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FAILURE ANALYSIS OF COMPOSITE LAMINATES WITH

A HOLE BY USING FINITE ELEMENT METHOD

by

MOUMITA ROY

Presented to the Faculty of the Graduate School of

The University of Texas at Arlington in Partial Fulfillment

of the Requirements

for the Degree of

MASTER OF SCIENCE IN MECHANICAL ENGINEERING

THE UNIVERSITY OF TEXAS AT ARLINGTON

December 2005

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To my parents, Mrs.Mamata Roy and Mr.Sadhan Chandra Roy, who have

made me what I am today.

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ACKNOWLEDGMENTS

This thesis could not have been written without Dr.Wen S.Chan, my supervising

professor. All his teachings and his open door policy have been of great benefit to

me throughout my term of study at UTA. My sincerest gratitude goes to him for

suggesting the subject of research, and for his valuable suggestions and

encouragement throughout the work and his guidelines for publishing this thesis. In

addition, the author would also like to thank Dr.Bo P. Wang and Dr. Seiichi Nomura,

for serving on her committee. I want to express my loving appreciation to my parents

and my brother for their support and encouragement throughout my life. I would also

like to thank my friend(s) here at UTA and elsewhere who were always a constant

source of inspiration and encouragement.

October 26, 2005

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ABSTRACT

FAILURE ANALYSIS OF COMPOSITE LAMINATES WITH

A HOLE BY USING FINITE ELEMENT METHOD

Publication No . ______.

Moumita Roy, M.S.

The University of Texas at Arlington, 2005

Supervising Professor: Wen S.Chan

Predicting the strength of a given laminate has been extensively studied.

There have been several fracture models proposed but still the immense tests of

composite laminates are still needed. Different material systems or lay-ups exhibit

different failure modes and failure mechanisms. This thesis focuses in investigation

of the failure mechanism of the [/02]s and [02/]s laminates with a hole. ANSYS

finite element models with and without a crack in vicinity of hole were developed to

investigate the effect of the stress distribution due to the presence of the angle ply

crack. The stress concentrations were obtained. It is found that

The stress concentrations increases as d/w increases for a given in [/02]s

and [02/]s laminates

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For a given d/w ratio, the stress concentrations increases as ply orientation,

increases

The high stress concentrations on 0o ply occurs at the location where the -

ply crack intercepts at the line perpendicular to the 0o

ply

Based upon the understanding the failure mechanism, a simple expression to

estimate the strength of laminates with a hole is established. A strength model

based upon the 0o

ply load carrying capability is proposed. Prediction is a good

agreement with the experimental results

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TABLE OF CONTENTS

ACKNOWLEDGEMENTSiii

ABSTRACTiv

LIST OF ILLUSTRATIONSviii

LIST OF TABLES....x

Chapter

1. INTRODUCTION....1

1.1 Applications of Composites in Aerospace Structures .1

1.2 Hole in Composite Structures ..2

1.3 The Objective of this Thesis.3

1.4 Outline of Thesis.4

2.FAILURE OF LAMINATE WITH A HOLE...5

2.1Introduction.....5

2.2Experimental Study.6

2.2.1 Failure Mode.6

2.2.2 Notched and Un-notched Strengths...8

2.3Predictive Model of Notched Strength9

2.3.1 Waddoups-Eisenmann-Kaminski (WEK) Model [6].........10

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2.3.2 Whitney-Nuismer (WN) Model [7]....12

2.3.3 Mar-Lin (ML) Criterion..13

2.4 Stress Analysis of Laminates with a hole..14

3.FINITE ELEMENT MODELING...19

3.1 Definition of Problem.19

3.2 Element Type and Size...20

3.3 Modeling and Meshing...22

4.RESULTS AND DISCUSSIONS...28

4.1 Model Validation28

4.2 [02/ ]s composite laminate....29

4.2.1 Composite plate with crack at + layer..30

4.2.2 Composite plate with crack at - layer...32

4.3 [

/ 02]s Composite laminate.36

4.3.1 Composite plate with crack at + layer..36

4.3.2 Composite plate with crack at - layer...38

4.4 Failure Strength Prediction Using Classical Lamination Theory.38

4.5 Conclusion.48

REFERENCES...50

BIOGRAPHICAL INFORMATION..52

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LIST OF ILLUSTRATIONS

Figure Page

2.1 Failure of Hole in laminate with 62% of 0o

ply.[Ref.4].6

2.2 Failure of Hole in laminate with 25% of 0o

ply.[Ref.4].....6

2.3 X-radiographs showing damages around hole of laminates with various d/w ..7

2.4 X-radiographs of fiber breakage of 0o

ply in the [02/]S laminates.....8

2.5 Effect of width on the strength.......9

2.6 Strength variations due to fiber orientation........9

2.7 WEK fracture model....11

2.8 (a) Schematic representation of the point-stress criterion, (b)Schematic

representation of the average-stress criterion, for a laminate containing a circular

hole................................................................................................................13

2.9 Axial stress distributions in a laminate with d/w = 0.85 subjected to uniform tensile

load....15

2.10 Axial stress distributions of notched [02/]sb laminates with various fiberorientations.16

2.11 The location of the peak SCFs along the laminate straight edge....16

2.12 In-plane shear stress, the inter-laminar normal and shear stresses distribution at the

0/+45 interface along the y-axis...17

2.13 The interlaminar stresses along the hole edge at the interfaces of 0/+45 (z = 2h)plies and +45/-45 (z = 1h) plies.18

3.1 SOLID 191 Geometry...21

3.2 The Element output definition notation.21

3.3 TARGE170 Geometry...22

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3.4 CONTA174 Geometry...22

3.5 Full and Sub Models with a crack..23

3.6 3-d modeling in which + crack is present in +layer for [,02]s laminate....24

3.7 Meshed volume & with applied boundary conditions...26

3.8 Meshed volume with applied boundary conditions for [02,90]s laminate....27

4-1 Aluminum plate with a hole under uniaxial loading.....28

4.2 Composite laminate under axial loading....29

4.3 Quarter model of composite plate with hole for d/w=0.5...30

4.4 Normalized stress distribution along the direction in the 0o layer when crack ispresent at + layer31

4.5 Normalized stress distribution with varying d/w ratio for [02/60]s ..33

4.6 Normalized stress distribution for [ 02/45]s for varying d/w ratio...33

4.7 Stress distribution along the y-axis in the zero degree layer when crack is present at -

layer....34

4.8 Stress distribution in 0 degree layer when crack is present at + layer for[ / 02]s laminate...36

4.9 Notched Strength in [02/ ]s laminate when crack is present in layer in therespective direction....42

4.10 Notched Strength in [ /02 ]s laminate when crack is present in layer in therespective direction.....45

4.11 Comparison of Notched Strength between [02/ ]s & [ /02 ]s when crack ispresent in layer next to 0 degree ....45

4.12 Comparison of Notched Strength between [02/ ]s & [ /02 ]s when crack is presentaway from 0 degree.48

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LIST OF TABLES

Table Page

3.1 Material Constants Used..19

4.1 Comparison of Calculated values to ANSYS values...30

4.2 Max. Stress in 0o layer with varying d/w ratio for [02/ ]slaminate for + crack ..35

4.3 Max. Stress in 0o layer with varying d/w ratio for [02/ ]slaminate for - crack....37

4.4 Max. Stress in 0o layer with varying d/w ratio for [ / 02]slaminate for + crack...40

4.5 Max. Stress in 0o layer with varying d/w ratio for [ / 02]slaminate for - crack....41

4.6 Notched Strength of [02/]s when crack is present at + directionin + layer....43

4.7 Notched Strength of [02/]s when crack is present at - direction

in - layer.44

4.8 Notched Strength in [ /02]s laminate when crack is present in layer..46

4.9 Notched Strength in [ /02 ]s laminate when crack is present in layer..................47

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CHAPTER 1

INTRODUCTION

A material containing two or more distinct constituents in a macroscopic scale is

called a composite material. These distinct constituents have significantly different properties

and composite properties are noticeably different from constituent properties. One of the

constituents is used to reinforce the other constituent(s). In case of unidirectional

continuously fiber-reinforced composites single fiber orientation in a layer but different

layers in a laminate may have different single fiber orientation in a given layer. This kind of

composite laminates reinforces the stiffness or strength along the fiber orientation. In this

thesis this type of composite laminate has been studied.

1.1 Applications of Composites in Aerospace Structures

Composite materials are one such class of materials that play a significant role in

current and future aerospace components. Composite materials are particularly attractive to

aviation and aerospace applications because of their exceptional strength and stiffness-to-

density ratios and superior physical properties. Among the first uses of modern composite

materials was about 30 years ago when boron-reinforced epoxy composite was used for the

skins of the empennages of the U.S. F14 and F15 fighters. Initially, composite materials were

used only in secondary structures, but as knowledge and development of the materials has

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improved, their use in primary structures such as wings and fuselages has increased.

Composite materials are used extensively in the Euro fighter: the wing skins, forward

fuselage, flaperons and rudder all make use of composites. The use of composite materials in

commercial transport aircraft is attractive because reduced airframe weight enables better

fuel economy and therefore lowers operating costs. The first significant use of composite

material in a commercial aircraft was by Airbus in 1983 in the rudder of the A300 and A310,

and then in 1985 in the vertical tail fin. In the latter case, the 2,000 parts (excluding fasteners)

of the metal fin was reduced to fewer than 100 for the composite fin, lowering its weight and

production cost. Composites also has deep-water applications like those in hydraulic cylinder

for underwater operations, small diameter low pressure pipes in composite materials, etc.

Due to the superior specific strength and stiffness characteristics and their adaptability to

given structural needs composite laminate materials are not only widespread in the fields of

aircraft and spacecraft constructions but also gain more spread in the classical fields of

mechanical engineering.

1.2 Hole in Composite Structures

Most structures are assembled by a number of individual structural elements connected to

form a load path. These connections or joints, in general, can be classified as adhesively

bonded, mechanical fastened (bolted or riveted) or combination of both. Mechanical joints

usually have some type of bolt holding the different pieces of the structure, which needs a

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hole to be drilled into the composite laminate. Besides this, hole in structural laminate can be

served as access area of any purpose. Predicting the notched strength of a given laminate has

been extensively studied in the last three decades. Most of both analytical solutions and exper-

mental investigations were concentrated for wide laminates with a hole. A thorough review

of strength prediction models has been conducted by Awerbuch and Madhukar [1]. A

systematic study of hole problems in laminate was given by Tan [2]. The damage

characteristics around hole for laminates with different percentage of 0 ply was

experimentally investigated by Chan [4]. The notched strength of the [02/]S laminates was

studied by Harn [5]. Numerous failure criteria and fracture models have been proposed by

Waddoups et al. [6], Whitney and Nuismer [7] . However, little work on laminates with a

large hole size was conducted.

1.3 The Objective of this Thesis

The objective of this research is to investigate the stress concentration around the hole

with various ratios of the hole size to the laminate width. The finite element analyses are then

conducted to aid understanding of the failure process of the laminate. The test observations

of failure process of laminate with a hole reveal the final failure of laminates occurring in the

00ply. With this in mind, a fracture model based upon the 0 ply load carrying capability is

proposed for predicting the notched strength.

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1.4 Outline of Thesis

A brief discussion on various works done in the field of composite laminate with a hole

is given in chapter two. In chapter three, modeling and meshing using ANSYS has been

described. In chapter four the results have been discussed and the conclusions are drawn

based on the results.

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CHAPTER 2

FAILURE OF LAMINATE WITH A HOLE

2.1 Introduction

Failure of structures with a hole is caused by excess stress/strain in the neighborhood

of the hole. The stress or strain concentration tangential to the hole edge has been

investigated extensively for isotropic materials for a century. The equations of the state of

stress have been developed for both infinite- and finite-width isotropic plates. The solution

for the finite width is often used the solution for infinite width of the plate by multiplying a

factor. This factor is termed as finite width correction factor. The finite width correction

(FWC) factors are function of d/w but not the material properties.

The analytical solutions to the stress or strain concentration for isotropic materials

with infinite plate size have been well-established couples of decade ago. The FWC factors

of isotropic material are usually applied to the infinite-geometry anisotropic or orthotropic

materials to obtain the finite-geometry data reduction. Recently, the FWC factors for

anisotropic plates have been reviewed in Ref[2]. It is a function of d/w and material

properties. However, the exact solution to the axial stress distribution in a finite-width

composite laminate with a hole has not been found yet.

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2.2 Experimental Study

2.2.1 Failure Mode

Chan[4] studied the damage characteristics around hole of laminates with various

percentages of 0o-ply. He found that the damage process of the hole is confined in areas

shown in figure 2.1 for laminates with 62% of 0o

ply.

Figure 2.1 Failure of Hole in laminate with 62% of 0o

ply.[Ref.4]

For laminates with 25% of 0o

ply the hole damage grows from the edge of the hole to the

edge of laminates as shown in Fig 2.2.

Figure 2.2 Failure of Hole in laminate with 25% of 0o

ply.[Ref.4]

(62/31/7)%S 90/0/45/0/45 22

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In study of [02/]s laminates, Harn[5] observed the fiber splitting of 0o

plies and the

delamination at the interfaces of 0o/+45

oand +45

o/-45

o. Figure 2.3 shows intermediate and

final failures of the [02/45]S laminates with different d/w ratios. As shown, the fiber splitting

within 00plies is due to the steep strain and stress gradients. Delamination was first observed

within the regions where the material is not continuous.

Figure 2.3 X-radiographs showing damages around hole of laminates with various d/w

The extensive delamination starts from the edge of the hole and propagates rapidly toward

the edge of the specimen. The laminate with a smaller hole size exhibits more extensive

delamination compared to laminate with a large size of the hole. At the final failure, the

breakage of 00ply is clearly shown along the direction of 45

0, which is the fiber orientation

of the neighboring ply. The phenomenon of the fiber breakage of the 00ply was also

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observed along the direction of the neighboring ply in other laminates, such as [02/15]S,

[02/30]S and [02/60]S as shown in Figure 2.4

Figure 2.4 X-radiographs of fiber breakage of 0o

ply in the [02/]S laminates

2.2.2 Notched and Un-notched Strengths

Harn [5] studied the effect of notched strength due to d/w ratio and ply fiber orientation.

Figure 2.5 shows the ratio of ultimate failure and onset of delamination strengths to the

unnotched strength of the [02/450]S laminate with various d/w ratios. Figure 2.6 shows It is

observed that both notched strength and delamination strength decrease as the hole size

increases. It is also shown that the difference between two strength ratios is smaller as the d/w

ratio increases. The strength variation due to fiber orientation of the laminate is shown in Fig2.6.

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Figure2.5 Effect of width on the strength

Figure 2.6 Strength variations due to fiber orientation

As shown, except laminate with=300, both notched and unnotched strengths decrease as

increases. The higher value is, the lower the delamination strength is.2.3 Predictive Model of Notched Strength

The notched strength of a composite laminate depends on laminate configuration, laminate

stacking sequence, hole size, and laminate width. Different lay-up or material system may

exhibit different failure mechanism. A brief review of fracture prediction models is described

as following:

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2.3.1 Waddoups-Eisenmann-Kaminski (WEK) Model [6]

This model is based on Mode I fracture analysis for a laminate containing either a

hole or a line crack. This model assumes that there exists a small but finite region of intense

energy at the edges of the hole or crack in the direction transverse the to the loading

direction. Figure 2.7 shows the WEK fracture model. These intense energy regions were

intuitively seen as the damage region ahead of the hole or crack in the laminate. Then the

model further assumes that failure strength of the laminate will occur at the vicinity of the

crack.

The strength of a laminate with no hole can be obtained from Eq. (2-1) by setting R

equal to zero,

)/( Raaf

KICN

= (2-1)

The ratio of notched strength for infinite-width laminate, N and unnotched, o is

given as

)/(

1

Raf eo

N =

(2-2)

where the functionf(ae/R) is tabled by Paris and Sih[10] ae is a parameter that is determined

from a set of coupon tests. The assumptions of the existence of an intense energy region of

length ac results in the following equation at failure.

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Figure 2.7 WEK fracture model

[ ] 2/1)( cNIC acK += (2-3)

For the case of unnotched laminates, the strength of the laminate can be determined

by letting c equal to zero. Combining these two equations yields2/1

+=

c

c

o

N

ac

a

(2-4)

This model involves two parameters, the unnotched strength o and the characteristics

length ac to be determined. It should be noted that ac was assumed to be independent of the

original crack length and thus considered a material parameter. The concept of the intense

energy zone region is a fractious region treated the laminate as a homogeneous material.

IntenseEnergy

Region

a

R

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2.3.2 Whitney-Nuismer (WN) Model [7]

The model hypothesized that the strength of the laminate with a hole can be evaluated

at a characteristics distance based on the point or average stress across the region from the

edge of the hole. Both criterions assume that fracture occurs when the stress at some

characteristics distance from the edges of the hole or the crack tip reaches the unnotched

strength.

Point-Stress Criterion.

In point stress criterion, it is assumed that failure occurs when the stress y at some

distance do away from the edges of the discontinuity is equal to or greater than the strength of

unnotched laminates. Figure 2.8(a) shows this criterion schematically. The ratio of the

notched to unnotched strength is given as

[ ])75)(3(322

8

1

6

1

3

1

2

1

++=

To

N

K(2-5)

whereodR

R

+=1 (2-6)

The appropriate value of do can be determined from the test data.

Average Stress Criterion.

The average stress failure criterion states that failure will occur if the average value of

the stress y over a distance ao is equal to the unnotched strength. Fig 2.8(b) shows this case.

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x x

Figure 2.8 (a) Schematic representation of the point-stress criterion, (b)Schematic

representation of the average-stress criterion, for a laminate containing a circular hole

dxxa

oaR

R

y

o

o )0,(1

+

= (2-7)

))(3(2

)1(28

2

6

2

4

2

2

2

2

+=

To

N

K(2-8)

where

oaR

R

+=2 (2-9)

2.3.3 Mar-Lin (ML) Criterion

This model is based on the classic LEFM. Mar and Lin proposed that the fracture of

laminates is governed by

(a)

Y

y

o

do

Y

y

o

ao

R R

(b)

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n

cN cH = )2( (2-10)

where c can be either the hole radius or half of the crack length and H c is the laminate

fracture toughness similar to KC or KIC. The exponent n is the order of the singularity of a

crack with its tip at the interface of two different materials. The value ofn is a function of the

ratio of the shear modulus and Poissons ratio of the matrix and fiber.

2.4 Stress Analysis of Laminates with a hole

This paper investigates the failure mechanism and notched strength of the [02/]S

laminates with hole size to width ratio greater than 0.5. It is found that the stress

concentration increases as the hole size increases for a given in the laminate. For a given

d/w ratio, increasing the stress concentration as the ply orientation decreases. The

experimental investigation reveals that delamination damage occurs around the hole is more

extensive for a small hole than for a large hole in the laminates, resulting in a higher notched

strength. For d/w = 0.5, damage around the hole for laminate with = 15o is isolated in the

area where the material is continuous. How-ever, for greater than 15 extensive

delamination occurs around the hole region. A fracture model based upon the 0 ply load

carrying capability is proposed for predicting the notched strength.

The typical axial stress distributions along boundaries and across the laminate width

in the two dimensional analyses are plotted in Fig. 2-9.

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Figure 2.9 Axial stress distributions in laminate with d/w = 0.85 subjected to uniform tensileload

In case of two-dimensional stress distributions the stress reaches the peak value along

the plate straight edge about one radius away from the hole center. The stress drops below the

applied tensile load at point A, the intersection of the y-axis and plate straight edge. The

highest stress concentration is located at point B, the junction of the y-axis and the hole edge.

The stress along the centerline starts from 0 at the hole edge, then drops, and increases to the

far field stress. The stress recovers to the applied stress about 4 times of the radius away from

the hole edge. The recovery of the far field stress along the plate centerline is closer to the

hole edge as the hole size decreases. The existence of a hole affects the stress distributions

tremendously. The axial stress distribution along the y-axis and along the plate straight edge

are plotted in Fig. 2.10

It is indicated that the peak stress concentration along the straight boundary shifts

toward the plate net cross-section as is increased. The location of the peak SCFs is

observed at the place where the continuous fibers tangential to the hole edge reach the edge of

the plate as shown in Fig. 2.11.

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Figure 2.10 Axial stress distributions of notched [02/]sb

laminates with various fiber

orientations

Figure 2.11 The location of the peak SCFs along the laminate straight edge

In case of three-dimensional stress distribution The distributions of the in-plane shear

stress, the inter-laminar normal and shear stresses at the 0/ +45 (z = 2h) interface along the

x-axis are shown in Fig. 2.12.The interlaminar stresses near the curved edge are prominent

than those near the straight edge. The distributions of interlaminar normal and shear stresses

are affected by the d/w. The in-plane shear stress,xy is zero on the plate straight edge, but

finite elsewhere except on the hole edge. For every d/w ratios, the value of the stresses

reaches the maximum near the hole boundary.

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Figure 2.12 In-plane shear stress, the inter-laminar normal and shear stresses distribution at

the 0/+45 interface along the y-axis

The sign of the interlaminar stresses also change the sign from the hole edge to the laminate

straight edge. As the hole size increases, the interlaminar normal and shear stresses become

more significant. The interlaminar stresses exist everywhere along the y-axis in the 0/+45

interface as d/w increases. The interlaminar stresses along the hole edge at the interfaces of

0/+45 (z = 2h) plies and +45/-45 (z = 1h) plies are plotted in Fig. 2.13. Among these

interlaminar stresses, is the dominant stress at z z=1h, 2h along the hole edge. The figure

also indicates that the peak stress ofz occurs at = 80 and 100 at the 0/+45 interface.

This suggests that the initiation of the delamination may occur at these locations.

The effect of angle ply orientation is that the unnotched strength decreases as is

greater than 30. However, the notched strength reaches a fairly constant as is equal or

greater than 30. The delamination occurs at the interface of 0/+45 plies and interface of

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+45/-45 plies. In Fig.2.13, the interlaminar normal and shear stresses near the hole

boundary are much higher than those near the plate edge. The high interlaminar stresses near

the hole edge may contribute to the initiation of delamination starting from the hole edge.

The delamination is found near the hole edge and bounded by 0 ply splitting.

Figure 2.13 The interlaminar stresses along the hole edge at the interfaces of 0/+45 (z = 2h)

plies and +45/-45 (z = 1h) plies

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CHAPTER 3

FINITE ELEMENT MODELING

3.1 Definition of Problem

A composite laminate plate with hole, with and without crack is analyzed .The test

cases are [02/ ]s and [/02]s plate where ranges from 15o, 30

o, 45

o, 60

o, 75

o. Different

diameter to width ratio d/w has been studied. The d/w ratios studied are 0.5, 0.6, 0.7, 0.8,

0.85. The crack is modeled in + and - direction for + and - layer, respectively. Full

model has been used for analysis. Sub-modeling of full model has been done for more

accurate results. The material constants used are listed in table 3-1[11]

Table 3.1 Material Constants Used

Material

System

E1 Msi E2= E3 Msi 23 12= 13 G23 Msi G12= G13 Msi

IM6/3501-6,Gr/Ep *

23.3 1.395 0.342 0.2965 0.5198 0.9161

* ply thickness = 0.005

Three-dimensional modeling using ANSYS has been done. There is only one

symmetric plane in the model, namely, x-y plane (z=0). Therefore, half of the laminate is

needed to model. The dimension of the model that has been used is 1.0 inch long, 0.1 inch

breadth and 0.2 inch thick where the thickness of each layer is 0.005 inch.

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3.2 Element Type and Size

The element type and size in the mesh affects the results in the model. For the same

number of elements in a model, the choice of tetrahedral and quadrilateral element can

usually lead to different results. The difference may or may not be significant for solving

some problems. Therefore, the comparison of finite element results and experimental result,

are important for determining the element type and size.

Twenty-node quadrilateral element SOLID191 is used in this study. The quadrilateral

element can be degenerated to the tetrahedral to be best fitted into the boundary of the hole.

Fig.3-1 shows SOLID191 geometry and Fig.3-2 shows SOLID191 element output definition

notations. The number of elements and nodes in the model varies for different cases under

consideration.

Gap elements have been used for crack at zero degree layers for 90o

half model where

two cracks were present. Surface-to-Surface Contact elements, TARGE 170 and

CONTA174, have been used. Figure 3-3 shows TARGE170 geometry and Figure 3.4 shows

CONTA174 geometry. Each contact pair is identified via same real constant number.

CONTA174 is a 8-node higher-order quadrilateral element that can be located on the

surfaces of 3-D solid or shell elements with mid-side nodes.

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Figure 3.1 SOLID 191 Geometry

Figure 3.2 The Element output definition notation

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modeling. The local model is a sub model of the global model which models either the entire

of the laminate or portion of the laminate. A typical model of this study contains 2500

elements and 4500 nodes. Both full and sub models have been analyzed for all the cases,

except 15o crack. For15o crack, the sub model requires a large size in order to include

both the hole and the crack.

Full model Sub modelFigure 3.5 Full and Sub Models with a crack

For this care, the benefits of using the sub model are not significant. A sub model is

selected for refining the meshes to capture the high stress gradient around the vicinity of the

hole and the crack. Hence the size of the sub model should be large enough to contain a crack

which is emanated from the hole.

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The cases under study are [02 ,]s and [, 02]s laminates with hole and crack

where different values of are 15o, 30o, 45o, 60o, 75o, 90o. Different d/w ratio is analyzed.

The different d/w ratios analyzed are 0.5, 0.6, 0.7, 0.8 and 0.85.

Crack is present at + or - layer along the fiber orientation. For ,15o, 30o, 45o, 60o

and 75o

half model of the complete model is modeled as it is been assumed that a crack is

present at both the angle ply, respectively.

For composite modeling the fiber orientation is assigned through real constant

commands R and RMODIF commands assign real constant values. Because of laminate

symmetric to its mid-plane of the thickness only half of the laminate is considered.

Figure 3.6 3-d modeling in which + crack is present in +layer for [,02]s laminate

In creating meshes for ANSYS model, the KEYPOINTS are first assigned as nodal

points. The area and volume commands were executed. Two volumes are created in each

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layer and are glued together in the same plane except the layer with a crack. Then the volume

of each layer is stacked according to the stacking sequence of the laminate. For the layer with

a crack, double points with the same co-ordinates remain to simulate the crack. A cylinder is

built, which is then subtracted from the entire volume, which creates a hole in the model. The

hole is located at the center of the laminate and is tangential to the crack if the crack is

required in the model.

The model is constrained on the surface of one end in x-direction. The middle line in

z-direction on this surface is fixed in y-direction to simulate the symmetric surface of the

laminate width. The bottom surface of the model is constrained along z-direction. After

solving the full model sub-modeling was developed. In sub-model, a cut boundary command

of ANSYS, CBDOF is utilized to translate the displacement of each node in the full model

along the cut boundaries into corresponding displacement in the sub-model. The

interpolations of displacements are applied to the new created nodes of the sub model. Fig3-7

shows meshed volume with boundary condition . Unlike the other laminate, only a quarter of

the model is needed for [02/902]s laminate because of symmetry even if the crack in 90o

layer

is present. For this case, symmetry condition is enforced for the quarter model. Fig 3.8

shows meshed and with boundary condition of this case.

Another 90-degree case has been studied in which another crack in 0 degree layer is

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Figure 3.7 Meshed volume & with applied boundary conditions

modeled with the 90degree crack. This crack is of length 0.02 in and is tangential to the hole

at 0 degree. For this case two types of elements are used namely, SOLID191 and Gap

Elements. Two type of Gap Elements have been used, namely TARGE170 and CONTA174.

The combination of these two types of elements creates a surface-to-surface contact. To

create a contact pair same real constant number is assigned to both TARGE170 and

CONTA174.

This gap element pair is used for the crack in 0 degree layer. The reason for

choosing this element is that it prevents the nodes to penetrate into each other. Atypical

inputs for creating ANSYS models are attached to Appendix A. 90 degree ply with the only

difference that their will a crack at 0 degree layer which is not glued to any of the volumes.

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Figure 3.8 Meshed volume with applied boundary conditions for [02,90]s laminate

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CHAPTER 4

RESULTS AND DISCUSSIONS

4.1 Model Validation

Two steps were adopted before we use to investigate the stress distribution around the

hole. First, the material constants of aluminum were used in the model to check the

connectivity of node in the model, the boundary conditions and the load application.

The aluminum plate has 0.1 inch wide and 0.05-inch diameter. Since accuracy of stress

concentration factor is not the primary concern in this study a coarse mesh of ANSYS was

used. A quarter model of aluminum block has been studied and an axial load of 10 psi is

applied to it. Fig 4.1 shows Aluminum plate with a hole.

Figure 4.1 Aluminum plate with a hole under uniaxial loading

The result got from ANSYS plots is that the maximum value of stress is 46.09psi

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and the minimum value of stress is 3.895psi. The calculated stress concentration factor, Kt

is Kt = 4.25. However, ANSYS results gives Kt = 4.61. Negative sign indicates compression.

Then, a composite laminate of [02/45]s without a hole is modeled. The purpose of this study

is to check the applicability of the anisotropic material input. The ANSYS results are used to

compare the results obtained from the MATLAB results calculated by lamination theory. A

composite laminate of [02/45o] s is being studied. Fig 4.2 shows the ANSYS results for this

case.

.

Figure 4.2 Composite laminate under axial loading

A comparison of the calculated MATLAB results and ANSYS values are listed in After

this a composite plate with a hole under uni-axial loading is considered. Fig4.3 shows the

plot of composite plate with hole under uni-axial loading. The results were in agreement with

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Table 4.1 Comparison of Calculated values to ANSYS values

4.2 [02/ ]s composite laminate

There are two cases studied, one in which a crack is present at + layer and the other

crack is present at - layer.

Figure 4.3 Quarter model of composite plate with hole for d/w=0.5

4.2.1 Composite plate with crack at + layer

There are five d/w ratios studied, d/w = 0.5, 0.6, 0.7, 0.8, 0.85.There are six angles that

are being considered, = 15o, 30o, 45o, 60o, 75o. Fig, 4-4 shows the stress distribution along

Layer x MATLAB Results x , ANSYS results

-45o

2.6969psi 2.5898psi

+45o

2.6969psi 2.5898psi

0o

17.303psi 17.553psi

0o

17.303psi 17.553psi

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normalized increases as the ratio of d/w increases. For d/w ratio of 0.8 and 0.85 the point of

intersection lies outside the laminate plate and consequently the maximum stress is reached

at the free edge

From Fig.4-6 we can see that with increase in d/w ratio the point of maximum

stress shifts towards the free edge and beyond a particular d/w ratio the maximum value is

reached at the free edge. The orientation of the breakage of the 0o

plies tends to be tangent to

the hole edge. Referring to Fig 2.3 the final failure of the laminate is not along the -

direction for +y region .For=45o, as shown in Fig 4.5 the trend of the normalized x is

similar to the case of=60o. The peak value of the normalized stress, x is listed in Table 4.2

for all of d/w ratios and various s. For any given , the peak value ofx increases as the

d/w ratio increases

4.2.2 Composite plate with crack at - layer

There are five d/w ratios studied, d/w = 0.5, 0.6, 0.7, 0.8, 0.85.There are six angles that are

being considered, = 15o, 30o, 45o, 60o, 75o. Fig, 4-7 shows the stress distribution along the

-direction in the zero degree layer when crack is present at - layer for d/w=0.5.From this

plot we can see that with increase in the maximum value of stress increases but if we

compare the results when crack was present at + layer the stress value is less in case of -

layer crack compared to + layer. The table shows that the peak value ofx increases as

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-2

0

2

4

6

8

10

12

14

16

-0.06 -0.04 -0.02 0 0.02 0.04 0.06

-y +y

NormalizedStress

d/w=0.5

d/w=0.6

d/w=0.7

d/w=0.8

d/w=0.85

Normalized stress=60 for different d/w ratis

Figure 4.5 Normalized stress distribution with varying d/w ratio for [02/60]s

-1

0

1

2

3

4

5

6

7

-0.06 -0.04 -0.02 0 0.02 0.04 0.06

normalizedstress

d/w=0.5

d/w=0.6

d/w=0.7

d/w=0.8

d/w=0.85

+y -y

0.

x

y

Figure 4.6 Normalized stress distribution for [ 02/45]s for varying d/w ratio

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increases for a given d/w ratio. For any given , the peak value ofx increases as the d/w

ratio increases.

-2

0

2

4

6

8

10

-0.06 -0.04 -0.02 0 0.02 0.04 0.06

-y width +y

NormalizedStress

2

2

2

2

2

2

Variation of sx/so for d/w=0.5 for -x

-x

y

Figure 4.7 Stress distribution along the y-axis in the zero degree layer when crack is present

at - layer

The pattern of stress distribution for the entire d/w ratio remains same. With varying d/w

ratio only the values of maximum stress varies. Table 4-3 shows the value of maximum stress

in zero degree layers with varying d/w ratio. The patterns remain same as that in the case of

+ crack i.e. with increase in d/w ratio the maximum stress value increases and with increase

in also the maximum value of stress increases. Comparing the results from table 4.2 & 4.3

for crack in + layer and - layer, respectively one can see that the value of maximum stress

is lower in case of - layer crack compared to +layer crack. From this we can conclude that

the farther is the crack layer from the 0 degree layer the lower will be the maximum stress

value in zero degree layer.

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Table 4.2 Max. stress in 0o

layer with varying d/w ratio for [02/ ]s laminate for + crack

+ Crack layer Maximum x/in Location from ANSYS Location of intersection0o layer (Max. stress location) point(Max. stress location)

d/w =0.5

+15 2.88 0.0244 0.0241+30 3.95 0.0213 0.0211

+45 4.22 0.0201 0.015

+60 4.57 0.0092 0.0+75 5.17 0.0043 outside plate

d/w =0.6

+15 3.79 0.0213 0.0189+30 4.21 0.0147 0.0153

+45 4.78 0.0067 0.00757

+60 5.86 0.0023 outside plate+75 6.12 0.0 outside plate

d/w = 0.7+15 4.05 0.0133 0.0138

+30 5.32 0.0067 0.0094

+45 5.97 0.0 0.0005


d/w = 0.8+15 6.75 0.0067 0.0086

+30 7.32 0.0067 0.0038


+75 10.837 0.0 outside plate

d/w = 0.85

+15 7.44 0.0067 0.006+30 8.24 0.0 0.0


+75 14.12 0.0 outside plate

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From the table we can see that the point of maximum stress is close to the calculated value

and this means that in the case of crack layer, respectively the maximum stress value

point remains same. The little difference in the results obtained from ANSYS is because of

different number of elements in various cases. For the meshing of 15o

and 30o

very fine

elements had to be used because of the geometry of the figure which was not the case for

+15o

and +30o.Even for the other angles the total number of elements were not same.

4.3 [/ 02]s Composite laminate

4.3.1 Composite plate with crack at + layerThere are five d/w ratios studied, d/w = 0.5, 0.6, 0.7, 0.8, 0.85.There are six angles

that are being considered, = 15o, 30o, 45o, 60o, 75o. Fig., 4-8 shows the stress distribution

along the y-axis in the 0o

layer when crack is present at + layer for d/w =0.5.

-0.5

0

0.5

1

1.5

2

2.5

3

-0.1 -0.05 0 0.05 0.1

-y width +y

NormalizedStress = 15

= 30

= 45

= 60

= 75

Variation of Normalized Stress with different ford/w=0.5

Figure 4.8 Stress distribution in 0 degree layer when crack is present at + layer for[ / 02]s laminate

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Table 4.3 Max. Stress in 0o

layer with varying d/w ratio for [02/ ]s laminate for - crack

- Crack layer Maximum x/in Location from Ansys Location of intersection0o layer (Max. stress location) point(Max.stress location)

d/w = 0.5-15 2.77 0.0256 0.0241

-30 3.68 0.0233 0.0211

-45 4.01 0.0133 0.0147-60 4.41 0.0008 0

-75 4.79 0.0005 outside plate

d/w = 0.6-15 3.16 0.0196 0.0189

-30 3.88 0.0164 0.0154

-45 4.50 0.0067 0.0076-60 4.97 0 outside plate

-75 5.27 0 outside plate

d/w = 0.7

-15 3.75 0.0164 0.0138

-30 4.21 0.0133 0.0096-45 4.85 0.0066 0.0005



d/w = 0.8

-15 4.22 0.0067 0.0086

-30 5.30 0.0048 0.0038-45 6.66 0 outside plate



d/w = 0.85-15 5.27 0.0048 0.0060

-30 6.46 0 0.00009

-45 8.17 0 outside plate-60 9.85 0 outside plate


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From this figure we can see that the pattern of stress distribution remains same as that in the

[02,]s but the value of maximum stress has decreased compared to that in the case of

[02,]s. Table 4-4 shows the maximum value of stress with varying d/w ratio.

4.3.2 Composite plate with crack at - layer

There are five d/w ratios studied, d/w = 0.5, 0.6, 0.7, 0.8, 0.85.There are six angles

that are being considered, = 15o, 30o, 45o, 60o, 75o.The pattern of stress distribution even in

this case remain same as in the other cases. Table 4-5 shows the maximum value of stress

with varying d/w ratio. The value of maximum stress in this case is larger than that in +

crack. This pattern is same as in the previous case of [02/]s in which the layer that was

farther from the 0 degree layer had lower value of stress compared to that which was next to

0 degree layer.

4.4 Failure Strength Prediction Using Classical Lamination Theory

A fracture model for prediction strength of laminate with a hole is proposed. This

model is based on the fact that final failure of the laminates is controlled by fiber breakage in

zero degree laminas. The model is described below:

T

ON

pred

N XCC 1= (4-1)

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where CN is called the notched correction factor,TX1 is the strength of the 0

oply and Co is the

axial stress ratio of the 0

o

ply stress to the laminate stress in the same direction. The x of 0o

plies in a laminate, which is calculated using CLT ,o

atelax

0

min, is related to the applied load,

which is x of the 0o

lamina,o

alax

0

min, through co-efficient C0. The C0 is defined as

o

o

atelax

alaxC

0

min,

0

min,

0

= (4-2)

CN, which is the notched correction factor, is different for different lay-ups in a specific

material system and it is determined by taking the inverse of stress concentration factor in 0o

ply. The calculated notched strength is shown in table 4.6. Comparisons of the experimental

and calculated results have been done for [02 /45]s laminate. From table we can see that for

all the d/w ratio except 0.5 the results obtained are close to that of the experimental results.

The reason for the difference in results in case of d/w ratio of 0.5 can accounted to the fact

that in case of a smaller d/w ratio delamination propagation is larger and in this model

delamination propagation has not been considered which in case of experiment al results has

been taken into account. Table 4-6 shows the notched strength of [02/]s when crack is

present at + direction in + layer. Table 4-7 shows the notched strength of [02/]s when

crack is present at - direction in - layer. Comparing both the tables we can see that the

notched strength in case of - layer crack is more compared to + crack. From this we can

conclude that if a crack is placed right next to 0 degree layer then its notched strength will be

lower compared to the case when the crack is placed away from the 0 degree layer.

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layer with varying d/w ratio for [ / 02]s laminate for + crack


d/w = 0.5+15 1.90 0.0233 0.0241

+30 2.11 0.0200 0.0211

+45 2.37 0.0133 0.0147+60 2.59 0.0067 0

+75 2.79 0 outside plate

d/w = 0.6+15 2.75 0.0200 0.0189

+30 3.20 0.0133 0.0154

+45 3.45 0.0067 0.0076+60 3.61 0 outside plate


d/w = 0.7

+15 2.94 0.0133 0.0138

+30 3.44 0.0067 0.0096+45 3.87 0 0.0005



d/w = 0.8

+15 3.36 0.0133 0.0086

+30 3.95 0.0067 0.0038+45 4.73 0 outside plate



d/w = 0.85+15 4.65 0.0067 0.0060

+30 5.73 0.0004 0.0000

+45 6.50 0 outside plate+60 7.84 0 outside plate


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layer with varying d/w ratio for [ / 02]s laminate for - crack


d/w = 0.5

-15 2.26 0.0256 0.0242-30 2.48 0.0196 0.0241

-45 2.69 0.0133 0.0147

-60 2.89 0.0066 0.0146-75 3.15 0 outside plate

d/w = 0.6

-15 3.21 0.0196 0.0189-30 3.68 0.0164 0.0154

-45 3.84 0.0067 0.0153


d/w = 0.7-15 3.50 0.0133 0.0138

-30 3.97 0.0067 0.0096

-45 4.42 0 0.0005


d/w = 0.8-15 4.00 0.0133 0.0086

-30 4.50 0.0066 0.0038



d/w = 0.85

-15 4.94 0.00667 0.0060-30 6.20 0.00483 0.0001



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In case of [ /02]s it can again be seen that for the case of crack which is placed right next

to 0 degree layer, which in this case is -

layer, the notched strength is lower compared to the

case where the crack is present away from 0 degree layer, which in this case is + layer.

Table 4-8 & Table 4-9 shows notched strength [ /02 ]s laminate when crack is present in

layer in the respective direction. Comparing these two tables again we can see that the

notched strength of + layer crack is more compared to - crack. Figure 4.13 shows Notched

Strength in [ /02]s laminate when crack is present in layer in the respective direction.

0

20

40

60

80

100

120

140

0.4 0.5 0.6 0.7 0.8 0.9

d/w

N

otchedStrength(ksi)

= +15

= +30

= +45

=+ 60

= +75

= 15

= 30

= 45

= 60

= 75

Figure 4.9 Notched Strength in [02/ ]s laminate when crack is present in layer in therespective direction

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Table 4.6 Notched Strength of [02/]s when crack is present at + direction in + layer

Laminate Lay-up d/w Co CN

pred

N

test

N

[02/15]s 0.5 0.92 0.35 112.170.6 0.92 0.25 81.66

0.7 0.92 0.25 79.57

0.8 0.92 0.15 45.480.85 0.92 0.13 43.38

[02/30]s 0.5 0.71 0.25 63.120.6 0.71 0.24 59.21

0.7 0.71 0.19 46.84

0.8 0.71 0.14 34.050.85 0.71 0.12 30.27

[02/45]s 0.5 0.59 0.24 48.79 640.6 0.59 0.21 43.06 480.7 0.59 0.17 34.48 39

0.8 0.59 0.14 28.78 250.85 0.59 0.10 20.66 20

[02/60]s 0.5 0.54 0.22 41.320.6 0.54 0.17 32.19

0.7 0.54 0.16 30.090.8 0.54 0.11 21.32

0.85 0.54 0.08 16.75

[02/75]s 0.5 0.53 0.19 36.450.6 0.53 0.16 30.360.7 0.53 0.14 26.96

0.8 0.53 0.09 17.14

0.85 0.53 0.07 13.15

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Table 4.7 Notched Strength of [02/]s when crack is present at - direction in - layer


pred

N

[02/15]s 0.5 0.92 0.36 116.660.6 0.92 0.32 101.95

0.7 0.92 0.27 86.08

0.8 0.92 0.24 76.340.85 0.92 0.15 49.93

[02/30]s 0.5 0.71 0.27 67.740.6 0.71 0.26 64.35

0.7 0.71 0.24 59.16

0.8 0.71 0.19 47.070.85 0.71 0.15 38.59

[02/45]s 0.5 0.59 0.25 51.340.6 0.59 0.22 45.740.7 0.59 0.21 42.43

0.8 0.59 0.15 30.930.85 0.59 0.12 25.20

[02/60]s 0.5 0.54 0.23 42.810.6 0.54 0.20 37.96

0.7 0.54 0.19 36.040.8 0.54 0.14 19.43

0.85 0.54 0.10 19.15

[02/75]s 0.5 0.53 0.21 38.790.6 0.53 0.19 35.480.7 0.53 0.17 24.15

0.8 0.53 0.12 21.81

0.85 0.53 0.09 14.53

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Table 4.8 Notched Strength in [ /02 ]s laminate when crack is present in layer

Laminate Lay-up d/w Co CNpred

N

[15/ 02]s 0.5 0.92 0.53 170.000.6 0.92 0.41 131.000.7 0.92 0.34 109.60

0.8 0.92 0.25 81.27

0.85 0.92 0.17 56.25

[30/ 02]s 0.5 0.71 0.47 118.310.6 0.71 0.31 78.00

0.7 0.71 0.29 59.160.8 0.71 0.25 63.02

0.85 0.71 0.17 43.48

[45/ 02]s 0.5 0.59 0.42 86.960.6 0.59 0.29 59.66

0.7 0.59 0.26 53.20

0.8 0.59 0.21 43.500.85 0.59 0.15 31.64

[60/ 02]s 0.5 0.54 0.39 72.89

0.6 0.54 0.28 52.360.7 0.54 0.25 46.72

0.8 0.54 0.14 26.870.85 0.54 0.13 24.06

[75/ 02]s 0.5 0.53 0.36 66.600.6 0.53 0.18 34.370.7 0.53 0.16 29.17

0.8 0.53 0.12 22.14

0.85 0.53 0.11 21.96

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Table 4.9 Notched Strength in [ /02 ]s laminate when crack is present in layer


pred

N

[15/ 02]s 0.5 0.92 0.44 142.680.6 0.92 0.41 131.00

0.7 0.92 0.34 109.600.8 0.92 0.25 81.27

0.85 0.92 0.17 56.25

[30/ 02]s 0.5 0.71 0.47 118.310.6 0.71 0.31 78.00

0.7 0.71 0.29 59.16

0.8 0.71 0.25 63.020.85 0.71 0.17 43.48

[45/ 02]s 0.5 0.59 0.42 86.960.6 0.59 0.29 59.660.7 0.59 0.26 53.20

0.8 0.59 0.21 43.50

0.85 0.59 0.15 31.64

[60/ 02]s 0.5 0.54 0.39 72.890.6 0.54 0.28 52.36

0.7 0.54 0.25 46.720.8 0.54 0.14 26.87

0.85 0.54 0.13 24.06

[75/ 02]s 0.5 0.53 0.36 66.600.6 0.53 0.18 34.37

0.7 0.53 0.16 29.170.8 0.53 0.12 22.14

0.85 0.53 0.11 21.96

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The notched strength of decreases with increase in for a constant d/w ratio. Similarly with

increase in d/w ratio for a constant the notched strength decreases. The pattern remains the

same as in previous case. Figure 4.11 and Figure 4.12 shows a comparison of the two lay-ups

.

Figure 4.12 Comparison of Notched Strength between [02/ ]s & [ /02 ]s when crack ispresent away from 0 degree

We can again see the same pattern as in the previous case. The reason for this behavior

can be accounted to the fact that if all the 0 degree layers are placed next t each other it can

withstand more load compared to the case where the 0 degree layers are scattered. Hence the

notched strength for the case where all 0 degree layers are placed together is higher.

4.5 Conclusion

With increase in in a given d/w ratio the stress concentration factor in 0 degree layer

for [02/]s and [/02]s laminates increases .Even with increase in d/w ratio for a given

0

20

40

6080

100

120

140

160

180

0.4 0.5 0.6 0.7 0.8 0.9

d/w

NotchedStrength

= 15

= 30

= 45

= 60

= 75

= 15

= 30

= 45

= 60

= 75

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value of the stress concentration factor in 0 degree layer for [02/]s and [/02]s laminates

increases. The location of the layer due the matrix crack also plays an important role to

determine the stress concentration factor. The closer the crack layer to the 0 degree layer the

higher the value of stress concentration factor in 0 degree layer for [02/]s and [/02]s

laminates. If all the 0 degree layers are attached to each other then the value of stress

concentration factor in 0 degree layer is lower compared to the case where it is on the outer

side of the laminates.

The notched strength of a laminate decreases with increase in d/w ratio for a constant

. Similarly, for a constant d/w ratio, with increase in the notched strength decreases. In

case of lay-ups, the notched strength is higher in the case where all 0 degree layers, which in

this case is the load carrying ply, are placed next to each other compared to the case where

they are scattered. Also, the notched strength is dependent on the location of the crack layer.

If the crack layer is placed next to 0 degree layer then the notched strength is lower compared

to the case where the crack layer is not placed to the 0 degree layer.

This study can provide a guideline to the engineers that for a given d/w ratio what

value of laminate can they select based on the stress permissible for that application.

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REFERENCES[1] Awerbuch, J. and Madhukar, M. , Notched Strength of Composite Laminates:

Predictions and Experiments---A Review, Journal of Reinforced Plastics and Composites,

Vol.4, January 1985, p.3.

[2] Tan, S. C., Stress Concentrations in Laminated Composites, Technomic Publishing Co., 1994.

[3] Whitney, J.M. and Nuismer, R.J. , Stress Fracture Criteria for Laminated Composites

Containing Stress Concentrations,Journal of Composite Material, Vol. 8,July 1974,

p.253.

[4] Chan, W.S. "Damage Characteristics of Laminates with a Hole," Proceedings of the American

Society for Composites, the 4th Technical Conference, Oct. 1989, pp. 935-943.

[5] Harn, F. E., Notched Strength of [02/]S Graphite/Epoxy Laminates with various Hole Size

Ph.D. dissertation, University of Texas at Arlington, Aug. 1997.

[6] Waddoups, M.E., Eisemann, J.R., and Kaminski, B.E., Macroscopic Fracture

Mechanics of Advanced Composite Materials, Journal of Composite Materials, VOL.5,

October 1971, p.446.

[7] Whitney, J.M. and Nuismer, R.J., Uniaxial Failure of Composite Laminates Containing

Stress Concentrations,Fracture Mechanics of Composites, ASTM STP 593, 1975, p.117

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[8] http://asm.matweb.com/search/SpecificMaterial.asp?bassnum=MA5086H116 last

visited on July 31,2005.

[9] http://casl.ucsd.edu/data_analysis/carpet_plots.htm last visited on July 31,2005.

[10] Paris, P.C. and Sih, G.C., Stress Analysis of Cracks, Fracture Toughness Testing and

Its Applications, ASTM 381, American Society for Testing and Materials, Philadelphia,

pp.30-85, 1965.

[11] Konish, H.J. and Whitney, J.M., Approximate Stresses in an Orthotropic Plate

Containing a Circular Hole,Journal of Composite Materials, Vol 9, pp157-166, 1975

[12] Tan, S.C., Laminated Composite Containing an Elliptical opening I. Approximate

Stress Analyses and Fracture Models,Journal of Composite Materials, Vol .21,

pp-925-948, 1987

[13] Tan, S.C., Stress Concentration in Laminated Composites, Lancaster:

Technomic,1994

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BIOGRAPHICAL INFORMATION

Moumita Roy was born at Calcutta(now Kolkata),India, in 1981. She received

her Master of Science in Mechanical Engineering from University of Texas at Arlington in

December 2005. Author completed her Bachelors degree in Ceramic Engineering from

Regional Engineering College-Rourkela in May 2002. Author has primary interests in the

fields of stress analysis, design and analysis of composite materials, finite element analysis.

She plans to pursue a P.HD in Mechanical Engineering.

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