failure analysis of a quasi-isotropic laminated composite ......with the increased use of laminated...

Failure Analysis of a Quasi-isotropic Laminated Composite Plate with a

Hole in Compression

Dr. O.H. Griffin, Jr.

by

Nirmal Iyengar

'

Thesis submitted to the Faculty of the

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Master of Science

in

Engineering Mechanics

Dr. Z. Gurdal, Chairman

June 1992

Blacksburg, Virginia

Dr. S.L. Hendricks

FAILURE ANALYSIS OF A QUASI-ISOTROPIC

LAMINATED COMPOSITE PLATE WITH A

HOLE IN COMPRESSION

ABSTRACT

The ability to predict failure of laminated composites in compression has been doggedly pursued

by researchers for many years. Most have, to a limited extent, been able to predict failure for a

narrow range of laminates. No means, as yet, exist for predicting the strength of generic laminates

under various load conditions. Of primary concern has been the need to establish the mode at

failure in compression. Even this has been known to vary for fiber and matrix dominated laminates.

This study has been carried out to analyze the failure of specimens with a hole made of laminates

with various quasi-isotropic stacking sequences. Different stacking sequences are achieved by

rotating a [±45/90/0]9 stacking sequence laminate as a whole with respect to the loading axis of the

specimens. Two- and three-dimensional finite element models, using commercial packages, were

ge~erated to evaluate the stresses in the region of the hole. Two different compressive failure

prediction techniques based on distinctly different failure modes have been used. The validity of

these techniques was measured against experimental data of quasi-isotropic specimens tested.

To investigate the applicability of the failure criteria for different laminated composite plates,

analyses were repeated for specimens with different stacking sequences resulting from the rotation

of the laminate.

The study shows the need for the use of three-dimensional analysis of the stress state in the

vicinity of the hole in order to be able to accurately predict failure. It also shows that no one mode

of failure is responsible for limiting the strength for all laminate orientations but rather the mode

changes with change in stacking sequence. The failure of the laminate with a hole was seen to be

very sensitive to the stacking sequence. Experimental data presented also shows that the peak

strength obtainable from the laminate analyzed, [±45/90/0]9 , is going to be in the off-axis

configuration rather than on-axis placement of the stacking sequence with respect to the loading

direction.

! ~

I I I I

I I I I

ACKNOWLEDGEMENTS

The author would like to thank the following people for their, direct or indirect, contribution to this

work:

Dr.Z. Gurdal, committee chairman, for taking on the author as his student and for

his advice and patience during the course of this work.

Ors. O.H. Griffin, Jr. and S.L. Hendricks for serving on the authors graduate

committee.

Many colleagues of the author who helped him through the learning curve on

PATRAN and ABACUS, notably and

and who were innocent victims of the authors

frustrations of trying to understand compressive failure, and who in spite of it all

provided valuable insights to the problem.

and for their hospitality during the defense.

The •permanent• occupants of ESM Computer Lab namely,

and who helped maintain a sense of humour when all was not

going right.

, the author's in-laws who provided invaluable family support

when it was most needed.

and who provided the author and his family with valuable

professional and domestic assistance whenever needed.

, of NKF Engineering, who showed the author that curiosity is a

trademark of a good engineer, and that the need to learn should never stop.

for their yoghurt, after all one has to eat, and healthy too.

Finally the last but, most definitely, not the least, to the authors wife , who

provided the moral and financial support through it all, his daughter , who

is still trying to figure out why she sees her father only every three weeks, and

their dogs, . and , who provided the family security and companionship

during the past few years.

iii

TABLE OF CONTENTS

1 INTRODUCTION................................................................................................................... 1

1.1 Compression Failure of Laminated Composites............................................................. 2 1.2 Failure of Panel with Stress Raisers............................................................................... 3 1.3 Review of Literature......................................................................................................... 3

1.3.1 Failure Response of Laminated Composites in Compression.................................. 4 1.3.2 Failure Criteria for Laminated Composites

with Stress Concentrations in Compression ................................................................ 10 1.3.3 Summary of Literature Review ................................................................................... 13

1.4 Compression Strength Test Data.................................................................................... 13 1.5 Scope of Investigation ...................................................................................................... 19

2 :'FINITE ELEMENT MODELS ................................................................................................ 21

2.1 Introduction ....................................................................................................................... 21 2.2 Model Description ............................................................................................................ 23

2.2.1 Two-Dimensional Model. ............................................................................................ 23 2.2.2 Three-dimensional Model. .......................................................................................... 25

2.3 Boundary Conditions ........................................................................................................ 27 2.3.1 Symmetry .................................................................................................................... 27 2.3.2 Traction Free Boundary Conditions........................................................................... 29

2.4 Comparison Between 2-D and 3-D Models .................................................................... 32

3 FAILURE CRITERIA ............................................................................................................. 40 3.1 Introduction ....................................................................................................................... 40 3.2 Mechanical Properties ...................................................................................................... 41

3.2.1 Characteristic Distance ............................................................................................... 42 3.2.2 Buckling Wavelength .................................................................................................. 43

3.3 Failure Prediction ............................................................................................................. 45

iv

TABLE OF CONTENTS (contd)

3.3.1 Kinking Model. ............................................................................................................ 45

3.3.2 Delamination Criterion .................................. ;················-'············································ 50

4 STRESS ANALYSIS ............................................................................................................. 52 4.1 Introduction ....................................................................................................................... 52

4.2 Two-Dimensional Model Stresses ................................................................................... 57

4.3 Three-Dimensional Model Stresses................................................................................ 63

4.3.1 Three-Dimensional Model Stress Correction ............................................................. 75

5 FAILURE ANALYSIS ........................................................................................................... 80

5.1 Introduction ....................................................................................................................... 80

5.2 Initiation of Kinking ........................................................................................................... 82

5.2.1 Two-dimensional Model. ............................................................................................. 82

5.2.2 Three-dimensional Model.. ......................................................................................... 83

5.3' Inception of Delamination ................................................................................................ 90

6 CONCLUSIONS AND RECOMMENDATIONS .................................................................... 93

7 REFERENCES ...................................................................................................................... 95

VITA..................................................................................................................................... 99

v

LIST OF FIGURES

Figure 1-1: Kink Band Formation .................................................................................................. 7

Figure 1-2: Laminate Configuration ............................................................................................... 14

Figure 1-3: Compressive Strength of [45/-45/90/0bs Laminate .................................................... 16

Figure 2-1: Plate Specimen Dimensions and Loading ................................................................. 22

Figure 2-2: Finite Element Model. ................................................................................................. 24

Figure 2-3: Comparison of cr33 and -r23 in Half and Quarter Symmetric models .......................... 28

Figure 2-4: 3-D finite element model - undefromed state ............................................................ 30

Figure 2-5: 3-D finite element model - deformed state ................................................................ 31

Figure 2-6: Through the thickness strain distribution for 2-D and 3-D models ........................... 33

Figure 2-7a: Comparison of strains between 2-D and 3-D models, 0=0 and =O ....................... 34 Figure 2-7b: Comparison of strains between 2-D and 3-D models, 0=90 and =O ..................... 34 Figure 2-7c: Comparison of strains between 2-D and 3-D models, 0=-45 and =O .................... 35 Figu~.e 2-8: Comparison of .stresses obtained from 2-D and 3-D Models at q, = 0.... ...... ......... 36 Figure 2-9: Comparison of stresses obtained from 2-D and 3-0 Models at q, = 45............ ...... 36 Figure 2-10: Comparison of stresses obtained from 2-D and 3-D Models at q, = 90 .................. 37 Figure 2-11: Comparison of stresses obtained from 2-D and 3-D Models at q, = -45 ................. 37 Figure 3-t: Model of a fiber under combined axial and shear forces .......................................... 44

Figure 3-2: Forces on fiber adjacent to hole ................................................................................ 49

Figure 4-1a: Distribution of cr11 around hole at mid laminae at = 0 .......................................... 53

Figure 4-1b: Distribution of -r12 around hole at mid laminae at = 0 .......................................... 53

Figure 4-2a: Distribution of cr11 around hole at mid laminae at = 90 ......................................... 54

Figure 4-2b: Distribution of -r12 around hole at mid laminae at = 90 ......................................... 54

Figure 4-3a: Distribution of cr11 around hole at mid laminae at q, = -45 ........................................ 55 Figure 4-3b: Distribution of -r12 around hole at mid laminae at = -45 ........................................ 55

Figure 4-4a: Distribution of cr11 around hole at mid laminae at= 45 ......................................... 56


Figure 4-5a: Distribution of cr11 around hole at mid laminae at = 15 ......................................... 58


vi

LIST OF FIGURES (contd)

Figure 4-6a: Distribution of a11 around hole at mid laminae at cl>= 0 .......................................... 59 Figure 4-6b: Distribution of t 12 around hole at mid laminae at cl> = 0................................... ....... 59 Figure 4-6c: Distribution of t 13 around hole at mid laminae at cl> = 0.......................................... 60 Figure 4-7a: Distribution of 0 11 around hole at mid laminae at cl>= 45 ......................................... 61 Figure 4-7b: Distribution of t 12 around hole at mid laminae at cl>= 45 ......................................... 61 Figure 4-7c: Distribution of t 13 around hole at mid laminae at cl>= 45 ......................................... 62 Figure 4-8: Variation in axial stress, a 11 with laminate orientation (cl>= Oto 45) ......................... 64 Figure 4-9: Variation in axial stress, a11 with laminate orientation (cl>= Oto -45) ........................ 64 Figure 4-10: Variation in shear stress, t 12 with laminate orientation (cl>= Oto 45) ...................... 65 Figure 4-11: Variation in shear stress, t 12 with laminate orientation (cl>= Oto -45) ..................... 65 Figure 4-12: Variation in shear stress, t 13 with laminate orientation (cl>= Oto 45) ...................... 66 Figure 4-13: Variation in shear stress, t 13 with laminate orientation (cl>= Oto -45) .................... 66 Figure 4-14a: Distribution of a11 around hole at mid interfaces at cl>= 0 ...................................... 67 Figure 4-14b: Distribution of t 12 around hole at mid interfaces at cl>= 0 ...................................... 67

z Figure 4-14c: Distribution of t 13 around hole at mid interfaces at cl>= 0 ...................................... 68 Figure 4-15a: Distribution of a11 around hole at mid interfaces at q, = 0 ...................................... 69 Figure 4-15b: Distribution of t 12 around hole at mid interfaces at cl> = 0 ...................................... 69 Figure 4-15c: Distribution of t 13 around hole at mid interfaces at cl> = 0 ...................................... 70 Figure 4-16: Variation in axial stress, a 11 , at midplane, with laminate orientation

(cl> = O to 45) ................................................................................................................. 72 Figure 4-17: Variation in axial stress, a11 , at midplane, with laminate orientation

(cl>= 0 to -45) ................................................................................................................ 72 Figure 4-18: Variation in shear stress, t 12, at midplane, with laminate orientation

(cl> = Oto 45) ................................................................................................................. 73 Figure 4-19: Variation in shear stress, t 12, at midplane, with laminate orientation

(cl>= Oto -45) ................................................................................................................ 73 Figure 4-20: Variation in shear stress, t 13, at midplane, with laminate orientation

(cl>= Oto 45) ................................................................................................................. 74 Figure 4-21: Variation in shear stress, t 13, at midplane, with laminate orientation

(cl>= 0 to -45) ................................................................................................................ 74

vii

LIST OF FIGURES (contd)

Figure 4-22a: Comparison of 't°12 and t 12 for 9=0 and ip:o .......................................................... 77 Figure 4-22b: Comparison of 't°12 and t 12 for 9=0 and' fP=15 ......................................................... 77 Figure 4-22c: Comparison of t 0 12 and t 12 for 9=0 and fP=15 ......................................................... 78

Figure 5-1: Compressive failure distribution based on fiber kinking (2-D FEM Model) .............. 81

Figure 5-2: Compressive failure distribution based on fiber kinking (3-D FEM Model).............. 84

Figure 5-3: Compressive failure distribution based on fiber kinking using corrected t 12 • •• • ••••••• 85

Figure 5-4: Initiation of matrix failure at mid laminae (3-0 FEM Model) ..................................... 87

Figure 5-5: Compressive failure distribution based on fiber kinking at the interfaces ................ 88

Figure 5-6: Compressive failure distribution based on fiber kinking at the interfaces

using corrected t 12 ••• •••••••••••••••••••• •• •••••••••• ••••••••••• •••••••• ••••••• ••••••• ••••• •• ••••• ••••• •• • •••• •• • •••• ••• 89

Figure 5-7: Inception of delamination ........................................................................................... 91

LIST OF TABLES

Table 1-1 : Experimental Results of [±45/90/0]38 Laminate Specimens ....................................... 15

Table 2-1: Material Properties of AS1/T3201 .............................................................................. 23

Table 3-1: AS1/T3201 Lamina Strengths .................................................................................... 42

viii

1 INTRODUCTION

With the increased use of laminated composite materials for primary structural components, there

exists a need to fully understand their capabilities. What makes laminated composites attractive

as structural materials is their ability to be tailored to achieve specific material properties and

strength for a given laminate configuration.

For a material to be successfully utilized for structural purposes it must satisfy certain fundamental

requirements; one being the ability to maintain structural integrity in the presence of discontinuities.

Discontinuities, in the form of holes, are necessary in structural panels to allow for joining, access

openings for maintenance, and piping and electrical penetrations. The existence of these

discontinuities produce areas of local stress concentration, and it is necessary to be able to predict

the response of these panels under various loading conditions if they are to be used for structural

purposes.

Experimental research [1-4] has shown the detrimental effect of discontinuities such as notches,

holes etc., on the load carrying capacity of laminated composites. The effect of holes on the load

carrying capacity is greater in laminated composites than in metals. Being made of brittle fibers,

the ability of high performance laminated composites to undergo local plastic deformation in an

area of high stress concentration is limited [1]. It has also been shown [3-10] that the failure

mechanism is more complex in compression loading than in tension. The modes of failure in

compression are particularly sensitive to the presence of holes, notches, and other similar stress

raisers. Once the mechanism of failure propagation can be identified and the load carrying

capability predicted, the material can then be tailored to accommodate designed discontinuities.

Also the effect of discontinuities, introduced subsequent to initial design, can readily be assessed.

Introduction 1

1.1 Compression Failure of Laminated Composites

Prediction of strength of a laminated composite depends on two factors, which are material strength

and stacking sequence. The material strength of a laminate is dependent not only on the properties

of its constituents, but also on a number of other faqors governed by the manufacturing process.

Flaws incorporated during manufacturing that can affect laminate strength are non-uniform fiber

distribution, fiber waviness, and matrix voids. Another factor whose effect on strength is not known

but at present is under investigation is fiber-matrix interaction. Due to the number of ingredients

influencing compressive strength there is no completely satisfactory account, as yet, of

compressive failure [6].

In order to assess the compressive strength of a laminate, it is necessary that the mode of failure

be understood. Studies of failure on plates in compression [1,2, 11-26], for a variety of laminates,

accompanied by experimental verification, have been presented over the years. None of the

predictions, however, seem general enough to be able to predict the failure over a range of

laminate configurations. There does seem to be some consensus on the nature of failure being

localized and was initially attributed to a loss of fiber stability [11-21 ].

Unlike their isotropic counterparts where stability, whether global or local, is based on geometric

considerations, stability of laminated composites is to be evaluated at global, macro and micro

levels [3]. Due to the makeup of fiber reinforced composites, inhibiting the loss of stability at the

global (orthotropic failure) and macro (lamina failure) levels, will precipitate it at the micro

(fiber/matrix failure) level. At the micro level the loss of stability is, as mentioned earlier, only to a

limited extent dependent on the geometry of the fiber. Research [2,22-24] has shown evidence that

fiber failure may not be due to this •1oss of stability• at the micro level, but rather a phenomenon

called fiber •kinking• based on observations of kink bands in the failed specimens.

The theory of compressive strength of a laminate being dependent on kinking has been studied

by many researchers [2,4,9,10,26]. Experimental evidence has shown [15,17,20,21] that the

kinkbands originate at a free edge and propagate towards the center of the plate eventually

resulting in catastrophic failure. Kinking is also said to initiate from a defect, notch, or stress raiser,

and grows in a manner analogous to a dislocation. Experimental evidence also seems to indicate

that the initiation of kinking and subsequent catastrophic failure, are often not very far apart, thus

making it difficult to chart the progress of this phenomenon [21 ]. Recent studies, however, indicate

Introduction 2

that failure in compression cannot be attributed to one mode for all laminates, as suspected earlier,

but is more likely to be progressive in nature [2,27]. These researchers (2,27] have recorded

delamination along with kinking in specimens in which the failure process was partly captured by

interrupting loading prior to catastrophic failure. The exact order and effect of one failure mode on

another has never been evaluated.

1.2 Failure of Panels with Stress Raisers

As mentioned earlier the presence of stress raisers is detrimental to the load carrying capacity of

a laminated composite panel in compression. The failure of these panels has, in general, been

assessed at the macro and micro levels. The studies on the macro level use a fracture mechanics

approach to obtain Stress Correlation Factors (SCF) or Stress Intensity Factors (SIF) for a variety

of laminates. Experimental studies [28] Were also carried out to evaluate the effect of finite width

and hole size. These studies were confined to the lo5s in load carrying capacity and did not

address the mode of failure. To provide a generalized assessment of the failure, researchers have

focused on the micromechanics in order to evaluate the initial mode of failure [4,9, 1 O].

~-

Most discussions on failure have been restricted to unidirectional loading. Many practical loading

situations are unlikely to be unidirectional and the existence of combined compressive and shear

loads is a reality. Even under unidirectional loading shear stresses generated due to complex

stresses existing at edges of stress raisers [9, 1 O] brings up the need to understand the effect of

combined loading on such panels with geometric discontinuities. The response of a laminated plate

to combined loading can be simulated by analyzing a plate with a hole. The hole or other

geometric discontinuities provide the complex stress state to be analyzed, and the failure process

can be scrutinized better by focussing on the region around the hole.

1.3 Review of Literature

It is apparent that the failure process in compression is very complex. Lack of knowledge of the

micromechanical behavior of a laminated composite plate under compression inhibits proper

understanding of the failure process. This, however, has not deterred researchers in attempting to

analyze the problem. Excellent surveys of existing literature have been presented by Gurdal (4],

Greszczuk (20], and Shuart (29] to name a few. However, in order to bring the task on hand into

Introduction 3

perspective the process is repeated. The literature survey is discussed in two parts, namely, the

failure response and the failure criteria of laminated composites in compression. In the former, the

modes of failure in compression with experimental evidence are discussed. In the latter, analytical

models and criteria used to predict failure of laminated composites with geometric discontinuities

have been discussed

1.3.1 Failure Response of Laminated Composites in Compression

For the past two decades, since it was first recognized, the failure of a laminated composite in

compression was attributed to short wavelength buckling (microbuckling) and had been the focus

of many researchers. Most analytical work done so far is based on the models proposed by Rosen

[11 ]. He proposed two modes under which microbuckling could take place, the extension mode and

the shear mode. Rosen's model were based on the stability of a beam (representing the fiber) on

an elastic foundation (representing the matrix). Analytically he presented the two failure modes as

follows:

~ension Mode: (1.01)

Shear Mode: (1.02)

where:

a,cr - Compressive strength of composite Ei. Em - Elastic modulus of fiber and matrix respectively V1 - Volume fraction of fiber

Gm - Shear modulus of matrix

h - Width of fiber

L - Buckling wavelength of fiber

m - Fiber buckle mode number

Rosen [11] neglected the second term of the latter equation on the basis of its magnitude; the

argument being that the buckling wave length to fiber diameter ratio was very large. Thus reducing

Introduction 4

the shear mode equation to :

(1.03)

Rosen [11] however recognized that the predictions obtained by his models overestimated the

strength of the laminates. He attributed this difference to the possible loss of shear modulus

resulting from plasticity of the matrix [6]. Schuerch [12] also obtained similar equations as Rosen,

but he extended his model to include inelastic microbuckling as he noted that the strain levels

exceeded the yield strain of his matrix.

Hayashi [13] introduced the concept of "shear instability" associated with compressive loading. He

proposed the idea that in the case of structural members with high flexural rigidity compared to

shear rigidity, buckling along with shear deformation will take place. The buckling strength is

therefore attributed largely to the shear modulus of the laminate. Foye [14] was also soon to come

to the conclusion that, for unidirectional composites, the ultimate strength in longitudinal

compression is limited by. their shear modulus. He obtained a modified stress strain law for an

element under shear, indicating that in compression the effective shear modulus would decrease

With the increase in load.

(1.04)

At failure the element would experience a complete loss of shear stiffness resulting in shear

instability or crippling failure. Noting that his predicted strengths were higher than experimental

values, he included in his analysis model the effect of voids and matrix fillers.

In order to fully understand the behavior of a fiber in an elastic matrix Hermann and Mason [15]

and Sadowsky et al. [16] experimented with single fibers isolated in a matrix to study short

wavelength buckling of the fiber. Hermann and Mason [17] found that predicted failure of their

model related well to Rosen's [11] extension mode. They also showed that the initial waviness of

the beam (fiber) had a significant impact on failure. Sadowsky et al. [16] took into consideration

residual thermal stresses, sensing that these stresses could cause waviness and fiber buckling and

came to the conclusion that shear deformation had negligible effect on the compressive strength.

Introduction 5

They also measured the strain during buckling and found that the predicted strains were more than

an order of magnitude greater than that measured experimentally. Crawford [17) was the first to

incorporate initial waviness of the lamina into the expression of stiffness. This resulted in

interlaminar stresses being generated. Based on this, he concluded that the laminate failure was

a consequence of these interlaminar stresses, rather than short wavelength buckling, though the

exact mode of such a failure was not suggested.

To further study the effect of the matrix in fiber buckling, Lager and June [18) tested Boron fibers

in two epoxy systems. They based their model on that of Rosen's [11 ], but replaced the Elongation

and Shear Modulii in Equations 1.01 and 1.03 by their respective Tangent Modulii. Based on

experimental data they then replaced the tangent modulii by an "effective" modulus which is

defined as a product of an "influence coefficient" and the material modulus. The value of the

influence coefficient was determined experimentally to be 0.63. They also established that the

strength of a composite in compression is largely dependent on matrix strength. The study also

showed that for fiber volume fractions less than 10%, failure correlated well with the extension

model while for large volume fractions the shear mode was observed. Since most of the

experimental work was still over predicting the failure of a laminated composite in compression De

F~rran and Harris [19] conducted experiments on polyester reinforced steel wire. They questioned

the validity of Rosen's premise, that planar buckling was occurring. They concurred with what Lager

and June [19) had alluded to, in that the buckling had an out of plane component and so

considered a three-dimensional helical buckling approach. The hypothesis, of microbuckling being

affected by interlaminar stresses, having been presented it was not until much later that Guynn and

Bradley [27] and Waas et al. [2] would provide experimental evidence that in- and out-of-plane

microbuckling does indeed occur. This type of microbuckling could be a result of the interlaminar

stresses.

To investigate the effect of matrix properties on the strength of a laminated composite in

compression Greszczuk [20,21,30) performed many experiments by varying the matrices and

volume fractions and considering different failure modes to come to the following conclusions. For

composites with a low-modulus resin, failure in compression is generally governed by

microbuckling, however, as the resin modulus is increased the transformation to a non

microbuckling failure takes place. His data showed the effect of fiber matrix interface strength on compressive strength indicating an interaction type failure. Nonlinear material and geometric effects

of the fiber and matrix were not considered.

Introduction 6

T' T' -····~

_______ ...

. 111 " 1111

T'

Figure 1-1: Kink Band Formation (After Ref. 22)

7 Introduction

At this point, with still no means for accurately accounting for the compressive strength Berg and

Salama [22) through their studies of fatigue of graphite epoxy in compression, introduced the

concept of kinking, to laminated composites. They showed the existence of kink bands due to

microbuckling, in the specimens failed as a result of compressive fatigue. They concluded that axial

matrix cracking was essential to the initiation of microbuckling instability. This longitudinal cracking

permits kink bands on conjugate planes to link up leading to catastrophic failure. They postulated

a kink process by which microbuckled fibers undergo shear deformation to the point that they break

off into bits (Figure 1-1) due to tensile failure of the fibers.

Based on work by Berg and Salama [22), and Weaver and Williams [23), Evans and Adler [24)

produced an exhaustive study on kinking. They minimized the plastic work done to obtain the kink

inclination and minimized the elastic strain energy to determine the kink boundary. They also

utilized a model for statistical fiber fracture and matrix enhancement to determine parameters and

an expression for critical kink formation stress. This expression confirmed the need for high matrix

strength to suppress kink formation. Chaplin [1] attributed the low compressive strength to the

presence of a defect from which local failure can propagate. He suggested use of fracture

mechanics approach observing that the failure occurs at one region and then rapidly propagates

across the specimen. Defects which act as stress concentrators are responsible for the initial ·:;;·

failure. Chaplin [1] compared the failure to shear instability that occurs in un-reinforced resin and

so felt the term microbuckling to be inappropriate.

Utilizing the above information Parry and Wronski [25) investigated the compressive strength of a

plate with a notch. They supported the Berg and Salama [22) claim that the principal mode of

failure at the notch was due to the formation of a kink band. Later, experiments carried out by

Rhodes et al. [26), and Waas et al. [2] also support this assertion. Rhodes [26) described the failure

sequence beginning with matrix failure at 85% load which was interfiber in nature. Following matrix

failure, upon further loading shear crippling in plies of the same orientation were initiated resulting

in specimen failure. Waas et al. [2] also observed failure at the hole edge in the load carrying ply

(0 degree) and at the location of the maximum compressive stress. They also found the

microbuckling to originate at the hole surface and persist well into the interior of the specimen

leading to catastrophic failure.

Departing from the traditional view of fiber failure, Shuart and Williams [31] conducted experiments

on angle ply laminates, with ±45 degree and ±45 degree dominated plies, with holes to conclude that the failure of the former was due to matrix shear and the latter was a combination of matrix

Introduction 8

failure and delamination. On close inspection of the damaged specimens they observed kink band

formation and transverse matrix cracking. Continuing work in compression failure of laminated

composites, Shuart [29] developed a model with the fibers in the lamina as a plate and the matrix

acting as an elastic foundation. He used linear analysis to determine stress, strain and mode shape

using short wave buckling criteria. He extended this study to include the effect of volume fraction '

on the buckling stress and then incorporated the effect of fiber imperfections in a non-linear

analysis. He considered,the failure of the lamina based on the buckling of the outer lamina, in-

plane matrix shearing and interlaminar shear strains from the fiber imperfections. He also provided

a method to evaluate the load carrying capacity of a O degree dominated lamina in compression.

Shuart [32] extended his previous non-linear model to include the effects of out-of-plane ply

waviness, in-plane fiber waviness and fiber scissoring. He used this model to predict the failure

of [±9/•9ls. composite laminates in compression. Using experimental data, he showed his model

to provided excellent agreement ,for laminates with fiber angles greater than 45 degrees, 9 > 45, while that for those with fiber angles less than 45 degrees, 9 < 45, was not as good. Again he found the dominant failure modes to be interlaminar shearing, in-plane matrix shearing and matrix

compression failure.

So far the failure of a laminated composite in compression had been attributed to one failure mode.

The possible interaction of two modes of failure were first recorded by Guynn and Bradley [27].

Guynn and Bradley [27] provided evidence, based on experiments on compressive failure of

AS4/PEEK laminate specimens with open holes, of the existence of two failure modes but

discounted their interaction. Based on their observations of specimens that were inspected on

interruption of the tests, prior to catastrophic failure, that the principal mode of failure was shear

crippling. They also found evidence of in- and out-of-plane microbuckling prior to the shear

crippling, which was also accompanied by local delamination. They concluded that this local

de lamination was a result of the shear deformation of the fibers, caused by microbuckling, however,

they did not believe that the delamination propagated to become macroscopic delamination, until

final compressive failure when large scale brooming was observed. Waas et al. [2] on the other

hand observed, depending on the specimens tested, the formation of delamination buckling prior

to actual final failure. The damage initiated by microbuckling led to delamination cracks and

suQ&equently to catastrophic failure. The growth of the delamination was in large part dependent

on the percentage of 0 degree plies. In laminates with a small percentage of 0 degree plies

delamination size grew till the area buckled leading to failure. The larger percentage of O degree

plies led to brittle failure while a small percentage of 0 degree plies led to a more ductile failure.

Beyond failure initiation by microbuckling, the delamination and buckling growth stage was

Introduction 9

influenced by the overall stiffness. Waas et al. [2] also presented initial 0 degree strains which

seemed to initiate failure.

1.3.2 Failure Criteria for Laminated Composites with Stress Concentrations in Compression

The failure criteria of plates with cutouts, holes, and notches have been studied for years, however,

most of them relate to tensile loading. Many of the criteria proposed have been based on failure

at the laminate level without any emphasis on the failure modes. Whitney and Nuismer [3]

continued work by Waddoups et al. [5] to obtain average and point stress criteria for the failure of

laminated plates with holes. They presented characteristic distances for both criteria, based on

work by Bowie [33] on isotropic material. They, however, postulated that the characteristic

distances were constant for various laminates and discontinuities. Nuismer and Labor [7] extended

the concepts established above to laminated plates under compressive loads. However, the

characteristic distance was found to be much greater in compression than was observed in tension.

They attributed this difference to the greater redistribution of stress prior to failure in compression.

Garbo and Rogonowski [8] conducted experiments and provided strength analysis for composite ·:~

laminates under compressive and tensile loading. They postulated that the failure of a composite

laminate can be correlated with analytical predictions of point stress at a characteristic distance

away from the edge of the stress concentration. Garbo and Rogonowski [8] also extended the

above theory to predict strength of a laminate, by utilizing the point stress criteria to evaluate the

strength of each of the laminae. They recommended that, since, in a laminated composite, the

peak stress does not occur at the same point at the hole edge, it is necessary to evaluate the

stresses adjacent to the hole to obtain the combination of stresses that will lead to failure.

Gurdal [4] expanded the work on failure criteria of laminated plates with cutouts, holes and notches

by combining micromechanics and the laminate point stress criterion. Using information provided

by Rhodes et al. [26], on the occurrence of kinking at the boundary of a stress raiser and the work

by Greszczuk [21,30] on point stress criteria for plates with holes, he recommended that the two

could be coupled to evaluate the failure strength of plates with notches. He also noticed that most

of the prior work had restricted itself to the effect of the local compressive stresses only. Supported

by Greszczuk's [21] observations that microbuckling occurs at a higher load than experimentally

obtained, Gurdal [4] suggested that the discrepancy may be caused by shear stresses induced by

the presence of the discontinuity. He then used an energy formulation to analyze a beam on an

Introduction 10

elastic foundation, subject to in-plane and shear loading and obtained equations for stress in the

fiber due to bending. Details of the formulation are presented in Chapter 3. The model assumed

failure due to bending rather than microbuckling instability, as had been the norm till then. The

shear loading in the model helped account for the shear stresses generated due to the presence

of the notch. To prove the validity of the model he used the same variational formulation to do a

stability analysis to obtain a buckling expression similar to that of Rosen's [ 11 J microbuckling model in the shear mode. Whereas Rosen [11 J neglected the second term Gurdal [4J retained it. He

argued that experimental evidence provided by Evans and Adler [24J had shown that the buckling

wave length to fiber diameter ratio did not justify ignoring the second term in Equation 1.02. He

used two-dimensional finite element models to obtain the stress distribution in notched plates and

using experimental data he evaluated the characteristic distance for the quasi-isotropic laminate

to be 1.5 mm for plates with notches. Similarly for plates with holes the distance was shown to be

2.5 mm. Gurdal [4J extended his model of plates with cracks to represent those with holes as more

data was available for that purpose. He supported his decision by correlating the analysis with

experiments conducted by Haftka and Starnes [9J for laminates with varying degrees of orthotropy.

The results, however, showed the predicted values to be 40% greater than the experimental

values. Gurdal and Haftka [1 OJ extended the earlier work to predict failure of a plate with a hole

}lnder combined loading. They modeled the plate so as to obtain predictions for various laminate

orientations. This was done by rotating the principal axis of orthotropy of the laminate with respect

to the load direction. The stresses around the hole were evaluated using techniques proposed by

Savin [34J and corrected for finite width from Hong and Crews [28J. The study clearly showed the

anisotropy of strength of a quasi-isotropic plate under compression. Gurdal and Haftka [1 OJ also

concluded that for a plate under combined in-plane and shear loading, rotating the laminate axis

of orthotropy in a direction opposite to the applied shear load, can increase its load carrying

capacity, thus suggesting a possible use of orthotropy for strength improvements of plates with

holes.

Some other attempts to predict the compression strength of laminated composites plates with

holes, using more deterministic approaches are described below. Bums et al. [35J analyzed the

compressive failure of notched angle ply laminated composites utilizing a three-dimensional finite

element model and compared the results with experimental data. A point stress method and

classical quadratic tensor and maximum stress failure criteria were used to evaluate the failure.

Burns et al. [35J showed that failure analysis at the hole edge gave extremely conservative

strengths. They concluded that the lack of consistent results was due to the point stress criteria

being used and that failure in a laminate was not dependent on point failure. They found that the

Introduction 11

in-plane shear stress, 't12, was dominant in the midplane with the out-of-plane shear stress, 't13,

dominant at the interface. Observing the points at which failure was initiated, they concluded, that

a combination of the two shear stresses were required to initiate failure. The characteristic distance

theory had limited success in predicting failure.

Chang and Lessard [36] considered the progressive failure of a laminated composite containing

a hole subject to compressive loading. The stress analysis was performed using a finite element

model considering the material and geometric non-linearities based on finite deformation theory.

Failure criteria were also formulated for first ply failure, based on a semi statistical method and a

damage progression model. Property degradation models for various failure have been provided.

Good correlation was found when compared with experimental data for the laminates in question.

The model considered ply thicknesses, clustering and finite widths, however the possibility of

delamination was not considered.

In a very different approach, Soutis et al. [37] use a modified Linear Elastic Fracture Mechanics

technique to predict failure of a Carbon Fiber-Epoxy laminate, [(±45/0J3]., with a hole. Based on

experimental observations, they inferred that failure is initiated by microbuckling at the hole edge

)Yhich they equated with a "crush zone". This crush zone was considered to grow stably under

increasing load till a critical length was reached, when unstable growth results in catastrophic

failure. Using a non-linear approach to crack growth they predicted good results for the

compressive strength of the laminate, though their prediction of length of the microbuckled region

(crush zone) was greater than observed. This technique did not in any way address progressive

failure and, being based on a fiber failure mode only, may have its limitations when utilized for

laminates without fiber dominant layers.

Considering the hypothesis that interlaminar stresses may play a part in compression failure of

laminated composites, and that the magnitude of these stresses [38] in the vicinity of a hole cannot

be ignored, Lagace and Saeger [34] proposed a failure criteria based on the inception of

delamination at a hole edge. They used a methodology based on an eigen function solution

technique to evaluate the interlaminar stresses near a hole for a plate in tension. They then used

Hashim-Rotem type quadratic failure criteria proposed earlier (Lagace and Brewer [39]) to

determine the load at the onset of delamination. They plotted the load at onset of delamination

versus the laminate orientation, ~. to show the anisotropic and asymmetric distribution of strength of the quasi-isotropic laminates. The failure criterion was based on a single point stress and no

consideration was given to progressive failure initiated by either microbuckling or matrix cracking.

Introduction 12

1.3.3 Summary of Literature Review

In summary, the literature review shows the extensive research done on the failure prediction of

laminated composites in compression. Most of the research, however, has been restricted to the

compressive strength of unidirectional laminates a~ has been based on the assumption of short

wavelength fiber buckling (microbuckling) in the plane of the lamina. Attempts at correlating

analytical with experimental data by incorporating manufacturing defects such as voids, fiber

waviness and scissoring and debonding have been studied. Research has also covered the failure

of the fiber as a result of kinking.

Based on evidence that the mode of failure in compression is related to the type of laminate (fiber

or matrix dominated), research on matrix initiated failures have also been done. Having accepted

the fact that compression failure of a laminate is initiated at a notch or similar stress raiser

researchers have developed models to predict compression strength of laminates with holes with

reasonable success. The influence of interlaminar stresses on compression strength has also to

a limited extent been covered, though it has long been suspected to play a part in the failure

mechanism.

Most of the above studies were based on planar failure models and those that studied interlaminar

stresses did not include in-plane failure modes. As a result, in spite of the extensive work being

done there is no general means of predicting failure of a laminated composite with a hole in

compression.

1.4 Compressive Strength Test Data

As discussed in the previous section, existing techniques for the prediction of strength of laminated

composite plates with holes were applicable to only specific laminate systems, with none being

general enough to be applicable for a range of laminate configurations. The above prediction

techniques are very sensitive to specific laminate configurations, whether unidirectional or angle

plies. In order to assess the capability of predictive models for laminates in compression, it was necessary to obtain experimental data with which the analytical predictions of failure could be

compared. Also in order that the generality of the analysis technique could be demonstrated, it was

decided that the use of a quasi-isotropic laminate was necessary. Earlier models had been used

for predicting the strength of unidirectional and angle ply laminates separately; the response of a

Introduction 13

Figure 1-2: Laminate Configurations

Introduction 14

.I

Table 1-1: Experimental Results of [±45/9010],. Laminate Specimens

Orientation Number of Off-axis Failure Loads Average Failure Angle Specimens of Specimens Loads

,

-80 2 6728.8 7105.3 7481.7

-67.5 4 6405.9 6640.2 6700.2 6875.0 6875.0

-45 5 6446.8 6941.2 6703.2 6647.0 6813.96 6667.2

-22.5 3 . 8296.5 7923.3 8078.1 8014.5

-15 5 8126.5 7800.9 . 7919.9 7941.9 7803.0 7926.9

-10 5 8155.2 8416.7 8256.8 8227.5 8218.9 8265.4

-5. 5 7561.6 n38.o 7597.0 7492.0 7857.0 7336.1

0 5 7200.0 7080.0 7199.8 7266.2 7122.8 7330.0

10 5 7042.0 6n9.8 6940.7 6872.6 7094.1 6914.9

22.5 5 6756.9 7078.8 6970.6 6693.4 7378.8 6945.2

45 5 6929.6 7189.6 6740.5 6861.3 6325.4 6396.8

Introduction 15

--T·--r·---]-1--r-

l l

i l

i :

: :

: :

l 1

l l

~ ....................... i ........................ ! ....................... j··· ··················1·······--...... o-······f···············-···

l l

0 co

0 C")

0 0 C") I

0 co I

0 O> I

G)

- ca c ·-E ca ..J en C") ...... 0 - 0 0) - It) oi::t' I - It) -&

-oi::t' .....

c:: 0 -..::::; - 0 as .... 0

.c -

.... c::n

- 0 Q) c G) ..

15> c::

-"' <

( G

) >

·c;; en G) .. c. E 0 CJ Cf) I .

I T""

G)

I .. ~

I c::n i!

16

quasi-isotropic laminate is a combination of the above two configurations. Further the generality

of the technique could be corroborated by applying it to various laminate orientations that will be

described later.

Results of a test program carried out earlier [40], for quasi-isotropic laminates of [±45/90/0b. and

[±45/90/0k. stacking sequences are presented here (See Table 1-1 ). Specimens in Reference [40]

were cut from a large panel with a [±45/90/0J. layup so that laminates with varying angles of

laminate orientation, cl> (as defined in Figure 1-2), were obtained. That is, a specimen cut at cl>= 15 degrees would have a stacking sequence of [60,-30,-75,15]. with respect to the principal axes

of the specimen. This layup is still quasi-isotropic, however it is suspected that the strength of the

specimen would be different than a specimen cut at cl> = O degrees because of the change in the stacking sequence which would affect the local stresses around the hole. The loading direction was

kept constant and the laminate orientation cl> changed; this was analogous to keeping the laminate orientation constant and varying loading angle thus simulating the effect of combined shear and

axial loading on the laminate. The counter clockwise direction was considered to be a positive

rotation and is referred to as the laminate orientation cj>. The specimens were 1.5 inches long, 1.0 inch wide and 0.133 inch thick (24 ply) with a circular hole of 0.20 inch diameter.

'The compressive strength obtained at off-axis laminate orientations (cl>* 0) were normalized by the strength at cl> = O and shown in Figure 1-3. On plotting the normalized strengths, P/P 0 versus laminate orientation, cj>, the curves obtained for the specimens with two laminate configurations tested were practically coincidental with one another. The only difference between the two laminate

configurations being the total number of plies, and therefore the total laminate thickness. For this

reason the following discussion has been based on the experimental data points obtained from the

testing of [±45/90/0b. laminate specimens only. The results of the experimental data shown in

Figure 1-3 are described below:

• The single roost distinctive feature of the failure envelope, at laminate orientations -90 <

cl> < 90, is that its is anisotropic and asymmetrical about cl> = 0. By anisotropic, it is meant that the strength varies as the laminate is rotated about the z-axis.

• The failure loads at cl> = 45 and cl> = -45 are approximately the same, within experimental scatter.

• The trend at the extremes indicates that the load at cl> = 90 would be close to or equal to that at cl>= 0.

• The failure load (strength) peaks at about cl> = -15 degrees and is approximately 12% greater than that at cl> = O and about 19% greater than the value at cl> = 15 degrees. This is rather unusual because based on CL T, the axial stiffness of the laminate does not

change if it is rotated in the +15 or -15 degree direction. At the lamina level the axial

Introduction 17

" I ~ If

1:

I I I

•

stiffness coefficients, A1• are the same for a lamina with a fiber angle of +0 as that with -a. The difference, however, in the two laminate orientations,~= 15 and -15 is the stacking sequences. In these orientations the former becomes a [60,30,-75,15}. laminate and the

latter [30,-60,75,-151. laminate respectively. As can be seen in these two laminate

configurations, for each 0 in the former there exists a -0 in the latter and vice versa. So the lamina stiffness coefficients of the two are the same. The relative through-the-thickness

locations of the two laminates are different. The stacking sequence seems be the only

feature that probably accounts for the difference in compressive strengths of the two

laminate orientations. It can be conjectured that the strength of the laminate in

compression is not only a function of the total laminate stiffness but also one of the relative

location, i.e. stacking sequence, of the various axial stiffness components. It is for this

reason that it is essential to determine the three-dimensional stress state for this laminate

in various orientations. A two-dimensional model is not able to capture these subtle

differences.

Another manner in which the stacking..sequence can be considered to have significant

impact on compressive strength is in the through-the-thickness location of the primary load

bearing laminae. This can be seen by comparing the orientations at angles in steps of 45

degrees, that is at ~ = 0, 45, 90 and -45 degrees. It seems to indicate that when the primary load carrying laminae, i.e. one in which a = ~. is at the top and bottom of the laminate during rotation, there is a great reduction in the strength. For example, when ~ = 0 the laminate configuration is [±45/90/0J. while in the ~ = ±45 orientations they are

[0/90/±45]. and [90/0/•451. respectively. In the former condition (~ = 0) the primary load

bearing plies are at the midplane of the laminate while in the latter (~ = ±45) they are on or near the surface. This could result in out-of-plane buckling due to the lack of transverse

support for the surface layers. This hypothesis is in part supported by experimental

observations of Guynn et al. [27), who suggest that the out-of-plane buckling does occur

and may be due to the influence of transverse stresses on the laminate.

The asymmetry of compression strength has tremendous significance to real life applications. In

order to optimize the design of a quasi-isotropic material for a structure that may experience

tension and compression loading it may be necessary to utilize the material in an off-axis

configuration. Sun and Zhou [41] have also presented research showing the anisotropic behavior

of tensile strength in off-axis configurations of quasi-isotropic laminates. The angle of laminate

orientation, ~. exhibiting maximum strength in tension, was different from that shown in Figure 1-3 for compression.

Introduction 18

1. I

1.5 Scope of Investigation

There seems to be adequate evidence supporting the hypothesis that compression failure of a

plate with a hole is initiated by fiber kinking at the hole surface [2,4,9, 10,27,37]. This kinking leads

to fiber breakage causing kink bands and eventually resulting in catastrophic failure. Research

[4,9, 1 O] shows the significant contribution of shear to this mode of failure. Evidence of this is more

apparent when considering laminates that have no O degree plies in the loading direction, i.e. all

plies are angle plies. Even under a uniaxial compressive load combined compressive and shear

stresses are developed due to the stiffness coupling in the laminae and the presence of geometric

discontinuities. Analyses to date have had reasonable success in predicting the failure strength of

specific laminated composites under a compressive load, though a more general approach is

desired. The objective of the present work is to evaluate two existing models of compression failure

and attempt to predict the strength of a laminated composite plate with a hole under varying

laminate orientations, ~·

As discussed earlier, to date most work on microbuckling and kinking has utilized a two-

dimensional approach based on the assumption that the response is planar. Based on inference

·from limited experimental data, as discussed in Section 1 .4 above, that the variation in normalized

strength, P/P 0 , was the same for the two laminate configurations tested regardless of their

thickness, and for the sake of simplicity, this study is based on the analysis of a [±45/90/0].

laminate. Also for the reasons discussed in Section 1.4 above, in this study a three-dimensional

linear finite element model of the above laminate is used to analyze the stress distribution around

a hole. Comparison between the stress distribution in this model and a similar two-dimensional

model is presented. The stresses from the finite element models, along with the kinking model

proposed by Gurdal [4], are used to predict failure of a quasi-isotropic laminate under compression.

The kinking model failure criteria is applied to the stresses at the edge of the hole to evaluate

failure of the laminate, at varying angles of rotation of the quasi-isotropic laminate. The analysis

is then compared with experimental data available as described in Section 1.3. As it was suspected

that transverse stresses may contribute to compressive failure, the kinking model was modified to

incorporate them. Also in order to fully evaluate the effect of the inter1aminar stresses, the onset

of delamination was evaluated based on a criteria proposed by Lagace and Saeger [34]. Even

though it is suspected [2,39] that the failure is progressive in nature, the present study ignores the

progressive nature of failure and does not use ply discount technique. Also interaction between the

two modes of failure will not be considered.

Introduction 19

In Chapter 2 the two finite element models, two-dimensional and three-dimensional, are described,

while Chapter 3 discusses the kinking and delamination models along with the failure criteria that

have been used. In Chapter 4 the stresses obtained from the two finite element models have been

discussed while Chapter 5 discusses the result of the failure analysis using criteria and models

presented in Chapters 2 and 3. Conclusions and recommendations have been covered in Chapter

6.

Introduction 20

2 FINITE ELEMENT MODELS

2.1 Introduction

In order to compare the effect of different analysis models on the failure prediction of a plate with

a hole loaded in compression, two finite element models were generated. The first is a two-

dimensional model which could account for in-plane stresses only. The second model is a three-

dimensional model capable of providing all six stresses at a point. The primary objective of creating

the two-dimensional model was to provide a basis by corroborating data obtained by Gurdal [4],

as to the effect of in-plane compressive and shear forces on the failure strength of the laminated

plate in compression. Due to discrepancies between the analytical and experimental values

observed by Gurdal and Haftka [1 O], the natural progression seemed to be the study of the effects

of out-of-plane stresses on the failure of the plate. It has long been suspected [18] that these

stresses may play a significant role in the mechanism contributing to failure of the abovementioned

plate due to the out-of-plane nature of the kink formation. Recently experimental evidence by

Guynn et al. [27] have substantiated this claim. In addition to the out-of-plane microbuckling/kink

band formation, researchers [2,27,37] have also detected the existence of delamination during the

failure process. The prediction of the inception of delamination also requires evaluation of the

transverse stresses which makes a three-dimensional analysis necessary.

Many techniques have been put forth for the evaluation of the complex stress state along the edge

of a hole in a plate. These vary from being based on an elasticity approach [42], to using

Lekhnitskii complex function based solution [43], and finally the three-dimensional finite element

approach. Of the above, the three-dimensional finite element approach was chosen in order to

compare and contrast with similar techniques (two-dimensional) utilized earlier. It will be shown

later that the use of this model not only provides the out-of-plane stresses but also provides in-

plane stresses that are quite different those obtained from a similar two-dimensional model.

Finite Element Models 21

y

c .£ x 0

~ .-

------- 1 .50 in --------

z

0.048 in

Figure 2-1: Plate Specimen Dimensions and Loading


Both finite element models were generated using a commercially available software, PATRAN [44)

(version 2.5), for pre- and post-processing while the finite element analysis was carried out using

ABAQUS [45) (version 4.8). The dimensions of the plate are as shown in Figure 2-1. The

distribution of stresses around the hole are described by angle a., with a. = 0 being at x = 0.1 O and y = 0.00. The laminate orientation, as described in Figure 1-2, is denoted by cj>. The properties of the laminae are assumed to be hornogenous and are based on those of a commercially available

graphite/epoxy (AS1/T3201) provided in Table 2-1. The layup of the plate is [±45/90/0). which

exhibits quasi-isotropy at the laminate level.

Table 2-1: Material Properties of AS1/T320f

E11 E22 E33 \>12 \>13 \>23 G12 G13 G23

20.0 1.30 1.25 0.30 0.30 0.49 1.03 0.90 0.90

* All rnodulii in msi

The above properties are not the exact properties of the specimens described in Chapter 1, but

rather reflects those of similar commercially available. material.

2.2 Model Description

2.2.1 Two-dimensional Model

The two-dimensional model consists of a 0.048 in thick plate in which the properties of the various

laminae were smeared to provide the global properties of the laminate. By virtue of the layup it is

quasi-isotropic in the x-y plane. The analysis of the two-dimensional model is based on Classical

Lamination Theory (CL T) and a plane stress state is considered. Since no through-the-thickness

variation of strains are allowed, a single element was used to model the laminate in the z direction. Eight node quadratic elements were used in this model, except at places where due to geometric

necessity (i.e. at the fine-coarse mesh interface), triangular elements were utilized. These elements

consist of quadratic lagrangian interpolation functions, so are capable of providing better


I 'i

'' '

r

Finite Element Models

-G> "C 0 E ... c G> E G> -G> G> ~ c ii: .. N

I N

~ ::J C> ii:

24

compliance to the plate in the vicinity of the hole, than linear lagrangian function elements. Since

interlaminar stresses are ignored in Classical Lamination Theory (CL T), considerable freedom can

be exercised in generating this model due to the numerous planes of symmetry which will be

elaborated on later. However to facilitate comparison, the nodal distribution of the two models in

the x-y plane has been kept the same. The nodal points were located so as to limit computation

time and also prevent distortion of the elements. The maximum aspect ratio (Vt) of the elements

was 41.667, which is well within the limits for these quadratic elements, as established by

Vidussoni [46].

The distribution of the nodes at the point of interest, surrounding the hole, has been kept fine and

then gradually expanded away from it. The nodes were distributed radially along the hole to a point

0.15 inches, beyond which the distribution was on a cartesian basis as shown in Figure 2-2. The

nodal distribution was chosen so as to be sensitive to the radial variation of stresses along the hole

edge.

2.2.2 Three-dimensional Model

The primary objective of this model was to study the magnitude and effect of the three-dimensional

stress state on the compression failure of a plate with a hole. The modeling procedure is thus

critical for proper evaluation of stresses. Three-dimensional finite element models are often not

given consideration as analytical tools as, at the present state of the art of computers, they are not

cost effective. Many techniques have been presented over the years, with an emphasis on

improved cost-effectiveness, to provide the three-dimensional information within the area of interest,

yet utilize a two-dimensional model away from it. A study by Dong [47] provides Qood overview on alternatives to complete three-dimensional modeling that can be utilized. Of these the Local-Global

technique has been the widely accepted [46,48] procedure. Bums et al. [35] recommends that for

Local-Global analysis of a notched composite laminate, the local-global interface should be at least

five times the laminate thickness away from the hole boundary. This interface for our model would

have to be 0.240 inches from the hole boundary or 0.340 inches from the center of the hole. Given

the dimensions of our specimen, this technique, would not have provided any substantial reduction

in size of the finite element model. Since the objective of this study, however, is principally to

evaluate the three-dimensional effect on compressive strength, the cost effectiveness of this

technique will not be belabored.


Limitations on the size of the model are imposed by the capability of the computing system to

process the data. Most of the limitations experienced were hardware based, primarily inadequate

swap diskspace, resulting in the use of a barely acceptable model. The number of elements

chosen, particularly in the region of the hole, was such so as to provide accurate representation

of the stress distribution, and satisfaction of boundary conditions. The greater the number of nodes

in the x-y plane the better the representation of the transverse stresses. Similarly the greater the

number of nodes through the thickness the more accurate satisfaction of the boundary conditions

at the hole edge. It can be see that as the degree of accuracy of the stresses obtained from the

finite element model increases the size of the model grows exponentially. Since both of the above

requirements cannot be satisfied due to hardware limitations the model has to be compromised.

Pagano [38) has shown that the effect of interlaminar stresses are confined to a distance of

approximately one lamina thickness away from the free edge, which in our case is the hole

boundary where the in-plane stresses are the greatest. As the stresses in this region are known

to be singular, a large number of nodes is needed in this region to properly represent the

distribution of transverse stresses. Too few nodes would cause the model to be too stiff and so

over/under estimate the magnitude of these stresses, while too many nodes would rapidly increase

the size of the model. -·'

In order to ensure the proper modeling of the stress gradients, and to satisfy the traction-free

boundary conditions as closely as possible, it is desirable to use two elements per layer through

the thickness. Hardware limitations permitted only one element per layer to be used to represent

the thickness of each lamina. Vidussoni [46] has shown that reasonably adequate representation

can be provided with one 20 node element per layer for cross ply laminates, though, two such

elements per layer would definitely provide more accurate results. This has been corroborated by

Thompson [48) though she used more elements. Depending on the local distribution, the difference

between using one or two elements per layer may not be great due to the fact that, in the event

of using two elements, the aspect ratio of the element for the same planar dimensions increases

as compared to the one element case. Thus the gain in sensitivity may not have the desired result,

as the change in aspect ratio effects the stiffness characteristic of the element [46]. Also, in order

to maintain adequate sensitivity in the x-y plane at least two elements have been accommodated

within one laminate thickness from the hole edge.

In the x-y plane, the distribution of the nodes is the same as that in the two-dimensional model and

is shown in Figure 2-2. At least two rings of elements are located within a region of one laminate


thickness from the hole edge. The density of radial distribution of the nodes was made to vary with

the distance from the hole edge. Away from the hole, where the stresses are assumed to

gradually reduce to far field values, larger aspect ratios (Vt) have been used. Twenty node brick

elements have been used all over except at the location where the distribution changes from radial

to cartesian where, appropriately, 15 node triangular wedge elements have been used.

2.3 Boundary Conditions

2.3.1 Symmetry

The use of boundary conditions to represent planes of symmetry on the three-dimensional model

is far more critical than that of the two-dimensional model. In a two-dimensional model of a quasi-

isotropic plate, two planes of symmetry can be exploited. As such a quarter plate model can be

used for the two-dimensional model. By way of definition a model with a midplane of symmetry

(symmetric laminate) is called a half symmetric model. The increase in the number of planes of

symmetry reduces the order of symmetry of the plate by a factor of two, e.g. a two-dimensional

symmetric laminate plate will be an eighth symmetric model due to assuming symmetry about the

planes through the three orthogonal axes. In a three-dimensional model the number of elements

and the total degrees of freedom being large, symmetry provides a necessary means of reducing

the size of the model. For all symmetric laminates the midplane automatically presents a plane of

symmetry. Burns [35] has shown that no other plane of true symmetry exists for a three-

dimensional models of quasi-isotropic laminates.

Despite the lack of true symmetry, in order to reduce the computational effort other symmetry

conditions were investigated. One such condition which proved to yield accurate stress distribution

was introduction of the symmetry about the x-z plane. For the 0 and 90 degree laminae the

coefficients of mutual influence 1111.i• which are defined as [35]

Finite Element Models

(2511-2513-551 ) cos38sin8- (2522-2512 -SH) cos8sin38 S11cos'8+ (2S12+S11 ) sin28cos26+S22sin'8

(2.01)

27

'tx z

0.01.-----.----...-....... ---....-...-....... -..----..-----.-----.

0

-0.011

.. .f~ .. l : ! l i ~ !

~

I ......... -............ 1 ................... .

; l

··········f···-·····--··+······-·····-···············+·····················-··-···+····-·······················

~'. I 11~--i I --& - Ctr. Sym. ~ -0.1 ----------------------

0 10

Angle a 1311

Figure 2-3a: Comparison of a In half and zz

quarter symmetric models

110 ~

0.1 ..-........ -.----...-........ -.--....-..-........ -..---.-..-................ ---.---..

0.011 . . ···················-----···-·--··-···········-··········· .. ······· ··········-·· ............................... ..

0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

411 10

Angle a 1311

Figure 2-3b: Comparison of 't In half and xz quarter symmetric models

110 ~


where:

'Y1i - Shear strains

£1 - Normal strains

S11 - Compliance coefficients

9 - Fiber angle are zero about the plane perpendicular to the loading axis. However, this is not the case with angle

ply laminae. Thus, an in-plane strain will result in shear strains in the off-axis plies. The effect of

imposing symmetry on the x-z plane causes erratic behavior of the through-the-thickness stresses,

az and 'txz• in the immediate vicinity of the hole along the x-axis.

Figures 2-3 show the distribution of these stresses along the periphery of the hole in three-

dimensional models, with and without assumed symmetry about the x-z plane. The stresses are

normalized by the average far-field stress, att• which is obtained by dividing the force reactions of the nodes at the supports, i.e. X=-0.75, by the laminate area at that location. The location of the

support is at a distance such that the stresses there are unaffected by the existence of the hole.

The stress distribution of both, the half and quarter symmetric models coincide over most of the

hole boundary except when approaching the plane of assumed symmetry (x-axis) due to the

imposed boundary constraint, Ui = 0.00. Similar behavior has also been shown by Burns [35], which -~

he describes to be of localized nature. This imposition of symmetry is therefore not recommended

where absolute accurate representation of the stress state is necessary throughout the edge of the

hole. Thus a quarter symmetry model of a quasi-isotropic laminate would be considered adequate

for a failure analysis of an axially loaded condition as the point of interest is along the y axis, i.e.

away from the axis of symmetry. The use of a half symmetry model would result in significantly

more computation time with no particular gain in accuracy.

2.3.2 Traction-free Boundary Conditions

In finite element modeling, isoparametric displacement elements satisfy equilibrium by means of

static geometric equations between the internal forces and the external loads. The finite elements

are based on a displacement based variational principle and as such continuity of displacements

is Satisfied. Displacements are thus generated at each point of integration. The strains at these

points are then generated, on an elemental basis, as derivatives of the displacements. The

stresses at each strain are then obtained by multiplying the strains by the stiffness coefficient

matrix. It is thus seen that the stresses are discontinuous at the interfaces due to the difference


,.. I

Finite Element M

odels

CD ... cu ... 0 "C CD E ... 0 """" CD "C c :s I -CD "C 0 E -cu c 0 ·-0 c CD E ·-"C I CD CD ... .c I-.. ~ I C\I CD ... :s en ·-LL

30

s1ap

ow 1

uawa

13 a

11u1

:1

"T1 -· cc c ~ CD N I c.n •• -f ::r ~ CD CD I c. -·

. ~": 3 CD :J tn -· 0 :J m - 3 0 c. CD -I c. a 0 ~ 3 CD c. en .. m .. CD

in material properties in the two adjacent layers and due to the lack of continuity of the derivatives

of the displacements across the element boundaries. This is acceptable for in-plane stresses and

strains, however, equilibrium conditions require interlaminar stresses to be continuous at the

interfaces. These have been taken care of by using an averaging interpolation technique. By using

these displacement elements it is not possible to satisfy the boundary conditions at the traction free

edges. They will not be identically zero [48]. It is for'this reason that, in order to attain convergence,

the number of elements used to model the thickness be increased in order to attain traction free

edges. It has been shown that in spite of system limitations and the small number of elements per

layer used in the models, satisfactory results can be obtained to provide valuable information

[35,48].

2.4 Comparison Between 2-D and 3-D Models

So far most of the work done [3,4, 1 O], on the ~nalysis of failure of a plate with a hole has been

restricted to the use of two-dimensional finite element models, assuming planar failure. Due to the

disparity between experimental and analytical values of strength obtained, it has become necessary

to evaluate the three-dimensional effect on the response of a laminate to in-plane loading. A two-

Climensional model was used to correlate the results obtained earlier [1 O] and compare with a

three-dimensional model that is used to build upon it. Prior to performing a failure analysis it is

necessary to understand the response of the respective models under the uniaxial load. Figures

2-4 and 2-5 show the "undeformed" and "deformed" shapes of the three-dimensional finite element

model. In Figure 2-5 an "exagerration factor" (maximum model dimension/maximum displacement)

of 1 O was used.

In order to generalize a failure theory for quasi-isotropic laminates, it is necessary to validate it for

all laminate orientations,~- To ensure this Gurdal & Haftka [10] recommended that the model of a quasi-isotropic laminate [±45/90/0]. be subject to a load that was rotated through 180 degrees.

This is analogous to keeping the load constant and varying the fiber angles of the laminae by the

same amount resulting in a variety of quasi-isotropic laminates, as described in Section 1.3. In this

case, they showed, that it was necessary to perform only one two-dimensional finite element

analysis, subsequent orientations of the laminate could be accomplished by transformation of the

loading axis. This process is efficient and economical and was considered for the three-dimensional

model as well. However, as will be shown later this procedure is not conducive for use with three-

dimensional analysis.


r

z

3h/8 ··································f····································1····:············~················l.-.......................... .

i + ! I ;

·································-'····································•····· ·····A·················::'=,,,::····································

i I ~ ~

h/4

h/8 ··································f····································l·················4r··············f··········· i i I i

..--~~~-'~ : I

1~~=1 * 0 .__.__.__.__.__,&,__..__..__..__..__"'-___ ~,..._...__,_..._~ ............ ~

-0.00045 -0.0004 -0.00035

Strain £ xx

-0.0003 -0.00025 lg24.pm

Figure 2-6: Through the thickness strain distribution for '" 2-D and 3-D models at x=0.50 and y=-0.33

(symmetric about z-axis)


O.OOO&r-....-....... --...--.,.--.--...-.--....-....... --...--..--.--...-.--....--.

Eu -0.0005

-0.001

., I i

" i I i I i

" i ••..•. ,. ........... r······························ I i

_:_j ____ _ --.- 3-0 Model (0)

- -ll - 2·0Model(O) -0.002 .__ ____ ..__ __ _..._.__....__._ __ ..__...a..._..._.__....__,

0 46 10 135

Angle a

Figure 2-7a: Comparison of strains between 2-D and 3-D models, 9=0 and '=0

-0.0005 ___ ....... __....__ ............ __..._...._ __ ....... __. ___ ............ __._...._ __ ..... 0 45 10 135 180

Angle a

Figure 2· 7b: Comparison of strains between 2-D and 3-D models, 9::90 and t = O


r

0.001 .--...--.---.--...-------.-----------.-----.---....-..---e-- 3-D Modi! (..CS) - -II - 2-0 Modi! (..CS)

0.0005

. ' : ········-r-···················:a:·········r······· . . . . i ~ i . . . . . . i \ i i ..

-0.0005

-0.001 L-.......__;,..__,.....,.&-...................... ..._...L-_.___.__,..._....L.. .............. ,__...._....,1 0 45 10 135 180

Angle a

Figure 2-7c: Comparison of strains between 2-D and 3-D models, 9=-45 and '=0


r

au

4r;:::c:::c:::z:==::::::i====~,....----...--."""."""r-----_,..-, _..__ 3-0 Model (0)

- -b - 3-0 Modi! (45)- •• -45

: Jrq::~:::=::··~+~··:::~::~::~:-= : ! !

1 ······························!·················· ············!············ .................. L. .......................... .

I I : !

0 ~ .... ~ ...... .e-tS::·-··-1··-··--··-·1 ~...._ ........ _..__.,__.._...._ ........ ......i.. ....... -....__._....a... ........ __ ..._....__~

0 45 80 135 180

. Angle ex if29.pll Figure 2·8: Comparison of stresses obtained by

rotation and transformation of 3·0 model· at • = O degrees

__._ 3-0 Model (45)- •· 0

- -b - 3-0 Model to>- t•45 -+- ~D Model (0)- + • Onmd. 45

. ~~:=~--~~:::~··-~:~~::= : : : I I . i oa... ............... 4=:tP-'. .. :'.'.':--r-.

:

i 0 80 135 180

Angle Cl

Figure 2·9: Comparison of stresses obtained by rotation and transformation of 3-0 model at • = 45 degrees


a.,

0.1------.-. ....... --------------------------------. . : ., . __ ;;,_;;_~;~~--J_L.fL.: ...

failure analysis of a quasi-isotropic laminated composite ......with the increased use of laminated...

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