mm- «‘.‘-‘ L“
‘ _' a“, :J ~‘
34?. ‘-

    

 

W54

       

v

 

3’
I:

 

  

‘A

 

j'u‘fi‘fl- h“?

      
   

4 .
-\.- .
5“,?»

        

  
  
  

 
    

 

 

  
    

   

      

P _—_y 2
,. 53.23““
. ._ n; A”
. ‘1 If
‘. 34 ~. 2 .1
4 ‘ .;~V'~“'“.:$‘-‘. ‘
I ‘x ‘1‘”. ”:23,
y 4: V". . ’
5 - ‘ g ‘
4 .
,
l
.‘
f ,
.
0
I

IN.

  
 

..
J . .~ --é ‘
s. *3”: 3v 1"" '9
‘ ‘ wig" 1'3.
‘ , rm‘ ’A‘f‘
> . v ..: ...
‘ z “p 0“ ~
r
'l
w
' r
'. ,
V r

:4

 

I
'J
‘3:
f:
'1
.r
V!
.‘I
=.

 

.r
7

     
 
 
 
  
  
     
         
   
 
 
   

  

I V ' *‘WZT
' M‘. . A ‘

 

 

 

 

 

            
 
 

 
  

 

      

         

      

     

 

          

     

 
 
  
 

   

   
 
    

 

 

, , .
, . ‘
, . - ‘
L g‘ ‘. ' AI ‘ I .i'
,.
' . . » . 1 .
_ . - . . '
‘ ~ .. '
, u v
v 'u- '- ‘ , ' ~
0‘ ' ‘ f
. ,. . . ‘h x.
‘ . _ ‘ , ‘ ’ . ,4 d-‘l ,... V a». a o v“
r, . ,3 ~ . * ' ' - .' ‘ 4 ' ‘ - . ‘,, " :vvg‘fiv-wbz v‘ .
' ‘ ‘ : > V _ - ”a ' ”r; .v‘ ""
~ . ‘ V ., 4w 1",. graflgy' .H ('1 «d a
. A . , .4- , ._m_ Nam '
‘. _ .‘ ‘ g ‘ _,,.;.--~‘...r-3;',;. ”$.17. ~ .. ..
- ; . , ‘ - - ‘ fwzr-wm «w.'r“..w""‘» 4:: :31..- r. s
w" "f’k‘. W".'—~--",'::':*;J&vr 5.: v-"".".—‘:""“ '
. . ‘ ‘A" 1» "Ha - ""‘
, A u-M ‘, a ’3’..-» “a u» A
l ‘ m—I’ —-..A""
g . . .l-u ' ”w
‘ . ' , " 7' t L"...-
‘ , . . . 4 ~' . ' .4 .,
. ‘ . . . ‘1 ' - I i 0' " w ”J . I
. . , , . ,- . u
. A . , , , , , ‘. Mun ,.
§ ‘ ‘ l -
. .
; . ~ ..
‘. . . -~
‘ ' I -
: . :3...- J.” "
. . A ' Kw
. - , , "fluff”
. ‘ v ' v V '

 

rm '—

'\ “”1”???“
J~ lllllllllllzlll‘ll lllll lllll ll" lol; l H

93 00605

 

 

l

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

LI: W ARV
Michigan State
University

 

 

 

This is to certify that the

thesis entitled

STRESS ANALYSIS IN CROSS-FLY LAMINATES
DAMAGED BY TRANSVERSE MATRIX CRACKS

presented by
Jung Ki Lee

has been accepted towards fulfillment
of the requirements for

M. S.

 

 

degree in MECHANICAL ENGINEERING

00"”GIX

Major professor

 

Date August 10, 1989.

 

0-7639 MS U is an Affirmative Action/Equal Opportunity Institution

 

a; ; ,fl, _

PLACE IN RETURN BOX to remove thie checkout from your record.
TO AVOID FINES min on or betone one due.

 

DATE DUE DATE DUE DATE DUE

l

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

g

MSU Is An Affirmdive ActionlEqual OppOI’tunlty Institution

 

STRESS ANALYSIS IN CROSS-PLY LAMINATES

DAMAGED BY TRANSVERSE MATRIX CRACKS

By

Jung Ki Lee

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering

1989

(U o 4 C: 495A

ABSTRACT
STRESS ANALYSIS IN CROSS-FLY LAMINATES
DAMAGED BY TRANSVERSE MATRIX CRACKS
By
Jung Ki Lee

Damage in composite materials has been a subject of great interest to both engineers
in industry and subsequently numerous researchers since the early 19705 when this
class of advanced materials began to assume greater utility in the defence, aerospace,
automotive and sporting goods industries, for example. Damage mechanisms in
cracked composite laminates have been investigated by using micromechanics model-
ling (ply discount scheme, shear lag analysis, self-consistent scheme, etc.) and inter-
nal variable characterization. Stress analysis techniques are crucial for modelling of
damage mechanisms in cracked composite laminates, however, due to phenomenolog-
ical complexities of the field, there are no exact solutions for the stress distributions
of cracked composite laminates. In this work, a finite element model for the damage
mechanisms of cracked composite laminates is proposed by reducing this class of
three-dimensional phenomenological problems to two-dimensional plane-strain prob-
lems. In particular, stress distributions in cross-ply laminates damaged by transverse
matrix cracks are investigated by employing the ANSYS finite element computer pro-

gram. Four types of problems for glass/epoxy and graphite/epoxy laminates are con-

sidered: (1) [O°/9O°]s laminate with an isolated crack; (2) [0°/90°]S laminate with in-

teracting cracks; (3) [O°/9O°3]s laminate with an isolated crack; (4) [0°/90°3]S lami-

nate with interacting cracks. Furthermore, the finite element analysis results for

[O°/90°]s glass/epoxy composite laminate with interacting cracks and [O°/90°]S

glass/epoxy composite laminate with an isolated crack, were compared with the varia—
tional calculus results based on the minimum complementary energy theorem. These
variational calculus results were obtained by employing a more generalized test func-
tion than that used by Hashin. Reasonable agreement is obtained between the finite
element analysis results and the variational calculus results. Important aspects of
damaged composite laminates are investigated and discussed by utilizing the finite el-

ement formulation developed herein.

To my family

iv

ACKNOWLEDGMENTS

I would like to thank my adviser Dr. Mukesh V. Gandhi for his significant contri-
butions to this research, technical guidance, and editorial advice.

I would like to thank Dr. Nicholas J. Altiero and Dr. Brian S. Thompson who
served on members of the thesis committee for the time and effort they devoted to
helping me on developing and proofreading this thesis.

Numerous people contributed the efforts to complete this thesis. In particular, I
would like to thank Mr. Mohammad Usman and Mr. Devang Desai for providing their
assistance and proofreading this thesis. Also, I would like to thank numerous Korean
students for helping and encouraging me during this research.

Finally, I would like to thank my parents, brother and sisters, my wife and my

friends for their concern and everlasting support.

TABLE OF CONTENTS

List of Tables ..................................................................................................... vii
List of Figures .................................................................................................... viii
Chapter 1 Introduction ................................................................................. 1
1.1. Concepts of damage in composite materials ................................. 1
1.2. Overview ....................................................................................... 6
1.3. Objective ....................................................................................... 10
Chapter 2 Description of the problem and approach ................................ 12
2.1. Description of present investigation .............................................. 12
2.2. Methodologies ............................................................................... 15
Chapter 3 Results and discussion on finite element analysis ..................... 21

Chapter 4 Comparison of the finite element analysis results with the
analytical results ......................... 43
4.1. Analysis of a cracked symmetric cross-ply laminate under
uniform tension ............................................................................. 43

4.2. Analytical results and comparison with the finite element

analysis results ............................................................................... 56
Chapter 5 Concluding remarks .................................................................... 73
List of references ............................................................................................... 75

vi

Table 2.1

Table 2.2

Table 4.1

Table 4.2

LIST OF TABLES

Stress concentration factor for homogeneous material near the

crack tip (Type 1) .......................................................................... 15
Material properties for the composite laminates .......................... 16
Ratio (SI/0'2 and 01/0 by the classical lamination theory .............. 55

Complementary energy (U c ’2) variation with

various power n in equations (4.13 d,e,f) ...................................... 56

vii

LIST OF FIGURES

Figure 1.1 Schematic representation of the damage accumulation process

in composite laminates .................................................................. 3
Figure 1.2 Typical saturation crack patterns observed in [O°/90°]S

laminate ......................................................................................... 3
Figure 1.3 Edge replica from a [0°/9O°2]S laminate at the Characterization

Damage State (CDS) ..................................................................... 4
Figure 1.4 Typical stiffness reduction for a [0°/90°2]S laminate .................... 4

Figure 1.5 Schematic "flow diagram" of damaged mechanics for
composite materials ....................................................................... 7

Figure 1.6 A typical one-dimensional model in formulation of

shear lag analysis ........................................................................... 8
Figure 2.1 Cross-ply laminate damaged by transverse matrix cracks ............ 12
Figure 2.2 Representative volume element .................................................... 12
Figure 2.3 [O°/90°]S laminate with an isolated crack ...................................... 13
Figure 2.4 [O°/90°]S laminate with interacting cracks .................................... 13
Figure 2.5 [0°/9O°3]S laminate with an isolated crack .................................... 14
Figure 2.6 [O°/90°3]s laminate with interacting cracks ................................... 14

Figure 2.7 Finite element modelling of type 1 composite laminate
(Number of elements = 1518, number of nodes = 1598) .............. 17

Figure 2.8 Finite element modelling of type 1 composite laminate
viii

Figure 2.9

Figure 2.10

Figure 2.11

Figure 3.1

Figure 3.2

Figure 3.3

Figure 3.4

Figure 3.5

Figure 3.6

Figure 3.7

Figure 3.8

(Enlargement at crack) ............................................................... 17
Finite element modelling of type 2 composite laminate
(Number of elements = 3450, number of nodes = 3595) ............ 18
Finite element modelling of type 3 composite laminate
(Number of elements = 3036, number of nodes = 3162) ............ 19

Finite element modelling of type 4 composite laminate

(Number of elements = 3675, number of nodes = 3833) ............ 20
Stress component O'y at the crack plane in 0° ply

[0°/90°]s glass/epoxy composite laminate (Type 1) ................... 25
Stress component 0y at the crack plane in 0° ply

[0°/90°]S graphite/epoxy composite laminate (T ype 1) .............. 25
Stress component 0), at the interface in 0° ply

[0°/90°]s glass/epoxy composite laminate (T ype 1) ................... 26
Stress component 0'), at the interface in 0° ply

[0°l90°]s graphite/epoxy composite laminate (Type 1) .............. 26
Stress component oy at the interface in 0° ply

[0°/90°]s glass/epoxy composite laminate (Type 2) ................... 27
Stress component 0'y at the interface in 0° ply

[0°/90°]s graphite/ep0xy composite laminate (Type 2) .............. 27
Stress component oy at the interface in 0° ply

[0°/90°3]s glass/epoxy composite laminate (Type 3) .................. 28
Stress component 0'), at the interface in 0° ply

[0°/90°3]S graphite/epoxy composite laminate (Type 3) ............ 28

Figure 3.9

Figure 3.10

Figure 3.11

Figure 3.12

Figure 3.13

Figure 3.14

Figure 3.15

Figure 3.16

Figure 3.17

Figure 3.18

Figure 3.19

Stress component 6 at the interface in 0° ply

Y

[0°/90°3]s glass/epoxy composite laminate (Type 4) .................. 29
Stress component try at the interface in 0° ply

[0°/90°3(]s graphite/epoxy composite laminate (Type 4) ............ 29
Stress component Ox and ny at the interface in 0° ply

[0°/90°]s glass/epoxy composite laminate (Type 1) ................... 30
Stress component <3x and ox), at the interface in 0° ply

[0°/90°]S graphite/epoxy composite laminate (Type 1) .............. 30
Stress component (5'x and ox), at the interface in 0° ply

[0°/90°]S glass/epoxy composite laminate (Type 2) ................... 31
Stress component (IX and oxy at the interface in 0° ply

[0°/90°]S graphite/epoxy composite laminate (Type 2) .............. 31
Stress component ox and ny at the interface in 0° ply

[0°l90°3]s glass/epoxy composite laminate (Type 3) .................. 32
Stress component 0x and ox), at the interface in 0° ply

[0°/90°3]s graphite/epoxy composite laminate (Type 3) ............ 32

Stress component (3'x and 0x at the interface in 0° ply

Y

[0°/90°3]s glass/epoxy composite laminate (Type 4) .................. 33

Stress component ox and (5x at the interface in 0° ply

Y
[0°/90°3]S graphite/epoxy composite laminate (Type 4) ............ 33

Stress component 0’), at the interface in 90° ply

[0°/90°]s glass/epoxy composite laminate (Type 1) ................... 34

X

Figure 3.20

Figure 3.21

Figure 3.22

Figure 3.23

Figure 3.24

Figure 3.25

Figure 3.26

Figure 3.27

Figure 3.28

Figure 3.29

Figure 3.30

Stress component 0 at the interface in 90° ply

Y
[0°/90°]s graphite/epoxy composite laminate (Type 1) .............. 34
Stress component 0y at the interface in 90° ply
[0°/90°]s glass/epoxy composite laminate (Type 2) ................... 35
Stress component oy at the interface in 90° ply
[0°/90°]s graphite/epoxy composite laminate (Type 2) .............. 35
Stress component oy at the interface in 90° ply
[0°/90°3]S glass/epoxy composite laminate (Type 3) .................. 36
Stress component O'y at the interface in 90° ply
[0°/90°3]s graphite/epoxy composite laminate (Type 3) ............ 36
Stress component 0), at the interface in 90° ply
[0°/90°3]S glass/epoxy composite laminate (Type 4) .................. 37
Stress component oy at the interface in 90° ply
[0°/90°3]s graphite/epoxy composite laminate (Type 4) ............ 37

Stress component ox at the midplane in 90° ply
[0°/9O°]s glass/epoxy composite laminate (Type 1) ................... 38
Stress component (3'x at the midplane in 90° ply
[0°/90°]s graphite/epoxy composite laminate (Type 1) .............. 38
Stress component ox at the midplane in 90° ply
[0°/90°]s glass/epoxy composite laminate (Type 2) ................... 39
Stress component <3x at the midplane in 90° ply

[0°/90°]s graphite/epoxy composite laminate (Type 2) .............. 39

xi

Figure 3.31

Figure 3.32

Figure 3.33

Figure 3.34

Figure 3.35

Figure 3.36

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

Stress component ox at the midplane in 90° ply

[0°/90°3]S glass/epoxy composite laminate (Type 3) .................. 40
Stress component ox at the midplane in 90° ply

[0°/90°3]S graphite/epoxy composite laminate (Type 3) ............ 40
Stress component ox at the midplane in 90° ply

[0°/90°3]S glass/epoxy composite laminate (Type 4) .................. 41
Stress component ox at the midplane in 90° ply

[0°/90°3]s graphite/epoxy composite laminate (Type 4) ............ 41
Stress component ox at the crack location in 90° ply

[0°/90°3]S glass/epoxy composite laminate (Type 4) .................. 42

Stress component 0'x at the crack location in 90° ply

[0°/90°3]s graphite/epoxy composite laminate (Type 4) ............ 42
[0°/90°n]s laminate between two cracks .................................... 43
[0°/90°n]S laminate with an isolated crack ................................ 54

Stress component 0'), at the interface in 0° ply

[0°/90°]s glass/epoxy composite laminate (Type 2) ................... 61
Stress component oxy at the interface in 0° ply

[0°/90°]s glass/epoxy composite laminate (Type 2) ................... 62
Stress component 0'x at the interface in 0° ply

[0°/90°]s glass/epoxy composite laminate (Type 2) ................... 63
Stress component <3y at the interface in 90° ply

[0°/90°]s glass/epoxy composite laminate (Type 2) ................... 64

xii

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

4.7

4.8

4.9

4.10

4.11

4.12

4.13

4.14

Stress component 0 at the midplane in 90° ply

Y

[0°/90°]S glass/epoxy composite laminate (Type 2) ................... 65

Stress component 0'x at the midplane in 90° ply

[0°/90°]s glass/epoxy composite laminate (Type 2) ................... 66
Stress component oy at the interface in 0° ply
[0°/90°]S glass/epoxy composite laminate (Type 1) ................... 67

Stress component ny at the interface in 0° ply
[0°/90°]s glass/epoxy composite laminate (Type 1) ................... 68

Stress component 0'x at the interface in 0° ply

[0°/90°]s glass/epoxy composite laminate (Type 1) ................... 69
Stress component 6), at the interface in 90° ply
[0°l90°]s glass/epoxy composite laminate (Type 1) ................... 70

Stress component oy at the midplane in 90° ply
[0°/90°]s glass/epoxy composite laminate (Type 1) ................... 71
Stress component ox at the midplane in 90° ply

[0°/90°]s glass/epoxy composite laminate (Type 1) ................... 72

xiii

CHAPTER 1
INTRODUCTION

1.1 Concepts of damage in composite materials

Damage can be loosely defined as the effect of micro-failure events on material
behavior, or, as a collection of permanent (irreversible) microstructural changes
brought about in the material by a physical process, resulting from the application of
loads. The identification of damage in composite materials consists of a number of
damage modes, such as matrix cracking, fiber breaks, fiber/matrix debonds, delamina-
tion cracking and interlaminar cracking.

One of the major differences between the mechanical response of fiber reinforced
composite laminates and that of more conventional structural materials, such as alu-
minum and steel, is that the damage that develops in such composites is generally
much more complex. Metals usually fail by crack initiation and its growth; in a man-
ner which is predictable through fracture mechanics analysis. Whereas composites
exhibit several modes of damage including matrix crazing, fiber failure, void growth,
matrix cracking, delamination and composite cracking. A particular structure may ex-
hibit any or all of these damage modes, a priori, it is not known which mode will domi-
nate and cause failure.

In the succeeding part of this section, it would be useful to discuss the physical
aspects of damage in composite materials. In advanced composites, e.g. graph-
ite/epoxy, the microstructural changes are in the form of cracks of various geometries
and orientations, as mentioned earlier. The development of the micromechanical

changes is conveniently observed in fatigue, since the rate of change is small

1

2
compared to that in, for instance, quasi—static fracture and impact failure. Based on

fatigue damage studies of advanced composite laminates, certain patterns that seem

to be valid for a large class of composite materials can be investigated. A schematic
representation of damage development under fatigue is shown in Figure 101111.12]. Al-

so, a typical stiffness reduction under fatigue is shown in Figure 1.48].

Damage deveIOpment consists of three stages (namely, initiation, growth, and
localization) leading to ultimate failure. The initial stage consists of primary trans-
verse matrix cracking along fibers in off—axis plies. The parallel cracks appear in the
cracking ply. According to the stresses in the ply and the constraints to cracking giv-
en by the neighboring plies, the crack number density increases. The cracking process
may continue until cracks in each ply have attained equilibrium or saturation spacing.
The saturation spacing is a property of the laminate and is independent of the load-
ing history. The state of damage given by the saturated and stable matrix cracking

pattern in a laminate has been termed the Characterization Damage State (CDS), a

well-established condition of saturated ply cracks (Figures 1.2 and 1.3)[3]’[4]. It ap-
pears to indicate the termination of the first stage of matrix cracking having an insig-
nificant interaction between cracks. Also, it is noted that after sufficient loading the
off-axis plies in a laminate reach a saturation state, where the distance between con-
secutive cracks in a particular ply is nearly uniform throughout the specimen. This
fact can be attributed to the fact that transverse cracking reduces the load-carrying
capacity of those plies only in a local region adjacent to the cracks. The material be-
tween adjacent cracks outside of these relaxed zones is capable of carrying some load
and hence contributing to the laminate stiffness. This mechanism causes the fiber
breaks to occur in a more localized configuration, in regions of stress concentration
created by the primary cracks.

The primary cracks also initiate longitudinal cracks. The longitudinal cracks are

generally perpendicular to the primary cracks. The longitudinal cracks are caused by

 

mo>z>u

l—Matrix Cracking 3-Delamination S-Fracture
Fiber Breaking Fiber Breaking

0° 0°

A
E

ll

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2-Crack Coupling, Interfacial 4-Delamination Growth, Fiber
Debonding, Fiber Breakage Brcalungmocahzed)
l
j
0 PERCENT OF LIFE 100

Figure 1.1 Schematic representation of the damage accumulation

process in composite laminatesm'm.

 

 

 

 

 

 

 

1 2 l l l l T 93:
.) K I 1 1' \. \. I. 90°
0°

 

 

Figure 1.2 Typical saturation crack patterns observed in [0°/90°]S

laminatew.

 

 

 

 

 

Figure 1.3 Edge replica from a [0°/90"2]s laminate
at the Characterization Damage State (CDS)[3].

 

 

 

1.00 "‘
0.99 fig
0.98 "
0.97 4
0.96 q
0.95 ""‘
0.94 "'
0.93 "
0.92 "
0.91 ‘
0.90 "'
0.89 T
0.88 _
0.87 l l l l l I l l 7
0 5 10 15 20 25 30 35 40 45

THOUSAND OF CYCLES
Figure 1.4 Typical stiffness reduction for a [0"/90"2]s laminatem.

 

 

 

mcrcco: az>nmm umn~r>zwoz

 

 

 

5
the tensile stress along the crack axis ahead of the primary cracks. The material in

the 0° ply adjacent to the crack tip is subjected to the tensile stress in its lowest
strength direction. So, the transverse crack can be considered a likely site for the nu-
cleation of longitudinal cracks.

Next, delamination takes place in the interior of a laminate. In regions where the
primary transverse matrix cracks and the longitudinal cracks intersect; a local out-of-
plane tensile stress condition is created by the transverse matrix cracks and the
longitudinal cracks. Hence, the centers of incipient delaminations frequently coincide
with crack interactions. They are also the points of maximum delamination opening.
These delaminations coalesce in regions between longitudinal cracks, isolating small
volumes of material in the 0° plies that become longitudinal splits.

Numerous fiber fractures occur during the damage development of transverse
matrix cracking, longitudinal cracks, and delamination. However, these fractures do
not occur in randomly distributed array but instead have a distinct, consistent pat-

tern. It suggests the involvement of the adjacent transverse cracks whose tips are a

fiber fracture initiator in the 0° ply. In addition, the fiber fractures that occur are segre-

gated into zones that are bounded by fiber fracture-free zones. Delamination appears

to isolate the 0° plies from the transverse cracks and thereby prevents related fiber
breaks. The formation of broken fibers in this way produces numerous weakened net
sections in the load-bearing ply. Such a crack structure, if repeated across the width
of a laminate, has serious implications for laminate failure. So, the effect of this order-
in g of fiber fractures cannot be ignored.

Many of these damage modes occur long before the ultimate failure, hence, there
can be many types of subcritical failures. These phenomena, in turn, contribute to the
final fracture which seems to be either controlled from rapid fiber failures or from se-
vere interactions of cracks leading to loss of material integrity.

In addition, numerous numerical and experimental investigations have

6
demonstrated the free edge effect in composite laminates subjected to remote ten-

sion. It has been suggested that high interlarrrinar stresses in the boundary layer play
a dominant role in the delamination failure mode of the composite laminates. At the
free edges of a laminate (sides of a laminate or holes), the interlaminar shearing
stress is very high (perhaps even singular) and would therefore cause the debonding

that has been observed in such regions.

1.2 Overview

For more than two decades, considerable effort has been expended in under-
standing the phenomena of damage in composite materials, and new theories of the
mechanisms still continue to emerge. The literature contains an important record of in-
formation on the subject of damage, both experimental and theoretical. Particularly,
the transverse ply cracking in composite materials and the development of a satisfac-
tory theory for cross-ply laminates damaged by transverse matrix cracking under
monotonic loading have been a subject of extensive research. A schematic "flow-rule"
of damage mechanics related to composite materials is shown in Figure 1512]. A large
body of information has been collected utilizing various non-destructive techniques in-
cluding edge replication, light and electron microscopy, X-ray radiography, ultrasonic
C-scan and also destructive techniques, e.g. specimen sectioning and de-ply for mi—
croscopy observations. The above are used to generate knowledge that constitutes a
damage mechanics analysis. Therefore they may be utilized, broadly speaking, in one
of the two common approaches to performing a damage mechanics analysis. One is

micromechanics modelling approach, and the other is internal variable characterization

approach.
As a method of micromechanics modelling analysis for cross-ply laminates dam-

aged by transverse matrix cracking, the formation of a shear lag model appears to

have been first proposed in a series of papers by Bailey and his co—workers[5]'[7].

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

/

MICROMECHANICS
MODELLING

 

 

 

\

 

 

 

 

DESTRUCTIVE NON-DESTRUCTIVE
EVALUATION
EVALUATION
X-RAY ULTRA_ THERMO
SECIIONING DEPLY - VIBRO-
RADIO SONICS AP
GRAPHY GR HY
DAMAGE
MECHANICS

 

INTERNAL

VARIABLE
CHARACTERIZATION

 

 

 

 

 

 

MECHANICAL PROPERTIES

 

 

 

 

STIFFNESS

 

 

 

 

l\

 

 

 

 

 

STRENGTH LIFE

 

 

 

 

 

FIGURE 1.5 Schematic "flow diagram" of damage mechanics for
composite materialsm.

 

 

8
Subsequent contributions to the theory have been given by numerous researchers.

Especially, Reifsnider and co-authors[4] evaluated stiffness reduction in terms of

shear lag analysis. Figure 1.6 shows a typical one-dimensional model in formulation

of shear lag analysis.

 

 
   
 
   
 
 

0° thickness Of 0°ply shear transfer
lregion
N «— 90° thickness of 90°p1y N
}shear transfer
00 thickness of 0°ply region

  

 

Figure 1.6 A typical one-dimensional model in formulation of
shear lag analysis.

 

The shear lag analysis is based on the following assumptions: (a) The normal
StI'ess in external load direction is constant over ply thickness; (b) Shear deforma-
tions in any given ply are restricted to a thin region in the vicinity of interfaces of that
ply with adjacent plies. Further, this region tends to be resin-rich, and thus is less
Stiff in response to shear loads than the central portion of the lamina. This thin region
is assumed to be a shear transfer region. However, the thickness of the boundary lay-
er must be assumed in somewhat arbitrary fashion and the transverse normal stress-
es Cannot be estimated.
An another method of micromechanics modelling analysis, the self-consistent
schemem, is very common. The method of self-consistent scheme was first devised
by I‘Iersheym and KrOnerUO] as a means to model the behavior of polycrystalline ma-

te ‘ . . . . .
r1313. Such materials are Just one phase medra. The extensron of the self-consrstent

Scheme to multiphase media was given by Hillnu and Budianskiuzl. As discussed by

9
Budianskiuz], the method has a very simple geometric interpretation. Specially, each

phase of the composite is alternatively viewed as being lumped as a single ellipsoidal
inclusion in an infinite matrix of the unknown effective properties of the problem. The
application of uniform stress or strain conditions at infinity allows the determination of
the average conditions in the inclusion. After this operation is performed for all phas-
es, the average conditions are known in all phases, in terms of the individual phase
properties and the effective properties. Thence, average conditions in the entire com-
posite are known and the effective moduli can be calculated from the averages.

The method of estimating the effective elastic properties of a cracked solid have
been adopted by Budianski and O’Connellm]. The model predicts the elastic proper-
ties for a material consisting of a homogeneous isotropic matrix in which flat elliptical

cracks are dispersed. And, analytical solutions for effective moduli of elastic bodies

with distributed cracks have been obtained by Laws and Dvorak[14]'[l7]. The cracks
are aligned and the total crack surface area has a first order effect on the stiffness.
These theoretical results apply only to elastic bodies with cracks of homogeneous and
predetermined dimensions.

In an important series of papers, Talreja[18]'[23] has presented a different ap-
proach for characterizing damage in composite materials. Talreja utilized vector fields
for characterization of damage in fatigue, or any other loading mode. Constitutive
equations are derived for isothermal small-deformation behavior following attainment
of a damage state. Talreja used a continuum model for damage characterization that
will predict the thermomechanical constitution of elastic composites. Damage is char-
acterized by a set of second order tensor valued internal state variables representing
locally averaged measures of specific damage states. However, in the formulation of
stiffness reduction, the phenomenological constants must be evaluated by conducting

a suitable set of experiments.

10

1.3 Objective

Historically, stress analysis has been closely related to the analysis on the loss
of stiffness in cracked composite laminates. Stress analysis techniques are crucial for

modelling of damage mechanisms in cracked cross-ply laminates with transverse ma-

trix cracking. Chen and Sih[24] analyzed stress distributions for a three-layered plate
with a crack in the center layer. They modeled the laminated composite as a multilay-
ered plate each layer being made of a different material. They assumed that the
stresses could vary in all three space coordinate directions in cracked composite lami—
nates and the problem would be as a three-dimensional one. They applied the mini-
mum complementary energy theorem in variational calculus such that the qualitative
three-dimensional character of the crack edge stresses was retained while approxi-
mations were made in a quantitative sense on the stress intensity factor. But, in this
analysis, they assumed that each layer was made of isotropic and homogeneous ma-

terials. So, the middle layer was assumed to be made of the material with elastic

properties (E1, v1) while the two outer layers possessed the same material proper—
ties (E2, v2). (Note: E1, B2 are Young’s modulus, and v1, v2 are Poisson’s ratios.)

Therefore, the results from Chen and Sih[24] are not sufficiently accurate for cracked

composite laminates.

Laws and Dvorak[14]'[17] analyzed stress distributions by employing the self-
consistent scheme, as an additional work on the loss of stiffness for cross-ply lami-
nates which have been damaged by transverse matrix cracking under monotonic load-
ing. But, one problem with this analysis is of conceptual nature for their model of a
cracked material is infinite in all directions while a cracked ply may be assumed to ex-
tend to infinity only in one direction and crack opening is significantly constrained by

adjacent plies.

11
Hashin[25]’[26] presented a variational approach to the problems of stress evalu-

ation which incorporates all of the important aspects of the problem and involves only

one assumption that normal ply stresses in the load direction are constant over ply

thickness. Hashin[25]'[26] thus constructed admissible stress fields which satisfy

equilibrium and all boundary and interface conditions and determined stresses on the
basis of the minimum complementary energy theorem. Hashinlstzm has good agree-

ment with experimental data for [0°/90°]s ply in predicting the stiffness reduction.

However, the primary perturbation stresses in cracked composite laminates are inca-
pable of evaluating the stress concentrations produced by transverse matrix cracks.
Hence, the objective of present investigation is to develop more generalized

variational calculus stress distributions which have stress concentrations at the crack

tips in 0° ply than those used by Hashin. In addition, it would be very useful to com-
pare variational stress distributions with stress distributions from numerical simula-
tion (finite element analysis) and to see what the stress distributions in cross-ply
laminates damaged by transverse matrix cracks would be physically possible. There-
fore, based on the stress distributions in cracked composite laminates, several impor-

tant physical damage aspects can be obtained and discussed.

CHAPTER 2
DESCRIPTION OF THE PROBLEM AND APPROACH

2.1 Description of present investigation

Consider a symmetric cross-ply laminate which is subjected to uniform in-plane
tensile loads. It is assumed that the 90° ply has well-established continuous in-
tralaminar cracks in the fiber direction. The cracks extend from edge to edge in z direc-

tion. Figures 2.1 and 2.2 describe the problem under investigation.

 

LOAD <—
2h

 

Matrix cracks 2

Figure 2.1 Cross-ply laminate damaged by transverse matrix cracks.

 

 

 

 

 

 

 

 

 

00
90°
, ........ ............ s? ...................... ' LOAD
LOAD<— , ' 0°
21. H” ll ll ...... - V ll

 

 

 

J1.

 

Figure 2.2 Representative volume element.

 

12

13
Finite element method has been employed to find stress distributions in cross-

ply laminates damaged by transverse matrix cracks. The following four types of prob-

lems are considered:

(1) [0°/90°]s laminate with an isolated crack;

 

 

 

 

 

 

0° t

N <— 90° 2t _> N
00

Figure 2.3 [0°/90°]S laminate with an isolated crack.

(2) [0°/90°]S laminate with interacting cracks;
Y

0° t

N <— 90° 2‘ —’ N
0° t

 

L
l‘ 'l‘ 'I
4t 4t
Figure 2.4 [0°/90°]S laminate with interacting cracks.

 

 

 

14
(3) [0°/90°3]s laminate with an isolated crack; and

 

Y W

 

 

Figure 2.5 [0°/90°3]s laminate with an isolated crack.

 

 

 

(4) [0°/90°3]s laminate with interacting cracks.

 

 

 

 

L -I- It T
I‘ '1‘ '1

4t 4t

Figure 2.6 [0°/90°3]s laminate with interacting cracks.

 

 

 

The above cracked composite laminates are subjected to axisymmetric tensile
loading at their ends. One of the main purposes of present analysis is to study the ef-
fect of transverse matrix cracks on the stress concentrations near the defective sites

Of the composite laminates.

15
2.2 Methodologies

Each type of composite laminate is modeled by employing the ANSYS finite ele—

ment computer program using a 2-Dimensional isoparametric solid element (Element

42)[27]. Plane-strain analysis using a linear quadrilateral element is used in all types
of composite laminates.

Usually the accuracy of solution in finite element analysis, especially near the

crack tip, is sensitive to mesh configuration. Guydish et. al[28] stated the following
important findings:

(1) It is necessary that the elements immediately adjacent to the crack tip be
very small in proportion to crack length;

(2) The most efficient mesh results from a smooth progression of node spacing
from the minimum space at the crack tips.

To check the validity of using present modelling for the cracked composite lami-
nates, the stress concentration factor for homogeneous, isotropic material with same
modelling as the present was compared to that from modelling of different methods.

Table 2.1 shows that the present modelling has good accuracy.

 

Table 2.1 Stress concentration factor for homogeneous material near the
crack tip (Type 1).

 

 

 

 

 

 

Method Stress concentration factor
Initial Mesh [29] about 4.0
Subdomain Mesh [29] about 15.0
Adaptive Mesh [29] about 23.0
Present Mesh 25.36

 

 

 

 

 

16
In the computational work, the following material properties were employed.

 

Table 2.2 Material properties for the composite laminates.

 

 

 

 

 

 

 

 

Property Glass/epoxy Graphite/epoxy
Axial Young’s modulus in fiber direction 41.7 208.3
EA Gpa
Transverse Young’s ITIOGUIUS 130 155 (65 1k)
ET Gpa
Axial shear modulus 3.40 1.65
G A Gpa
Transverse shear modulus 4.53 230
GT Gpa
Associated axial Poison’s ratio 0 30 0255
VA ' .
Assocrated Ears/inverse Porsson 5 ratio 0.42 0.413

 

 

 

 

 

 

From Table 2.2, for glass/epoxy the unidirectional properties are chosen as those re-
ported by Highsmith, et al.[4] for scotch-ply specimens. For graphite/epoxy, the prop-
erties are chosen as those reported by Hashin for graphite(T300)/epoxy.[25] For com-

putational convenience, the property of transverse Young’s modulus, Er, will be used

as 15.5 Gpa instead of 6.5 Gpa.
For the analysis of type 1, due to symmetry, only one quarter of the composite

laminate was considered. The dimensions of the composite laminate were taken as:
length=6.0; thickness in 0° ply = 1.0; thickness in 90° ply=1.0. Keeping in view the

application of Saint-Venant’s principle in composite materialsBO], the total length in

l7
finite element modelling was chosen as 10.0. Figures 2.7 and 2.8 show the finite ele-

ment modelling of type 1 composite laminate.

 

 

 

TYPE 1 CDPPOSITE LRHINRTE

 

 

 

Figure 2.7 Finite element modelling of type 1 composite laminate.
(Number of elements = 1518, number of nodes = 1598)

 

Crack Tip

Figure 2.8 Finite element modelling of type 1 composite laminate.
(Enlargement at crack)

18
For the analysis of type 2, one quarter of the composite laminate was consid-

ered. The dimensions of the composite laminate were taken as: length between two
interacting cracks =4.0; thickness in 0° ply =1.0; thickness in 90° ply =1.0. The total
length in finite element modelling was chosen as 10.0. Figure 2.9 shows the finite ele-
ment modelling of type 2 composite laminate. In Figure 2.9, in order to include the up-
per crack in the finite element modelling, each node at the crack location was num-
bered twice so that one number was used in the lower element connectivity, and the

second number was used in the upper element connectivity at the crack location.

 

p'

—'
.- u:

-—l
Sou-m:
..---d-.-lll

 

TYPE 2 COMPOSITE LRMIHRTE

 

 

 

Figure 2.9 Finite element modelling of type 2 composite laminate.
(Number of elements = 3450, number of nodes = 3595)

19
For the analysis of type 3, one quarter of the composite laminate was consid-

ered. The dimensions of the composite laminate were taken as: length = 6.0; thick-
ness in 0° ply = 1.0; thickness in 90° ply = 3.0. The length in finite element modelling
was taken as 10.0. Figure 2.10 shows the finite element modelling of type 3 compos-

ite laminate.

 

 

TYPE 3 COMPOSITE LRMIHRTE

 

 

 

Figure 2.10 Finite element modelling of type 3 composite laminate.
(Number of elements = 3036, number of nodes = 3162)

20
Finally, for the analysis of type 4, one quarter of the composite laminate

was considered. The dimensions of the composite laminate were taken as: length be-

: l. ; thickness in 90° ply =

4.0; thickness in 0° ply

tween two interacting cracks

3.0. The total length in finite element modelling was chosen as 13.0. Figure 2.11

shows the finite element modelling of type 4 composite laminate.

 

 

 

COMPOSITE LAMINATE

TYPE 4

 

 

Figure 2.11 Finite element modelling of type 4 composite laminate.

(Number of elements = 3675, number of nodes = 3833)

CHAPTER 3
RESULTS AND DISCUSSION ON
FINITE ELEMENT ANALYSIS

The results of the finite element analysis are presented in this section. Axial,

shear, and transverse stress distributions at the interface in 0° ply, and axial stress

distributions at the interface and transverse stress distributions at the midplane in

90° ply will be mainly discussed for all four different types of composite laminates
(glass/epoxy and graphite/epoxy laminates). (Note: x, y coordinates in the following
figures are set up differently from those in Figures 2.1-2.6. The new x, y coordinates
are shown in the following figures.)

Figures 3.1-3.2 show the axial stress distribution in x—direction at the crack

plane in 0° ply for type 1 composite laminate. It is observed that an axial stress con-
centration occurs at the crack tip. The stress concentration in glass/epoxy laminate is
stronger than that in graphite/epoxy laminate. The same phenomenon may be ob-
served for all four different types of composite laminates. The existence of this phe-

nomenon is due to the fact that the ratio Eaxial / E = 417/130 = 3.21 in

transverse

glass/epoxy laminate is much lower than E = 208.3 / 15.5 =13.44

axial / E transverse
in graphite/epoxy laminate.

Figures 3.3-3.10 show the axial stress distribution along y~direction at the in-
terface in 0° ply for four different types of composite laminates (glass/epoxy and
graphite/epoxy laminates). It is easily observed that an axial stress concentration oc-

curs at the crack tip. The above result predicts that the transverse matrix crack can be

21

22
considered a likely site for the nucleation of longitudinal cracks and/or delamination

cracking. It may also be noted that the stress concentrations for type 3 and 4 compos-
ite laminates are much stronger than those for type 1 and 2 composite laminates.
From the theory of fracture mechanics, it seems very reasonable. But, for type 2 and 4
composite laminates, exactly symmetric axial stress distributions between two inter-
acting cracks are not obtained. For this case, more discrepancies may be found in
graphite/epoxy laminates rather than in glass/epoxy laminates. Furthermore, it is ob-
served that as the Poisson’s ratio(v) in 90° ply increases, the axial stress concentra-
tion becomes stronger for all four different types of composite laminates (glass/epoxy
and graphite/epoxy laminates).

Figures 3.11-3.18 show the transverse and shear stress distributions along y-
direction at the interface in 0° ply for four different types of composite laminates
(glass/epoxy and graphite/epoxy laminates). Initially, type 1 and 3 composite lami-
nates are considered. It is observed that near the crack tip, the transverse stress
is tensile, then it drops rapidly to zero, changes to compressive, and tends to ze-
ro again. It is also observed that near crack tip, the shear stress is positive, and it
drOps rapidly to zero, and changes to negative, and stays negative. It may be noted
that at almost the same place, in the vicinity of the crack tip, the transverse and shear
stresses reach the maximum negative limit. But, it is observed that the magnitude of
tensile transverse stress is much larger than that of tensile shear stress.

The above results predict that the possible debonding which might occur due to

the tensile transverse stress (ox(1,y), ox(3,y)) near the crack tip is unlikely to prop-
agate in the fiber direction because of the change of sign of ox(1,y), and ox(3,y). And

it is noted that a short crack transversing a few fibers in 0° ply cannot propagate too
far in the original crack direction before it turned to propagate along the fiber direction.
Next, type 2 and 4 composite laminates are considered. It is observed that near

the lower crack tip, the transverse stress is tensile, and it drops rapidly to zero,

23
changes to compressive, and reaches zero again at the midpoint between two inter-

acting cracks. The transverse stress distribution is almost symmetric about the mid-
point between two interacting cracks. It is indicated that near the lower crack tip, the
shear stress is tensile, and it drops rapidly to zero, changes to compressive, and
reaches zero again at the midpoint between two interacting cracks. After the mid-
point, it changes tensile and reaches the maximum tensile, drops to zero, and be-
comes rapidly compressive. Also, the shear stress distribution is almost anti-sym-
metric about the midpoint between two interacting cracks. It may also be noted that
the magnitudes in transverse and shear stresses for type 3 and 4 composite lami-
nates are larger than those for type 1 and 2 composite laminates.

Figures 3.19-3.26 show the axial stress distribution along y—direction at the in-

terface in 90° ply for four different types of composite laminates (glass/epoxy and
graphite/epoxy laminates). For glass/epoxy laminates, an axial stress concentration
may be found at the crack tip for all four different types of composite laminates.

The axial stress concentration in 90° ply is not so strong as that in 0° ply. How-
ever, the stress concentrations for type 3 and 4 composite laminates are much stron-
ger than those for type 1 and 2 composite laminates. For graphite/epoxy laminates, a

strong axial stress concentration may not be seen near the crack tip. It is expected

that this is due to the small ratio of Eaxial / Etransverse in 90° ply for graphite/epoxy

laminates.

Figures 3.27-3.34 show the transverse stress distribution in the midplane for
four different types of composite laminates (glass/epoxy and graphite/epoxy lami-
nates). First, glass/epoxy laminates are considered. For type 1 and 3 composite lami-
nates, a kind of compressive stress concentration occurs near the crack location. This
is due to a very strong tensile axial stress concentration at the crack tip. So, the com-
pressive transverse stress concentration for type 3 composite laminate is stronger

than that for type 1 composite laminate.

24
It is observed that near the crack location, the transverse stress is maximum

compressive, and it goes to zero, changes to tensile, and reaches maximum tensile, in
the vicinity of the crack location, and then it goes to zero. However, the maximum
tensile transverse stress is not critical. For type 2 composite laminate, it is also ob-
served that near the lower crack location, the transverse stress is maximum compres-
sive, and it goes to zero, changes to tensile, and reaches the maximum tensile within
the vicinity of the lower crack location, and continues to the neighborhood of the up-
per crack location, and goes to zero, and then becomes compressive, and reaches the
maximum compressive. The transverse stress distribution is almost symmetric about
the midpoint between two interacting cracks. For type 4 composite laminate, it is ob-
served that near the lower crack location, the transverse stress is the maximum com-
pressive, and it goes to zero, changes to tension , and then reaches the maximum ten-
sile at the midpoint between two interacting cracks. This is due to the fact that the
maximum compressive transverse stress occurs in the neighborhood of the crack tip,
rather than at the midplane, as shown in Figure 3.35. So, the compressive transverse
stress concentration for type 4 composite laminate at the midplane is not as strong as
that for type 3 composite laminate.

Similar results as for glass/epoxy laminates are obtained for graphite/epoxy lam-
inates. The distinctive difference is that there is hardly any tensile transverse stress

region in graphite/epoxy laminates.

 

 

UUUUUUU

con-flucoccod

 

-CIOOCCOCOC...‘

 

 

 

 

 

20

15

IL

-L

 

 

5.4

 

X

Figure 3.1 Stress component oy at the crack plane in 0° ply.

[0°/90°]s glass/epoxy composite laminate (Type 1).

 

 

 

coo-ccdcoouooq

 

 

 

 

 

 

15

‘—

I'—

i—
1 .50
X

T—

T
.2

.-.J

 

—T

 

O

f
1 .75

2.00

1

1.00

[O°/9O°]s graphite/epoxy composite laminate (Type 1).

5
Figure 3.2 Stress component oy at the crack plane in 0° ply.

 

--.q

.......

 

 

 

 

 

 

 

U

U

l ' r i
4.0 5.0 0.0

I

I

1.0 2.0 3.0

I

I

 

 

0.0

 

Y

Figure 3.3 Stress component 0", at the interface in 0° ply.

 

 

rooopoood

 

 

 

 

 

 

 

 

[0°l90°]s glass/epoxy composite laminate (Type 1).

acoo‘ocnnboooq

truth

""""

IIIL
V0-0;

00-1

-1

v-0-L

 

I I

r 7 T I

T

I

1.0 2.0 3.0 4.0 5.0 8.0

r' T

 

 

 

Y

Figure 3.4 Stress component try at the interface in 0° ply.

[0°/90°]s graphite/epoxy composite laminate (Type 1).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

20 -
1’. 1
...... ;I___§I_, ......
. 5 x :
b8 i i ;
>2 ‘0 ''''' 0""n-=t—.-'J """ i """
"‘ Z l i
1“ * s s s
L """" : ''''' i J
o . i . ,L . 1 .
0.0 1.0 2.0 3.0 4.0
Y

Figure 3.5 Stress component oy at the interface in 0° ply.
[0°/90°]s glass/epoxy composite laminate (Type 2).

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

15 .
. 1’. 5
5.1.1- s
8 10-1» ------ § § " ----- J ......
b : E '
> . . : l
>.‘ r 1
=2. = =
e ------ f- ----- J ------ a
i— fi f '7
2.0 3.0 4.0
Y

Figure 3.6 Suess component 0’, at the interface in 0° ply.
[0°/90°]s graphite/epoxy composite laminate (Type 2).

 

baa-q

‘-"---

 

ocod

 

 

 

eeeeeeeeeee

1]....

 

 

  
 

-..-fi

.-..‘....

I I I I
I I 1 I I I I I

1.0 2.0 3.0 4.0 5.0 6.0

 

0"--v

I
I

fl

 

 

0

0.0

Figure 3.7 Stress component 0’), at the interface in 0° ply.

[0°/90"3]s glass/epoxy composite laminate (Type 3).

 

.1

UUUUUUU

I
I
I
0
I
I
'cuoorouc1
I
I
I
I
I
0.0-.0-..

 

 

 

...........

 

 

 

1

I I I
I i I
I 1 I 1 I I

I

 

0

0

0

3

8

0
---‘----

I

0

0

I

0
ococ'coco

 
 

 

1.0 2.0 3.0 4.0 5.0 6.0

I

--l

I

 

 

 

0
0.0

[0°/90°3]s graphite/epoxy composite laminate (Type 3).

Figure 3.8 Stress component oy at the interface in 0° ply.

 

 

 

 

 

 

 

 

:50 ...... J” _ ......
: I fix :
68 i :
f\\ .— I
>.‘ 20 ...... I ......... : """" I
n I
"S - I
° :

 

 

L

 

.4
.J

 

 

d
o
. . !

.—1-->----

o
o
o
N

‘<o
:"
o

3.0

Figure 3.9 Stress component 0), at the interface in 0° ply.
[0°l90°3]s glass/epoxy composite laminate (Type 4).

 

 

 

 

 

 

 

 

 

 

   

 

 

:55— .
l" - :
30 ...... I—é ...-.: ......
8 25 ...... : i :x ..... i ......
b :
\ 20 """ ."—_'. """ I """
$3 . . :
9‘15 ------ ' ------- 5 ------------- .
e .
1O ------------- : ------- , ----- 4
5 :-. ; -..-J
o . 4'— . % . f .
oo 10 2.0 3.0 4.0
Y

Figure 3.10 Stress component 0), at the interface in 0° ply.
[0°/9O°3]s graphite/epoxy composite laminate (Type 4).

 

I T T j

 

LIALLI

 

 

 

 

 

l

 

 

 

T—

T
2.0

r

 

1 r I 1*
are 410 5.0 6.0
y

Figure 3.11 Stress components (I" and ox), at the interface in 0° ply.
[0°l90"]s glass/epoxy composite laminate (Type 1).

 

 

 

 

 

 

 

 

 

 

 

 

‘05 I
I
i
’3
5
k
'I
b
k
>:
‘2
b
-005 1 I I 1 I I I I I I I
0.0 1.0 2.0 3.0 4.0 5.0 6.0

Y

Figure 3.12 Stress components (71‘ and ox), at the interface in 0° ply.
[0°/90°]s graphite/epoxy composite laminate (Type 1).

 

 

 

 

 

 

 

 

 

a.(1.y)/a..-a.,(1.y)/o..

 

 

Figure 3.13 Stress components ox and ox), at the interface in 0° ply.
[0"/9O°]s glass/epoxy composite laminate (Type 2).

 

 

 

 

 

 

 

 

 

 

 

 

 

1.5 tr‘,‘==
I .
I .___3
b 3 40+
\ 1.0- g :
’8 . 4+.
. 0.5 i =
i
A
:9 0.0 I _______________
‘3? 0
b i xy
-05 tfi r fiT
00 1.0 20

 

Figure 3.14 Stress components (I" and ox), at the interface in 0° ply.
[0°l90°]s graphite/epoxy composite laminate (Type 2).

 

 

12.5

1

10.0

 

lJJ

l

 

 

 

 

IJLI

 

Ux(3.Y)/O’..' xy(3IY)/aeo

 

 

-2.5

 

010 1I0 210 310 4.r0 530 3.0
Y
Figure 3.15 Stress components a" and on at the interface in 0° ply.
[0°l90°3]s glass/epoxy composite laminate (Type 3).

 

 

Li;

 

 

 

 

 

lLJ

J

l

0x(3.y)/0.'0xy(3.y)/0.

 

 

-1

 

I I l 1 I I I I r I j I
0.0 1.0 2.0 3.0 4.0 5.0 6.0
Y

Figure 3.16 Stress components <3" and oxy at the interface in 0° ply.
[0°/90°3]s graphite/epoxy composite laminate (Type 3).

33

 

12.5 *

 

10.0

 

 

7.5

 

 

 

 

 

0x(3.y)/a..-oxy(3.y)/0.

 

 

2.5
0.0 l‘ --- , 9.. -------
0.0 1.0

 

Figure 3.17 Stress components (3x and ox), at the interface in 0° ply.
[0°/90°3]s glass/epoxy composite laminate (Type 4).

5.0

 

 

 

 

2.5- i s "

 

 

“Q

 

 

 

.0
O
l
D

 

vx<a.y)/a..-a,.y<a.y>/o..

 

 

-205 i— F F I
0.0 1 .0 2.0

Y

Figure 3.18 Stress components 0x and ox), at the interface in 0° ply.
[0°/90°3]s graphite/epoxy composite laminate (Type 4).

 

 

----+

 

 

J

 

"'1

 

  

 

 

 

 

 

----d

0-0-3--..

 

 

 

 

aw . m"
6 Va
. NH
0 m
is wm
r .m.
.rmw a
4 mm
r S
.ranmm.
..a m“ w
0 av.
.. mw
I0 mm
II a
. mwfly
0 wk.
00 “W
mm
m
3
H

 

 

 

 

 

 

 

 

 

 

 

 

 

0.
o o g 6
. . . I
.
M . n ..... raw
. o u 5
. . .
c c c I
‘ a IIIIIII II o
u. 4
O '
ICHIIIIJIOO
. 1v
.
O '
"IIII "IIIILIIOO
. a 2
o c I
. .
.I c o
u 0-- O 00000 l1.
. .
. . I
n u D
u - u
1w J 0.0
o o o
8 o A
b\9 5 0

Y

Figure 3.20 Stress component 0’, at the interface in 90° ply.

[0°/90°]s graphite/epoxy composite laminate (Type 1).

 

 

 

 

 

y I
J t :
:-'"-': '
. I
:— . _. n---“ ......
: f X '
i 2 i
’ I

 

 

 

 

 

 

 

 

 

 

 

o I f I— i I i— W
0.0 1.0 2.0 3.0 4.0
Y

Figure 3.21 Stress component 0), at the interface in 90° ply.
[0°/90"]s glass/epoxy composite laminate (Type 2).

 

 

‘<

 

 

 

 

 

 

 

 

   

 

 

I
I
h
I I I
d I I I
I I I
O o I rL I f T f I

 

0.0 1 .0 2.0 3.0 4.0

Figure 3.22 Stress component a, at the interface in 90° ply.
[0°/90°]s graphite/epoxy composite laminate (Type 2).

 

 
 

 

 

 

 

 

 

‘ I Id. I
. . . .
. c . . I
. . . .
. .- L IIIII fIIIIiI
. . .
. . .
. . . I
. o .
I no IIIII 'IIIJII
. .
. . I
. .
. .
III. IIIII .IIIIIAT
. .
. .
. .
L IIIII I
. .
. .
. .
. .
I. IIIII .-
. .
. .
.

 

1.0 2.0 3.0 4.0 5.0
Y

6.0

 

 

.l

80>».nvxb

 

0.0

 

 

 

 

 

        

 

 

\).

3
.m . . 0
1W” “ . H 8
o w . n n I
m ....... .. ...... . ....... . ...... - o.

. . . 5
.m n u n r
we ....... .. ...... u ...... ... ...... no.
.ni . 4
ms .

o . ..
.mnmu. . o
c I.
mm .3
MW 0
gym. . In‘o
ole . I
mw ” 0
mm. . u. ...... . ............. -...
m s u u .
0] I I n
c 3 q d — d d J m d m
mm 0 3 0. 5 0

2 1 1 Q 0
SW. I x
.
fl b\A>nvb
3
m
H

Y

Figure 3.24 Stress component <3y at the interface in 90° ply.

[0°/90°3]s graphite/epoxy composite laminate (Type 3).

37

 

 

    
 

 

 

 

 

 

 

 

 

 

 

 

 

v»
. 0. m
m u . 4 ma
. u u _. m
u o . W
IIIIIII .IIIII . . O .
u. -u ...... .. ....... ..3 mm
H u u . mm
.. u n .m .w
T 2 I” IIIIIII u IIIIII . IowVI.m m.
:KIULI._ ......... m " m m
., . u .. .... ....
I ......... n IIIIII I“ IIIIIII I0. Gym.
" u n .. m k
u n u . m a
n . . mm.
_ . _ 0. wk.
0. 5 o. 5 00 u 3
w 7 5 z 0. mm
x s mu.
.5333 .5
2”
n

 

rcucufioccou-

 

 

 

 

 

 

 

 

 

 

 

 

d

2.5
2.0-m----

 

 

4.0

Figure 3.26 Stress component 0’ at the interface in 90° ply.

[0°/9O°3]s graphite/epoxy composite laminate (Type 4).

 

 

 

 

 

 

38

 

 

 

 

 

 

 

 

 

 

 

 

o o m
.. 6 w... m u 6. ..v.... a
. n w D.“ n r 60.. m
. ......... l ......... 5 w m .......... ........ .... m .m
. 5 . 5
. I .m . . r .mlm
_ . u o C n o C m
i IIIIIIII v IIIIIIII 3 IIIIIIII LI..." .m m IIIIIIIII 4 IIIIIIII I IIIIIIII I4" Md”
r ”min m m.
. an m m. 0.Y m m
W3 m m 3 ..m. w
fin t w t y
IIIIIIII O. a v.. 0 a M
fi . . 2 OXX . 1 OX up».
u . r t o a I t I,
. . n w _ . n .m
1 IIIIIII A. .u IIIIIIII 41-10.” m W t IIIIII .fi IIIIIIII .u IIIIIIIII Io. m .m.
u I mew ' u I w. my
I w 4 w 1 AN ml“. 4 ”[1 0. ML.»
5 0. e 0.0 a w M M 6 00 a W
o 0 1| 0 1|
0 . . «W. W _ . «W... W.
£2.83. N. 363.83. a
3 3
m. m
0‘ 0%
F F

 

 

 

 

 

 

 

 

 

 

 

--I

-------d

 

 

 
 
 

 

 

 

 

 

 

 

 

 

g . """" 1 """ ‘
k i i
>. 9 ;
8 :
X I
b -..-.I
I. i
I
:
I
l

 

0.0 1.0 2.0 3.0 20

Y
Figure 3.29 Stress component ox at the midplane in 90° ply.

[0°l90°]s glass/epoxy composite laminate (Type 2).

 

     

 

 

 

 

 

 

 

 

 

 

°'5 : T I
. a s s

0.0-{P - I I I ‘
be i i i
\ I I I
A I I I
>.~ 4 .
9.. l w’. s

e -0.5-4 ------- 7": - - - - .. ------ I
s-D: :
. é"“:‘ a

-100 f z I I f-: I % i
0.0 1.0 2.0 3.0 4.0
Y

Figure 3.30 Stress component <3x at the midplane in 90° ply.
[0°/90°]s graphite/epoxy composite laminate (Type 2).

_”

 

     

 

 

 

 

 

 
  

 

 

 

1 1 O. o
. . 6 1y
. . p
o
. ............. . ....... n ...... 0. w
o o 5
o
. n .m
o o
IIIIIIIIIIII u :x “U. .m
. p
. ..................... . m
1 IIIIIIIIIIII u .1” V) m
”y u . m
IIII I III..- . . ........... 0. fl
. . 2 OX
.
. u r m
IIIIIIIIIIII . . o C
I. ccccccc . IIIIIIII . n
. o p 1 m
H . r m
. .T . 0 o
_ d d .1410. c
3 o. 3 o s a
0 0 0 L L m
. _ . S
8 1
{2.83.6 a
3
H

[0°l90°3]s glass/epoxy composite laminate (Type 3).

 

 

 

 

 

 

 

 

 

 

 

D
. . . . . 8
. o . o o I
. . . .
O
........ .----u----+---u.---rs
. . . . r
- o o c .
vIIIIuIIIIHIIII. . . .Ioo
_ o :X 4
. . I
_ u ......... : .......... o
. :II‘W
VIII? r
I ........... o.
. . 2
n u j
..----T--...m
. .
. . T
u c o.
o
.5 0. 5
1 2 2
. . _

Y

Figure 3.32 Stress component (Ix at the midplane in 90° ply.

[0°/90°3]s graphite/epoxy composite laminate (Type 3).

41

 

 

 

 

 

0.5 I j .
a 0.0— ------------ : ...... ' .......
b 1 I
f\\ l I
>.~ « r :
O 7 I
Ԥ - :
e -o.5- ------- —-; ----« ----- «
q 5 2 ix ;
.1.0 T i 'I g f 1
0.0 1.0 2.0 3.0
Y

 

 

 

 

 

  

 

 

4.0

Figure 3.33 Stress component <3x at the midplane in 90° ply.
[0°/90°3]s glass/epoxy composite laminate (Type 4).

 

 

 

 

 

 

 

 

 

 

 

 
 

 

 

0.5 . . I
1 : :
0.0-4- ' ------ ' ----- :
J ; .
> . :
>.~ + ' :
o 17 I
‘1‘ l = '

b -o.5-4 ------- '—: -----1 ------
13’ § :
I A" 5
-1.0 ' P L
I U 1 r r T
0.0 1.0 2.0 3.0 4.0

Y

Figure 3.34 Stress component (rx at the midplane in 90° ply.
[O°/9O°3]s graphite/epoxy composite laminate (Type 4).

42

 

 

  
 

 

 

 

 

 

 

 

 

 

0.0 . I
-o.5- --------- % ------ 4 --------
.3 4 5 I
’\‘ . I
o. -1.0dbooooooao: ........ I: .......
35 J 11’ . i
X ___' I
b 2 :
-1 5d ......... J ........ q
:_:D;._..x:
I s : :
-2.o . ' f r
0.0 1.0 2.0 3.0
X

Figure 3.35 Stress component <3x at the crack location in 90° ply.
[0°l90°3]s glass/epoxy composite laminate (Type 4).

 

 

 

 

 

 

 

 

 

 

 

0.25 r .r
4 l l
0.00- --------- E -------- E ------- -
8 .4 i i
b - ... ........ ' ....... ' ......
\ 0.25 . i - -
A
Q “l ' i
x r I
I? -o.5o-j ----- H E -------
...o,75.. ..... z-g—g‘ .-.-; .......
-1.00-+ ‘ t l , 1
0.0 1.0 2.0 3.0

Figure 3.36 Stress component a" at the crack location in 90° ply.
[0°l90°3]s graphite/epoxy composite laminate (Type 4).

CHAPTER 4

COMPARISON OF THE FINITE ELEMENT ANALYSIS RESULTS
WITH THE ANALYTICAL RESULTS

4.1 Analysis of a cracked symmetric cross-ply laminate under uniform tension

For the analytical work, the variational principle of minimum complementary poten-

tial energy is usedlzs]. Consider a symmetric cross-ply laminate which is subjected to
uniform in-plane tensile loads, as shown in Figures 2.1 and 2.2. Hashin’s primary

stress analysis does not include stress concentration effect at the crack tip, especially
in the 0° ply. The finite element analysis shows a strong axial stress concentration at
the crack tip in the 0° ply. It is also noted that the axial stress concentration at the crack

tip in the 90° ply is not so strong as that in the 0° ply. Therefore, based on the finite ele-

ment analysis, it is necessary to obtain more refined stress distributions including the

stress concentration effects at the crack tips in the 0° ply, for cross-ply laminates dam-

aged by transverse matrix cracking.

 

 

 

r T i

a a

Figure 4.1 [0°l90°n]s laminate between two cracks.

 

 

 

43

44
Figure 4.1 shows a representative element between any two consecutive

cracks. Let the 90° ply be ply 1 with thickness t1, and let the 0° ply be ply 2 with

thickness t2, as shown in Figure 4.1. For all cases, the thickness of a single lamina is

chosen as t = 0.203 mm.

This problem can be solved in two steps: (1) When there are no intralaminar

cracks in 90° ply, the ply stresses are determined by the classical lamination theory.
The stress component 6xc(m) (m = ply index 1,2) in each ply is constant through the
respective plies; (2) When there are cracks, the cracks introduce stress perturbations

which are denoted by Oijpfln). The superposition of the two solutions will then pro-

vide the stress field in the laminate with intralaminar cracks in the 90° ply as given by
oijon) = Gijcon) + cijp(m), (4.1)
where i, j are restricted to x, y.
Now, the stress component ch(m) (m = ply index 1,2) in each ply may be ex-
pressed in the form

O.xc(1) = 0.1 and O.XCCZ) = 02. (4.2a)

Furthermore, in order to simplify the problem, perturbation stress in the 90° ply,
6xp(1), is assumed to be constant through the ply thickness. Hence, the stress may
be expressed as

exp“) = —01 (DICK), (4.2b)

where ¢1(x) is an unknown function. In order to obtain more refined stress distribu-

tions including stress concentration effects in the 0° ply, it is necessary that

45

perturbation stress in the 0° ply, oxpa), should be functions of x and y. This stress
may be chosen as,

pr<2> =—oz(h-y>“¢2<x). (4.2c)

where h = t1 4» t2 , and ¢2(x) is an unknown function. The choice of the perturbation

stress in the 0° ply, pr(2), is motivated by the knowledge of the series expansion of
the crack tip stresses. The equilibrium equations for the plies are

acxpm‘) /ax + aoxyPO“) / ay =0, and (4.3a)
P(m) P(m) =
aoxy /ax + Boy /ay 0. (4.3b)

The stresses oil-Con) disappear since they are constant through respective plies. The

form of the perturbation stress components for each ply of the cracked laminate will be
chosen as a product of a function of the transverse variable y multiplied by a function

of x and y (in some cases, only x) and are given as

0,190“) = t‘m) "<y) Q(m)(x). (4.4a)
oxypm) = -f(m) '(y) R(m)(x), and (4.4b)
cymm) = f(m)(y)S(m)(x), (4.40)

where prime designates differentiation with respect to the variable y[24]. The function

f(m)(y) describes the stress distribution through a given ply of the cracked laminate.
The equilibrium equations (4.3a,b) yield the following equations involving the
variable x only:

R(m)(x) = 8Q(m)(x) / ax, and (4.5a)

S(m)(x) = aR(m)(x) / ax. (4.5b)
The equations (4.2b,c) and (4.4a) yield the relations

46
f“) "<y) = —ol, #2) "(y> = «rah-y)“. (4.6a)

thx) = we. and Q(2)(X) = cm). (4.61»
It is noted that stress variations through the ply thickness depend on the function

f(m)(y). It is easily seen that this problem is not well-posed, and there is no unique
solution to this problem. One of the main purposes of this work is to find out a more
physically reasonable solution using the finite element analysis.

Now, the perturbation stresses take the forms

oxyP“): [Gly+a1(x)]¢1’(x), (47a)
1 II
of“)=—[—2—oly2+al(x>y+a2(x>]¢1 (x). (47b)
0 Pa): [0 L (h- )“+1+a (x)]¢ ’(x) and (4 8a)
xy 2M1 Y 3 2 . -
1 , 1 n
oypm = - [02 —— (h-y)n+2 + a3(x)y + a4(x) m (x). (4.8m

n+1 n+2

where al(x), a2(x), a3(x) and a4(x) are unknown functions.
The applied load per unit specimen width, N, is simply given by
N = 2( ‘5me t1 + 6x00) t2)
= 2( olt1 + <52 t2). (4.9)

Therefore, equilibrium condition in x direction for perturbation stresses is obtained

as

h
2
t1

1
= 2(— 014510.) t1 - Wazoo; If”) =0. (4.10)

From equation (4.10), the following relation between ¢1(x) and (1)1(x) is given as

47

()—-(+1)31——tl— <) (411)
¢2x— n 62:2 ¢1X- .

n+1

The origin of the system of coordinates is set up at the center of the distance 2a

between any two typical cracks, as shown in Figure 4.1. The perturbation stresses in

the region lxl S a, |y| S h must satisfy the following interface and boundary condi-
tions:

1. from symmetry condition,

oxyP(1)(x,0) = 0, (4.12a)
2. from continuity condition,

Oxyp(1)(x,t1) = Oxyp(2)(x,t1), (4.12b)

0yp(1)(x,tl) = Gyp(2)(x,t1), (4.120)
3. from traction free condition at the free surfaces,

oxyp(2)(x,h) = o, (4.12d)

oyp(2)(x,h) = 0, (4.12e)
4. from traction free condition at the crack surfaces,

cxP(1)( i a,y) = —01, and lyl 3 t1 (4.120

oxyP(1)( i a,y) = 0. lyl 3 t1 (4.12g)

Let ¢1(x) be ¢(x), then using equations (4.3)-(4.6) and (4.12), the perturba-

tion stresses can be written in the forms

exp“) = —O’1¢(x), (4.13a)

oxme = my) '00, (4.1313)
pa): __1_2 3.12 _1__ 2 "

0'), 61( 2 y + n+2 + 2 t1)¢ (x), (4.13c)

48

t

0x139) = 61(n+1) 711:1 (h—y)“¢(x), (4.13d)
2

oxypa) = Gltl( :1 )MI (1) ’(x), and (4.13e)
t2

1 h- n+2 ”
o p(2)=0'1t1— ( y) 4) (x). (4.131)
y n+2 t2n+1

 

From equations (4.13a,b), the crack surface boundary conditions (4.12f,g) as-
sume the form
¢(: a): 1,and ¢'(i a)=0. Iyl 311 (4.14)

From equations (4.13d,e), it is seen that

n+1

expo-Rm,” = 01(n+1)t-2t-1— (h-y)“, and t1 5 lyl Sh (4.1521)

oxyp(2)(i a,y) = 0. t1 s Iyl s h (4.15b)

From equations (4.14)-(4.15), it is found that equation (4.14) is valid for the entire

thickness of the cracked laminates. Thus,

¢(: a)= l,and ¢'(:a)=o. 0 s |y| Sh (4.16)
A consistent approach of developing mathematical theories of plates is to use the

principle of variational calculus, furthermore, it is convenient to use the theorem of
minimum complementary potential energym]. Suppose a body B which has volume
V and boundary A, is in equilibrium under the action of surface forces Ti assigned over

a part A1 of the surface A, and on the remaining part A2 of A the displacements “i

are assumed to be known.

If the Gij are the stress components of the equilibrium state, then

49

aoij / an = 0 in B, (4.173)
Gijnj = Ti on A1, and (4.171))
ui = fi on A2. (4.17c)
Now, introduce a set of functions 0 ’ij such that
oij = Oij + Soij, (4.17d)
with the properties
as 'ij /axj = 0, in B, (4.17c)
o 'ijnj = Ti, on A1, and (4.171)
0 are arbitrary on A2. (4.17g)

1.]

It follows from these equations that the variations Soij satisfy the conditions

(scij)nj = O, and on A1, and (4.171)
Soij are arbitrary on A2. (4.17j)

It may be observed that the Oij are associated with the equilibrium state of the

body and hence they satisfy the Beltrami-Michell compatibility equations, but we do

not assume that Soij satisfy any such conditions. We define the complementary ener-

gy U by the relationship,

UC =J WdV—J TiuidA, (4.18)
V A2

where W is the strain energy density function. The theorem of minimum of comple-

mentary energy states that the complementary energy UC has an absolute minimum

50

when the stress tensor Oij is that of the equilibrium state and the varied states of

stress fulfill the equations (4.17h,i,j).

Suppose that the body B is an elastic material without cracks, there is another
elastic body B which has same shape A and same volume V with the body B, and

both bodies, B and B’, have space variable compliances Sijkl and are subjected to the

same external mixed boundary conditions. In the body B, the complementary energy

will be

1
U 1=—I su onco CdV—I T-Cu-dA. (4-19)
c 2 V ljkl 1] kl A2 1 1

In the cracked body B’, if we let oijp be perturbation stresses due to cracks,
l
2 _ __ a a _ a (4.20
v A2
where
a _ c a _ c (4.21)
Cij — Cij + Oijp, and Ti — TI + Tip.

It is useful to record a theorem due to Hashinns] which is given as
2_
Uc= _.[vsijk1°ijp°klpdv° (4.22)
where,
U1-—1-IS oco CdV (423
c‘zvijklijkl' -)

Next, we consider the cracked laminate region in Figure 4.1. Because of symme-

try, it is sufficient to take the region -a S x S a, 0 S y S h with unit thickness in z

direction. Further symmetry consideration shows

¢(X) = ¢(-X)- (4.24)

51
It is necessary to calculate the strain energy density W of a transversely isotropic

composite materials for the calculation of complementary energy,

2 2 2
o v v o o
11 2] 12 13
2W=o--e--=— - o o — o o +— +—
1111 E1 E2 11 22 E2 11 33 G12 G12
2 2
V 022 V 023
0 0 +— - Jo 0' +—
E2 11 22 E2 E3 22 33 G23
v v 0332
— o 0 ——120 o +—
52 11 33 E3 22 33 1.32

2 2 2
011 022 +633 _ 2011(022 + 033)V12

 

 

 

=— +
E1 E2 E1
2V 0232 0122 + 0132
G G __ + , (4.25)
Ez 22 33 G23 G12

I
where l is in fiber direction and 2,3 are transverse directions. UC 2, the second term

in right hand side in equation (4.22) for this region, is given by

’ 3. I1 21 h
UC 2 = 2! I Wldy dx + 2 I J Wzdy dx, (4.26a)
-a 0 -a II

 

where,
(1)2 (1) <1) 12 12
2W - i _ 20x1) Gyp VT ... £12 + .(LXPLU d (4 26b
1- Br ET Er GT ’an . )
13(2) 2 9(2) 19(2) 13(2) 2 DO)2
6 20 o v o 6
2w: Ji— — x Y A + —Y—— +—fl-— . (4.26c)

 

EA EA ET GA
Inserting equation (4.13) into equations (4.26b,c), integration of equation (4.26a)

with respect to y and introduction the nondimensional variable
1] = — , (4.27)

yields

52

, P (12¢ (12¢ 2
Uc 2:014 [ (31¢2+ C2¢(W)+ C3(d—‘2)

 

 

d4) 2
+C4(——-dn ) ldn. (4.28)
where
a
p = -—1', (4.293)
C t12 1 t3 (n+1)2 (4 29b)
=—"'" + 2 °
1 Br EA t2 2n+1
2v t t t 2v t1t(n+1)
II 1 2
= —-—-— 4.29
2 Br 3 + +2) EA —A—2{(2n+3)(n+2) } ( C)
1 2 2t1t2 t2 2
c =— — 2 +—— + —
3 ET { 15 t1 3(n+2) n+2) }
1 12 1
+ —— —— —- , d 4.29d
E1- t {(2n+5)(n+2)2 } an ( )
2
t t t
C4= -‘- ( 12) (4.292)

 

3GT G A 2n+3

Equation (4.28) yields the Euler equation which minimizes equation (4.28) as

(12¢ d d2 (12¢
{2c ¢+c2(%7)}- dn _(2C4—d_1:|—)+—d—TTZ{C2¢+2C3(—d-TP)}

 

 

 

 

 

-C
=21—.+<°2(;‘;34) 224‘; 13:)11-0
thus,
4 2
d 4) c2C -BC4 d¢ c1
(1114 +( )d dnz +( C3 )(1) = 0. (4.30)

The solutions to equation (4.30) are of the form

53
(I) = eiancoan, eiansian, (4.31)
where
_1_ _1_

__C1_4 .1. __C1_4-;
-( C3) cos 2 e, 13_( C3) sm 2 e, and (4.322)

 

2
'C
[ane = \/4(_(é_13_) /(_9_C_3.4_) — 1, (4.3211)

provrded that 4(—%— 13) )(Czc——3—- C4 )2 and(——— C4 )(0.

By using the symmetry condition (4.24) and the boundary condition (4.16), the solu-
tion (1) can be expressed in the form

4) = Alcoshomcoan + Azsinhomsian, (4.33)
where

2(acoshapsian + BsinhapcosBp)
Ar asinZBp + BsinhZap ’ and (43421)

 

2 cosha sin - asinha cos
A2: ([3 p 59 . p BP) . (4.34b)
asm2Bp + Bsrnh2ap

 

In order to determine the stresses (4.13), it is necessary to calculate

. 1 d (D
(p (x) = — — = —[ (aAl + BA2 )sinhomcosfln
11 d n
+(onA2 - BA1)coshansian],and (4.35a)
2
., 1 <1) 1 2 2
q) (x) = 7m = 't—2[{(Ot —B )A1+ 2(XBA2}COShOtT]COSBT|
1 1

+ {(012 = (32);).2 - 20113141 }sinhansin13n]. (4.351»

Inserting equations (4.33)-(4.35) into equations (4.13), we can obtain all

54
perturbation stresses. Thus, the total stresses will be written as

ox“) = 01+ 0X13“), (4.36a)
oxy“) = oxyP“), (4.36b)
Gym = prfl)’ (4.360)
0x9) = 02+ pr(2), (4.36d)
oxya) = nyPQ)’ and (4.36e)
oya) = Gym). (4.360

Now, it is interesting to consider a case when the cracks are far apart. Figure

4.2 shows a representative element when the cracks are remote from each other.

 

 

 

 

 

3’
0° t2 h
N <— 90° 2‘1 —’ N
00 t; h
Figure 4.2 [O°/9O°n]S laminate with an isolated crack.
In this case, the boundary conditions will be expressed in the form
¢(O)=1,and ¢’(0)=0. OSIyISh (4.37)
The perturbation stresses will be written as
- a o
(I) = e Om(coan + — smBn ), (4.38a)

B

’ _ _1_
(D (X) 11
u 1
¢ (X)= ":2
1

Inserting equations (4.38) into equations (4.13), we have the total stresses from

equations (4.36).

Prior to the calculation of perturbation stress components, it

calculate 01 and 0‘2 by employing the classical lamination theory. Table 4.1 shows

the ratio 01/02 and 01/ o, (o = N / 2h), for [O°/90°]s and [0°/9O°3]s composite lam—

55

 

 

d9 1
dn _— II B
2
d ¢ 1 ((12+Bz)
dnI — 112 c

2 2
(01 +5 )e'omsinfin, and

'm‘msinfln — Bcoan).

(4.38b)

(4.38c)

inates with the material properties of glass/epoxy and graphite/epoxy in Table 2.2.

 

Table 4.1 Ratio 01/02 and <51 / oby the classical lamination theory.

 

 

 

 

 

 

 

 

 

 

Glass/epoxy Graphite/epoxy
[00/ 900]s [00/903]S [0° /90°]s [0°/ 9093]S

0.05720 0.03919 0.01693 0.01491
61 /62 0.18911 0.12874 0.22938 0.20157

=0.3025 =0.3044 = 0.0738 = 0.074

0.05720 0.03919 0.01693 0.01491
01/0 0.12316 0.06158 0.12316 0.06158

=O.4645 =0.6364 = 0.1375 = 0.2421

 

 

is necessary to

 

56
4.2 Analytical results and comparison with the finite element analysis re-

sults

The stress distributions from the principle of variational calculus using the theo-
rem of minimum complementary potential energy, are presented in this section. Axial,
shear, transverse stress distributions at the interface in 0° ply, and axial stress distri-
butions at the interface and axial, transverse stress distributions at the midplane in
90° ply will be mainly discussed and compared to the results from the finite element
analysis.

Prior to the discussion of stress distributions, it is useful to investigate variation

of the complementary energy (UC 2) in equation (4.26a), with various power n in

equations (4.13 d,e,f). Table 4.2 shows the variation of the complementary energy (U c

I I
2). It is observed that as the power 11 increases, the complementary energy (U c 2)

increases. This phenomenon can be predicted easily, since as the power 11 in equa—
tions (4.13 d,e,f) increases, the perturbation stress components become more intensi-

fled.

 

Table 4.2 Complementary energy (Uc ’2) variation with
various power 11 in equations (4.13 d,e,f).

 

 

 

 

 

 

Power 11 Complementary energy
11 = 0 1.0
n = 5 1.072
n = 10 1.255
n = 15 1.44
n = 20 1.619

 

 

 

 

 

57
For the purpose of comparison between the finite element analysis results and

the variational calculus results, [0°/90°]s glass/epoxy composite laminate with inter-
acting cracks and [O°/90°]s glass/epoxy composite laminate with an isolated crack
were investigated. (Note: x, y coordinates in the following figures are set up different-
ly from those in Figures 4.1-4.2. The new x, y coordinates are shown in the following
figures.)

Figures 4.3-4.8 show the variational calculus results and the finite element anal-

ysis results for [O°/90°]s glass/epoxy composite laminate with interacting cracks

(Type 2). In this case, the dimensions of the composite laminate in Figure 4.1 were

taken as: a = h; t1 : t2 = 1 : 1. Figures 4.9-4.14 show the variational calculus results

and the finite element analysis results for [O°/9O°]s glass/epoxy composite laminate

with an isolated crack (Type 1). In this case, the dimensions of the composite lami-

nate in Figure 4.2 were taken as: length in the positive x direction = 6 t1; t1 : t2 = 1 : 1.
In Figures 4.3-4.14, the case n = 0 stands for Hashin’s primary results.

Figures 4.3 and 4.9 show the axial stress distributions at the interface in 0° ply.
It is observed that, as the power 11 in equation (4.20) increases, the axial stress con-
centration from variational calculus becomes stronger and gets closer to the value
from the finite element analysis, as shown in Figures 4.3 and 4.9. But, some discrep-
ancies can be observed between the variational calculus results and the finite element
analysis results.

Figures 4.4 and 4.10 show the shear stress distributions at the interface in 0°
ply. It is indicated that, as the power 11 in equation (4.20) increases, the maximum
compressive shear stress position moves closer to the crack locations, like the ten-
dency in the finite element analysis results, as shown in Figures 4.4 and 4.10. But,
there is a remarkable difference very near the crack locations between the variational

calculus results and the finite element analysis results. As we mentioned earlier, the

58
tensile shear stress in the finite element analysis can be explained from the fact that

a short crack transversing a few fibers in 0° ply can propagate in the vicinity of the
crack tips, even though the tensile shear stress changes into compressive shear

stress, rapidly. In order to explain the difference, one important investigation should

be mentioned, here. The function f(m)(y) in equations (4.4) (Note: it follows x, y coor-

dinates in Figures 4.1-4.2), must be reconstructed separately, in the bulk and surface

layer or interface region[24]. But, it is extremely difficult to set up the function f(m)(y)

in equations (4.4) (Note: it follows x, y coordinates in Figures 4.1-4.2) separately in

the present problem. So, for the convenience of calculation, the function f(m)(y)

(Note: it follows x, y coordinates in Figures 4.1-4.2) was not separated in the

present analysis. Thus, when the shortcoming in selecting the function fimhy)
(Note: it follows x, y coordinates in Figures 4.1-4.2) is taken into account, the finite
element analysis result has reasonable agreement with the variational calculus re-
sult.

Figures 4.5 and 4.11 show transverse axial stress distributions at the interface

in 0° ply. In Figures 4.5 and 4.11, it is noted that the transverse axial stress distribu-
tions from variational calculus have good agreement in the magnitude and the tenden—
cy with those from the finite element analysis, except for those very near the crack 10-
cations. However, as we mentioned before, the tensile transverse axial stress very
near the crack locations from the finite element analysis can be explained from the
point that the tensile transverse axial stress before changing into compressive
stress can break few fibers in the transverse direction.

Figures 4.6 and 4.12 show the axial stress distributions at the interface in 90°
ply. It is observed that there are large amounts of discrepancies between the finite el-
ement analysis results and the variational calculus results, as shown in Figures 4.6

and 4.12. However, as in Figures 4.7 and 4.13 which show the axial stress

59
distributions at the midplane in 90° ply, it is indicated that in the region of midplane

which belongs to the bulk area, the axial stress distribution from the finite element
analysis has very good agreement with Hashin’s primary result. And, it is noted that
the maximum value of axial stress at the midplane in 90° ply is at the midpoint be-
tween two interacting cracks. This phenomenon can be observed from both the finite
element analysis results and the variational calculus results. The maximum tensile
axial stress at the midplane is responsible for new crack generation. Therefore, a new
crack will be likely to occur midway between two existing cracks.

It would be useful to discuss several analytical models which have been pro—

posed to predict the multiple transverse cracking in cross-ply composite laminates. It
is noted that Bailey et al.[5]'[7] assumed that new cracks always occurred midway be-
tween any two consecutive cracks. Manders et al.[32] and Fukunaga et al.[33] used a
statistical model. Manders et al.[32] proposed a simple statistical model which accu-
rately fit the data and predicted a dependence of strength in 90° ply on size. It was
shown that a model incorporating a Weibull distribution of strength in 90° ply was a
good description of the crack spacings. Fukunaga et al.[33] investigated the failure
characteristics of cross-ply laminates based upon the statistical strength analysis.
The strength in 90° ply was assumed to obey a two-parameter Weibull cumulative
distribution function. It was assumed that a new crack occurred midway between two

existing cracks at 50% failure probability. Laws and Dvorakun supposed that a trans-
verse crack would propagate when it was energetically favorable and that the location
of this transverse crack was associated with a probability density function. Based on
simple statistical fracture mechanics they suggested a choice for the required proba-
bility density, namely that it was proportional to the stress in the transverse ply.

From the above discussion, it is observed that a statistical model is better for

prediction the multiple transverse cracking in cross-ply composite laminates.

60
However, a new crack will still be likely to occur midway between two existing cracks.

Figures 4.8 and 4.14 show transverse axial stress distributions at the mid-

plane in 90° ply. Hashin’s primary transverse axial stress distribution shows that
the maximum stress occurs at the midpoint between two interacting cracks, as shown
in Figure 4.8. But, as the power 11 in equation (4.2c) increases, the maximum trans-
verse axial stress positions are moving closer to each interacting crack. From the fi-
nite element analysis, it can be expected that each maximum stress may influence
the other, so the value between two maximum stress positions reaches the maximum
stress, too. Therefore, as in the result from the finite element analysis, the maximum
stress value may spread into wider region. And, in Figure 4.14, it is observed that as
the power 11 in equation (4.2c) increases, the maximum tensile transverse axial
stress position moves near to the crack location, like the tendency in stress distribu-
tion from the finite element analysis. In Figures 4.8 and 4.14, it is indicated that the
transverse axial stress distributions from the finite element analysis have more simi-
lar magnitude and tendency to those from the variational calculus, in the region of mid-

plane which belongs to the bulk area, than in the region of interface.

61

AN 255 035.5: axououma—w 6.7362 in 6c 5 38..an 05 an ab Egon—=8 mmobm mé 2:3".

 

 

 

 

 

 

 

 

 

 

x
.. .4. .N a o
p p b b 00°
lllll'l‘mltltlflli‘llllullllllrlxlr
------xvsx .lflnm .........
\\ \\ // // 10.0
\\\\\\ / / Ill
\ \ // I
\\ \ /// . (AD
\ / / fiv
\ / 92 ....
\ Ila / n 1A
5... 1 m . r/x
own: III M W . a
m. .... c 1!... .. m . ed.
2 u c 1......
m u c ....... a
o u = II...

 

 

62

AN 0&5 035:3— »xoaoxmmflw magi: SE 5 E 3.?an 05 3 axb “5:358 32% 1v 05E"—

 

 

 

 

 

 

 

 

 

 

 

>
Sun.

ON I

2. I

o. u

n 3 x0
«A
(
/
D

/nm.1.nV\\ .
100.0

 

 

63

AN on»: 085:5 >300?me m_ooo\60_ SE 60 E 3.0.—85 05 3 x0 2.28088 39:0 We 95$..—

P")

i—v-

 

 

 

ON
0..
0p

2m...

II
C

II
1:

II
1:

ll
1:

II
c

 

 

 

 

 

 

 

 

 

00.—l

00.0

100.—

100.N

100.0

100....

 

 

00.0

0 / (K‘ ‘1)xa

 

 

 

2w...
0N":

 

 

 

 

 

 

 

 

 

TQN

10.n

 

 

0.0

AN 003.0 03552 >33}.me ...—60262 in 600 5 85:85 05 an ab 595058 32.0 06 23E

.o/(Ii“1)'(o

65

.a 696 323 $3653» aroma. in .8 s 6532:. 65 a 4.6 28858 ”.85 5 2sz

 

 

 

 

 

 

 

 

 

 

 

 

.Am Rib 08552 .3253...» ...—6860. in 600 5 05—05:. 0... .a «b 52.2.58 3.2.0 0... 2:0."—

 

 

 

 

 

 

 

 

x
.. . .5 . m _ o
. p p . n.911
5 N41
5.
~ 0.91
. 0
t x
. W
1001 ....A
. (
. /
. 0
..nd-
W
10.0
a

 

 

 

 

 

 

67

.2 on»: 03555 zxonohmflw ”.6862 SE 60 5 85.8.5 2.. .a so E25058 82.0 a... 2:0."—

 

 

x
L P '1- 00°
2w... III ///././...I.
/fl0/ 10.0
I'llll /
n— fl 1111 /// MU
op nu llll / 10.0.. «A
(
....... /
m n .. D
0 "II
10.0—
f

 

 

 

 

 

 

 

 

 

0.0N

.2 on»... 08552 32.0)me ...—6860. SE e0 5 850.35 05 .a 0.6 Becca—E8 22.0 0.... 0.5»...—

 

 

 

 

 

 

 

 

 

 

 

 

 

x
n
... ......N...88.o-
\an ..
i ,, ..nwdl
1% .
..|.| 11111.... [Eon-I.“ 00.0
E: I... .mud
omuc III M W
n. .1. c 1111 - W 1 4
o. u c 1...... n. 4 .
W n C 1111111 filomo
0 n.. C III .2 w.

 

 

05.0

0/(K“1)"‘0

69

.2 0%.... 0.0580. 32.0333 0.60%.... ...... 60 5 80.025 0... ... xb 52.2.58 32.0 ...v 05%....

 

 

 

 

 

 

 

 

 

 

n. 1 c 1111 - 1 .

2 o. 1 c 111.. .0. n .80
n 1 c ....... .4. fl

0 1 c .84
.4

 

 

 

 

.o/(1(“1)xa

70

.2 62.... 2.552.246.6220 0.02.2 ......

.00 5 0053.5 0... .0 ab 82.2.58 22.0 a... 0.53".

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

71

.A ~ 2.55 03582 x5352» ....ocQoa in .8 E 05328 05 3 >0 50:38.8 “moan 2% 2:3"—

>
m n .v n
»

 

can: III

 

m. .. c ......i m

2.: A
....... x

 

 

 

 

 

 

 

 

 

 

'72

A _ 255 85:82 xxoaohmm—w ”Tomi: SE 56 E 0532:. 05 3 xb Egon—E8 may—um :6 Bani

 

 

 

 

 

 

 

 

 

x

m 0 ¢ n N F o
L _ L — b L . — p _ L noPl

ENE E w u r
om u c II... ~ «.7.

9 u c 1!... u .

Or a C .ll..|l h GAY!

n n... ....... .
o .l.. c mdl

 

 

 

 

.o/(fi‘o)xa

CHAPTER 5
CONCLUDING REMARKS

The finite element analysis has been presented to analyze the stress distribu-
tions in symmetric cross-ply laminates damaged by transverse matrix cracks. Four

types of problems for glass/epoxy and graphite/epoxy laminates were considered: (1)
[0°/90°]s laminate with an isolated crack; (2) [O°/9O°]S laminate with interacting
cracks; (3) [O°/9O°3]s laminate with an isolated crack; (4) [O°l90°3]S laminate with in-
teractin g cracks.

The finite element analysis yielded the following important results: (1) Strong

axial stress concentrations occurred at the crack tips in 0° ply. This phenomenon ex-
plains that the transverse matrix crack can be considered a likely site for the nucle-
ation of longitudinal cracks and/or delamination cracking. The axial stress concentra-

tion in glass/epoxy laminates is stronger than that in graphite/epoxy laminates. Axial

stress concentrations in [O°/90°3]s laminates with an isOlated crack or interacting
cracks become stronger than those in [0°/90°]s laminates with an isolated crack or in-

teracting cracks; (2) Axial stress concentrations occurred at the crack tips in 90° ply.

But, the axial stress concentrations were not so strong as those in 0° ply; (3) Near
the crack tips, the transverse stresses and shear stresses are tensile, and changed
into compressive, rapidly. These tensile stresses play an important role in damage

development in cracked laminates; (4) At the midplane of the laminates, a sort of

compressive transverse stress concentration occurred, especially for the [O°/90°]s

laminate with an isolated crack and [0°l90°]s laminate with interacting cracks; (5)
73

74

Axial stress at the midplane in 90° ply reaches the maximum tensile stress at the
midpoint between two consecutive cracks. This phenomenon explains that a new

crack will be likely to occur midway between any two adjacent cracks.
Furthermore, the finite element analysis results for [0°/90°]s glass/epoxy com-

posite laminate with interacting cracks and [0°/90°]s glass/epoxy composite laminate
with an isolated crack, were compared to variational calculus results using the mini-

mum complementary energy theorem. These variational calculus results which includ-

ed the stress concentration effects at the crack tips in 0° ply, were obtained by em—
ploying a more generalized test function according to more generalized perturbation
stress functions, than that used by Hashin. Although it should be separated into the
bulk area and surface layer or interface region in order to obtain more exact stress
distributions in variational calculus, the perturbation stress function in variational cal-
culus was not separated in this analysis. Thus, some discrepancies between the finite
element analysis results and the variational calculus results can be found.

However, as mentioned earlier, it will be extremely difficult to separate the per-
turbation stress function according to the specifically interesting areas. Hence, taking
into account the shortcoming in selecting the perturbation stress function, we can see
reasonable agreement between the finite element analysis results and the more gen-
eralized variational calculus results.

Generally speaking, as the power 11 in more generalized perturbation stress
function in variational calculus increases, the stress distributions from the finite ele-

ment analysis and the variational calculus become closer to each other, except for the
axial stress distribution at the midplane in 90° ply. In the axial stress distribution at

the midplane in 90° ply, result from the finite element analysis has very good agree-

ment with Hashin’s primary stress distribution.

LIST OF REFERENCES

LIST OF REFERENCES

[1] K.L. Reifsnider, E.G. Henneke, W.W. Stinchcomb and LC. Duke, Mechanics
of Composite Materials, Recent Advances, Editors Z. Hashin and CT Herakovich,
Pergamon, New York, 1983, pp. 399-420.

[2] R. Talreja, Damage Mechanics of Composite Materials, Department of Solid
Mechanics, The Technical University of Denmark, Lyngby, Denmark, Report S 32,
May, 1986.

[3] RD. Jamison, K. Schulte, K.L. Reifsnider and W.W. Stinchcomb, Characteriza-
tion and Analysis of Damage Mechanisms in Tension -Tension Fatigue of Graph-
ite/Epoxy Laminates, Effects of Defects in Composite Materials, ASTM STP 836,
American Society for Testing and Materials, 1984, pp. 21-55.

[4] AL. Highsmith and KL. Reifsnider, Stiffness-Reduction Mechanisms in Com-
posite Laminates, Damage in Composite Materials, ASTM STP 775, K.L. Reifsnider,
Ed., American Society for Testing and Materials, 1982, pp. 103-117.

[5] KW. Garrett and J .E. Bailey, Multiple Transverse Fracture in 90° Cross-Ply
Laminates of a Glass Fibre-Reinforced Polyester, J. Mater. Sci., 12, 1977, pp. 157-
168.

[6] A. Parvizi, K.W. Garrett and IE. Bailey, Constrained Cracking in Glass Fibre-
Reinforced Epoxy Cross-Ply Laminates, J. Mater. Sci., 13, 1978, pp. 195-201.

[7] A. Parvizi and 1.15.. Bailey, On Multiple Transverse Cracking in Glass Fibre
Epoxy Cross-Ply Laminates, J. Mater. Sci., 13, 1978, pp. 2131-2136.

[8] R. M. Christensen, Mechanics of Composite Materials, Wiley, New York,
1979. pp. 59-61.

[9] A. V. Hershey, The Elasticity of an Isotropic Aggregate of Anisotropic Cubic
Crystals, J. Appl. Mech., 21, 1954, pp. 236-240.

[10] E. Kroner, Berechnung der Elastischen Konstanten des Vielkristalls aus den
Konstanten des Einkristalls, Zeitschrift fiir Physik, 151, 1958, pp. 504-518.

75

76

[11] R. Hill, A Self-Consistent Mechanics of Composite Materials, J. Mech. Phys.
Solids, 13, 1965. PP. 213-222.

[12] B. Budianski, On the Elastic Moduli of Some Heterogeneous Materials, J.
Mech. Phys. Solids, 13, 1965, pp. 223-227.

[13] B. Budianski and R. T. O’Connell, Elastic Moduli of a Cracked Solid, Int. J.
Solids Structures, 12, 1976, pp. 81-97.

[14] N. Laws, G..J. Dvorak and M. Hejazi, Stiffness Changes in Unidirectional
Composites Caused by Crack Systems, Mechanics of Materials, 2, 1983, pp. 123-
137.

[15] G..J. Dvorak, N. Laws and M. Hejazi, Analysis of Progressive Matrix Crack-
ing in Composite Laminates I: Thermoelastic Properties of a Ply with Cracks, J. Com-
posite Materials, 19, 1985, pp. 216-234.

[16] G..J. Dvorak and M. Hejazi, Analysis of Progressive Matrix Cracking in Com-
posite Laminates II: First Ply Failure, J. Composite Materials, 21, 1987, pp. 309-329.

[17] N. Laws and G..J. Dvorak, Progressive Transverse Cracking in Composite
Laminates, J. Composite Materials, 22, 1988, pp. 900-916.

[18] R. Talreja, Estimation of Weibull Parameters for Composite Material Strength
and Fatigue Life Data, Fatigue of Fibrous Composite Materials, ASTM STP 723,
American Society for Testing and Materials, 1981, pp. 291-311.

[19] R. Talreja, in H. Lilholt and R. Talreja, Eds., Damage Models for Fatigue of

Composite Materials, Proceedings of the 3rd Ris¢ International Symposium on Met-
allurgy and Materials Science, 1982, pp. 137-153.

[20] R. Talreja, Fatigue of Composite Materials: Damage Mechanisms and Fa-
tigue-Life Diagrams, Proc. R. Soc. Lond. A378, 1981, pp. 461-475.

[21] R. Talreja, A Continuum Mechanics Characterization of Damage in Composite
Materials, Proc. R. Soc. Lond. A399, 1985, pp. 195-216.

[22] R. Talreja, Transverse Cracking and Stiffness Reduction in Composite Lami-
nates, J. Composite Materials, 1985, pp. 355-375.

[23] R. Talreja, Stiffness Properties of Composite Laminates with Matrix Cracking
and Interior Delamination, Engineering Fracture Mechanics, 25, 1986, pp. 7 51-762.

[24] ER Chen and G.C. Sih, Stress Intensity Factor for a Three-Layered Plate
with a Crack in the Center Layer, Engineering Fracture Mechanics, 14, 1980, pp. 195-
214.

77

[25] Z. Hashin, Analysis of Cracked Laminates: A Variational Approach, Mechan-
ics of Materials, 4, 1985, pp. 121-136.

[26] Z. Hashin, Analysis of Stiffness Reduction of Cracked Cross-Ply Laminates,
Engineering Fracture Mechanics, 25, 1986, pp. 771-778.

[27] Swanson Analysis Systems, Inc., ANSYS Engineering Analysis System Us-
er’s Manual, Vol. 1-2, June 1985.

[28] 1.]. Guydish and J.F. Fleming, Optimization of the Finite Element Mesh for the
Solution of Fracture Problems, Engineering Fracture Mechanics, 10, 1978, pp. 31-42.

[29] A. Tezuka and O. Okuda, An Adaptive Mesh Refinement for the Finite Ele-
ment Method (Trial by y-method), JSME Int. J., 31, 1988. PP. 50-55.

[30] C. O. Horgan, Saint-Venant End Effects In Composites, J. Composite Materi-
als, 16, 1982, pp. 411-422.

[31] I. S. Sokolnikoff, Mathematical Theory of Elasticity, McGraw-Hill, New York,
1978. PP. 387-390.

[32] P. W. Manders, T. W. Chou, F. R. Jones, and J. W. Rock, Statistical Analysis

of Multiple Fracture in 0°/90°/0° Glass Fibre/Epoxy Resin Laminates, J. Mater. Sci.,
18, 1983. PP. 2876-2889.

[33] H. Fukunaga, T. W. Chou, P. W. M. Peters and K. Schulte, Probabilistic Fail-
ure Strength Analysis of Graphite/Epoxy Cross-Ply Laminates, J. Composite Materi-
als, 18, 1984, pp. 339- 356.

"Illlllllllllllllllllf