variations of G'
have similar tendency in both stitched beam and unstitched
beam. Nondimensional energy release rate G' dramatically
increases with crack length for smaller delaminations, and
reaches maximum Gmax at certain crack length a' and then G'
goes down as the delamination size increases. Whether a
crack will propagate or not depends on fracture toughness
Gc. Whether it will be a stable crack propagation or an
unstatble propagation depends on the rate of change of
energy release rate G with delamination size a. Referring
to Figure 2.10, we can note that in the beginning the crack
34
2 S'2 7T4h\6a2+ 2a) 288Â£'2(20a'6 + 12a5)
(4a +2)2+ 9(4a3 + 2a'2 )2 ;r4Â£ 2h'2 (4a + 2)2
n2hi28(%a + 2) SKh (4a2 + 4a ) 2Kh
+ 3(4a2 + 2a )2 5'(4a+2)2 + B
24K8 (&a3 + 6a2) n4U4 360K'2a4
7r2B hi (4a +2)2 24a4 + ;t45'2/j2
In the following section the results of the analytical
model are discussed in term of the nondimensional
parameters.
2.3 Results and Discussion
The results presented in this section are based on
Equation 2.32 and 2.33. In all numerical examples h'=0.02
and B' =0.08.
Figure 2.8 shows the variation of buckling load Pcr' as
a function of semicrack length a' for various values of
stitch parameter K'. From Figure 2.8 we can observe that the
nondimensional buckling loads Pcr' is not affected by
stitching when the range of the nondimensional crack length
a' is from 0.05 to 0.1. When a' is greater than 0.1, Pc/ for
21
The delaminated composite beam is composed of two
parts: One part is the uncracked portion and the other is
the cracked portion. The cracked portion is modeled as two
sublaminate beams, called top sublaminated and bottom
sublaminate. When the compressive load P reaches the
buckling load Pcr/ the cracked portion of the beamlike
structure will deform. Therefore let us consider the
deformation of the delaminated portion first. After the
compressive load P exceeds the buckling load Pcr, the
cracked portion of the beam will deform as shown in
Figure 2.2.
Z
/'
/'
o A
o A
P
X
2a
\4
Figure 2.2 Deformed shape of sublaminates
The Strain energy density of cracked portion is
(2.1)
19
2.1 Basic Assumptions
When we model the postbuckling behavior of a
delaminated composite beam with stitching the following
assumptions are made:
1.A stitch is modeled as a linear cohesive spring. The
stiffness of the spring is a function of stitch
density, Young's modulus E of the stitch, diameter
of the stitch yarn, and thickness of the stitched
composite beam. Stitches are smeared averaged on the
whole length of the beam, and thus the smeared
stitch stiffness should account for the stitch
spacing also.
2. A delamination is assumed to be preexisting before
the compressive loading is applied and is located at
the center of the beam both in length direction and
in thickness direction. Therefore the beamlike
structure is symmetric (see Figure 2.1).
3. The deformation shape of the stitched composite beam
under postbuckling behavior is assumed to be the
same as in an unstitched beam under postbuckling.
4. The laminated beam is approximated as a homogenous
beam with an equivalent Young's modulus.
86
From equation 4.1a the relation between contact force
F and velocity V can be derived as follows:
M^ = F{q) (4.18a)
at at
where
dq = v (4.18A)
dt
Substituting Equation 4.18b into Equation 4.18a and a new
equation Equation 4.18c relating the contact force F(q) and
the impact velocity V can be obtained:
M^
dq dt
= ~F{q)
(4.18c)
Substituting Equation 4.18b into Equation 4.18c and
integrating both sides of the equation, the relation between
V and F(q) can be derived as:
M(V2 V2) = f F(q)dq (4 18 d)
9i
Substituting Equation 4.17 into Equation 4.18d the velocity
can be expressed in following form
v, =
(4.19)
46
Three stitch yarn materials were considered: Glass 1
(3570 denier), Kevlar (1600 denier) and Glass 2 (5952
denier) bobbin yarn. The stitch densities are 4X1/4" and
8X1/8". For example 4X1/4" stitching pattern means a
number of 16 stitches per square inch where the pitch is
1/4 inch and the distance between two adjacent stitch lines
is 1/4 inch.
Stitch
L
h H
Figure 3.1 A delaminated beamlike structure with stitche
3.2 Finite Element Model
r
The commercial finite element software ABAQUS [54] was
used to simulate the stitched composite with through the
width delamination. This specimen is modeled by two
2
imperfections such as air entrapment or resin starvation
may cause initial delamination in the composite. Impact on
laminated composites by a foreign body during service may
also cause delamination while the surface of the composite
structure remains undamaged to visual inspection. This kind
of delamination can randomly occur somewhere in the
structure. No matter where it occurs it may decrease the
overall stiffness and loadcarrying capacity of the
composite structure. The growth of delamination cracks
under the subsequent application of external loads leads to
the rapid deterioration of mechanical properties and may
cause catastrophic failure of the composite structure. One
of the most dominant forms of damage is delamination due to
lack of reinforcement in the thickness direction. There are
some approaches that can be adopted in order to improve the
strength in the thickness direction. Using fully integrated
3D composites such as weaves and braids is one of the
methods; however, due to their complexity, limited shape
ability, and processability, their applications have been
limited. Inserting a pin in the thickness direction can
also improve reinforcement in the thickness direction, but
currently it is difficult to use this technique in large
structures. Another method for providing a reinforcement in
the thickness direction is to use stitching through the
31
NB
l 2 h J
(2.26)
here As is the area of cross section of a stitch (2 bobbin
yarns); Es is Young's modulus of stitch material (bobbin
yarn); 2h is the total of thickness of uncracked beam; N is
the stitch density (number of stitches/unit area), and B is
the width of the beam in the ydirection.
After using a procedure similar to that for an
unstitched beam, we derive the buckling load Pcr of a
delaminated beam with stitches as follows:
%Eln2  3K(2a)
(2.27)
From Equation 2.27 we can see that the first part of Pcr is
the same as the buckling load of an unstitched beam, and
the second part of the equation is contributed by the
stitches. Therefore stitching indeed can improve the
buckling load.
Now we can extend the result to the entire stitched
beam. The deformation of the beam undergoing postbuckling
is shown in Figure 2.7.
38
In summary stitching always increases the critical
load for buckling in delaminated beams. It reduces the
energy release rate at the cracktip thus preventing or
delaying the onset of delamination propagation. Both these
effects are very much dependent on the stitch parameter K'
which is given by (see Equation 2.26):
K' =
AES NB
2 h E
(2.34)
From Equation 2.34 one can see that K is proportional to
the stitch axial stiffness and stitching density, but
inversely proportional to the equivalent Young's modulus of
the parent laminate. Thus one needs to use high density,
high stiffness stitches for stiffer laminates.
8
using an analytical approach and verified the model using
experiments and finite element methods. He pointed out that
the buckling load reduces significantly due to the
existence of multiple delaminations.
Naganarayana and Atluri [1718] used a multiplate
model, in conjunction with a 3noded quasicomforming shell
element, to model the delaminated plates. The Jintegral
technique was used for delamination growth prediction in
term of pointwise energy release rate distribution along
the delamination edge. The effects of structural parameters
such as delamination thickness, size and shape on the post
buckling behavior and on the delamination growth were
examined.
Shen and Williams [19] studied a postbuckling
analysis of laminated plates subjected to combined axial
compress and uniform temperature loading. The analysis used
a perturbation technique to determine buckling loads and
postbuckling behaviors in antisymmetrically angleply and
symmetrically crossply laminated plates. Klug et al [20]
adopted Mindlin plate finite element to perform the post
buckling analysis and to compute energy release rate at the
delamination front with aid of the crack closure method in
composite laminates which have elliptic delamination shape.
The results show that this method is efficient and accurate
106
rr A\E\ .. P0a
Ex\ ~ (U\ Ul) +
a
(A. 5)
r AEi _;y
ri2 (m2 mi)+ _
a 2
(6)
We consider the segment shown in Figure A.3 for deriving
the expressions for Fzi and Ci (i=l,2):
Vx v
A
n nr
4 4 4
Po
X
Figure A.3 Free body diagram
in an arbitrary section
The moment and force resultants M and V in an
arbitrary cross section can be expressed as:
Â£,7,^ = A7 = A7, +V]xp0x^ (A7)
1 1 dx 1 1 y 2
(V/ + ^r) = V = V] (A. 8)
ax
Here TÂ¡ is moment of inertia and Gj is shear modulus of part
<1>.
114
Composites," Journal of Composite Materials 29(2), 1995:
156178.
[31] Kwon, Y.S., and Sankar, B.V., "IndentationFlexure and
LowVelocity Impact Damage in Graphite Epoxy Laminates,"
Journal of Composites Technology & Research 15, 1993: 101
111.
[32] Kwon, Y.S., and Sankar, B.V., "IndentationFlexure and
LowVelocity Impact Damage in Graphite/Epoxy Laminates,"
NASA Contractor Report 187624, 1992.
[33] Sankar, B.V., "LowVelocity Impact Response and Damage
in Composite Materials," Fracture of Composites, E.
Armanios, Ed., Transtech Publications, Ltd., Zurich,
Switzerland, 555582.
[34] Sawyer, J.W., "Effect of Stitching on the Strength of
Bonded Composite Single Lap Joints," AIAA Journal 23(11),
1985: 17441756.
[35] Su, K.B., "Delamination Resistance of Stitched
Thermoplastic Matrix Composite Laminates," Thermoplastic
Matrix Materials 5(2), 1991: 279300.
[36] Dexter, H.B., and Funk, J., "Impact Resistance and
Interlaminar Fracture Toughness of ThroughtheThickness
Reinforced Graphite/Epoxy," AIAA paper 861020CP, 1986.
[37] Pelstring, R.M., and Madan, R.C., "Stitching to
Improve Damage Tolerance of Composites," 34th International
SAMPE Symposium, Anaheim, CA., 1989: 15191528.
[38] Dransfield, K., Baillie, C., and Mai, Y., "Improving
the Delamination Resistance of CFRP by StitchingA Review,"
Composites Science and Technology 50, 1994: 305317.
[39] Shu, D., and Mai, Y., "Delamination Bucking with
Bridging," Composites Science and Technology 47, 1993:25
33.
[40] Shu, D., and Mai, Y., "Effect of Stitching on Inter
laminar Delamination Extension in Composite Laminates,"
Composites Science and Technology 49, 1993: 165171.
89
has to be stopped when the delamination propagated all the
way to the ends of the beam.
Table 4
.1 Various parameters used
in numerical examples
EXAMPLE
Mo
Vo
T
POSITION OF
Guc
NO.
(kg)
(m/s)
(J)
DELAMINATION
(hi/H)
(N/m)
1
5
1.5
5.625
0.5
530
2
5
1.5
5.625
0.5
300
3
1.25
3
5.625
0.5
530
4
2.5
1.5
2.813
0.5
530
5
2.5
1.5
2.813
0.5
300
6
5
1.5
5.625
0.25
530
7
5
1.5
5.625
0.25
300
8
2.5
1.5
2.813
0.25
530
9
2.5
1.5
2.813
0.25
300
Note: M0
energy).
(Impactor
Mass),
V0 (Impac
t velocity) T
(Kinetic
From the static analysis we obtain the loaddeflection
relation (Fq) and the delaminationdeflection relation (a
q). Sample Fq relations for Example 1 are shown in Figure
4.7. The Fq relation is linear until the crack begins to
propagate. It may be noted that the load at which the crack
begin to propagate is almost the same for all three cases
(no stitches, 16 ssi, and 64 ssi stitches). After that the
curves take different shapes depending on the stitch
density. The maximum load that the beam can carry very much
depends on the stitch density. The 64 ssi beam carries about
50% more load than the unstitched beam. The unloading was
assumed to be linear and hence the unloading curve was a
28
The strain energy U of the entire beam is the sum of
strain energy UÂ¡ in the cracked portion of the beam and
strain energy U2 in the uncracked portion of the beam, and
it has the following form:
U=Ul+U2
_ "3P2a 4EI27T4} P\La)
v 2EA Aa3 j 2EA
(2.23)
Strain energy release rate (G) at the crack tip is obtained
by differentiating total strain energy U with respect to
half of crack length a [53]. Then the expression for G can
be derived as:
G = 
2 B da s
1 f 4AES2 SEI;r2S(Sa + 2L)
~ 2B [(4a + 2L)1 {Aa + 2aLf
\6EI27v4(\2a2 +4 aL) 12Â£/2;r4
+ A(4a3 + 2a2 L)2 Aa4 J (2'24^
2.2.2 Delaminated Beam with ThroughtheThickness Stitches
A delaminated beam with stitching is depicted in
Figure 2.5.
80
Force and moment equilibrium equations:
Fx2 +Fx4 + FxS
(4.2)
+ F* f
(4.3)
+^z4 +FzS =
(4.4)
C2+C4 + C!+F.JFIt = 0
(4.5)
Compatibility equations at the joints:
h2
U2 =5 +y^5
(4.6)
K
4 = w5y^5
(4.7)
to
ii
ta.
n
(4.8)
to
II
II
O
(4.9)
Boundary conditions:
u] = u3 =0
(4.10)
o
II
ro
II
(4.11)
w] =w3
(4.12)
W6 =Fx6=C6= 0
(4.13)
113
[20] Klug, J., Wu, X.X., and Sun, C.T., "Efficient Modeling
of Postbuckling delamination Growth in Composite Laminates
Using Plate Elements," AIAA Journal 34(1), 1996: 178184.
[21] Graves, J.M., and Kantz, J.S., "Initiation and Extent
of Impact Damage in Graphite/Epoxy and Graphite/Peak
Composites," Proceedings of the AIAA/ASME/ASCE/AHS 29th
Structures, Structural Dynamics, and Materials Conference,
AIAA, Reston, Virginia, 1988: 967976.
[22] Clark, G., "Modeling of Impact Damage in Composite
Laminate," Composites 20(3), 1989: 209214.
[23] Grady, J.H., and Sun, C.T., "Dynamic delamination
Crack Propagation in a Graphite/Epoxy Laminate," Composite
Materials: Fatigue and Fracture, ASTM STP 907, 1986: 531.
[24] Grady, J.E., and Depaola, K.J., "Measurement of Impact
Induced Delamination Buckling in Composite Laminates,"
Dynamic Failure; Proceedings of the 1987 SEM Fall, Society
for Experimental Mechanics, Bethel, CT, 1987: 250255.
[25] Choi, H.Y., Wu, H.T., and Chang, F.K., "A New Approach
toward Understanding Damage Mechanism and Mechanics of
Laminated Composites Due to LowVelocity, Part I
Experiments," Journal of Composite Materials 25, Aug. 1991:
9921011.
[26] Sankar, B.V., and Hu, S., "Dynamic Delamination
Propagation in Composite Beams," Journal of Composite
Materials 25, Nov. 1991: 14141426.
[27] Hu, S., "Dynamic Delamination Propagation in Composite
Beams under Impact," PhD dissertation, University of
Florida, Gainesville, Florida, 1990.
[28] Razi, H., and Kobayashi, A.S., "Delamination in Cross
Ply Laminated Composite Subjected to LowVelocity Impact,"
AIAA Journal 31(8), 1993: 14981502.
[29] Sun, C.T., and Yih, C.J., "QuasiStatic Modeling of
Delamination Crack Propagation in Laminates Subjected to
LowVelocity Impact," Composites Science and Technology 54,
1995: 185191.
[30] Wang, H., and VuKhanh, T., "Fracture Mechanics and
Mechanisms of ImpactInduced Delamination in Laminated
115
[41] Jain, L.K., and Mai, Y., "On the Effect of Stitching
on Mode I Delamination Toughness of Laminated Composites,"
Composites Science and Technology 51, 1994: 331345.
[42] Jain, L.K., and Mai, Y. "Determination of Mode II
Delamination Toughness of Stitched Laminated Composites,"
Composites Science and Technology 55, 1995: 241253.
[43] Chen, V.L., Wu, X.X., and Sun, C.T., "Effective Inter
laminar Fracture Toughness in Stitched Laminates,"
Proceedings of the 8th annual technical meeting of American
Society of Composites, Technomic Publishing Co., Lancaster,
Pennsylvania, 1993: 453462.
[44] Kang, T.J., and Lee, S.H., "Effect of Stitching on the
Mechanical and Impact Properties of Woven Laminate
Composite," Journal of Composite Materials 28(16), 1994:
15741587.
[45] Wu, E., and Liau, J., "Impact of Unstitched and
Stitched Laminates by Line Loading," Journal of Composite
Materials 28(17), 1994: 16401658.
[46] Sharma, S.K., and Sankar, B.V., "Effect of Through
theThickness Stitching on Impact and Interlaminar
Fracture Properties of Textile Graphite/Epoxy Laminates,"
NASA Contractor Report 195042, 1995.
[47] Sankar, B.V., and Sonik, V., "Modeling EndNotched
Flexure Tests of Stitched Laminates," Proceedings of the 8th
annual technical meeting of American Society of Composites,
Technomic Publishing Co., Lancaster, Pennsylvania, 1995:
172181.
[48] Dickinson, L.C., "TransLaminarReinforced (TLR)
Composites," Department of Applied Science, PhD
dissertation, College of William and Mary, Williamsburg,
Virginia, 1997.
[49] Sankar, B.V., and Dharmapuri, S.M., "Analysis of a
Stitched Double Cantilever Beam," Journal of Composite
Materials 32(24), 1998: 22032225.
[50] Jain, L.K., Dransfield, K.A., and Mai, Y., "Effect of
Reinforcing Tabs on the Mode I Delamination Toughness of
Stitched CFRPs," Journal of Composite Materials 32(22),
1998: 20162041.
55
1 MJ
2 B Aa
>
5
Figure 3.5 Strain energy change
due to crack growth
3.5 Results and Discussion
In this section the results of the nonlinear FE
analysis are presented. The effects of stitching are
understood by analyzing the loadend shortening diagrams,
P8 curves, for various types of stitches. The second
quantity of interest is the energy release rate G at the
crack tip. The G is also plotted as a function of 8 for
various stitches. Another important and interesting
information is the amount of axial strain in the stitches,
CHAPTER 2
AN ANALYTICAL SOLUTION FOR POSTBUCKLING OF A DELAMINATED
COMPOSITE BEAM WITH STITCHING
In this chapter an analytical solution for post
buckling behavior of a delaminated composite beam with
stitching is derived based on RayleighRitz method. The
effect of stitching on buckling load and energy release
rate is discussed. Nondimensional parameters for buckling
load, energy release rate [G) crack length (a), and stitch
stiffness (K) are derived. Although finite element methods
can model a stitched composite more realistically,
analytical methods have several advantages. Analytical
methods provide a closedform solution to all the desired
quantities such as energy release rate, and critical
buckling loads. Further, analytical methods allow
introduction of nondimensional parameters, and the results
expressed in terms of these dimensionless parameters are
applicable over a wide range of the actual problem
variables such as beam thickness, delamination length, and
stitch density.
18
CHAPTER 5
CONCLUSIONS AND FUTURE WORK
5.1 Summary
In this dissertation analytical and finite element
methods are used to investigate the effects of throughthe
thickness stitching on the mechanical behavior of
delaminated composite laminates. In particular the buckling
and postbuckling behavior and lowvelocity impact response
of delaminated composites are studied. These two loading
modes were selected as they are very detrimental
to effective functioning of composites containing
delaminations.
The analytical modelsuse the RaleighRitz method in
which stitches are modeled as smeared foundation springs.
The finite element models have the capability of modeling
I
individual stitches. The lowvelocity impact response
calculations use the static loaddeflection behavior of the
stitched composite beam.
From the results presented in the previous chapters
the following conclusions can be reached:
101
[10] Davidson, B.D., "Energy Release Rate Determination for
Edge Delamination under Combined InPlane, Bending and
Hygrothermal Loading. Part IITwo Symmetrically Located
Delaminations," Journal of Composite Materials 28(14),
1994: 13711392.
[11] Davidson, B.D., and Krafchak, T.M., "A Comparison of
Energy Release Rates for Locally Buckled Laminates
Containing Symmetric and Asymmetrically Located
Delaminations," Journal of Composite Materials 29(6), 1995:
701713.
[12] Kyoung, W., and Kim, C., "Delamination Buckling and
Growth of Composite Laminated Plates with Tranverse Shear
Deformation," Journal of Composite Materials 29(15), 1995:
20472068.
[13] Lee, J., Gurdal, Z., and Griffin Jr., O. H., "Post
buckling of Laminated Composites with Delaminations," AIAA
Journal 33(10), 1995: 19631970.
[14] Yin, W.L., and Fei, Z., "Delamination Buckling and
Growth in a Clamped Circular Plate," AIAA Journal 26(4),
1988: 438445.
[15] Whitcomb, J.D., and Shivakumar, K.N., "StrainEnergy
Release Rate Analysis of Plates with Postbuckled
Delaminations," Journal of Composite Materials 23, July
1989: 714733.
[16] Suemasu, H., "Effect of Multiple Delaminations on
Compressive Buckling Behaviors of Composite Panel," Journal
of Composite Materials 27(12) 1993: 11721192.
[17] Naganarayana, B.P., and Atluri, S.N., "EnergyRelease
Rate Evaluation for Delamination Growth Prediction in a
MultiPlate Model of a Laminate Composite," Computational
Mechanics 15, 1995: 443459.
[18] Naganarayana, B.P., and Atluri, S.N., "Strength
Reduction and Delamination Growth in Thin and Thick
Composite Plates under Composite Loading," Computational
Mechanics 16, 1996: 170189.
[19] Shen, H., and Williams, F.W., "Postbuckling Analysis
of Imperfect Laminated Plates under Combined Axial and
Thermal Loads," Computational Mechanics 17, 1996: 226233.
76
force. The contact force F is a function of the beam
deflection. The contact force F will be a linear function of
g if there were no stitches or delamination propagation.
However in the present case it will be a nonlinear function.
Once F(q) is determined, then the equation of motion
Equation 4.1a4.1c can be numerically integrated to obtain
q(t). From q(t) one can compute the impact force history
F (t) using the Fq relation. In the following section we
discuss the procedure for determining the Fg relation in a
stitched delaminated beam.
4,2.1 Relation between Contact Force and Beam Deflection
The problem to be solved in this section is depicted in
Figure 4.2. Because of symmetry only one half of the beam
will be analyzed. The problem is to find the relation
between transverse force F and deflection g at the center of
the beam in the delaminated stitched beam. Further the
energy release rate G at the crack tip needs to be computed
also. In the numerical simulation the crack will be
propagated by a small distance (symmetrically on both sides)
if the G exceeds the Mode II fracture toughness GIIC of the
parent laminated material system.
First we will provide an overview of the procedures to
be followed. Since the structure is symmetric we can analyze
the right half of the beam as shown in Figure 4.3. The crack
Load P (lb)
64
Figure 3.8 Compressive load vs. end shortening
at crack length (a=l in.)
10
45
Method no restriction is placed on the location of the
crack in the thickness direction. Further the stitches do
not have to be smeared as continuous springs; each stitch
can be modeled individually, if necessary. In this chapter
the FE model is used to understand the effects of stitch
parameters, especially stitch density and stitch yarn
material properties, on the loadend shortening relation
and the energy release rate of a delaminated composite
under axial compression.
3.1 Specimen
The specimen considered for finite element analysis is
made up of 48 unidirectional AS4 graphite/epoxy plies. The
material properties for each lamina are: Â£=19.4X106 psi,
Â£=1.3X106 psi, Gi=0.7X106 psi, vir=0.28. The length L of
the specimen is 5 inches (127mm). The width B is 1 inch
(25.4mm) and the thickness H is 0.27 inch (6.86mm). A
single delamination (length of 2a) is located at a distance
h from the top surface of the beam. The specimen is
depicted in Figure 3.1. The specimen is under axial
compression. The FE analysis is used to simulate the
compression test. In particular we are interested in the
loadend shortening relationship and energy release rate as
a function of the load.
49
we denote the axial and transverse displacements by u and
w, then relative displacements are (see Figure 3.3):
A u = uxu2, A w = wxw2 (34)
AW2 t W1
 /W
u2 2
Figure 3.3 Node specification
for SPRING2 element
The procedure for determining the spring constants of
the stitches is as follows. Figure 3.4 shows a stitch where
the beam surfaces have relative displacements in both x and
z directions. The initial length of the stitch is the total
thickness of the beam, H. The relative displacements in the
x and z directions are Au and Aw, respectively. The
elongation of the stitch AL is related to the displacement
components by
o
1
Au = ALcos?
Aw = AL sin d
(3.5)
51
several buckling mode shapes. Thus a linear bifurcation
analysis has to be performed to determine the buckling mode
shapes. In the present analysis only the first mode shape
was considered in the initial imperfection. The
imperfection amplitude was 5% of the eigenmode calculated
by ABAQUS.
Top beam
1
Aw AL
u
A u
Bottom beam
Figure 3.4 Stitch deformation
The nonlinear analysis used the Riks algorithm in
order to determine the successive equilibrium positions. To
analyze a postbuckling problem, it must be turned into a
problem with continuous response instead of bifurcation.
This effect can be accomplished by introducing an initial
imperfection into a "perfect" geometry so that there is
some response in the buckling mode before the critical load
is reached. The postbuckling problem is a geometrically
nonlinear static problem. Static equilibrium states during
85
The strain energy density in terms of force and moment
resultants is given by:
U
L
P2
~EA
M2
El
~GAy
(4.16)
where P, M and V are the axial force, bending moment and
shear force resultants; EA, El and GA are the equivalent
axial, flexural and shear rigidities of the beam cross
section.
4.2.2 Impact Response
After computing the Fq relation for a beam, the impact
equations (Equation 4.1c) can be solved numerically. The F
q relations were stored in a spread sheet program (Excel) .
The expression for F(q) on the RHS of Equation 4.1a can be
approximated by a linear interpolation in each small
integration step as:
F{q) = Frf Fa F\ (??,) (4.17)
(02
where qÂ¡ and q2 are impactor initial and final displacements
and Fj and F2 are initial and final contact force in each
increment.
58
delamination length approaches the total length of the
beam, and global buckling takes over the subdelamination
buckling. During global buckling the delamination is under
more shearing mode than opening mode, and the stitches are
not that effective in suppressing the relative sliding
motion of the delaminated surfaces.
The effects of stitching on energy release rate G are
shown in Figure 3.10 through 3.12. Figure 3.10 corresponds
to a stitch density of 16 ssi. It is seen that G is almost
negligible until the sublaminate buckling begins.
Thereafter the G rises rapidly with the end shortening of
the beam. The stitches are able to delay the point when G
begins to rise. For example, in Figure 3.10, G begins to
rise at 50.03 in. in an unstitched beam, whereas it is
delayed until 50.55 in. in stitched beams. Further, the G
is greatly reduced compared to the unstitched beam. For
5=0.1 in. the value of G is reduced from about 800 lb./in.
in an unstitched beam to about 100 200 lb./in. in
stitched beams. Further it can be seen that stitches with
higher axial rigidity (see Table 3.1) cause the greatest
reduction in G.
Similar results for 64 ssi stitching are presented in
Figure 3.11. Both stitch densities are compared with
116
[51] Glaessgen, E.H., Raju, I.S., and Poe, C.C., "Fracture
Mechanics Analysis of Stitched StiffenerDebonding," AIAA
Journal 20(22), 1998: 26332647.
[52] Sankar, B.V., and Sonik, V., "Pointwise Energy Release
Rate in Delaminated Plates", AIAA Journal 33(7), 1995:
13121318.
[53] Anderson,T.L., Fracture Mechanics Fundamentals and
Applications, 2nd Ed., CRC Press, Boca Raton, Florida, 1995.
[54] ABAQUS /Standard User's Manual (Version 5.7).
[55] ABAQUS Theory Manual (Version 5.7).
41
a'
Figure 2.10 Variation of G' with a' for a given (8'=0.1)
7
located delamination is higher than an offcenter
delamination.
Lee et al. [13] proposed a finite element method based
on a layerwise laminated composite plate theory for
solving the postbuckling problem. In their study a contact
algorithm, which overcame the physically unallowable
overlapping between delaminated surfaces, was used.
There are also several works concerning two dimension
postbuckling analysis. Yin and Fei [14] presented an
elastic postbuckling analysis of a delaminated circular
plate under axissymmetric compression along its clamped
boundary. They found that certain features of the post
buckling behavior are qualitatively similar to the buckling
of an axially loaded beam plate containing a one
dimensional delamination. Whitcomb and Shivakumar [15] used
virtual crack closure technique to calculate the total
energy release rate in a locally postbuckled laminate with
embedded delamination. They found that there is a large
variation of energy release rate along the delamination
front for square and rectangular plates, and delamination
growth depends on current delamination aspect ratio, the
strain level, and the size of the delamination.
Suemasu [16] investigated an effect of multiple
delaminations of composite panels on the buckling loads by
110
We can derive expressions for Fxi, Fzi, and C (i=5,6) as
shown below. The procedure for deriving these expressions
is similar to that one in part <1>.
^=^(3.) <^2i)
^.=^(".5) (4.22)
4'z 2v43' /43Z 2/43'
w. If/ w, iy,
F7, = r + ^liT + +
4j L 2^3 /43 Z, 24
C5 =
^5 ,Â£373 L
w
EJ, L
2 A.
r + (rL + ~)Â¥s +TTr + ( 7^ +
4/4
2/4,
4/4,
C6 =
W5
2/4,
/ Â£3/3
+ ( +
w, ,Z7Z,
Vs + T~r + (7a + TT^Vs
4/4, 2/4, Z, 4/4,
(/4.23)
(24)
(25)
(/4.26)
A3 is area of cross section, E3 is Young's modulus and G3 is
shear modulus in part <3>, and A3 is given by:
1  Z2
G3A3 12Â£3Z3
9
compared to threedimensional finite method for calculation
of the energy release rate.
1.2.2 Research on LowVelocity Impact Response and Damage
of Delaminated Composite Materials
Lowvelocity impact response and damage caused by low
velocity projectile impact has received attention from some
researchers. When the lowvelocity foreign body drops on a
structure such as the surface of an airplane wing, it may
cause damage such as a delamination which is invisible on
the surface and difficult to detect. Therefore
understanding the lowvelocity impact response of a
delaminated composite is very important, and a method is
needed to predict the damage due to lowvelocity impact.
Graves and Kantz [21] used the maximum shear stress failure
criterion to estimate the damage area. Clark [22] provided
a model only for qualitative prediction of the delamination
size. Grady and Sun [23], and Grady and Depaola [24]
provided an estimation of the delamination growth due to
impact. Choi et al.[25] adopted a dynamic finite element
method associated with failure analysis to predict
threshold of impact of damage and initiation of
delamination. Sankar and Hu [26] and Hu [27] used a finite
element method to analyze dynamic delamination growth in a
47
separate beams, top beam and bottom beam. The nodes of each
beam are distributed uniformly along the axial direction.
Type B21 element, which considers transverse shear effect
is adopted. The stiffness matrix of this type is given in
ABAQUS Theory Manual [55]. The element size is chosen as
0.0125 inch. In an undelaminated portion the connection
between top and bottoms is achieved by using the EQUATION
command. The boundary conditions that were implemented are
depicted in Figure 3.2.
Stitch
Lx
Figure 3.2 Boundary condition in a delaminated beam
with throughthethickness stitches
The relations between the top and bottom beam
displacements for the uncracked portion of the composite
are as follows:
42
K'
Figure 2.11 Variation of G' with K' for a given a'
(a=0.2)
95
Table 4.3 Contact force vs.crack length in various examples
EXAMPLE
hi/H
CASE
Fi (N)
3max (nun)
Fmax (N)
Example 1
T=5.6125J
Gnc=530N/m
0.5
No stitch
553
56.4
645
16 ssi
562
50.4
678
64 ssi
577
44.4
713
Example 2
T=5.6125J
Gnc=300N/m
0.5
No stitch
416
101.4
431
16 ssi
423
101.4
465
64 ssi
438
59.4
679
Example 3
T=5.6125J
GIIC=530N/m
0.5
No stitch
553
56.4
645
16 ssi
562
50.4
678
64 ssi
577
44.4
713
Example 4
T=2.813J
Gnc=530N/m
0.5
No stitch
545
27.4
545
16 ssi
546
27.4
546
64 ssi
548
27.4
546
Example 5
T=2.813J
Gnc=30 ON/m
0.5
No stitch
416
49.4
478
16 ssi
425
44.4
496
64 ssi
438
39.4
517
Example 6
T=5.6125J
Gnc=53ON/m
0.25
No stitch
568
47.4
708
16 ssi
571
46.4
711
64 ssi
578
44.4
721
Example 7
T=5.6125J
Gnc=300N/m
0.25
No stitch
428
74.4
646
16 ssi
431
67.4
667
64 ssi
437
60.4
692
Example 8
T=2.813J
Gnc=530N/m
0.25
No stitch
547
27.4
547
16 ssi
547
27.4
547
64 ssi
549
27.4
549
Example 9
T=2.813J
Gnc=30 ON/m
0.25
No stitch
428
43.4
507
16 ssi
431
41.4
511
64 ssi
437
40.4
518
59
unstitched beam in Figure 3.12. It may be seen that 64 ssi
stitching offers slightly higher reduction in G.
As mentioned earlier monitoring the strains in the
stitches is an important consideration in analyzing
stitched specimens. Because stitch yarns could break if the
strains exceed the allowable limit and this could trigger a
domino effect of successive stitch failures and dynamic
delamination propagation, and lead to catastrophic failure.
The strains in various stitches as a function of end
shortening are shown in Figure 3.13. It should be noted
that the crack length is 1 inch and the stitch pattern is
4x1/4". Stitch 3 (Figure 3.13) is at the center of the beam
and thus undergoes large strains. Stitch 1 is actually at
the crack tip and hence there is no strain at all. Stitch 2
is in between stitches 1 and 3. The variation of stitch
strain with end shortening is strikingly similar to that of
G. discussed earlier.
The strains of stitches located at the center of beam
in various stitch materials and stitch patters are shown in
Figure 3.14. It may be seen that the strains are sensitive
to the stitch material as well as the stitch density. In
general the strains are significantly less for higher
stitch density. Further, stitches with lower axial
stiffness (ASES) undergo larger strains.
93
for midplane delaminations. Thus the stitches play a very
useful role in preventing crack propagation. This situation
is similar to the effectiveness of stitches for various
Guc s.
The stitches become more effective at higher impact
energies when the propensity for crack propagation is also
higher. Comparing Examples 1 and 4 in Figure 4.9 one can see
this phenomenon. In Example 4 the impact energy was very low
so that stitching was not necessary. However in Example 1
the effectiveness of stitch density could be inferred.
From the above discussion we can arrive at the
following conclusions:
1. Static simulations of delaminated stitched beam
provide an estimate of the apparent fracture
toughness of the stitched laminates. The stitch
density significantly affects the increase in
apparent fracture toughness. The percentage
increase is more for laminates with lower inherent
fracture toughness.
2. The impact force at which the delamination begins
to grow is not dependent on stitching. However
after the delamination growth is initiated stitches
come into play, and the extent of delamination
growth depends on the stitching parameters. In
general the extent of crack propagation at the end
72
The impact problem is depicted in Figure 4.1. The force
acting on the beam is not known a priori, and has to be
calculated as part of the solution of the problem. A major
difference between the impact loading and the compressive
loading discussed in the earlier chapters is that the
delamination is under pure Mode II (shearing mode)
conditions. Previous experimental studies by Sharma and
Sankar (46] have found that under pure Mode II the stitches
try to plough through the matrix, and the resistance offered
by the matrix is responsible for the increase in apparent
fracture toughness. Thus the stitch model has to be modified
to account for this phenomenon.
M0, V,
F(t)
Impactor
V7
+
F(t)
M0: Impactor mass
V0: Impact velocity
t: Time Variable
Figure 4.1 A structure under impact
92
(hl/H). Thus it does not depend on impact parameters such as
impact energy.
In general the extent of crack propagation at the end
of the impact event is the least in the 64 ssi beam and the
highest in the unstitched beams. The result for 16 ssi beams
are somewhere in between. This can also be observed readily
from the bar charts in Figures 4.9 and 4.10. However the
amount of delamination propagation depends also on the
impact energy, Gllc and hl/H. Stitching is very effective in
the beam with lower inherent fracture toughness. In
Example 2 (Gllc = 300 N/m) the crack propagates all the way
to the ends of the beam in the unstitched and 16 ssi
specimens, whereas the crack propagated up to a=59.4 mm in
the 64 ssi beam. In Example 1 {Gllc = 530 N/m) the stitches
were able to reduce the delamination extension by about 6 
12 mm.
Stitches are also more effective when the crack is in
the middle plane of the laminate (hl/H= 0.5) compared to the
cases wherein the crack is near the top surface of the beam
(hl/H=0.25). This can be explained as follows. In an
undelaminated beam the shear stresses are higher at the
midplane (1.5 times the average shear stress) compared to
the plane at 1/4 distance from the top. Thus when the
delamination is at the midplane the tendency for propagation
is much higher. In fact the energy release rate G is higher
Maximum load P (lb)
65
Crack length a (in.)
Figure 3.9 Maximum load vs. crack length
88
problem, i.e., aq relation, we can translate that into at
relation, and thus the propagation of delamination can be
followed.
4.3 Numerical Example, Results, and Discussion
The specimen dimension and material properties used in
the impact simulation are as follows: initial uncracked
length L0=74.2 mm, initial crack length a0=27.4 mm, beam
width B=25.4 mm, equivalent Young's modulus Â£eg=90.375 GPa,
equivalent shear modulus Gxy=6.8GPa. Stitching material is
3570 denier glass yarn. Nine different examples were
studied. In these examples, the impactor mass, impact
velocity, position of the delamination in the thickness
direction and the Mode II fracture toughness Gllc were
varied. The parameters used in these examples are listed in
Table 4.1.
In Table 4.1 H is the total thickness of the beam
and hj is the distance of delamination from the top surface
(impact surface). For each example three different cases 
beam without stitching, 16 ssi stitches, and 64 ssi stitches
 were considered. Thus a total of 27 impact simulations
were performed. The simulations were stopped when the
contact force becomes equal to zero denoting the contact
between the impactor and the beam has ceased. In two cases
(Example 2, no stitches and 16 ssi stitches) the simulation
90
straight line joining the point of unloading and the origin.
This assumption is validated by the Mode II experiments
conducted by Sharma and Sankar [46].
Another interesting result that can be deduced from the
static loaddeflection curve is the apparent fracture
toughness of stitched laminates. The area enclosed by the
loaddeflection diagram (Fig.4.7) denotes the work of
fracture. Since we know the extent of delamination
propagation we can compute the apparent fracture toughness
from:
AW
here AW is the work done and Aa is the new delamination
surface created. The apparent fracture toughness for various
cases is presented in Table 4.2 along with that for
unstitched laminates. The numbers in parentheses are the
percentage increase in apparent fracture toughness. It may
be seen that the percentage increase in Gllc is higher for
laminates with lower fracture toughness.
The results for each impact analysis include the
complete impact force history (Ft) and the delamination
propagation history (at). A sample impact force history is
shown in Figure 4.8. In general, stitched beams carry more
impact force, provided the impact energy is sufficient to
cause delamination propagation. For low impact energies, the
10
composite beam. The delaminated beams were modeled as two
offset beams and spring elements were used to connect the
beams in the uncracked portion. The dynamic energy release
rate was computed from the strain energy in the cracktip
element. Razi and Kobayashi [28] performed finite element
analysis and experiments to study damage growth and
distribution in crossply laminate beams and plates. A
quasistatic finite element analysis coupled with a Gu> Guc
criterion was proposed for the delamination growth, and Gu <
Giia criterion was used to arrest crack propagation. Sun and
Yih [29] investigated the quasistatic characteristics of
impact responses and the impactinduced delamination growth
of composite laminates subjected to lowvelocity impact.
They used a springmass model to predict contact force
history and the peak force was associated with the energy
release rate. Thermal loads were included in computing
energy release rate. Wang and VuKhanh [30] also used the
finite element method to model carbon fiber/PEEK crossply
damaged laminates. They found that a delamination occurred
in the form of a Mode IIdominated unstable crack growth
and subsequent arrest.
Kwon and Sankar [3132] performed experiments to study
the lowvelocity impact damage of quasiisotropic and
crossply graphite/epoxy composite laminates. They found
26
Based on Equation 2.13 and 2.19 the relation between the
compressive load P and the end shortening Sc will be as
shown in Figure 2.3.
Figure 2.3 The relation between compressive
load P and end shortening 80
In Figure 2.3 Scr is end shortening at critical buckling load
Pcr and dd is end shortening at an arbitrary compressive
load Pj.
Based on Figure 2.3, the strain energy Ui of the
cracked portion can be written as:
V>=\P',S',+\(.PtP'r)(.8aS',)
_ 3Px2a 4nAEl1
~ 2EA Aa3
(2.20)
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
EFFECT OF THROUGHTHETHICKNESS STITCHING ON
POSTBUCKLING AND IMPACT RESPONSE OF
COMPOSITE LAMINATES
By
Huasheng Zhu
December 1999
Chairman: Bhavani V. Sankar
Major Department: Aerospace Engineering, Mechanics, and
Engineering Science
Fiber composite materials have very high specific
strength and stiffness. This structural efficiency enables
composite materials to be used in a wide variety of
applications. However, laminated composites have also some
disadvantages such as poor interlaminar strength, low
impact resistance, and poor delamination resistance. During
the manufacture of fiberreinforced composite laminates,
imperfections can cause initial delamination, and the
impact on laminated composites by foreign objects during
service can cause delamination propagation. Delamination is
one of the most dominant forms of damage due to lack of
reinforcement in the thickness direction. An effective
v
94
of an impact event is small for higher stitch
densities.
3. Stitching is effective when the impact energies are
higher and the propensity for delamination growth
is also higher.
4. Stitching is more effective when the delamination
is at the center.
73
In the following sections we discuss the assumptions
made in simplifying the impact problem, the stitch model and
the impact simulation procedure. The results focus on the
extent of delamination propagation due to impact in
specimens with different stitch densities.
4.1 Basic Assumptions
The following assumptions are made to simplify the
impact simulation:
1. The velocity of impact is low compared to the
velocities of wave propagation in the composite
beam.
2. The projectile is assumed to be rigid compared to
the target. Therefore the impactor can be treated as
a rigid body and its equation of motion is greatly
simplified.
3. The target, laminated beam in the present case, is
highly flexible. The deflection of the beam is
expected to be much higher than the local
indentation, and hence the Hertzian indentation
effects can be neglected.
4. The impactor mass is much greater that that of the
beam, and hence the impact duration will be very
long compared to the fundamental period of vibration
of the beam. Therefore the target can be represented
87
where V2 and V2 are initial and final velocities in each
step.
The corresponding impact time can be derived using
Equation 4.18b by assuming that the impact velocity in each
step varies linearly:
V(q) = Vl +Vl ~V' (qq,) (4.20a)
?2 "?i
here g2 and q2 are impactor initial and final displacements
in each step. Integrating Equation 4.18b and then
substituting Equation 4.20a we derive the following
integral equation:
,y,tx
(?2 )
Integrating Equation 4.20b an expression for time t2 can be
obtained as:
t2 /,
+ iiin
(Vivj V,
(4.20 c)
here tj and t2 are initial and final time in each step.
Equation 4.20c provides the qt relation for the impact
problem. From that using Fq relations, we can obtain the F
t relation or the impact force history. Since we know the
delamination length at each displacement in the static
Contact force F(N)
97
Center deflection q (m)
Figure 4.7 Contact force F vs. center deflection q
(GIIC=530J/m2)
43
8'
Figure 2.12 Variation of G with 5 for a given a (a =0.2)
37
will propagate rapidly in an unstable manner. For a'>0.1,
dG'/da' is negative and hence the crack propagation will be
stable.
In Figure 2.11 the relation between nondimensional
energy release rate G' and nondimensional stitch stiffness
at a given crack length (a'=0.2) is examined and the effect
of nondimensional end shortening 5' is shown. It
demonstrates that energy release rate G' decreases with
increasing the stiffness K' of stitching. The more non
dimensional end shortening 8' is, the more nondimensional
energy release rate G' has.
The variation of G' with S' for various stitch
stiffnesses is shown in Figure 2.12. The results correspond
to a0.2. The relations were very similar for other crack
lengths also. From Figure 2.12 one may note that the G'
remains close to zero in the beginning, but increases after
certain critical S'. The variation of G' is approximately
quadratic in S'. Further, it may be noted that for a given
S', G' decreases with the K'. This is brought out clearly in
Figure 2.12, where G' is plotted as a function of K' for
various S'. One can see that the dramatic decrease in G' with
increasing K'.
108
where
_ 1 a2
A, 1
G,4 \2EJX
Free body diagram of portion <2> is shown in Figure
A. 4:
Po
Figure A.4 Free body diagram in part <2>
The procedure for deriving the expressions for Fxi, Fzi,
and Ci (i=3,4) is similar to that one in part <1>, therefore
the derivation is omitted here.
Poa A2E2
f*3 = ^ + ^~Mw3 ua)
2 a
Poa a2e2
= + u3)
2 a
1 p0h2a3 y/3a \Â¡/xa
Fz4 = (Â£JLi w3+1* + w.+ ll)
A2a 24E2I2 3 2 4 2 '
F = L (ME + Wj Vt _
A2a 24 E,1
21 2
(15)
(A. 16)
(17)
2
2
(18)
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REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchemainstance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID EXSTEZBTA_ZXH2OK INGEST_TIME 20170724T21:03:00Z PACKAGE AA00029809_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES
25
2 4P(2a)2 32/
EAn2 A
(2.15)
In order for d2 > 0, we have:
P > P.
8 n2El
(2a)2
(2.16)
here Pcr is the critical buckling load.
This solution set describes the behavior of the
cracked portion of the beam where the compressive load is
greater than the buckling load Pcr. There is a deformation
in the z direction. The relation among compressive load P
and axial displacement u0 and deformation shape w are:
3 Px 8 n2Ix
2AE A(2a)2
(2.17)
w = d sin2
TJX
2 a
(2.18)
here d is obtained from Equation 2.15 for a given P. End
shortening 8C at x=2a can be expressed as shown as:
u
0 \x=2a
' 3 P
k2AE
8n2l >
A(2a)2 J
(2a)
(2.19)
91
impact force history will be identical in stitched and
unstitched beams, because the stitches come into effect only
when there is sufficient energy to propagate the
delamination.
Table 4.2 Comparisons of Gllc
CASE
Guc(N/m)
Gnc for 16ssi
Guc for
64ssi
1
530
595 (12%)
707
(33%)
2
300
344 (15%)
430
(43%)
The results presented in Table 4.3 show for each case
impact energy, Gllc of the parent material system, the
contact force F at which the delamination began to
propagate, the final crack length amax, and the maximum
contact force Fmax during the impact event. The extent of
delamination propagation is also shown in the bar charts in
Figures 4.9 and 4.10. Figure 4.9 considers the examples
wherein the Gllc 530 J/m2' and Figure 4.10 corresponds to
Giic= 300 J/m2.
There are many interesting observations that can be
made from the results presented in Table 4.3. The contact
force at which the delamination propagates is almost the
same in unstitched and stitched beams. This initiation force
depends only on the Gllc and the position of the crack
BIOGRAPHICAL SKETCH
Huasheng Zhu was born on July 17, 1966, in Shenyang,
People's Republic of China. She received a B.E. degree from
Northwestern Polytechnical University in the Department of
Aircraft in China in July 1987. Then she worked at Shenyang
Institute of Aeronautical Engineering four years. She came
to the United State for her graduate study in August 1991.
She obtained her M.S. degree in Aerospace Engineering from
the Department of Aerospace Engineering, Mechanics, and
Engineering Science, University of Florida, in May 1994 and
continued her Ph.D. study at the same department. She will
receive her Ph.D. in December 1999.
117
11
that the impact force history and delamination radius could
be predicted using the results from a static indentation
flexure test. Sankar [33] reviewed the necessary concepts
for understanding lowvelocity impact response and damage
in fiber composite materials and gave several algorithms
for predicting the impact force history to different
degrees of approximations. Sankar [33] presented a semi
empirical method for predicting impact damage in
composites.
1.2.3 Research Related to Delaminated Composite Materials
with Stitches
In transversely stitched composites the delamination
is not completely separated but is held together by the
stitches. Therefore the propagation of cracks can be
prevented in stitched composite materials. Several studies
have investigated the effect of stitching on mechanical and
impact properties, failure loads, interlaminar strength,
and fracture toughness.
In general there are three types of stitches as shown
in Figure 1.1: modified lock stitch, chain stitch, and
standard lock stitch. Some parameters of stitches are:
Stitch density, stitch yarn, and yarn material density. We
define the stitch density in a composite laminate by the
16
response of undamaged composite laminates. Sankar and
Dharmapuri [49] presented an analytical mode to describe
delamination growth in laminated DCB (Double Cantilever
Beam) specimen with stitches and proposed a simple method
to compute the energy release rate due to delamination in
the stitched specimen. Jain and Mai [50] studied the effect
of reinforced tabs on the Mode I fracture toughness of
stitched composites. They indicated that reinforcing
aluminum tabs along the length of stitched DCB specimens
can alter the failure mechanism of the stitch threads and
hence erroneous values for Mode I fracture toughness.
Glaessgen et al [51] used finite element method in
conjunction with crack closure technique to study the
effect of stitching on Mode I and Mode II energy release
rate for various debond configurations.
1.3 Objectives and Scope of the Present Study
From the above sections we can realize that one of the
most dominant damage in composites is delamination growth
due to lack of reinforcement in the thickness direction.
Stitching is found to be one of the effective reinforcing
methods in the thickness direction of laminated composites.
However it is not clearly understood how stitching can
delay the delamination propagation of composite laminates
4
EFFECT OF STITCHING ON LOWVELOCITY IMPACT RESPONSE...71
4.1 Basic Assumptions 73
4.2 Analytical Model 75
4.2.1 Relation between Contact Force and
Beam Deflection 76
4.2.2 Impact Response 85
4.3 Numerical Examples, Results, and Discussion 88
5 CONCLUSIONS AND FUTURE WORK 101
5.1 Summary 101
5.2 Suggestions for Future Work 103
APPENDIX DERIVATION OF FORCEDISPLACEMENT RELATIONS
IN THE BEAM ELEMENT 104
REFERENCES 112
BIOGRAPHICAL SKETCH 118
IV
14
composite materials. The advantages and disadvantages of
adopting a stitch in a composite structure and the
necessary technigues of manufacture were examined. Shu and
Mai [3940] used analytical approaches to investigate the
effect of stitching on buckling load and energy release
rate of thin film structure. They assumed that the stitches
followed a Winkler elastic foundation type of stress
separation relation. The results shown that the strength of
composite laminates under edgewise compression can be
significantly increased. Jain and Mai [41] proposed two
analytical models to study the effect of stitch parameters
such as stitch density, stitch thread diameter, and
matrix/stitch interfacial shear stress etc. on improving
Mode I delamination toughness. They found that a large
matrix/stitch interfacial shear stress, high stitch density
and a small stitch thread diameter were desirable in order
to maximize the delamination growth resistance. Jain and
Mai [42] also used an analytical method to study the effect
of stitching parameters on Mode II energy release rate in a
laminated composite and found that throughthickness
stitching can improve Mode II fracture toughness also. Chen
and Sun [43] used a finite element method to propose an
effective energy release rate for measuring Mode I fracture
toughness of a stitched composite laminate. Kang and Lee
103
effective when the impact energies are higher and when
the delamination is at the center.
5.2 Suggestions for Future Work
The present research assumed a single central
delamination. The models, especially the FE models, can be
extended to beams with multiple delaminations. Further, FE
models can handle laminated plates with stitching and other
forms of translaminar reinforcements. Since it will not be
possible to model individual stitches in larger plates,
eguivalent stitch parameters such as foundation spring
constants may have to be used. More complex constitutive
relations based on experimental observations can be used to
model the stitch yarn or other translaminar reinforcements.
Since stitching results in increased cost and weight,
optimization studies can be undertaken to determine the
effective stitch materials and stitch pattern in the design
of laminated composite structures.
CHAPTER 1
INTRODUCTION AND LITERATURE REVIEW
1.1 Background
Fiber composite materials have very high strength and
stiffness but low density. This structural efficiency
enables these composite materials to be used in a wide
variety of applications, especially in aerospace
structures. However, advanced composites also have some
disadvantages such as poor interlaminar strength, low
damage resistance, low damage tolerance, and poor
delamination resistance. Since most of the commonly used
advanced composites are made by laying up fibrous
reinforcement in a surrounding matrix and the mechanical
properties between the plies are dominated by a relatively
weak matrix, this weakness causes low damage resistance and
low damage tolerance and finally results in low
compressionafterimpact strength.
A delamination is an interface crack or a debond
between two adjacent plies. It is one of the critical
failure modes in composite materials. During the
manufacture of fiberreinforced composite laminates,
1
cr
39
0.0 0.1 0.2 0.3 0.4
a'
Figure 2.8 Variation of Pcr' with a' for different K'
in the range 0
15
[44] did an experiment to study the effect of stitch on
mechanical and impact properties of woven laminated
composites. The mechanical properties were improved at an
optimum stitch density compared with unstitched woven
laminated composites. The damage area caused by repeated
impact was far smaller in stitched woven composites than
that in unstitched composites. Wu and Liau [45] also used
an experimental method to investigate the behavior of
stitched laminates under low velocity impact. They also
found that stitching could significantly reduce the
delamination area. With increasing the stitch density the
damage mechanism changed from delamination in unstitched
laminates to the plastichinge type of local deformation at
the impact location in stitched laminates. Sharma and
Sankar [46] performed experiments to study the effects of
stitching on CAI (CompressionAfterImpact) strength,
fracture toughness in Mode I and Mode II. They found that
CAI strength, Fracture toughness of Mode I and Mode II can
be improved significantly by stitching. Sankar and Sonik
[47] used Finite Element and analytical models to study the
effect of stitching on Mode II fracture toughness.
Dickinson [48] adopted a micromechanics approach to
characterize the effects of stitching on the elastic
constants, inplane strength and interlaminar mechanical
30
Z
2a
H M
Figure 2.6 Deformed shape of crack portion with stitch
Strain energy U of the delamated portion of the beam
with stitching includes two parts: strain energy Dnostitch
from the delaminated portion of the beam without stitching
and the energy Ustitch contributed by the stitches. The total
strain energy U is of the following form:
^ ^ nostitch ^ stitch
= Unostitch + \ \K (2w)2 dx (2.25)
here K is foundation constant that represents the stiffness
of the smeared stitches. The foundation constant K is
related to the actual stitch parameters as:
22
here
Â£ x ~I
dx 2\dx )
du 1 (chv\2
+
(2.2)
dx 2\dx,
Here Uo is the midplane displacement in the x direction, w
is the displacement of top sublamated beam in the z
direction. sx is the normal strain of top sublaminated beam
in x direction. The crosssectional area of the
sublaminates is represented by A and E is the Young's
modulus. Assume the forms of u0 and w as: u0=cx,
w=dsin"7ix/2a, where c and d are unknown coefficients to be
determined.
The strain energy of the delaminated portion of the
beam can be derived as:
2 a
U=Â¡Uldx
0
+
3 7T2d4 2ElnAd2
32(2 af (2a)2
+
(2.3)
Energy release rate G (lb/in.
66
Figure 3.10 Energy release rate vs. end shortening
(a=l" h/H=l/4)
79
displacements at the junction of all three elements is
depicted in Figure 4.5.
Figure 4.4 Free body diagrams of part <1>, <2>, <3>
Figure 4.5 Compatibility of displacement
in axial direction at joint position
The force and moment eguilibrium eguations, the
compatibility eguation and boundary conditions are as
follows:
60
From the above discussion we can arrive at the
following conclusions:
1. Stitching can improve the buckling loads of
delaminated beams. The increase is not sensitive
to the three types of yarns used in this study.
The yarns had axial rigidities in the range of
13,000 N 223,000 N and they seem to perform
equally well. However, the maximum load increases
with the stitch density. This is consistent with
the experimental observation of Sharma and Sankar
[46] .
2. Throughthethickness stitches significantly
reduce the energy release rate at the crack tip.
They also increase the end shortening required to
cause sublaminate buckling when the energy release
rate rises rapidly. By reducing the energy release
rate stitching prevents the crack propagation
which may lead to catastrophic failure of the
specimen. The axial rigidity of the stitches also
affects the energy release rate. The Higher axial
rigidity the greater is the reduction in G.
3. The maximum strain in the stitches depends
strongly on the axial rigidity of the stitches and
the stitch pattern. The stitches with small axial
Impact load F(N)
98
Figure 4.8 Impact load F vs. time t
CHAPTER 4
EFFECT OF STITCHING ON LOWVELOCITY IMPACT RESPONSE
In the previous two chapters we noted that a
delaminated beam is susceptible to buckling failure under
axial compression and that stitching can greatly improve its
compressive loadcarrying capacity. During sublaminate
buckling the stitches are predominantly under tension, and
the fracture mode is close to Mode I or the opening mode.
Another type of loading that can be detrimental to
delaminated beam is lowvelocity foreign object impact.
Composite structures are prone to low velocity impact damage
due to dropped hand tools, runway debris, hail stones etc.
If the composite structure has prior delaminations, then the
impact force can cause them to propagate, leading to
catastrophic failure. Translaminar reinforcements such as
stitching are expected to prevent delamination propagation
during foreign object impact.
In this chapter we study the effects of stitching on
the impact response of delaminated beams. Several
assumptions are made to simplify the problem so that the
focus can be on the effects of various stitching parameters.
71
75
stitches per inch. In terms of stitch density the
distributed traction can be written as:
p0 = NBDh
where N=l/(nxs) is the number of stitches per unit area. As
the delamination propagates new stitches come into action in
the freshly created delamination areas, and they offer
additional shear resistance. This assumption is consistent
with the experimental observations of Sharma and Sankar
[46] .
4.2 Analytical Model
The equation of motion of the impactor along with the
initial conditions can be written as:
M,^r = F(q)
at
(4.1 a)
<7(0) = 0
(4.1 b)
% =v
(4.1 c)
here M0 is the impactor mass, q is the impactor displacement
which is same as the transverse deflection of the target
beam at the point of impact, V0 is the initial velocity of
the impactor or the impact velocity, and F (q) is the contact
81
Fxi and Fzi are external forces in the axial direction and
transverse direction respectively. C is moment of a beam.
Odd indices i (i=l,3,5) denote the left end node of each
element and even indices (i=2,4,6) correspond to the right
end nodes. Expressions of Fxi/ Fzi and C are derived in the
Appendix.
After implementing the aforementioned element
equilibrium conditions and displacement compatibility
conditions at the nodes, a compact set of 5 equations are
obtained for the five displacements wlf u5, w5, y/5f and as
shown in Equation 4.14:
"*n
*12
*13
*14
*15'
7,'
*21
*22
*23
*24
*25
5
7
*31
*32
*33
*34
*35
<:
W5
> = 
*4.
*42
*43
*44
*45
^5
7
.*51
*52
*53
*54
*55.
/5.
(4.14)
The stiffness coefficients kj (i=1,5, j=l,5) and the
generalized forces F2...F5 on the right hand side of Equation
4.14 are given in Equation 4.14a:
k]2 k15 k21 k23 k25 k32 k5! k52 0
1 1
=
B,a B2a
k k 
/v13 n 31
1 1
Bx a B2a
20
2.2 Analytical Model
In this section we first derive the buckling load Pcr
and energy release rate (G) in an unstitched composite beam
using the RayleighRitz method and then obtain the buckling
load Pcr and energy release rate (G) for a stitched
composite beam. A single crack is located at the center of
the structure.
2.2.1 Delaminated Composite Beam Without Stitching
A composite beamlike structure without considering
throughthethickness reinforcement is shown in Fig. 2.1.
4
2L
Figure 2.1 A delaminated beam
The length of the beam is 2L, B is the width, 2h is the
thickness of the beam, 2a is the delamination length and P
is the compressive load.
100
120
100
80
&
ji
o 60
M
O'
u
<0
20
0
Initial crack length of a specimen
Final crack length of an unstitched specimen
Final crack length of a specimen with stitching (16 ssi)
Final crack lenth of a specimen with stitching (64 ssi)
Figure 4.10 Crack growth in different examples
(GiIC=300J/m2)
61
rigidity undergo large axial strains during
sublaminate buckling.
6
a laminate with a thin delaminated sublaminate through a
procedure that is based on large deflection theory of the
delaminated layer.
Chen [78] proposed a shear deformation theory and
obtained energy release rate by a variational energy
principle. He found that the energy release rate is larger
with transverse shear effect included for the same applied
load. Therefore the initiation of delamination growth will
occur at a lower applied load than that evaluated with the
classical lamination theory. He also found that the
magnitude of the transverse shear effects depends on the
delamination location and size.
Davidson and Krafchak [9], Davidson [10], Davidson and
Krafchak [11] used crack tip element analysis to determine
total energy release rate and individual mode I and II
energy release rate in different loading conditions or
different delamination cases.
Kyoung and Kim [12] presented an energy method to
obtain an analytical solution for determining the buckling
load and the growth of delaminated beamplate structures.
In their study the delamination was arbitrary located. They
also found that the shear effects decrease the buckling
loads and increase energy release rate. The results from
their analysis show that the buckling load of a center
53
propensity for propagation along the plane of delamination
as the interlaminar fracture toughness is much lower for
most laminated composites. In particular Mode I (opening
mode) interlaminar fracture toughness is about onehalf of
Mode II (shearing mode) fracture toughness. For example,
the values for a typical AS4/35016 graphite/epoxy
composite are [46] : Gjc = 300 N/m and Guc = 670 N/m. Thus
the effectiveness of stitching can be judged by the
reduction in energy release rate, G, at the crack tip.
There are three different methods to compute G: (i)
computing stress intensity factor at the cracktip; (ii)
Strain Energy Density Method; (iii) Virtual crack growth
method. The first method requires detailed three
dimensional stress field near the crack tip, which can only
be obtained by using threedimensional finite elements with
a fine mesh. Since we are using structural elements in the
present analysis, the stress intensity factor approach
cannot be used. The strain energy density method developed
by Sankar [52] requires computation of energy densities
just behind and ahead of the crack tip. However in the case
of stitched composites, the stitch forces affect the energy
densities significantly, and hence a very fine mesh is
needed near the crack tip. Further ABAQUS does not provide
the energy density values readily, and they need to be
27
We can extend the results to the entire composite
beam. After the compressive loads P reaches the buckling
load Pen the beamlike structure will deform as shown in
Figure 2.4.
H H
*
2L
*
Figure 2.4 Deformed shape of the entire delaminated beam
The relation between the compressive load (P) and end
shortening (S) is:
EAS
L
(P
(2.21)
P =T~j(8 +T) (P>Pcr)
2a + L Aa
Aa
(2.22)
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ii
ABSTRACT v
CHAPTERS
1 INTRODUCTION AND LITERATURE REVIEW 1
1.1 Background 1
1.2 Literature Review 3
1.2.1 Research on Postbuckling of Delaminate
Composite 4
1.2.2 Research on Low Velocity Impact Response and
Damage of Delaminated Composite Materials...9
1.2.3 Research Related to Delaminated Composite
Materials with Stitches 11
1.3 Objectives and Scope of the Present Study 16
2 AN ANALYTICAL SOLUTION FOR POSTBUCKLING OF
A DELAMINATED COMPOSITE BEAM WITH STITCHING 18
2.1 Basic Assumptions 19
2.2 Analytical Model 20
2.2.1 Delaminated Composite Beam
without Stitching 20
2.2.2 Delaminated Beam with ThroughtheThickness
Stitches 28
2.3 Results and Discussion 34
3 FINITE ELEMENT ANALYSIS OF A DELAMINATED COMPOSITE
WITH STITCHING 44
3.1 Specimen 45
3.2 Finite Element Model 46
3.3 PostBuckling Analysis 50
3.4 Energy Release Rate 52
3.5 Results and Discussion 55
iii
cr
40
a'
Figure 2.9 Variation of Pcr' with a'
for different K'
in the range 0.1
105
F P
1 X2 *2
F V
1 Z 2 v 2
C, = M\
Now we consider the internal force P in the axial direction
shown in Figure A. 2.
dx
l 4
Po
>P+dP
Figure A.2 Free body diagram
in axial direction
The differential equations of equilibrium are:
A,E.
d2u dP
dx dx
= Po
(A3)
M4)
Here Aj is area of cross section of part <1>, E: is young's
modulus of part <1>. Substituting boundary condition at
x=0, u=Ui, at x=a, u=u2 into Equations A. 3 and A. 4 and
consider the equations in Equations A.l and A.2, we have
following expressions of Fxl and Fx2:
35
an unstitched beam decreases as the nondimensional crack
length a' increases. This is understandable because a
structure buckles more easily with increasing crack length.
When we look at stitched beams in the range 0.1
Fig. 2.9) the buckling load is always higher than that of
unstitched beam, and further the buckling load Pcr' increases
as the stiffness of stitching K! increases. In a stitched
beam an interesting phenomenon occurs: the buckling load Pcr'
decreases as the crack length a' increases in the range
0.1
not decrease due to increasing crack length a', but in
contrast Pcr' increases to some extent depending on the
stiffness of stitching. The reason is that the longer
crack, the more stitches involved in the structure and the
stitches hold the structure together more tightly,
therefore the buckling load Pcr can be improved in the
stitched beam even with longer delamination. A similar
behavior was obtained by Shu and Mai [39] for the case of
thin film delamination with stitches. Figure 2.9 shows this
phenomenon in detail.
In Figure 2.10 the relation between nondimensional
energy release rate G' and nondimensional crack length a' and
the effect of K' on energy release rate G' for a
48
h, h.
, u =yY', + y^
(3.1)
wt wb = 0
(3.2)
Â¥, = n
(3.3)
In the above three equations t and b denote the top and
bottom beam respectively, u is the nodal displacement in
the x direction, w is the nodal displacement in the z
direction and i// is the rotation.
Throughthethickness stitches are modeled as linear
springs using SPRING2 elements in ABAQUS. Two spring
elements are used to model each stitch. One spring
represents the resistance offered by the stitch to relative
transverse displacements of the top and bottom beams. The
second spring element represents the relative displacement
in the axial (longitudinal) direction. These spring
elements connect the two corresponding nodes on the top and
bottom beams. The relative displacement across a SPRING2
element is the difference between the ith component of
displacement of the spring's first node and the ith
component of displacement of the spring's second node. If
33
Strain energy release rate G (when P>Pcr) is
j_du_
2 B 8a
1 [576Â£2(20a6 +12a5I) \92KI(4a2 +4 aL)
2B { (4a + 2L)2 EAk4 + A(4a + 2L)2
48K8(8a3 + 6 a2L)
n2 (4a + 2L)2
48K1
720K2a4}
A
EAn4 J
(2.31)
Let us define the nondimensional parameters as:
Pcr=Pcr /EA, a=G/Eh, a'=a/L, B'=B/L, h'=h/L,8 '=8/L, K'=K/E. Pcr'
is nondimensional buckling load; G' is the nondimensional
energy release rate; a' is the nondimensional half crack
length of the structure; B' is the ratio of the width of the
structure to the half length of the beam; h' is the ratio of
the thickness to the length of the structure; 8' is the
nondimensional end shortening of the beam along the axial
direction; K' is the nondimensional stiffness of the
stitches. After tedious algebraic manipulations we can
derive the following relations:
Per
n2h \2K a
Â¡T
6 a
n~Bh
(2.32)
Maximum Strain of stitch
70
Figure 3.14 Maximum strain in a stitch
vs. end shortening (a=l" h/H=l/4)
REFERENCES
[1] Chai, H., Babcock, C.A., and Knauss, W.G., "One
Dimension Modeling of Failure in Laminated Plates by
Delamination Buckling," International Journal of Solids and
Structures 17(11), 1982: 10691083.
[2] Yin, W.L., and Wang, J.T.S., "The Energy Release Rate
in the Growth of a OneDimension Delamination," Journal of
Applied Mechanics 51, Dec. 1983: 939941.
[3] Whitcomb, J.D., "Finite Element Analysis of Instability
Related Delamination Growth," Journal of Composite
Materials 15(9), 1981: 403426.
[4] Sheinman, I., and Soffer, M., "Postbuckling Analysis
of Composite Delaminated Beams," International Journal of
Solids and Structures 27(5), 1991: 639646.
[5] Kardomateas, G.A., and Schmueser, D.W., "Buckling and
Postbuckling of Delaminated Composites under Compressive
Loads Including Transverse Shear Effects," AIAA Journal
26(3), 1988: 337343.
[6] Kardomateas, G.A., "Large Deformation Effects in the
Postbuckling Behavior of Composites with Thin
Delaminations," AIAA Journal 27(5), 1989: 624631.
[7] Chen, H., "Shear Deformation Theory for Compressive
Delamination Buckling and Growth," AIAA Journal 29(5),
1991: 813819.
[8] Chen, H., "Transverse Shear Effects on Buckling and
Postbuckling of Laminated and Delaminated Plates," AIAA
Journal 31(1), 1993: 163169.
[9] Davidson, B.D., and Krafchak, T.M., "Analysis of
InstabilityRelated Delamination Growth Using a Crack tip
Element," AIAA Journal 31(11), 1993: 21302136.
Ill
99
60
Example 1 Example 4 Example 6 Example 8
Initial crack length of a specimen
Final crack length of an unstitched specimen
Final crack length of a specimen with stitching (16 ssi)
Final crack length of a specimen with stitching (64 ssi)
Figure 4.9 Crack growth in different examples
(Gllc=530 J/m2)
4
1.2.1 Research on Postbuckling of Delaminated Composites
In many cases, when delaminated composites are under
external compressive loads the delaminated structure will
have the capability of carrying loads beyond their buckling
loads and may fail in the postbuckling regime. Therefore
bifurcation analysis may not be adequate in describing the
whole damage process of delaminated composites up to final
failure; a postbuckling analysis may be required.
The initial development and growth of delamination in
compressed composites are affected by various geometrical
parameters, loading conditions, material properties and so
on. The different combinations of these parameters can
result in different types of buckling and delamination
growth behavior. Most studies were based on the static
postbuckling solution of homogeneous or laminated plate
theory. The energy release rate (G) is considered as the
critical parameter and different methods were used to
calculate the energy release rate. An assumption that uses
Griffithtype fracture criterion with a specific constant
critical fracture energy release rate (Gc) was adopted in
these studies. When the energy release rate G is greater
than critical fracture energy release rate Gc the
delamination in the composite may propagate.
84
using the procedure explained in the next paragraph. If the
value of G exceeds the Mode II fracture toughness Gllc, the
length of the delamination is increased by a small amount.
The numerical value of the extension is arrived by a trial
and error method until G equals Gllc for that deflection
increment.
In order to compute the G we used the strain energy
density method derived by Sankar [52] This method is very
much suitable for the present model as the force and moment
resultant ahead and behind the crack tip can be obtained in
closedform from the solution of the differential equations
of equilibrium and the energy density values thus calculated
does not have any numerical errors. Consider the three
elements surrounding the crack tip as shown in Figure 4.3.
There are two (Elements 1 and 2) behind the crack tip and
one ahead of the crack tip (Element 3). The G is derived as:
(4.15)
here UL represents the strain energy density (strain energy
per unit length of the beam), the superscripts (1) and (2)
denote the cross sections immediately behind the crack tip
and (3) denotes the cross section immediately ahead of the
crack tip and B is the width of the beam in the ydirection
32
The relation between compressive loads P and end
shortening 8 is:
P =
AE
~T
8
(P
AE 41 tv2 24 Ka\
(8 + + r)
2 a + L Aa EAn2
iP>P.)
(2.29)
2a
\4 N
K
2L
Figure 2.7 Deformed shape of the delaminated
beam with stitches
When compressive loads P is greater than the buckling
load Pcr of the stitched beam total strain energy U of the
entire delaminated beam with stitching is:
U =
P\La) 
3 P2a
2EA
a ,2EIn2 12Ka\2
+ )
EA a 7T
2EA
(2.30)
5
Simple analytical and finite element models seem to be
popular in analyzing onedimensional delaminations. Chai et
al. [1] used an analytical method to obtain energy release
rate by differentiating the strain energy with respect to
delamination length. Yin and Wang [2] used the Jintegral
method to obtain an expression for energy release rate in
the postbuckled delaminated plate. Whitcomb [3] adopted
the crack closure technique and analyzed the effects of
various parameters on energy release rate in a beamlike
structure using geometrically nonlinear finite elements.
Sheinman and Soffer [4] investigated the effects of
bendingextension coupling and initial imperfection on the
postbuckling behavior of a composite beam with various
delamination geometries. It was found that the coupling
effect significantly reduced the buckling load and
increased the postbuckling deformations, and the global
postbuckling deformation was shown to be very sensitive to
initial imperfections.
Kardomateas and Schmueser [5] proposed a one
dimensional beamplate model that accounts for the
transverse shear effect. Analytical solutions for the
critical instability load and the postbuckling deflections
were obtained with aid of the perturbation technique.
Kardomateas [6] also modeled the postbuckling behavior of
CHAPTER 3
FINITE ELEMENT ANALYSIS OF A DELAMINATED COMPOSITE WITH
STITCHING
As discussed in Chapter 2 delaminations can greatly
reduce the compressive load carrying capacity of laminated
composite structures. Throughthethickness stitching seems
to be helpful in increasing the critical load at which a
delaminated beam becomes unstable under compressive
loading. In Chapter 2 an analytical solution for post
buckling of a stitched composite with a delamination was
given. Several assumptions were made in order to obtain a
closed form solution. In this chapter finite element
analysis is used to investigate the effect of stitching on
the postbuckling behavior of a stitched composite.
Although analytical models provide insight into the effects
of stitching, more detailed behavior of a structure can be
obtained using finite element models. Several restrictions
of the analytical approach can be easily removed in FE
models. For example, the analytical model assumes the beam
is homogeneous and orthotropic, Whereas FE models can
handle laminated composites. Unlike in the analytical
44
50
here 9 is the inclination of the stitch as shown in Fig.
3.4. The tensile force in the stitch is given by
F = ^AL (3.6)
H
here As and Es are the area of cross section and Young's
modulus of the stitch yarn. It should be remembered that
there are two yarns in each stitch (see Figure 1.1). The x
and z components of the force F are: Fx=Fcos6 and FzFsin6.
Substituting for F from Eq. 3.6 we obtain
A F A F
Fx = ^ALcosGAu
x H H
A F A E
Fz =^LJLALsm6=Aw (3.7)
H H
From Equation 3.7 it may be seen that the stiffness of the
spring in the FE model should be equal to ASES/H. In fact
the stiffness of both the springs, horizontal as well as
vertical, happens to be the same.
3.3 PostBuckling Analysis
The postbuckling analysis requires that an initial
imperfection be specified for the beam. Usually the
imperfection shape is a certain linear combination of
Energy release rate G (lb/in.
67
End shortening 5 (in.)
Figure 3.11 Energy release rate
vs.end shortening (a=l" h/H=l/4)